阅读说明：本技术 预测生物反应器中的细胞培养性能 (Predicting cell culture performance in a bioreactor ) 是由 W·L·约翰森 M·R·卡卡莫·贝伦斯 于 2020-03-27 设计创作，主要内容包括：一种对生物反应器进行建模的计算方法组合了代谢通量的动力学的机制模型和通量平衡分析来预测细胞培养性能。这些机制模型包括描述细胞外环境的过程变量的影响,该细胞外环境例如温度、酸度、渗量和/或代谢物浓度。该方法基于机制模型约束通量速率,并鉴于合适的代谢目标计算通量速率。该方法基于用户输入来模拟生物反应器的过程变量的时间演化,并计算性能量度以显示给用户、控制生物反应器、和/或训练生物反应器的人工智能模型。(A computational method of modeling bioreactors combines a mechanistic model of the dynamics of metabolic flux and flux balance analysis to predict cell culture performance. These mechanistic models include the effects of process variables describing the extracellular environment, such as temperature, acidity, osmolarity, and/or metabolite concentration. The method constrains the flux rate based on a mechanism model and calculates the flux rate in view of appropriate metabolic targets. The method simulates the time evolution of a process variable of a bioreactor based on user input and calculates performance metrics for display to a user, controlling the bioreactor, and/or training an artificial intelligence model of the bioreactor.)

1. A method of computationally modeling a bioreactor, wherein the method comprises:

receiving a plurality of current values of process variables that describe virtual contents in at least a portion of the modeled bioreactor and that include virtual cellular biomass in a virtual extracellular solution;

calculating, by processing hardware of the computing system, new values of the process variables during the simulation time period at least in part by:

generating constraints on flux rates describing metabolic flux of the virtual cellular biomass by modeling one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics,

calculating the flux rates of the metabolic fluxes by performing a flux balance analysis influenced by metabolic targets and generated constraints on the flux rates,

calculating a rate of change of at least some of the process variables based at least in part on the calculated flux rate, an

Updating one or more of the current values of the process variables at least in part by integrating one or more of the calculated rates of change for a virtual time step within the simulation time period;

calculating, by the processing hardware, a metric of the computationally modeled bioreactor based at least in part on the calculated new values of the process variables; and

generating, by the processing hardware and based on the metric of the modeled bioreactor, one or more of: i) information displayed to a user via a user interface, ii) control settings for a real-world bioreactor, or iii) a training set for an artificial intelligence model of a bioreactor.

2. The method of claim 1, wherein the process variables include at least one of: temperature, acidity, or one or more variables indicative of the total volume of these virtual contents.

3. The method of claim 1, wherein the process variables include one or more variables indicative of the concentration of the virtual cellular biomass and one or more variables indicative of the concentration of extracellular metabolites in the virtual extracellular solution.

4. The method of claim 3, wherein the one or more variables indicative of extracellular metabolite concentrations in the virtual extracellular solution include one or more variables indicative of concentrations of at least one of: oxygen, carbon dioxide, or ammonia.

5. The method of claim 3, wherein the one or more variables indicative of extracellular metabolite concentrations in the virtual extracellular solution include variables indicative of concentrations of one or more of: glucose, asparagine, glutamine or glycine.

6. The method of claim 3, wherein the one or more variables indicative of extracellular metabolite concentration in the virtual extracellular solution comprise a variable indicative of a concentration of at least one target metabolite.

7. The method of claim 3, wherein the process variables additionally include one or more variables indicative of one or more intracellular metabolite concentrations in the virtual cellular biomass.

8. The method of claim 1, wherein the plurality of constraints on the flux rates include an upper limit of at least one of: i) glucose uptake, ii) glutamine uptake, iii) asparagine uptake, or iv) oxygen uptake.

9. The method of claim 1, wherein the plurality of constraints on the rates of change include a lower limit on the rate of energy expenditure for cell maintenance.

10. The method of claim 1, wherein the effect of at least some of the current values of the process variables on the one or more of metabolic reaction kinetics comprises an effect of a value of at least one of:

i) temperature, ii) acidity, or iii) osmolarity.

11. The method of claim 10, wherein the first and second light sources are selected from the group consisting of a red light source, a green light source, and a blue light source,

wherein modeling the one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics comprises calculating an effect of a value of at least one of: i) temperature, ii) acidity, or iii) osmolality, the correction factor indicating a decrease in the uptake rate due to non-ideal conditions.

12. The method of claim 1, wherein the one or more effects of the at least some of the current values of the process variables on metabolic reaction kinetics include an effect of stress on a lower limit of energy expenditure rate for cell maintenance.

13. The method of claim 12, wherein modeling the effect of stress comprises calculating the effect based at least in part on cumulative effects of temperature variations on the virtual cellular biomass.

14. The method of claim 1, wherein the one or more effects of the at least some of the current values of the process variables on metabolic reaction kinetics includes an effect of a concentration of at least one of metabolic byproducts.

15. The method of claim 1, wherein modeling the one or more effects comprises calibrating the modeled effects with experimental data from real-world cell culture.

16. The method of claim 1, wherein the metabolic goal comprises minimizing a ratio of linear combinations of at least some of the flux rates and flux rate squares in the flux balance analysis.

17. The method of claim 1, wherein the metabolic target comprises minimizing a linear combination of at least some of the flux rates in the flux balance analysis.

18. The method of claim 17, wherein the linear combination is a sum of fluxes in the flux balance analysis.

19. The method of claim 1, wherein the metabolic goal comprises maximizing a rate of energy expenditure for cell maintenance.

20. The method of claim 1, wherein the metabolic target comprises maximizing the growth rate of the virtual cellular biomass.

21. The method of claim 1, further comprising:

receiving into the modeled bioreactor one or more variables indicative of flow rate and composition of one or more virtual input streams, and wherein

Updating the at least some of the current values of the process variables includes interpreting a flow rate and a composition of the one or more virtual input streams.

22. The method of claim 21, further comprising:

receiving one or more variables indicative of a property of a virtual output filter;

calculating a composition of one or more virtual output streams from the modeled bioreactor using properties of the virtual output filter, and wherein

Updating the at least some of the current values of the process variables includes interpreting a composition of the one or more virtual output streams.

23. The method of claim 21 wherein updating at least some of the current values of the process variables includes solving mass balance equations.

24. The method of claim 1, further comprising:

receiving one or more control variables, an

The affected set of the current values of the process variables is updated based at least in part on the received control variables.

25. The method of claim 1, wherein

The metric for the modeled bioreactor includes a value of at least one of the process variables at different virtual times within the simulation time period.

26. The method of claim 1, wherein

The measure of the computationally modeled bioreactor is indicative of the effectiveness or efficiency of the computationally modeled bioreactor in producing the target product.

27. The method of claim 1, wherein

The method includes generating the control setting, and wherein

The method further includes controlling an input to the real-world bioreactor based on the generated control settings.

28. The method of claim 1, wherein

The method includes generating a training set of the artificial intelligence model for the bioreactor.

29. The method of claim 1, wherein

The virtual cellular biomass comprises virtual Chinese Hamster Ovary (CHO) cells.

30. The method of claim 1, wherein receiving the plurality of current values comprises at least one of

i) Receiving a value from a user via a graphical user interface;

ii) receiving values based on measurements of the real bioreactor;

iii) loading a value from the computer memory and based on a previously calculated value; or

iv) loading predetermined default values from the computer memory.

31. The method of claim 1, wherein the modeled bioreactor is a spatially homogeneous bioreactor and the at least a portion of the modeled bioreactor is the entirety of the spatially homogeneous bioreactor.

32. The method of claim 1, wherein:

the modeled bioreactor is a spatially heterogeneous bioreactor;

the at least part of the modeled bioreactor is a first part of the modeled bioreactor;

the process variables are first process variables;

the method further includes receiving a plurality of current values of second process variables describing a second virtual content in a second portion of the modeled bioreactor; and

the calculation of the new values of the first process variables during the simulation time period is based in part on the received current values of the second process variables.

33. The method of claim 32, wherein calculating new values for the first process variables during the simulation time period includes calculating a gradient of at least one of the first process variables based on values for corresponding ones of the second process variables.

34. The method of claim 32, further comprising:

one or more velocities associated with the virtual contents of the first portion of the modeled bioreactor are determined.

35. The method of claim 34, wherein

Determining the one or more velocities is based at least in part on computational fluid dynamics.

36. A non-transitory computer-readable medium storing instructions for computationally modeling a bioreactor, wherein the instructions, when executed by one or more processors, cause the one or more processors to:

receiving a plurality of current values of process variables that describe virtual contents in at least a portion of the modeled bioreactor and that include virtual cellular biomass in a virtual extracellular solution;

calculating, by processing hardware of the computing system, new values of the process variables during the simulation time period at least in part by:

generating constraints on flux rates describing metabolic flux of the virtual cellular biomass by modeling one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics,

calculating the flux rates of the metabolic fluxes by performing a flux balance analysis influenced by metabolic targets and generated constraints on the flux rates,

calculating a rate of change of at least some of the process variables based at least in part on the calculated flux rate, an

Updating one or more of the current values of the process variables at least in part by integrating one or more of the calculated rates of change for a virtual time step within the simulation time period;

calculating, by the processing hardware, a metric for the modeled bioreactor based at least in part on the calculated new values for the process variables; and

generating, by the processing hardware and based on the metric of the modeled bioreactor, one or more of: i) information displayed to a user via a user interface, ii) control settings for a real-world bioreactor, or iii) a training set for an artificial intelligence model of a bioreactor.

37. The non-transitory computer readable medium of claim 36, wherein the process variables include at least one of: temperature, acidity, or one or more variables indicative of the total volume of these virtual contents.

38. The non-transitory computer readable medium of claim 36, wherein the process variables include one or more variables indicative of a concentration of the virtual cellular biomass and one or more variables indicative of a concentration of an extracellular metabolite in the virtual extracellular solution.

39. The non-transitory computer readable medium of claim 38, wherein the one or more variables indicative of extracellular metabolite concentrations in the virtual extracellular solution include one or more variables indicative of a concentration of at least one of: oxygen, carbon dioxide, or ammonia.

40. The non-transitory computer readable medium of claim 38, wherein the one or more variables indicative of extracellular metabolite concentrations in the virtual extracellular solution comprise variables indicative of concentrations of one or more of: glucose, asparagine, glutamine or glycine.

41. The non-transitory computer readable medium of claim 38, wherein the one or more variables indicative of extracellular metabolite concentration in the virtual extracellular solution include a variable indicative of a concentration of at least one target metabolite.

42. The non-transitory computer readable medium of claim 38, wherein the process variables additionally include variables indicative of one or more intracellular metabolite concentrations in the virtual cellular biomass.

43. The non-transitory computer readable medium of claim 36, wherein the plurality of constraints on the rate of change include an upper limit of at least one of: i) glucose uptake, ii) glutamine uptake, iii) asparagine uptake, or iv) oxygen uptake.

44. The non-transitory computer-readable medium of claim 36, wherein the plurality of constraints on rate of change include a lower limit on rate of energy expenditure for cell maintenance.

45. The non-transitory computer readable medium of claim 36, wherein the one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics include an effect of a value of at least one of:

i) temperature, ii) acidity, or iii) osmolarity.

46. The non-transitory computer readable medium of claim 45, wherein

Modeling the one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics includes calculating an effect of a value of at least one of: i) temperature, ii) acidity, or iii) osmolality, the correction factor indicating a decrease in the uptake rate due to non-ideal conditions.

47. The non-transitory computer-readable medium of claim 36, wherein the one or more effects of the at least some of the current values of the process variables on metabolic reaction kinetics include an effect of stress on a lower limit of energy expenditure rate for cell maintenance.

48. The non-transitory computer-readable medium of claim 47, wherein modeling the effect of stress comprises calculating the effect based at least in part on a cumulative effect of temperature variations on the virtual cellular biomass.

49. The non-transitory computer readable medium of claim 36, wherein the one or more effects of the at least some of the current values of the process variables on metabolic reaction kinetics includes an effect of a concentration of at least one of metabolic byproducts.

50. The non-transitory computer readable medium of claim 36, wherein modeling the one or more effects comprises calibrating the modeled effects with experimental data from real-world cell culture.

51. The non-transitory computer-readable medium of claim 36, wherein the metabolic goal comprises minimizing a linear combination of at least some of the rates of change.

52. The non-transitory computer readable medium of claim 36, wherein the metabolic target comprises minimizing a linear combination of at least some fluxes in the flux balance analysis.

53. The non-transitory computer readable medium of claim 52, wherein the linear combination is a sum of fluxes in the flux balance analysis.

54. The non-transitory computer readable medium of claim 36, wherein the metabolic goal comprises maximizing a rate of energy expenditure for cell maintenance.

55. The non-transitory computer readable medium of claim 36, wherein the metabolic goal comprises maximizing a growth rate of the virtual cellular biomass.

56. The non-transitory computer-readable medium of claim 36, wherein the instructions further cause the one or more processors to:

receiving into the modeled bioreactor one or more variables indicative of flow rate and composition of one or more virtual input streams, and wherein

Updating the at least some of the current values of the process variables includes interpreting a flow rate and a composition of the one or more virtual input streams.

57. The non-transitory computer-readable medium of claim 56, wherein the instructions further cause the one or more processors to:

receiving one or more variables indicative of a property of a virtual output filter;

calculating a composition of one or more virtual output streams from the modeled bioreactor using properties of the virtual output filter, and wherein

Updating the at least some of the current values of the process variables includes interpreting a composition of the one or more virtual output streams.

58. The non-transitory computer readable medium of claim 56, wherein updating the at least some of the current values of the process variables comprises solving mass balance equations.

59. The non-transitory computer-readable medium of claim 36, wherein the instructions further cause the one or more processors to:

receiving one or more control variables, an

The affected set of the current values of the process variables is updated based at least in part on the received control variables.

60. The non-transitory computer readable medium of claim 36, wherein

The metric for the modeled bioreactor includes a value of at least one of the process variables at different virtual times within the simulation time period.

61. The non-transitory computer readable medium of claim 36, wherein

This measure of the modeled bioreactor is indicative of the effectiveness or efficiency of the modeled bioreactor in producing the target product.

62. The non-transitory computer readable medium of claim 36, wherein the instructions

Cause the one or more processors to generate the control setting, and

the one or more processors are further caused to control the input to the real-world bioreactor based on the generated control settings.

63. The non-transitory computer readable medium of claim 36, wherein the instructions

Causing the one or more processors to generate a training set of the artificial intelligence model for the bioreactor.

64. The non-transitory computer readable medium of claim 36, wherein

The virtual cellular biomass comprises virtual Chinese Hamster Ovary (CHO) cells.

65. The non-transitory computer readable medium of claim 36, wherein receiving the plurality of current values comprises at least one of:

i) receiving a value from a user via a graphical user interface;

ii) receiving values based on measurements of the real bioreactor;

iii) loading a value from the computer memory and based on a previously calculated value; or

iv) loading predetermined default values from the computer memory.

66. The non-transitory computer readable medium of claim 36, wherein the modeled bioreactor is a spatially homogeneous bioreactor and the at least a portion of the modeled bioreactor is the entirety of the spatially homogeneous bioreactor.

67. The non-transitory computer readable medium of claim 36, wherein

The modeled bioreactor is a spatially heterogeneous bioreactor;

the at least part of the modeled bioreactor is a first part of the modeled bioreactor;

the process variables are first process variables;

the instructions further cause the one or more processors to receive current values of second process variables describing a second virtual content in a second portion of the modeled bioreactor; and

the calculation of the new values of the first process variables during the simulation time period is based in part on the received current values of the second process variables.

68. The non-transitory computer readable medium of claim 67, wherein calculating new values of the first process variables during the simulation time period includes calculating a gradient of at least one of the first process variables based on values of corresponding ones of the second process variables.

69. The non-transitory computer-readable medium of claim 67, wherein the instructions further cause the one or more processors to determine one or more velocities associated with the virtual contents of the first portion of the modeled bioreactor.

70. The non-transitory computer-readable medium of claim 69, wherein

Determining the one or more velocities is based at least in part on computational fluid dynamics.

Technical Field

The present disclosure relates generally to methods of computer simulation modeling of bioreactors and related bioprocesses for the purpose of improving bioprocess design and/or operation.

Background

Biotechnology uses living cells to produce valuable biochemical compounds. Different cells can be engineered to produce a variety of biochemical products of interest for pharmaceutical, agricultural, or other uses. Practitioners construct and use bioreactors to grow cell cultures that produce the product of interest along with other metabolic byproducts. Maximizing the target product volume and mass for a given production cost is one of the main objectives in bioreactor design.

The bioreactor can be configured and controlled to operate in different modes for different purposes. A first objective of the process may be to grow a sufficient amount of cell culture without using more than the necessary amount of substrate nutrients. The type of cellular biomass grown in a bioreactor may depend on the application, with some popular biological cultures (e.g., Chinese Hamster Ovary (CHO) cells) being used for many different applications. In one mode of operation of the bioreactor, known as batch mode, growth of the biological culture can be carried out without any liquid flow into or out of the reactor, especially if the bioreactor control system can maintain good growth conditions. Good growth conditions may require appropriate temperatures, acidity and substrate metabolite concentrations. As the growing biomass consumes substrate metabolites, the input flow can help replenish metabolites required for growth, which forms a fed-batch configuration of the bioreactor. Metabolic byproducts generated during growth may compromise the optimality of growth conditions. The addition of input and output flow streams, forming a continuous bioreactor configuration, can mitigate the negative effects of metabolic byproducts and provide more control over the duration of the cell growth phase. A filter may be added to the output stream to prevent the cellular biomass from being removed, which forms the perfusion configuration of the bioreactor.

Once a sufficient amount of biomass is produced in the bioreactor, the next stage of operation may prioritize the production and collection of the target metabolites. Such target products may include therapeutic antibodies or other biologicals. Inducing cells to transfer metabolic energy from growing biomass to producing a product of interest often involves the introduction of a stress factor. Stress may include a decrease in optimal host temperature that allows the cell to consume more energy in non-growing metabolism, which may result in an increase in the target yield.

Optimizing all phases of bioreactor operation faces significant challenges. Many decisions may include when to transition from growth to production, when to start harvesting, when and in what amounts to supply and remove metabolites, how to control the bioreactor environment. Typically, many expensive and time-consuming in vivo experiments are performed to select the operating conditions. Alternative methods for optimizing and controlling bioreactor operation would bring considerable benefits to the industry and the public it serves.

Disclosure of Invention

Computational methods for modeling bioreactors are used to predict cell culture performance to improve and/or optimize bioreactor design and/or operation. In one aspect, the method includes receiving a plurality of current values of process variables that describe virtual contents of a modeled bioreactor and the virtual contents include virtual cellular biomass in a virtual extracellular solution. The method also includes calculating, by processing hardware of the computing system, new values of the process variables during the simulated time period. The method calculates the new value at least in part by: generating constraints on flux rates describing metabolic flux of the virtual cellular biomass by modeling one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics; calculating the flux rates of the metabolic fluxes by performing a flux balance analysis influenced by metabolic targets and generated constraints on the flux rates; calculating a rate of change of at least some of the process variables based at least in part on the calculated flux rate; and updating one or more of the current values of the process variables at least in part by integrating one or more of the calculated rates of change for a virtual time step within the simulation time period. The method also includes calculating, by the processing hardware, a metric of the computationally modeled bioreactor based on the calculated new values of the process variables, and generating, by the processing hardware and based on the metric of the modeled bioreactor, one or more of: i) information displayed to a user via a user interface, ii) control settings for a real-world bioreactor, or iii) a training set for an artificial intelligence model of a bioreactor. For any of the methods described herein, the method can optionally further comprise culturing the mammalian cell in a bioreactor using one or more parameters identified in the modeled bioreactor. For example, a mammalian cell can encode a therapeutic protein and can produce the therapeutic protein when cultured in a bioreactor. Examples of mammalian cells include CHO cells and BHK cells. For the sake of brevity, "bioreactor" may also be referred to herein as a "reactor," and these terms will be understood to be interchangeable herein unless specifically stated otherwise.

In another aspect, a non-transitory computer-readable medium stores instructions for computationally modeling a bioreactor, wherein the instructions, when executed by one or more processors, cause the one or more processors to receive current values of process variables describing virtual contents of the modeled bioreactor, and the virtual contents include virtual cellular biomass in a virtual extracellular solution. The instructions also cause the one or more processors to calculate, by processing hardware of the computing system, new values of the process variables during the simulated time period. The one or more processors calculate the new value at least in part by: generating constraints on flux rates describing metabolic flux of the virtual cellular biomass by modeling one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics; calculating the flux rates of the metabolic fluxes by performing a flux balance analysis influenced by metabolic targets and generated constraints on the flux rates; calculating a rate of change of at least some of the process variables based at least in part on the calculated flux rate; and updating one or more of the current values of the process variables at least in part by integrating one or more of the calculated rates of change for a virtual time step within the simulation time period. The instructions also cause the one or more processors to calculate, by the processing hardware, a metric for the modeled bioreactor based on the calculated new values of the process variables, and generate, by the processing hardware and based on the metric for the modeled bioreactor, one or more of: i) information displayed to a user via a user interface, ii) control settings for a real-world bioreactor, or iii) a training set for an artificial intelligence model of a bioreactor.

Drawings

FIG. 1 illustrates an exemplary process and corresponding computational model for computationally modeling a bioreactor.

FIG. 2 is a graph depicting an exemplary effect of temperature variation on stress variables in an embodiment of the modeling process of FIG. 1.

FIG. 3 illustrates an exemplary computer system in which the modeling process of FIG. 1 may be implemented.

Fig. 4A and 4B illustrate panels of an exemplary user interface for receiving user input.

FIG. 5 illustrates an exemplary user interface for displaying output to a user.

Fig. 6 illustrates an exemplary bioreactor control system.

FIG. 7 is a flow chart depicting an exemplary method of computationally modeling a bioreactor.

Fig. 8 shows a comparison between the bioreactor model and experimental observations.

Fig. 9A and 9B show the relationship between a metabolic model of a heterogeneous bioreactor modeled with a homogenous portion and a velocity field that can be calculated using Computational Fluid Dynamics (CFD).

Detailed Description

Much of the description herein focuses on metabolic (biological and biochemical) models of bioreactors configured, for example, to produce certain biological products. Although the description primarily regards the bioreactor as a homogeneous mixture of extracellular solution and cellular biomass, adjustments to the metabolic model can be readily made to account for spatial heterogeneity in the bioreactor. For this purpose, the metabolic model can be integrated with the computational model of the transport physics in a computer simulation. Thus, some portions of this description describe model integration, demonstrating the general applicability of metabolic models to bioreactors that do not require thorough mixing.

FIG. 1 illustrates an exemplary process 100 for computationally modeling a bioreactor with a corresponding computational model 104 represented in a virtual space 110 in a series of time steps (including the depicted time step 102). It should be appreciated that time step 102 may be repeated any suitable number of times during modeling process 100. The virtual contents of the modeled bioreactor 120 may include a virtual extracellular solution 124 and a virtual cellular biomass 130 disposed within the virtual extracellular solution 124. For convenience, the modeled bioreactor 120 may be referred to herein as a "bioreactor", the virtual extracellular solution 124 may be referred to herein as an "extracellular solution" (or simply "solution"), and the virtual cellular biomass 130 may be referred to herein as a "cellular biomass" (or simply "biomass"). A set of fluxes 132a-132i may represent, at least in part, a state of the biomass 130. Virtual input stream 140 and virtual output stream 142 may replenish and deplete, respectively, the virtual contents of the modeled bioreactor 120. Virtual filter 144 may selectively facilitate, allow, limit, or prevent some of the contents from flowing out of modeled bioreactor 120. For example, to model a bioreactor in a perfusion mode of operation, the filter 144 may allow flow of the extracellular solution 124 while restricting or preventing flow of the cellular biomass 130. In contrast, excluding or zeroing out the virtual input stream 140 and the virtual output stream 142 models the bioreactor in batch mode of operation. In general, varying the flow and/or composition of the virtual input flow 140 and the virtual output flow 142 may allow for modeling of batch, fed-batch, continuous, and/or perfusion configurations or modes of operation of the bioreactor 120.

In some embodiments, the modeling process 100 can be configured to model spatial heterogeneity of the bioreactor. Spatial heterogeneity may refer to, for example, spatial differences in the composition of the extracellular solution, the concentration or composition of the cellular biomass, or other process parameters (e.g., temperature) at different locations within the bioreactor. In a well-mixed bioreactor, such differences may be small, and a homogeneous model may simulate a well-mixed bioreactor with sufficient accuracy. On the other hand, when modeling a poorly mixed bioreactor, computationally considering spatial heterogeneity can greatly improve the accuracy of the model. To this end, the metabolic model described herein may be coupled with a physical model of mass and heat transfer within the reactor. In some embodiments, a Computational Fluid Dynamics (CFD) model may cooperate with the metabolic models discussed herein. In general, heterogeneity can be modeled in view of a trade-off between accuracy and computational complexity.

To model spatial heterogeneity using modeling process 100, modeled/virtual bioreactor 120 may model a portion of a heterogeneous bioreactor. Thus, in the context of a spatially heterogeneous bioreactor model, bioreactor 120 may be considered part of a heterogeneous bioreactor. That is, the simulated heterogeneous bioreactor may be divided into sections in any of a number of suitable ways depending on, for example, the degree of heterogeneity and the coupled physical model. In some embodiments, at least 2, 5, 10, 20, or other suitable number of portions of different sizes (including ranges between any two of the listed values) may represent different regions of the bioreactor (e.g., inlets and outlets, a central portion, etc.). In some embodiments, the bioreactor volume may be integrated into at least 100, 1000, 10000, 100000, 1000000 or any other suitable number of portions (including ranges between any two of the listed values) represented by finite volume elements, which may also serve, for example, as finite volume elements of a CFD model.

It may be assumed that the modeled portion represented by bioreactor 120 is substantially homogeneous. Thus, the extracellular solution 124 may model a portion of the extracellular solution corresponding to the modeled portion of the heterogeneous reactor, and the biomass 130 may model a corresponding portion of the biomass distributed throughout the heterogeneous bioreactor. The input stream 140 may model mass transport (i.e., flow of extracellular solution and biomass) from one or more adjacent portions of the heterogeneous bioreactor into the portion of the heterogeneous bioreactor simulated by the bioreactor 120. Similarly, the output stream 142 can be a model of mass transport (i.e., flow of extracellular solution and biomass) out of a portion of the heterogeneous bioreactor (represented by bioreactor 120) into an adjacent portion of the heterogeneous bioreactor. In this way, a set of homogeneous bioreactor models (each representing a portion of a heterogeneous bioreactor) may be connected to model the heterogeneous bioreactor. The input stream and the output stream for replenishing and depleting the heterogeneous bioreactor may be added to an input stream 140 and an output stream 142 of a portion of the heterogeneous bioreactor having an input from or an output to the exterior of the heterogeneous bioreactor. In this case, the input stream 140 and the output stream 142 may be considered to take into account, at least in part, the input and output boundary conditions of the heterogeneous bioreactor model assembled from the homogeneous bioreactor model.

The computational (e.g., CFD) model may be configured to compute the input and output streams (e.g., streams 140, 142) for each portion of the heterogeneous bioreactor model (e.g., bioreactor 120). In some embodiments, each portion of the heterogeneous bioreactor modeled by the process 100 may correspond to a finite volume element of the CFD model as discussed above. In general, however, the spatial discretization for the CFD model need not be the same as the spatial discretization for the reactor portion of the application simulation process 100. For example, the CFD model may calculate the velocity field within the modeled heterogeneous bioreactor at one spatial resolution. The velocity field may be combined with the bioreactor composition calculated by the metabolic model at different spatial resolutions to update the input stream 140 and output stream 142 for each of the bioreactor sections represented by the metabolic model.

The calculation of the velocity field may depend on the physical configuration of the bioreactor (e.g., mixing mechanism). In some bioreactors, natural convection may be the primary mixing mechanism. That is, temperature gradients due to metabolic processes or reactor temperature control may drive convection. In other embodiments, the injected fluid (e.g., a liquid substrate or gas) may drive convection. Additionally or alternatively, an impeller or other suitable mixing mechanism may propel the mixing process. A computational or empirical model may be used to calculate the velocity field at a suitable time scale. For example, after calculation, the velocity field may be assumed to be substantially constant until a substantial change occurs in the physical properties of the bioreactor contents (e.g., changes in viscosity due to biomass growth) or the manner in which the mixing process is driven (e.g., changes in input or output flow rates imposed by the control of the bioreactor).

Just as with spatial resolution, the temporal resolution of the metabolic and physical models need not be the same (at least in view of the above discussion). That is, time step 102 of modeling process 100 describes a metabolic process model. Aspects of integrating the modeling process 100 applied to portions of a heterogeneous bioreactor in which a mixing process (e.g., by CFD) or other physical process within the heterogeneous reactor is modeled are referenced herein as desired.

The modeling process 100 and the computational model 104, implemented in software (running on the hardware of the computing system), may simulate the operation of the virtual bioreactor 120 over a simulation time period consisting of a series of time steps 102. The modeling process 100 includes: receiving (from a user or a previous time step 102) a current value 150a of the process variable at the beginning of the time step 102; and generating updated values 150b of at least some of those process variables through a series of calculation steps at the end of time step 102. The current value 150a and the updated value 150b may be collectively referred to herein simply as "values" of the process variable in the context of no problem with respect to the precise timing of any given time step 102. The process variables may be a set of variables describing the computational model 104 of the bioreactor. In some embodiments, the process variables may reflect values of the control variables 152, process conditions, experimental conditions, media composition, bioreactor parameters, model constants, and/or various other terms describing parameters of the computational model 104. The process variables may include variables describing or indicating various physical, chemical, and biochemical properties of the extracellular solution 124, such as volume, temperature, acidity or pH, osmolarity, osmotic pressure, mass, density, and chemical and biochemical composition of the solution. Additionally, the process variable may be indicative of the amount or concentration of cellular biomass 130 disposed in the extracellular solution 124 within the modeled bioreactor 120. Additionally or alternatively, the process variable 150 may indicate the concentration or absolute amount of various inorganic or organic metabolites disposed in the extracellular solution 124 or within the biomass 130. The dissolved metabolites (whose extracellular or intracellular concentration is indicated by process variable 150) may include gases, salts, ions, metals, minerals, sugars, amino acids, fatty acids, and/or other organic acids and/or their conjugate bases. More specifically, the dissolved metabolites may include oxygen, carbon dioxide, ammonia, glucose, sucrose, lactose, lactate, glutamine, glutamate, alanine, asparagine, aspartate, and/or various other chemicals or biochemicals. The process variable may also indicate the concentration of one or more metabolites of interest (or simply, a product of interest), which may include antibodies, peptides, therapeutic proteins, or other biologicals. The process variable may also indicate the concentration of one or more of various metabolic byproducts (or simply, byproducts) that may be produced by the virtual cellular biomass 120 along with the target product. The target product and by-products may be collectively referred to as metabolites (or simply, products).

To determine the value 150b of the process variable that is updated from the current value 150a of the process variable, the modeling process 100 may calculate the velocity or rate of at least some of the fluxes 132a-132i associated with the cellular biomass 130. The fluxes 132a-132i may correspond to the growth 132a of cellular biomass 130, the uptake 132b, 132c of substrate metabolites, the removal or secretion 132d, 132e of metabolic target products and byproducts, and/or fluxes representing various intracellular metabolic pathways 132f-132 i. Intracellular metabolic pathway fluxes 132f-132i may represent metabolic conversions of several metabolites into several other metabolites, as indicated by the topology represented by the fluxes in fig. 1. For example, flux 132f may represent a reaction or metabolic pathway in which two metabolites combine to produce another metabolite, flux 132g may represent a reaction or metabolic pathway in which one metabolite produces another two metabolites, flux 132h may represent a reaction or metabolic pathway in which two metabolites combine to produce another two metabolites, and flux 132i may represent a reaction or metabolic pathway in which two metabolites combine to produce three metabolites. In some embodiments, growth flux 132a may be modeled as the output of another flux (which additionally represents an intracellular metabolic process). Collectively, the fluxes 132a-132i may represent a metabolic network used to model a given type of cell that produces the cellular biomass 130. The metabolic network for the modeled bacterial cells, such as e.coli, may be different from the metabolic network for the modeled mammalian germ cells, such as CHO cells. The fluxes 132a-132i in fig. 1 are for general illustration, and the metabolic network may contain fluxes containing from zero to over twelve inputs and/or outputs. Furthermore, the metabolic network may comprise 5, 20, 100, 500, 2500 or any other suitable number of metabolic pathways.

Intracellular metabolites disposed in cellular biomass 130 modeled by fluxes 132a-132i of the metabolic network may include some or all of the metabolites modeled in the extracellular solution: oxygen, carbon dioxide, ammonia, other gases, salts, ions, metals, minerals, glucose, sucrose, lactose, lactate, glutamine, glutamate, glycine, alanine, asparagine, aspartate, and/or various other chemicals or biochemicals. Intracellular metabolites involved in fluxes 132a-132i may also include various enzymes, proteins, amino acids, and/or ADP (adenosine diphosphate) and ATP (adenosine triphosphate) molecules involved in energy transfer. The variable indicative of the intracellular metabolite concentration may be included in the process variable.

The flux of each metabolic pathway can be quantified by a flux rate that characterizes the rate of conversion of the metabolite associated with the flux. The flux rate may also be referred to herein as flux velocity. Certain flux rates may represent, be equivalent to, indicate, or otherwise relate to the rate of change of some process variable with respect to time. For example, the rate of growth flux 132a may represent the rate of change of a process variable indicative of the concentration or total amount of cellular biomass 130 disposed in the extracellular solution 124. Under at least some of the modeled bioreactor operating conditions, the rate of metabolite uptake flux 132b or 132c can represent the rate of change of a process variable indicative of the amount or concentration of the corresponding metabolite in the extracellular solution 124. For example, the rate of oxygen uptake may represent the rate of change of a process variable indicative of extracellular oxygen concentration, while the rate of glucose uptake may represent the rate of change of a process variable indicative of glucose concentration. The rate of flux corresponding to the production of the target product or byproduct by the cellular biomass 130 may represent the rate of change of the process variable related to the amount or concentration of the corresponding target product or byproduct in the extracellular solution 124.

As described above, in addition to the extracellular solution 124 and the cellular biomass 130, the computational model 104 may also include a description of one or more virtual input streams 140 and/or one or more virtual output streams 142. Herein, for simplicity, the one or more input streams 140 may be referred to as a singular "virtual input stream" or "input stream", and for simplicity, the one or more output streams 140 may be referred to as a singular "virtual output stream" or "output stream". The description of the streams 140, 142 may include variables that indicate the composition and/or flow rate associated with the input stream 140 and the output stream 142. In some embodiments, the absolute value of the sum of the volumetric flow rates of the liquid input stream 140 and the absolute value of the sum of the volumetric flow rates of the output stream 142 are equal, which results in a constant volume of extracellular solution. Embodiments in which the flow rates of the input stream 140 and the output stream 142 are not balanced may result in a change in the volume of the extracellular solution 124. The virtual input stream 140 and the virtual output stream 142 may affect the rate of change of process variables, including process variables indicative of the composition of the extracellular solution 124, for example, by replenishing substrate metabolites and/or consuming the virtual contents of the modeled bioreactor 120.

Returning to the modeling process 100, the current values 150a of the process variables may be used as inputs to a dynamics model 160. The kinetic model 160 may calculate several constraints 164 on flux rates (i.e., the velocities of at least some of the fluxes 132a-132i associated with the biomass 130). For example, the constraint 164 may specify an upper limit on flux rate representing substrate metabolite uptake. Thus, constraints 164 may constrain the rate of change of process variables indicative of the amount or concentration of substrate metabolites associated with the constrained flux. For example, the constraints may represent a maximum uptake rate of oxygen, glucose, asparagine, and/or glutamine and/or glycine, and, correspondingly, a maximum rate of change of process variables indicative of the amount or concentration of oxygen, glucose, asparagine, glutamine and/or glycine in the extracellular solution 124 in the absence of the influence of the input stream 140 and the output stream 142.

The dynamics model 160 can calculate the effect of the current value 150a of the process variable on the metabolic reaction kinetics, which in turn affects the constraints on the flux rate. For example, the maximum uptake rate may depend on other process variables that indicate or describe physical or chemical properties of the extracellular solution 124, as calculated by the kinetic model 160. As more specific examples, process variables indicative of extracellular solution temperature, acidity or pH and/or osmolarity may affect the maximum uptake rate.

The kinetic model 160 can calculate the effect of the current value 150a of the process variables indicative of temperature, acidity and/or osmolality on the maximum metabolite uptake rate by weighting or multiplying the optimal maximum uptake rate by a process condition factor (which results in a deviation from optimal conditions). Under ideal process conditions, the optimal maximum uptake rate may correspond to an upper limit for the uptake rate. The process condition factors may then serve as correction factors that indicate a decrease in the rate of uptake due to non-ideal conditions. The process condition factor may be defined in various functional forms. For example, for a given metabolite uptake, the process condition factor may have a uniform value for the optimal process condition, and as the deviation from the optimal process condition increases, the uniformity may decrease. The process condition factors may take different or similar forms for different metabolite uptake rates, and the optimal conditions may be different or similar for different metabolite uptake.

In some embodiments, the kinetics model 160 calculates the effect of current values 150a on the metabolic reaction kinetics, which are indicative of the concentration of one or more metabolic byproducts. For example, concentrations of ammonia and lactate may reduce the maximum uptake rate of glucose, glutamine, asparagine, and other substrate metabolites.

In some embodiments, the kinetic model 160 models the effect of the current value 150a of the process variable on the rate constraints of the intracellular metabolic flux and corresponding energy expenditure rate responsible for cell maintenance. The cell maintenance flux may model a metabolic pathway that consumes some of the energy available to the cellular biomass 130 in the form of ATP without growing the cellular biomass 132. In some embodiments, the cell maintenance flux can be tightly coupled with the flux that produces the product of interest. In other words, calculating the constraint on flux rate production for the target product may include calculating a lower limit on energy expenditure rate for cell maintenance.

To calculate the lower limit on the rate of energy expenditure for cell maintenance, the kinetic model 160 may take into account the influence of the relevant current value 150a of the process variable. In some embodiments, a "stress" variable may be defined to quantify the effect of a process variable on the lower limit of the rate of energy expenditure for cell maintenance. In turn, the stress may be a function of the change in extracellular solution temperature over time and the temperature difference experienced or sensed by the cellular biomass 130. In some embodiments, stress is assessed as the cumulative effect of temperature differences integrated over time. Additionally or alternatively, in some embodiments, different stress variables help calculate the minimum energy expenditure rate for cell maintenance, and/or stress variables result in states or changes in temperature, osmolality, concentration of byproducts, and/or high cell density of the biomass 130.

The calculation of the effect of the current process variable 150a on the flux rate constraint 164 by the dynamical model 160 may be based on experimental data. The mathematical formula used for the calculation may be derived from theoretical models, may be empirically derived, and/or may combine empirical factors with theoretical derivations. In any case, the formula for calculating the effect of the current process variable 150a may include coefficients or model parameters that may be obtained by a calibration procedure (e.g., by fitting calibration data). In some embodiments, the calibration data comprises experimental data. Additionally or alternatively, the computationally generated data may be used for at least some of the calibrations. Fitting, regression, and/or optimization algorithms may help calculate kinetic model parameters from the calibration data.

In some embodiments, the kinetic model updates at least some of the process variables based on the control variables 152 prior to calculating the effect of the process variables on the flux rate constraints 164. The control variables may include a temperature setting, an acidity setting, or any other suitable variable that may represent a virtual control setting for bioreactor 130. The modeling process 100 may receive the control variable 152 prior to beginning the simulation (with the relevant time within the simulated time period), or may receive the control variable 152 at any time within the simulated time period.

As discussed above, constraints 164 calculated by the kinetic model 160 may define upper or lower limits on the rate of some of the fluxes 132a-132i that are used as models of the metabolic pathways of the virtual cellular biomass 130. The modeling process 100 may then use the constraints 164 to perform a flux balance analysis 170 that simultaneously calculates flux rates 174 for all modeled fluxes within the metabolic network of the cellular biomass 130. The flux balance analysis 170 may determine flux rates that ensure that the intracellular metabolite concentrations in the modeled biomass 130 are substantially at steady state. For example, if one of the fluxes depletes certain intracellular metabolites while producing other metabolites, the other fluxes may balance the first flux by: consuming or removing all metabolites produced by the first flux and producing or ingesting all metabolites consumed by the first flux from the extracellular solution 124. Metabolites taken up by the flux into the cellular biomass 130 may be removed from the extracellular solution 124, while metabolites removed by the flux from the cellular biomass 130 may be deposited into the extracellular solution 124. Thus, while a flux equilibrium assay may keep the rate of change of the concentration or amount of an intracellular metabolite at essentially zero, it may have an effect on the rate of change of extracellular metabolites, which may include substrate metabolites, target products and by-products, and related process variables.

Additionally or alternatively to facilitating calculation of the change in extracellular metabolite concentration, the flux balance analysis 170 may determine the growth rate of the cellular biomass 130. In some embodiments, the determined growth rate may be negative as a model of cell death or depletion of cellular biomass 130. Death or loss of the modeled cellular biomass 130 may occur as follows: constraints on flux rates include a lower limit on the rate of energy consumption for cell maintenance, and metabolic fluxes are unable to produce energy at a desired rate given other flux rate constraints. In this case, the modeling process 100 may ignore the minimum maintenance energy constraint for the flux balance analysis 170 and may calculate the death rate based on the deficiency in maintaining the energy production rate.

In addition to the constraints 164 calculated by the kinetic model 160, the flux balance analysis 170 may also use metabolic targets 172 (or simply, metabolic targets) defined by the modeling process 100 to represent the biological targets of the virtual biomass 130. Without the metabolic objective function 172, the flux balance analysis 170 may not yield a unique solution, but may yield a solution space with many or an infinite set of possible flux rates that maintain steady state of intracellular metabolites. Thus, the effect of the metabolic objective function may be to limit the flux balance analysis 170 to obtain a unique solution. In other embodiments, the solution of the flux balance analysis 170 may not be unique, even if the metabolic objective function 172 is included.

Various metabolic objective functions 172 may be used in different embodiments. For example, the metabolic objective function 172 may result in a flux balance analysis 170 solution that yields a maximum in the growth rate of the cellular biomass 130. Alternatively, the metabolic objective function 172 may result in a flux balance analysis 170 solution that yields a minimum of a linear combination of metabolite uptake rates. The linear combination may be a sum or a weighted sum of some or all of the uptake flux rates. In some embodiments, the metabolic objective function combines multiple requirements, limitations, or constraints. For example, the metabolic objective function 172 may combine maxima on growth rates while minimizing linear combinations of uptake flux rates. In another embodiment, the metabolic targets 172 minimize a linear combination of uptake flux rates while maximizing flux associated with cell maintenance energy.

Some or all of the flux rates 174 calculated by the flux balance analysis 170 may correspond to the rate of change of some of the process variables without the effect from the input stream 140 and the output stream 142. The rate 174 may be an input to a rate integration module 180 that may use the rate to calculate an updated value 150b of the process variable. The updated value of the process variable 150b may then become the current value of the process variable 150a for the next time step.

The rate integration module 180 may calculate the time rate of change of the process variable using the flux rate 174 provided by the flux balance analysis module 170. In some embodiments, the time rate of change of some of the process variables directly corresponds to some of the flux rates 174. In other embodiments, input and output flow variables 182 describing the input flow 140 and output flow 142 facilitate calculating the rate of change of the process variable. The input and output flow variables 182 may contain information about the flow rates and compositions of the input flow 140 and the output flow 142. When modeling the bioreactor 120 in batch mode of operation, the effect of the input stream 140 and the output stream 142 can be neglected. When modeling the bioreactor 120 in the fed-batch mode of operation, the flow variable 182 corresponding to the input flow 140 may have an effect on the calculation of the updated value of the process variable 150 b. When modeling the bioreactor 120 in a continuous or perfusion mode of operation, the flow variables 182 corresponding to both the input flow 140 and the output flow 142 may each have an effect on the calculation of the updated value 150b of the process variable. In some embodiments, only the flow variable 182 corresponding to the output flow 142 has an effect on the rate of change of the process variable 150 when, for example, the contents of the modeled bioreactor 120 are virtually harvested without supplementation.

The flow variables 182 defining the input flow 140 and the output flow 142 (both of which contribute to the calculation of the updated set of process variables 150 b) may include the total volumetric flow rates (which may or may not be equal) of the flows 140, 142, as well as the flow variables defining the composition of the input flow 140 and the output flow 142. The composition of the input stream 140 may be directly defined by the input flow variables (i.e., independent of the calculations performed in the modeling process 100), while the calculation of the composition of the output stream 142 may depend on the current values 150a of the process variables describing the composition of the extracellular solution 124. Additionally or alternatively, the flow variable 182 that has an effect on the calculation of the output flow 142 composition may define the selectivity of the output filter 144. For example, the glucose concentration of the input stream 140 may be directly defined as the input flow variable, while the glucose concentration of the output stream 142 may be calculated from an output flow variable indicative of the glucose selectivity of the output filter 144 and a process variable indicative of the glucose concentration in the extracellular solution 124. Similarly, for other metabolites or cellular biomass 130, the calculation of the concentration in the output stream 142 may depend on process variables indicative of the respective concentrations in the extracellular solution 124, as well as output flow variables indicative of the respective values of the output filter 144 selectivity.

When modeling bioreactor heterogeneity, the input stream 140 and the output stream 142 may represent flows that connect different bioreactor sections, each as a model of a homogeneous bioreactor (e.g., bioreactor 120), as discussed herein. Thus, the input and output flow variables 182 may be calculated in view of the velocity field obtained from the physical model (e.g., CFD model).

The rate integration module 180 may construct a differential or difference equation using the flux rate 174, the input/output flow variable 182 that combines the input flow variable and the output flow variable, and the current value 150a of the process variable. The rate integration module 180 may then calculate the updated value of the process variable 150b using a numerical differential or differential equation solver. The time step 102 of integrating the rate may be equal to a few seconds, minutes, hours, days, or any other suitable time period, depending on the implementation. Once the updated values 150b of the process variables are calculated, the modeling process 100 may replace one or more of the current values 150a of the process variables with the updated values 150 b. The replaced current value 150a may be discarded or saved in computer memory for further computation. The modeling process repeats in the next time step (e.g., the same duration as time step 102) using the new set of current values 150a, and repeats until the simulated time period expires.

In a heterogeneous bioreactor, the current value 150a and the updated value 150b of the process variable can be considered as samples or representative values (for each bioreactor section) of the spatial distribution of the process variable. Thus, the differential or difference equation constructed by the integration module 180 may be a partial differential or difference equation having a partial time derivative, as described below. The integration module 180 may construct a spatial partial derivative (or difference) based on the process variable values in the neighboring regions. The integration module 180 may combine the spatial partial derivative of the process variable value with the velocity field (e.g., calculated by the CFD model) to calculate an updated value 150b of the process variable. In an example embodiment, the dot product between the gradient vector and the local velocity of the process variable may represent the difference between the input stream 140 and the output stream 142 for a limited volume region representing a portion of the heterogeneous bioreactor.

In some embodiments, some of the process variables are updated based on the received values of the control variables 152. In some embodiments, the dynamical model 160 updates the value of the process variable using the control variable and then uses the process variable to calculate constraints 164 for the flux rate. Other portions of the modeled process may also update the process variables based on the control variables 152. The control variables may include temperature settings at different times of the simulated time period, times to change the filters 144, times to change the rates and/or compositions of the input stream 140 and the output stream 144, or any other suitable variable that may represent a virtual control setting for the bioreactor 130. In some embodiments, the rate integration module 180 may construct and solve differential or differential equations that represent the gradual response of the process variables to the corresponding control variables. For example, as the temperature control variable changes from one time step 102 to the next, the process variable for the extracellular solution temperature may be adjusted over multiple time steps by the rate integration module 180. In other embodiments, the process variable immediately accepts the value of the control variable, thereby modeling a faster response than the virtual time step 102.

The foregoing discussion illustrates exemplary embodiments of the dynamical model 160 and provides details of these exemplary embodiments. As discussed above, the kinetic model 160 generates a plurality of constraints 164 on flux rates by modeling one or more effects of at least some of the current values 150a of the process variables on the metabolic reaction kinetics. For example, the modeled effects may include the effect of temperature, acidity, and/or osmolality on the upper limit of the metabolite uptake rate. The model may define an upper limit on the flux rate associated with a given metabolite uptake for some ideal conditions, which may include ideal temperature, ideal acidity, and/or ideal osmolality. The dynamical model 160 may calculate the effect of non-ideal conditions by multiplying the upper limit for ideal conditions by a correction factor. A correction factor indicative of a decrease in uptake rate due to non-ideal conditions may be defined as a function of one or more variables such as temperature, acidity, and/or osmolarity. In some embodiments, the correction factor or includes the effect of the concentration of metabolic byproducts (e.g., lactate or ammonia).

For example, under ideal conditions, the upper limit on the uptake rate for glucose can be modeled by the following equation:

wherein when conditions for glucose uptake are ideal, and when the glucose concentration in the extracellular solution is sufficiently large that any further increase does not result in a significant increase in the rate of glucose uptake, v_GLCApproach toThe saturation value of (c). In equation 1, [ GLC]Is the glucose concentration, andis a kinetic model parameter describing the dependence of the glucose uptake flux rate on glucose concentration. When conditions are not ideal for glucose uptake, a correction factor (e.g., C)_GLC(T，pH，π，[LAC]，[NH3]) < 1) can be multiplied by the glucose uptake rate, wherein the correction factor can depend on a number of factors including temperature T, acidity pH, osmolality π, lactate concentration [ LAC ]]And/or ammonia concentration [ NH3]：

The correction factor may depend on 1, 2, 3, 4, 5, or any other suitable number of current values of the process variable. Correction factor C_GLC(T, pH) may represent a correction dependent on temperature and acidity, while the correction factor C_GLC(T，π，[NH3]) A correction depending on the temperature, the amount permeated and the ammonia concentration can be expressed. Correction factors that depend on multiple variables may combine multiple correction factors that depend on one or more variables. E.g. C_GLC(T，pH，π，[LAC]) The following four individually defined correction factors can be combined: e.g. correction factor for temperatureCorrection factor for acidityCorrection factor for osmolarityAnd a correction factor for lactate concentrationThe combination of constitutive factors can take various mathematical expressions that can properly represent the dependence of reaction kinetics on environmental factors in a mechanistic model. For example, in some embodiments, correction factors that depend on multiple variables combine constituent correction factors by multiplication, as illustrated by the following equations:

in another embodiment, correction factors that depend on multiple variables combine constituent correction factors in another manner:

whereinAre the kinetic model 160 parameters for the corrections. In the above equation, ifBoth the correction factor for temperature and the correction factor for acidity will have to be zero so that the total correction factor is also zero.

The above description of the correction factor as a correction factor for glucose uptake is for illustrative purposes only. Correction factors for glutamine uptake, asparagine uptake, or additional metabolite uptake may have similar forms. In general, for any metabolite M, the correction factor may depend on a number of process variables, including, for example, temperature, acidity, osmolality, lactate concentration, ammonia concentration, and/or any other metabolite (generally designated M2) that may affect the maximum uptake flux rate for M. The resulting correction factor C_M(T，pH，π，[LAC]，[NH3]，[M2]) The following individually defined correction factors may be combined: correction for temperatureFactor(s)Correction factor for acidityCorrection factor for osmolarityCorrection factor for lactate concentrationCorrection factor for ammonia concentrationAnd a correction factor for M2 concentration

The correction factor for a single process variable may take various mathematical forms that represent the mechanical dependence of the flux rate constraint on the environment. These correction factors may be derived from first principles or observed empirically. The correction factor for temperature may take the form:

wherein the temperature is between a minimum and a maximum, thus T_min＜T＜T_max，d_tempIs a positive coefficient of dependence, andthe correction factor is normalized to have the maximum unity value. The correction factor for acidity may take the form:

wherein the acidity expressed by the pH is between a minimum and a maximum, thus the pH_min＜pH＜pH_maxAnd is andthe correction factor is normalized to have the maximum unity value. The correction factor for the amount of bleed may take the form of the following function:

wherein pi > 0 is the seepage amount, pi_SAnd pi_0.5To determine the coefficient of dependence on osmolarity, andthe correction factor is normalized to have the maximum unity value. The correction factor for the effect of metabolite concentration may take the form of a saturation equation:

wherein [ M2]Concentration of metabolites to limit uptake flux, andto determine the uptake flux rate limit value pair for M [ M2]Coefficient of dependence of (c).

In an exemplary embodiment, the maximum uptake rates for glucose, glutamine, asparagine, and oxygen, respectively, can be calculated in the following manner:

wherein the correction factor due to non-ideal temperature and acidity is C_GLC(T，pH)、C_GLN(T，pH)、C_ASN(T, pH), andand glucose [ GLC]Glutamine [ GLN ]]Asparagine [ ASN ]]Lactate [ LAC ]]Glutamate [ GLU ]]Ammonia [ NH ]₃]Aspartic acid salt [ ASP ]]Has an effect on some uptake rates, and kinetic model 160 parameters used to calculate the limits for the uptake flux rates described above are summarized in table 1 below in terms of the relevant kinetic mechanisms. In some embodiments, the input to the equation is the extracellular concentration of the metabolite, and in some embodiments, the input to the equation is the intracellular concentration. In other embodiments, a combination of extracellular and intracellular concentrations of the metabolite may be an input to the calculation of the constraints 164 on flux rates. In some embodiments, additional kinetic model 160 parameters quantify the dependence of uptake flux rate on temperature, acidity, and combined correction factors.

TABLE 1 kinetic model parameters

Additionally or alternatively to calculating constraints on metabolite uptake flux rates, the kinetic model 160 may calculate constraints on one or more flux rates of metabolic processes that determine energy expenditure for cell maintenance and/or production of target products. The constraint on metabolite uptake rate may be an upper limit on uptake rate and the constraint on maintenance energy may be a lower limit on the relevant flux rate. The maintenance energy represents the modeled metabolic demand of the virtual cellular biomass 130, which can reflect the energy used by real cells to live in a real environment. Energy can be expressed by intracellular concentration or amount of ATP. When the energy or ATP produced by the cellular biomass 130 exceeds that required for maintenance, the cellular biomass can grow at a positive growth rate. On the other hand, when the cellular biomass 130 is unable to produce the amount of ATP or energy required for maintenance, the cellular biomass 130 may begin to diminish or nearly die. The modeling process 100 may account for the reduction in cellular biomass 130 due to cell death or depletion by assigning a negative growth rate to the equation for growth.

The lower bound of cell maintenance energy required to prevent reduction of cellular biomass may depend on various environmental factors represented by one or more process variables. The kinetic model 160 may calculate stress variables or factors S to represent at least some of the environmental factors that affect the need to maintain energy. The modeling process 100 may then calculate the minimum maintenance energy as the minimum rate of ATP consumption, as follows

Wherein v is_ATP，minIndicates maintenance energy requirement in the absence of stress (or S ═ 0), and k_S，mntIs a coefficient that regulates the effect of stress variables. While the above equation represents maintaining a linear dependence of energy demand on S, the dynamical model 160 may use other functional forms, including polynomials, square roots or other fractional powers, logarithms, exponents, or combinations of the foregoing, or any other suitable relationship. The stress variable itself may be calculated based on temperature, acidity and/or any other suitable process variable and its change over time. Stress variables may include dependence on metabolite (e.g., metabolic byproduct) concentrations. However, in some embodiments, the kinetic model 160 may include some metabolite concentrations aloneThe effect in the minimum maintenance energy is calculated as shown in the following exemplary equation:

wherein k is_mntIs when a significant lactate concentration [ LAC ] is present in the extracellular solution (or in some embodiments, the intracellular solution)]＞＞K_mnt，LACAnd ammonia concentrationMaximum additional maintenance energy or ATP consumption rate approached, and K_mnt，LAC、Kinetic model parameters to determine the effect of lactate and ammonia on the minimum maintenance energy, respectively. While the above equation shows a single additional term to account for the effects of both metabolites, a single additional term or any other suitable functional dependence may account for metabolite concentrations or other environmental factors that affect the minimum maintenance energy or minimum ATP consumption rate for cell maintenance.

After calculating the lower bound for the sustaining energy, the kinetic model 160 may additionally calculate a minimum rate of production of the target product. The product of interest may be an antibody or another suitable biologic. The kinetic model 160 may calculate a lower constraint for the production of the target product by: a proportional relationship between minimum maintenance energy and product production is used and correction factors are applied that account for non-ideal conditions for product production. For example, the kinetic model 160 may calculate the minimum rate of product production, where the product of interest is an antibody, by the following equation:

wherein C is_ANTI(T, pH) is a correction factor,this correction factor accounts for the reduction in antibody production due to non-ideal temperature and acidity; [ LAC]Is lactate concentration; k_i，ANTI/LACKinetic model parameters reflecting the inhibition of lactate on product formation; and β is a proportionality constant that is related to the ATP consumption rate and the antibody production rate for maintenance. Correction factor G_ANTI(T, pH) may have a similar functional dependence on temperature and acidity as a correction factor for metabolic uptake rate. For example, other correction factors may be included in the calculation of the lower limit of antibody or additional target product production, which account for additional inhibition of the metabolite.

The kinetic model parameters (such as those used in equations 1-15 and listed in table 1) may be obtained using a calibration procedure (or simply, calibration). Experiments performed for calibration purposes may provide experimental calibration data. The experiment used for calibration may be a perfusion experiment, a small-scale batch experiment or chemostat experiment, and/or any other suitable experiment. Additionally or alternatively, the calibration data may be modified from published literature, theoretical calculations, and/or any suitable combination of sources. The calibration procedure may apply various regression, fitting, or optimization techniques and/or algorithms (including, for example, Levenberg-Marquardt, differential evolution, and/or genetic algorithms) to the calibration data to find the kinetic model parameters. The calibration procedure may combine global and local optimization algorithms and may use one or more of various suitable cost functions or objective functions. The calibration procedure may calculate the kinetic model parameters separately and prior to execution of the modeling process 100. However, in some embodiments, one or more portions of the calibration procedure are integrated into the kinetic model 160. Further, in some embodiments, different sets of kinetic model parameters are applied at different stages of the simulation, as determined, for example, by the current values of the process variables 150 a.

The experimental calibration data may include concentrations of the metabolite sets at different time points. The cumulative calibration error to be minimized may be defined as:

wherein [ M ] is_k]_sim(t) is the simulated concentration at time t of the kth metabolite, [ M [_k]_exp(t) is the experimental concentration at time t of the kth metabolite, and w_kIs the weight of the error of the kth metabolite in the cumulative error. In this embodiment, the k-th metabolite error is calculated as the sum of the squares of the relative concentration errors at different times. In other embodiments, the error may be based on a combination of absolute concentration error and relative concentration error. The cumulative error is a weighted sum of the errors for each metabolite. The weights for different metabolites may emphasize the importance of some metabolite concentrations in the calibration while suppressing or excluding other metabolite concentrations, and may be experiment dependent. For example, the concentration of glucose may be more relevant to experiments corresponding to the growth phase of a biological process, while the concentration of antibodies may be more relevant to experiments corresponding to the production phase. In general, various suitable experimental and error function equations may be suitable for calibrating different kinetic model parameters.

The following discussion sets forth and provides details of exemplary embodiments of the flux balance analysis 170. The flux balance analysis 170 uses the constraints 164 (such as, for example, the maximum metabolite uptake rate, the minimum energy expenditure rate for cell maintenance, and the associated minimum rate of target product production) generated by the kinetic model 160 to calculate a complete set of flux rates 174 for the metabolic model of the cellular biomass 130. Different metabolic models may take into account different metabolites and different metabolic pathway arrangements. Each metabolic pathway flux may represent a reaction or set of reactions that convert one set of metabolites into another set of metabolites. The proportion of metabolic pathway flux that consumes or produces a metabolite forms a stoichiometric coefficient for that flux. Table 2 below illustrates an exemplary network of metabolic fluxes (as a model for CHO cells) configured for production of antibody products.

TABLE 2 exemplary network of Metabolic pathway fluxes

The subscripts distinguish cytosolic or cellular metabolites (c), extracellular metabolites (e) and mitochondrial metabolites (m) as part of the cellular process. Table 3 below lists and describes the metabolites involved in the metabolic flux in Table 2.

TABLE 3 metabolite definitions

The flux balance analysis 170 of the modeling process 100 may calculate flux rates for each of the metabolic pathways. Continuing with the example of the metabolic network described in table 2, the flux balance analysis 170 may determine the flux rate for each of the 34 metabolic pathways, subject to a number of conditions, limitations, constraints, or goals. One constraint on the calculation of the flux may be the requirement to maintain a steady state of the cellular biomass 130 metabolite concentration. For example, metabolic pathway 33 in table 2 can convert hydrogen, reduced nicotinamide adenine dinucleotide and pyruvate to the non-reduced forms of nicotinamide adenine dinucleotide and lactate in equal proportions as indicated by a uniform stoichiometric coefficient. Other metabolic flux sets may then produce hydrogen, reduced nicotinamide adenine dinucleotide and consume the unreduced forms of nicotinamide adenine dinucleotide and lactate. As another example, the metabolic pathway 14 of table 2 may produce hydrogen by taking hydrogen from the extracellular solution 124 or consume lactate by removing lactate into the extracellular solution 124, but may not accomplish both simultaneously. There may be no pathway for pyruvate uptake from the extracellular solution 124 or pyruvate removal into the extracellular solution 124, resulting in a need that pyruvate may need to be balanced by intracellular pathway flux. In general, intracellular metabolite concentrations can be balanced at steady state by intracellular pathway flux or interaction with extracellular solutions. Equation ofSteady state conditions for flux balance analysis 170 are described, whereIs a stoichiometric matrix of metabolic networks, and velocity vectorsIs a vector of all flux rates. However, steady state conditions may not be sufficient to find a unique solution for the flux balance analysis 170, and thus, the flux balance analysis 170 may require other constraints 164 to limit the set of possible solutions.

Some of the metabolic pathways explain the removal of metabolites from the cellular biomass 130 or the uptake of metabolites from the extracellular solution 124 into the cellular biomass 130. The flux balance analysis 170 steady state solution may depend on constraints on the rate of uptake or removal of flux. For example, as discussed above with reference to the kinetic model 160, the uptake flux of a substrate metabolite may be constrained by the concentration of the corresponding metabolite in the extracellular solution 124 and non-ideal environmental conditions. For example, for the metabolic network described in table 2, oxygen may be present in significant amounts in the extracellular solution as indicated by metabolic pathway 5 of table 2 (ignoring extracellular oxygen input). However, the maximum rate of oxygen uptake may be constrained by equation 12, which takes into account the maximum rate of oxygen uptake for ideal conditions and the non-ideal condition correction factor indicated by the current value 150a of the process variable. In some embodiments, at least some of the metabolic flux rates corresponding to metabolic pathways 3-17 of table 2 are constrained by rate constraints 164 calculated by the kinetic model 160. In other embodiments, the rate constraints 164 calculated by the kinetic model 160 apply to intracellular metabolic pathways that may be indirectly affected by extracellular conditions and uptake rates. For example, the glutamine constraint of equation 10 can limit the flux rate of metabolic pathway 27, the glucose constraint of equation 9 can limit the flux rate of metabolic pathway 34, and the asparagine constraint of equation 11 can limit the flux rate of metabolic pathway 25.

The flux balance analysis 170 may not be able to calculate a unique set of flux rates for steady state conditions under the constraints 164 generated by the kinetic model 160, and additionally may obtain at least one metabolic target 172 defined for the cellular biomass 130. The metabolic targets for the virtual cellular biomass 130 in the computer simulation model 104 may correspond to real biological targets of in vivo cell culture. Various metabolic objective functions may constrain the flux balance analysis 170 in different embodiments (representing different biological alternatives). The metabolic objective function 172 may maximize the growth rate of the cellular biomass 130, minimize a linear combination of metabolite uptake rates, maximize the flux rate of a pathway related to energy or ATP production, or generate another limit for the set of flux rates. In general, the metabolic objective function 172 may calculate a flux rate represented by a velocity vector (e.g., a velocity vector that may be represented by a metabolic network)(flux rate set) representation) cost or value of any set. In some embodiments, the flux balance analysis 170 minimizes costs or values, while in other embodiments, the flux balance analysis 170 maximizes costs or values. In addition, the metabolic objective function may depend on various current values 150a of the process variables, their history, and various other variables describing the cellular biomass 130 and its environment at the current time and previously simulated times.

Taking into account the flux velocity v₁，v₂，...v_N-1，v_N(these flux rates are velocity vectorsElement(s) of a set of N metabolic pathways or reactions, the metabolic objective function 172 may take a variety of mathematical expressions, as exemplified in table 4 below. The table also provides an explanation and motivation for the metabolic objective function.

TABLE 4 Metabolic goals

For a given metabolic objective function 172, the flux balance analysis 170 may not be able to find a solution that satisfies all of the constraints generated by the kinetic model 160. Specifically, if the constraint on the minimum rate of sustained flux fails to satisfy the objective function 172 (which maximizes the ratio of growth or growth to some function of other flux rates), the flux balance analysis 170 may switch to a different metabolic target 172 and possibly ignore the constraints on the minimum sustained flux rate and minimum production rate of the target product. For example, the flux balance analysis 170 may switch to a metabolic target 172 that maximizes the maintenance flux rate. After finding a steady state solution that maximizes the maintenance flux rate, the flux balance analysis 170 can calculate a death rate or loss rate for the cellular biomass 130 based on the difference between the ignored minimum maintenance flux rate and the achieved maintenance flux rate while satisfying the remaining constraints 164 (e.g., maximum uptake rate). The death rate equation, which represents the rate of change of biomass 130 relative to the biomass concentration, can be calculated using the following

v_d＝c_d(v_ATP，min-v_ATP) (equation 17) of the process,

wherein v is_ATP，min-v_ATPIs a positive difference between the minimum maintenance flux rate and the calculated insufficient maintenance flux rate, and c_dAre scaling factors, which are parameters of the modeling process 100.

The following discussion sets forth and provides details of exemplary embodiments of the rate integration module 180. Once the flux balance analysis 170 calculates the flux rate 174 (including the growth rate or death rate of the cellular biomass 130), the rate integration module 180 can use the calculated flux rate 174 to calculate the updated value 150b of the process variable. The rate integration module 180 may mass balance the virtual contents of the bioreactor 120. Additionally or alternatively, the rate integration module 180 may calculate the consumption of the bioreactor 120 inputs (including substrate metabolites) and/or the production of the target product.

The rate integration module 180 may first calculate the time rate of change of the process variable and then solve or numerically integrate the differential equation or differential equation in discrete time form for the process variable. The equation for the time rate of change of the process variable may include contributions from the input stream 140 and the output stream 142. Some exemplary equations for a set of process variables are shown below. For example, a general equation for the time rate of change of metabolites in extracellular solution can be written as

Wherein [ M ] is_e]Is the concentration of the metabolite in the extracellular solution 124 (i.e., the process variable under consideration) [ BIOM]As is the concentration of biomass in the extracellular solution 124,to target the flux of extracellular metabolite M in metabolic pathway i_eStoichiometric coefficient of (v)_iFlux rate calculated by flux balance analysis 170 for flux i, V is the volume of extracellular solution 124, F_INIs the volumetric flow rate of the input stream 140,is the metabolite M (and extracellular metabolite M) in the input stream 140_eSame substance), F_OUTIs the volumetric flow rate of the output stream 142, and T_MIs the transport or selectivity factor of the filter 144 for the metabolite M. Expression formulaIs all flux pairs M in the metabolic network_eAnd when (respectively) only the flux is taken up or only the flux is removed (excreted)Quantity influence M_eCan be reduced to-v_MOr v_M. In embodiments of modeling process 100 having multiple input streams, the summation of multiple terms representing multiple input streams may be substitutedThe volume V may be a process variable, the rate of change of which is given by

And in embodiments wherein the volume is kept substantially constant, F_IN＝F_OUT. In some embodiments, there is no filter 144 for the output stream 142 and for each metabolite T_MWhile in other embodiments, the filter does not substantially filter a given metabolite, resulting in T for a given metabolite_M＝1。

For a heterogeneous reactor, equation 18 may represent the rate of change of metabolites in a given portion of the bioreactor, and the equation may be adjusted to account for the heterogeneity and mixing mechanics, as discussed herein. In some embodiments, particularly embodiments with relatively fine spatial discretization, the additional terms representing the input stream 140 and the output stream 142 may be replaced by the dot product of the metabolite concentration gradient and the velocity field. In a sense, this method computes the full derivative with respect to time based on the partial derivative with respect to time and the partial derivative with respect to the spatial dimension. In other embodiments, additional terms representing the input stream 140 and the output stream 142 may be the result of adding flow contributions from adjacent portions of the heterogeneous bioreactor according to the velocity field. In some bioreactor volume discretizations, one portion may comprise an input stream entering from outside the reactor, while another portion may comprise an output stream from the reactor.

Returning to the discussion of the homogeneous bioreactor model, although, as discussed, general applicability to the heterogeneous bioreactor is not lostEquation 18 may be modified to accommodate one or more particular metabolites. For example, consider an embodiment of the modeling process 100 in which the cellular biomass 130 is modeled by a metabolic pathway in Table 2, where the input stream 140 and the output stream 142 have a flow rate F_IN＝F_OUTThe glucose concentration in the input stream 140 is FAnd the perfusion configuration is T_GLC＝T_ANTI1 and T_BIOMThe equation for the time rate of change of glucose concentration, biomass concentration, and antibody product concentration may be 0:

wherein v is₁₅Flux rate, v, for pathway 15 for glucose uptake₁₆Is the flux rate of pathway 16 for antibody removal, and v₁₇Is the flux rate of pathway 17 for biomass removal. While there may be no distinction between intracellular and extracellular biomass in some embodiments, the definitions in table 2 distinguish between biomass production pathway 19 and biomass removal pathway 17, where the extracellular biomass concentration represents cellular biomass 130 in extracellular solution 124.

The rate integration module 180 can calculate the time rate of change of the cellular biomass 130 based on the death rate or depletion rate calculated in the flux balance analysis 170, as follows

Wherein [ BIOM]Is the concentration of cellular biomass 130 in the extracellular solution 124, and v_dIs the death rate or wear rate calculated in the flux balance analysis 170. While biomass 130 may only account for virtual living cell biomass in which the metabolic process continues, in some embodiments, the concentration or amount of dead cells or dead biomass is another of these process variables that is updated by modeling process 100. From [ BIOM]Any reduction in the value of the virtual living biomass 130 represented may contribute to a corresponding increase in the value of the process variable representing dead biomass. Although dead biomass may not have a metabolic process that continues, it may still affect the properties of the extracellular solution 124.

Throughout the duration of the time period simulated by the modeling process 100, the rate integration module 180 may calculate the total amount of substrate metabolites supplied by one or more input streams 140 and consumed by the cellular biomass 130 and the total amount of products removed or collected by one or more output streams 142. For example, the following calculations can be used for the collected antibody products

Wherein ANTI is the molar amount or mass of antibody product collected, T_ANTIFor time-varying transmission of antibody through the output filter 144, F_OUTIs the time-varying flow rate of the output stream 142, and [ ANTI]Is the molar concentration or density of the antibody in the extracellular solution 124. The flow rate and transmission may be varied over time to represent different operating scenarios or analog output filter performance, and may be represented by process variables that are updated by the rate integration module 180. The operating protocol may represent the accumulation phase of the product in the extracellular solution 124 and the harvest phase that is initiated once the product concentration is at a sufficient level. In a real bioreactorIn operation, the separate accumulation and harvest stages may allow for a reduction in the cost of separating the product from the output stream 142. Filter transfer function T_ANTIMay represent a step change in delivery when the filter 144 is replaced or a gradual change in delivery due to retention of some of the filtered metabolites or other components of the output stream 142.

In some embodiments, the rate integration module 180 may calculate the stress on the cellular biomass 130 from changes in temperature, pH, and/or other process variables. These changes in the process variable may depend on the control variable 152. In some embodiments, the control variables 152 are processed by the flux balance analysis 170 and/or the rate integration module 180. Stress may accumulate over time, which makes rate integration module 180 suitable for calculating an updated process variable that is indicative of stress. Additional process variables may aid in the calculation of stress. For example, the temperature memory variable may indicate the time evolution of the effect of the varying temperature on the cellular biomass 130 to maintain the energy demand.

Rate integration module 180 may use the calculated time rate of change for the metabolite concentration to calculate an updated value for the corresponding process variable for the concentration based on the current value of the process variable in the following manner:

wherein [ M ] is_e]_n+1May represent an updated value of the extracellular concentration of metabolite M, [ M ] at the (n +1) th time step 102 of modeling process 100_e]_nThe current value after the nth time step 102 may be represented,is the calculated derivative, and Δ t_n+1The duration of the (n +1) th time step.

In the case of a non-homogeneous reactor,the full derivative for a portion of the bioreactor may be represented, which is calculated, for example, from the concentration gradient and velocity field, as discussed above. The velocity field may be updated at each time step 102, and the update frequency may be greater than time step 102 or less than time step 102. The velocity field may be updated in response to some triggering condition in the simulation. For example, a change in the flow rate of the simulated stream supplied to the bioreactor, a change in the simulated mixing of the bioreactor, or a substantial change in the physical properties of the bioreactor contents may trigger a new calculation of the velocity field.

In some embodiments, the duration of each time step 102 remains constant, while in other embodiments, the modeling process may vary the duration of time step 102 for different values of n. The duration of each time step 102 may vary, for example, depending on the calculated flux rate 174, the flow rate of the input stream 140 or the output stream 142, the control variables 152, and/or requirements regarding the speed or accuracy at which the modeling process 100 is performed.

The rate integration module 180 can implement process variable update formulas other than equation 24 using one or more of a variety of techniques for integrating the derivative or rate of change of the process variable. The rate integration module 180 may use first, second or higher order methods including the Euler method, the Runge-Kutta method, the backward differential equation, exponential integration techniques or other suitable methods.

The iterative portion of the modeling process 100 may end when the process variable corresponding to the simulated virtual time indicates the end of the simulated time period. In an embodiment, the new/updated values of the process variables calculated by the modeling process 100 in each iteration of the time step 102 may be stored or saved for further calculation or processing. For example, values computed at different virtual times or in different iterations of the virtual time step 102 may be combined to create a data set that indicates the time evolution of the process variable.

Upon completion of the iterative portion performed for the sequence of time steps 102, the modeling process 100 may include additional modules for calculating one or more metrics of the computationally modeled bioreactor 120 based on the calculated new values of the process variables. One of the one or more metrics may be a numeric value or set of numeric values in a vector, array, or other suitable data structure, a text value or set of text values in a suitable data structure, or a Boolean value or set of Boolean values in a suitable data structure. The metric may indicate the quality of the simulation and the confidence of the output, or it may be a measure of the performance of the hypothetical real bioreactor corresponding to the simulation. For example, the measure may indicate the efficiency of converting a substrate metabolite into a target product, wherein the efficiency is calculated from financial cost and/or time aspects. Additionally or alternatively, the metric may indicate the total amount of target product produced and/or the product quality.

The modeling process 100 may include additional modules for generating information based on the metrics of the modeled bioreactor (which information is displayed to a user via a user interface), control settings for a real-world bioreactor, which is operationally similar to the simulated bioreactor 120, and/or a training set of artificial intelligence models for hypothetical real-world bioreactors.

FIG. 2 is a graph 200, graph 200 depicting an exemplary effect of temperature variations on a calculated stress variable in an embodiment of modeling process 100. The graph 200 has a horizontal time axis 202 representing time increasing from left to right as simulated by the modeling process 100. The vertical temperature axis 204 corresponds to a solid trace, labeled T, and a dashed trace_SOLThe temperature of the extracellular solution 124; the dashed trace is labeled T_MEMRepresenting the memory temperature, and another vertical axis 206 corresponds to the dotted trace labeled stress, which represents the stress variable discussed above.

Still referring to FIG. 2, the simulation implemented by the modeling process 100 may begin with a process variable that is specific to a level T₁Temperature T of the extracellular solution 124_SOLAt the level ofT₁No substantial stress is introduced to the dummy biomass 130. Horizontal T₁May correspond to an optimal temperature for growth of cellular biomass 130. At a certain time (in t)_sIndicated), the temperature of the solution 124 may drop to a lower level T₂This level is below the optimal temperature for the cellular biomass 130. The drop in temperature may occur in response to one of the control variables 152 and may be abrupt, as shown in graph 200, or may be more gradual. T is_SOLMay depend on the modeled control mechanism for bioreactor 120, including the speed and quality of mixing of the modeled bioreactor contents. Even for the sharp changes in temperature shown in graph 200, the memory temperature T sensed by biomass 130_MEMCan still change more gradually (accounting for the biological response to temperature variations) and can follow the time evolution represented by the following equation:

wherein k is_tempWhich is a model parameter that corresponds to the rate at which the cellular biomass 130 adapts to temperature variations. The rate integration module 180 may calculate T by integrating equation 25_MEM(T) and can be calculated by using the following equation for T_MEMAnd T_SOLThe area between the two is numerically integrated to calculate the stress

The results are indicated by the dotted line on the graph 200 indicating stress.

FIG. 3 illustrates an exemplary computer system 300 in which the methods and techniques described herein may be implemented. Computer system 300 of fig. 3 includes a computer 310. Components of computer 310 may include, but are not limited to, one or more processors 312, which are communicatively coupled to a system memory 330 via a system bus 320. The one or more processors 312 may include one or more single-core or multi-core Central Processing Units (CPUs), Graphics Processing Units (GPUs), or any other suitable processor architecture. For example, system memory 330 may include Read Only Memory (ROM) components and Random Access Memory (RAM) components. The RAM component of system memory 330 may be any of several types of RAM, including Static RAM (SRAM), Dynamic RAM (DRAM), Synchronous Dynamic RAM (SDRAM), double data rate SDRAM (DDR SDRAM), or another suitable type of RAM. The system bus 320 may include one or more types of bus structures, including a memory bus or memory controller, a peripheral bus or local bus, and the like, and any suitable bus architecture may be used. By way of example, and not limitation, such architectures include Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, enhanced ISA (eisa) bus, and Peripheral Component Interconnect (PCI) bus also known as Mezzanine bus.

The one or more processors 312 may also be communicatively coupled to one or more peripheral devices and network interfaces 340 and an internal non-volatile memory 342 via the system bus 320. For example, the internal nonvolatile memory may be, for example, a Hard Disk Drive (HDD) or a Solid State Drive (SSD). Additionally or alternatively, an external nonvolatile memory device 344 may be communicatively connected to the computer 310 via the peripheral and network device interface 340. Peripheral and network interfaces 340 may include various connectors or adapters for communicative coupling with devices outside the packaging or housing of computer 310.

Peripherals and network interface 340 may include one or more Universal Serial Bus (USB) interfaces, one or more video connections including, but not limited to, VGA (video graphics array), DVI (digital visual interface), and/or HDMI (high-definition multimedia interface). The peripherals and network interface 340 may be connected to one or more user interface devices 350, including but not limited to a display 352, a keyboard 354, and a mouse 356. In some implementations, some or all of the user interface devices 350 are integral to the computer 310 and may be communicatively connected to the one or more processors 312 via the bus 320. Additionally or alternatively, the display 352, keyboard 354, and mouse 356 may be integrated as a touch screen.

In addition to or in the alternative to connecting to the user interface device 350, the peripheral device interface 340 may be communicatively connected to one or more external non-volatile memory devices 344. Non-volatile memory device 344 may include a HDD, SSD, and removable storage drives for optical, magnetic, or electronic media, including, but not limited to, Compact Discs (CDs), Digital Versatile Discs (DVDs), magnetic tape, floppy disks, or flash memory. Additionally or alternatively, peripheral devices and network interface 340 may be communicatively connected to output devices including, but not limited to, printers, plotters, speakers, or any other suitable visual, audible, tactile, or haptic output devices.

Peripherals and network interface 340 may include one or more network adapters to communicatively connect to one or more networks (not shown), which may include a Local Area Network (LAN) and/or a Wide Area Network (WAN), such as the internet. The connection may be a wired connection or a wireless connection as for example using radio signals or optical signals. Through the connected network, the computing system 300 may be communicatively connected to other computing systems, peripheral devices, or other devices.

Collectively, the one or more processors 312, bus 320, system memory 330, peripheral devices and network interfaces 340, internal non-volatile memory 342, external non-volatile memory devices 344, and user interface device 350 may be referred to as processing hardware of computing system 300. The processing hardware implementing the modeling process 100 of fig. 1 may omit some of the elements depicted in fig. 3, or include additional elements.

In operation, computing system 300 may load program instructions into system memory 330 for execution by one or more processors 312. For example, the system may load program instructions into the system memory 330 from the internal non-volatile memory 342 or from the external non-volatile memory device 344. Program instructions loaded into system memory may include an operating system 362, as well as various application programs 364. In some embodiments, computing system 300 may communicatively couple other computing systems with computing system 300 to load program instructions for execution by processors of the other computing systems.

The application 364 loaded into the system memory 330 of the computer 310 may include a bioreactor simulator 365 containing instructions to implement, at least in part, the modeling process 100 of fig. 1 using the processing hardware of the computing system 300. In some implementations, other computing systems communicatively connected with the computing system 300 may implement portions of the modeling process 100 using processing hardware of the other computing systems. The application 364 loaded into the system memory 330 of the computer 310 may additionally include a user interface implementer 366 that configures the hardware of the computing system 300 to implement portions of the modeling process 100 in various ways or to facilitate or enhance implementation. For example, the user interface implementer 366 may configure the hardware to receive the plurality of current values 150a of the process variables, the control variables 152, and/or the input and output flow variables 182 at the start of the simulation or in some embodiments during the simulation. Additionally or alternatively, the user interface implementer 366 may configure the hardware to display information to the user via a user interface presented, for example, on the display 352.

The user interface implementer 366 may configure the hardware to display information to the user in a variety of ways. In this context, displaying information may mean presenting information in visual form on display 352, or presenting information on printed paper, or presenting information in audio form through a speaker, or presenting information in tactile or haptic form through a tactile or haptic device, or presenting information in various other suitable ways. The user receiving the displayed information may be a scientist, a bioreactor operator, or any other person interested in the displayed information. In some implementations, the user is another machine that is capable of processing the displayed information and initiating an action based on the processed information. In some implementations and/or scenarios, the user receiving information through at least one of the user interface devices 350 and the user entering information through at least one of the user interface devices 350 are different users.

Returning to the application 364 (which may include the bioreactor simulator 365 and the user interface implementer 366), in operation, the application 364 may access and modify the program data 368 portion of the system memory 330. Program data 368 may include data received at the beginning of a simulation as well as all new data calculated throughout the simulation. Program data 368 can include, for example, all or some of the following examples: a current value of a process variable 150a, an updated value of a process variable 150b, a control variable 152, and/or input and output flow variables 182. The program data 368 may include the metrics or values of the time step 102, as well as other time parameters of the simulation. Other time parameters may include, for example, a virtual start time, a virtual end time, and various virtual transition times, such as t_s(i.e., temperature fluctuation time in fig. 2). Additionally, the program data 368 may include one or more metrics of the computationally modeled bioreactor 120 (based on the calculated new values of the process variables) and the generated output information (based on the one or more metrics). The generated output information may include information intended to be displayed to the user via a user interface (which is presented in one or more of the user interface devices 350), or a training set of artificial intelligence models for the bioreactor of interest, for example, for storage in the internal non-volatile memory 342 or the external non-volatile memory device 344.

In some embodiments, the bioreactor simulator 365 and the user interface implementer 366 are part of a single application. In other embodiments, bioreactor simulator 365 and user interface implementer 366 are separate applications that share the hardware resources of computer 310, either simultaneously or sequentially. For example, the user interface implementer 366 may present a user interface on the user interface device 350 to receive data necessary for the bioreactor simulator 365 and store the data in the internal non-volatile memory 342 or the external non-volatile memory device 344. At various times, the bioreactor simulator 365 may implement the bioreactor modeling process 100 using data stored by the user interface implementer 366 (or by another application) and store any generated metrics or other output data in the internal non-volatile memory 342 or the external non-volatile memory device 344. At a later time, the user interface implementer 366 can retrieve the output of the bioreactor simulator 365 and generate an appropriate output for the user.

Fig. 4A and 4B illustrate two panels of an exemplary user interface 400 that may be presented on the display 352, for example, by a user interface implementer 366 running on the computer 310. The user interface 400 may enable a modeling process (e.g., the modeling process 100 of fig. 1) to receive at least some of the initial values of the process variables (e.g., the current values 150a for the first time step 102). To this end, the example user interface 400 may include one panel 410 for receiving experimental condition variables from a user and another panel 412 for receiving media composition variables from a user, wherein the user enters information via the user interface device 350 (e.g., keyboard 354 and mouse 356). The experiment condition panel 410 in FIG. 4A may contain input cells 422a-422m (where a user may input a numerical value for a specified initial value of a process variable) and buttons 424A-424m that may be activated by the user by clicking a button on the mouse 356 to cause the corresponding value to be loaded into the program data 368.

Some of the values entered in the experimental conditions panel 410 may set the values of the control variables 152 in the modeled process 100. For example, when the process variable may change based on virtual control, rather than changing due to updating the process variable by integrating the calculated rate of change, the filter change 422d, the harvest time 422f, and the glucose injection time 422m may indicate a virtual time within the simulated time interval.

The media composition panel 412 in fig. 4B enables the modeling process 100 to receive initial process variable values from a user via input units 432a-432h (these variable values indicate metabolite concentrations in the batch media or the batch mode extracellular solution 124 of the bioreactor 120), and/or metabolite concentrations in the input stream 140 during the perfusion mode of operation of the bioreactor 120 via input units 442a-442 h. Once the simulation time interval corresponding to the perfusion mode begins, the concentration of the metabolite in the input stream 140 may begin to affect the process variable corresponding to the metabolite concentration in the extracellular solution 124 by changing the corresponding rate of change calculated by the rate integration module 180.

Fig. 5 illustrates an exemplary user interface 500 for displaying information to a user. For example, the user interface implementer 366 of fig. 3 may present the user interface 500 on the display 352. The user interface implementer 366 may generate the information needed to display the user interface 500 based on the metrics calculated by the bioreactor simulator 365 and display the information in the panels 510a-510 e. The displayed metrics may include traces 512a-512f that represent values of a plurality of process variables during the simulated time period. The user may select the variable to be displayed through another user interface provided by the user interface implementer 366. Additionally or alternatively, the user may select to display another set of traces 514a-514f that represent values from a previously simulated time period. The traces 512a-512f, 514a-514f from the two simulations may reflect different process conditions, which may have been input by a user, for example, through the user interface 400.

In one example, panels 510a-510e of user interface 500 display a trace 512a of a biomass variable, a trace 512b of a lactate variable, a trace 512c of a glucose variable, a trace 512d of an ammonia variable, a trace 512e of a permeate variable, and a trace 512f of an antibody variable, respectively. The comparison provided by the respective 514a-514f traces from the previous simulations may provide the user with useful information about the expected changes in hypothetical bioreactor operating performance without the need for in vivo experiments with real-world bioreactors. The permeate traces 512e, 514e may indicate, for example, the virtual time evolution of the concentration or amount of antibody products harvested from the output stream 142 over the course of several days. A sharp increase from the zero baseline of the permeate traces 512e, 514e may correspond to starting virtual harvesting at virtual times that are input into the input unit 422f for the harvesting time of the user interface 400A. The start of harvesting may also correspond to the following: the change in the simulated operation of the bioreactor 120 from batch mode to perfusion mode, or the change in the virtual filter 144 that controls the amount of target antibody product in the output stream 142, and the corresponding change in the updated calculation of the antibody concentration variable by the rate integration module 180. Thus, as the simulation of the harvest begins, traces 512f, 514f begin to fall, which are indicative of the antibody concentration in the extracellular solution.

Traces 512a, 514a for biomass may indicate two different regions: a high growth rate region followed by a slower growth region. The transition between these two regions may correspond to, for example, the introduction of stress by: at time t_sThe temperature of the extracellular solution is changed from T₁Down to T₂(as shown in fig. 2) and is based on values entered in the input elements 422g-422i of the user interface 400A. Changes corresponding to temperature variations of the metabolic byproduct traces 512b, 514b, 512d, 514d for lactate and ammonia may indicate a decrease in lactate production and an increase in ammonia production, as calculated by the flux balance analysis 170 of the modeled process 100. The variability of the first half of the glucose traces 512c, 514c may correspond to the introduction of a glucose injection into the input flow stream 140, depending on the values entered in the input cells 422l-422m of the user interface 400A. In an exemplary embodiment, the traces presented in the panels 510a-510f imply interdependencies of process variable values that are calculated by the applications 365, 366 running on the computing system 300 using the modeling process 100.

Fig. 6 illustrates a bioreactor control system 600 in an embodiment in which one or more programs (e.g., a bioreactor simulator 365 and/or another application) for modeling a bioreactor cause processing hardware of a computer 610 to generate control settings for a real-world bioreactor 620. For example, the computer may be computer 310 in FIG. 3, and the control settings may be communicated through one of the interfaces of the peripheral and network interfaces 340. In addition to real-world bioreactor 620, bioreactor control system 600 also includes a controller 630 and one or more sensors 640. Controller 630 may be communicatively coupled with the one or more sensors 640 that sense, measure, and/or convert physical, chemical, and biological properties of bioreactor 620 and any connected systems. The sensor 640 may measure the following: for example, the weight of bioreactor 620, the level of liquid within bioreactor 620, the temperature at various locations, acidity, turbidity, optical density, osmolality, oxygen levels, the concentration of various metabolites, the chemical composition of solid liquid or gas samples, cell counts, and any other suitable indicator of the performance or operation of bioreactor 620 and/or components fluidly or mechanically connected to bioreactor 620.

Controller 630 may control various mechanical and/or electrical components including one or more heaters or heat exchangers, one or more mixers, converters, pumps, valves, and/or other devices for altering the physical, chemical, and biological properties of bioreactor 620 and its contents via one or more communication, electrical, or mechanical connections. In some embodiments, controller 630 controls one or more quantities (e.g., temperature, glucose concentration, etc.) associated with bioreactor 620 using respective proportional-integral-derivative (PID) controller hardware, firmware, and/or software. For example, the controller may cause a change in the input flow rate or composition of the flow into bioreactor 620, and/or a change in the temperature of the contents of bioreactor 620. The controller may cause changes to the operating mode of bioreactor 620 including, for example, switching from batch mode of operation to continuous flow operation to perfusion mode, changing from high growth rate mode under substantially optimal conditions for cell growth to stressed mode for increasing production of the product of interest, changing from production mode to harvest mode, and/or any other desired changes. Controller 630 may take any of the actions described above in response to one or more control settings communicated to controller 630 by computer 610. Additionally or alternatively, the controller 630 may communicate signals from the sensor 640 to the computer 610. The computer 610 may use the signals from the sensors 640 to implement the modeling process 100. For example, computer 610 may adjust one or more modeling parameters of model 104 based at least in part on sensor signals received from controller 630. In some embodiments, the sensor 640 is in direct communication with the computer 610. In some embodiments where the computer 610 is the computer 310 of fig. 3, a user may input a value indicative of the sensor signal through the user interface device 350 or through the external non-volatile memory device 344. In some embodiments, controller 630 implements Model Predictive Control (MPC) to control mechanical and/or electrical components.

Whether or not signals from sensors 640 are used in implementing the modeling process 100, control settings generated by the computer 610 and based at least in part on the computationally modeled metrics of the bioreactor 120 may alter the operation of the real-world bioreactor 620 through the controller 630. For example, the modeling process 100 implemented on the computer 610 may calculate a metric indicative of an optimal time to change the mode of operation and send corresponding control settings to the controller 630. In some embodiments, computer 310 and computer 610 are different devices communicatively connected to each other, and the control settings are generated by computer 310. For example, the control settings may be transmitted to the computer 610 over a network via one of the peripheral devices and the network interface 340, or via one of the external non-volatile memory devices 344. In some embodiments, the control settings for the real-world bioreactor 620 are part of a real-time model-based control of the bioreactor 620. In other embodiments, the control settings are based on metrics generated from a model at a previous time as part of a computer simulation experiment associated with the operation of the real-world bioreactor 620.

FIG. 7 is a flow diagram depicting a computer-implemented method 700 of computationally modeling a bioreactor, according to an example embodiment. For example, the method 700 may apply the computational model 104 of FIG. 1 and may include execution of the modeling process 100 of FIG. 1. Further, the method 700 may be implemented by one or more processors 312 of the computer 310 (e.g., when executing the bioreactor simulation program 365), or another suitable computing device or computing system. Method 700 may be suitable for modeling spatially heterogeneous bioreactors using, for example, the techniques described above.

The method 700 includes receiving a plurality of current values of a process variable (block 710). These values may be received from a user via a graphical user interface, as illustrated in fig. 4A and 4B and the related discussion. As illustrated in fig. 6 and the related discussion, in some embodiments, the received values may be based on measurements of the real bioreactor. Additional or alternative ways of receiving or obtaining the current value of the process variable may include loading a value (e.g., a previously calculated value) from a computer memory and/or loading a predetermined default value from the computer memory.

For example, the process variable received at block 710 may describe a virtual content of a bioreactor, such as the virtual cellular biomass 130 in the virtual extracellular solution 124 of the bioreactor 120 modeled in fig. 1. The process variables may include, for example, one or more variables describing the temperature, acidity, and/or indicating total osmolarity of the extracellular solution (e.g., extracellular solution 124), the biomass (e.g., biomass 130), and/or the input and output streams (e.g., input stream 140 and/or output stream 142). Additionally or alternatively, the process variable may be indicative of an amount or concentration of a virtual biomass (e.g., biomass 130) and/or an amount or concentration of any of the metabolites in a virtual extracellular solution (e.g., solution 124), and/or input and output streams (e.g., input stream 140 and/or output stream 142). Metabolites having amounts or concentrations described by process variables may include oxygen, carbon dioxide, ammonia, lactate, glucose, asparagine, glutamine, glycine, and/or any other suitable metabolite. The variable may describe the extracellular and/or intracellular (inside the virtual cell) amount or concentration of the metabolite. In some embodiments, the process variable describes the metabolite concentration in different organelles (cellular organs or cellular aptamers). Further, when method 700 models a heterogeneous bioreactor, the process variables may describe each of the modeled bioreactor sections.

To calculate new values for the process variables during the simulated time period, the method 700 may generate a plurality of constraints (e.g., describing the virtual cellular biomass 130) for the flux rate of the metabolic flux for each given time step (e.g., multiple iterations of time step 102) at block 720. The metabolic flux may be the flux depicted at 132a-132i in fig. 1, the flux listed in table 2, or any other suitable description of a metabolic pathway and/or biochemical reaction during cellular metabolism. Metabolic flux may describe the metabolic network of different types of cells, including bacterial cells, animal cells and/or fungal cells. Cell types modeled by cellular biomass may include stem cells or germ cells (e.g., Chinese Hamster Ovary (CHO) cells).

The method 700 may generate the constraints for the flux rate by: at least some of the current values of the process variables (e.g., current values 150a) are modeled for one or more effects on metabolic reaction kinetics using a kinetic model (such as, for example, kinetic model 160). The constrained flux rates may include glucose uptake, glutamine uptake, asparagine uptake, oxygen uptake, and/or any other flux rate that is directly or indirectly limited by extracellular conditions (e.g., conditions of extracellular solution 124) at the higher end and/or at the lower end. For example, the limiting conditions may include temperature, acidity, and/or osmolarity. Constraints on metabolic flux rates may include upper and/or lower limits on flux rates for biomass growth or energy expenditure for cell maintenance. Modeling the one or more effects of the current values of the process variables on metabolic reaction kinetics can include calculating the effect of temperature, acidity, and/or osmolality on the upper limit of metabolite uptake rate by multiplying the upper limit under ideal conditions with a correction factor that indicates a decrease in uptake rate due to non-ideal conditions. Correction factors can be constructed in view of experimental observations on real-world cell cultures that are subject to changes in conditions. These and other experimental observations may be included in a calibration procedure for a computational model, such as computational model 104.

The effect of the current value of the process variable on the metabolic reaction kinetics may include the effect of stress on the lower limit of the rate of energy expenditure for cell maintenance. The stress may be increased to induce a higher production rate of the product of interest. In one embodiment, as illustrated in fig. 2, temperature fluctuations induce stress, requiring the cellular biomass to consume more energy during non-growth processes. Other ways of inducing stress in the cellular biomass may be to alter the acidity or osmolarity of the extracellular solution. The effect of the current value of the process variable on the kinetics of the metabolic reaction may include the effect of metabolic byproducts. Lactate concentration and ammonia concentration may be among the factors that affect the constraints on flux rates calculated by the kinetic model. Modeling the above-described effects may include calibrating the modeled effects with experimental data from real-world cell cultures.

The method 700 may continue at block 730 with calculating a flux rate of a metabolic flux (e.g., for the virtual biomass 130) by performing a flux balance analysis (e.g., the flux balance analysis 170). The flux balance analysis may be affected by a metabolic target (e.g., metabolic target 172) and a number of generated constraints on flux rates (e.g., constraint 164). Some exemplary metabolic targets are listed in table 4 above. By minimizing the ratio of the flux rate and a linear combination of at least some of the squares of the flux rate, a broad range of metabolic targets can be generated. Each weight in the linear combination may be positive, negative, or zero. Thus, the generated metabolic targets may, for example, effectively maximize growth per total flux ratio, or maximize growth relative to substrate metabolite consumption. For example, in a scenario where maintenance energy is insufficient, non-negative growth may not be achieved, and maximizing maintenance energy may be a metabolic goal used in flux balance analysis. Maximizing the growth rate of the virtual biomass when the maintenance energy is sufficient may be a metabolic goal used in flux balance analysis.

At block 740, the method 700 may calculate a rate of change of at least some of the process variables based at least in part on the flux rate calculated at block 730. Calculating the rate of change may include receiving one or more variables indicative of: flow rate and composition of the input flow (e.g., flow variable 182 corresponding to input flow 140), the output flow (e.g., flow variable 182 corresponding to output flow 142), and/or the output filter (e.g., filter 144).

The method 700 may then update the current value of the process variable (e.g., the current value 150a) at block 750 at least in part by: integrating one or more of the rates of change calculated for the durations of the virtual time steps. This may end the iteration for a given virtual time step. For example, a value updated at the end of one time step may become the current value used for the next time step. For example, block 750 may be performed by rate integration module 180 of fig. 1 when calculating an update value 150b from current value 150 a. The method 700 may calculate (using, for example, the rate integration module 180) the update value at least in part by considering variables indicative of the flow rate and composition of the input and/or output streams. At block 750, the method 700 may include balancing the flow variables in the differential or differential equations discussed above and solving the equations by integration. Additionally or alternatively, the mass balance equation may include the effect of variables on the composition of the virtual output stream that indicate the properties of the virtual output filter.

When method 700 models a heterogeneous bioreactor, the method can include updating the process variables for each portion of the modeled bioreactor based at least in part on the calculated velocity field. The velocity field may be calculated, for example, using a CFD model. The method 700 may update a process variable for a portion of a bioreactor in view of a process variable for an adjacent portion of the bioreactor. In some embodiments, the method 700 may define a gradient for at least some of the process variables in at least some of the modeled portions of the bioreactor.

Before continuing to block 760, the method 700 may iterate 750 for a portion or all of the simulated time period, which is divided into a sequence of time steps. In other implementations, block 760 occurs in parallel with iterations associated with one or more of blocks 720 and 750. At block 760, the new values of the process variables calculated during any virtual time step or steps are used to calculate a metric for the modeled bioreactor, or more generally, a model of the system containing the virtual bioreactor (e.g., model 104). The metric of the computationally modeled bioreactor may be indicative of the effectiveness or efficiency of the computationally modeled bioreactor in producing the target product. For example, the metric may indicate total product output, yield with respect to energy and/or substrate metabolites, productivity of the reactor, and/or quality of the target product. In some implementations, a plurality of metrics is calculated at block 760. Additionally or alternatively, the one or more metrics may include a value of at least one of the process variables at different virtual times within the simulated time period, as illustrated, for example, in fig. 5 and related description. For a heterogeneous bioreactor, the metrics calculated for each portion of the bioreactor may contribute to the metrics for the entire bioreactor. However, in some embodiments, the bioreactor metric may be based on one or more portions of the bioreactor including the output stream flowing out of the bioreactor. Additionally or alternatively, at block 760, the method 700 may generate a metric indicative of heterogeneity within the bioreactor.

At block 770 of method 700, information may be generated and displayed to a user via a user interface presented, for example, on a display (e.g., display 352). The method 700 may generate the displayed information based at least in part on the metrics calculated at block 760. In some embodiments, block 770 additionally or alternatively includes generating control settings for a real-world bioreactor (e.g., bioreactor 620) as depicted, for example, in fig. 6 and the accompanying description. In some embodiments, the method 700 is integrated into a model-based control scheme of a real-world bioreactor (e.g., using model predictive control). When modeling a heterogeneous bioreactor, the method 700 may display the heterogeneity at a user interface or generate one or more control settings based on, for example, a calculated heterogeneity metric. For example, when the measure indicative of heterogeneity exceeds a threshold, blending may be initiated or enhanced.

Additionally or alternatively, block 770 may include generating a training set of artificial intelligence models for the bioreactor by: the metric calculated at block 760 is used as a ground round label for the bioreactor model, which is described by a parameter set. Multiple executions of the method 700 may allow for the accumulation of multiple data sets of model parameters and corresponding metrics. The data set may be used to train an artificial intelligence model based on: neural networks, convolutional neural networks, decision trees, clustering algorithms, and/or other suitable machine learning techniques. The trained artificial intelligence model can then estimate or predict new metrics based on the bioreactor model parameters without having to perform the computational modeling method 700 again.

FIG. 8 illustrates an example evolution of process variables and metrics calculated using the method 700. Data sets for initial process variables (including pH, temperature, media composition) and media feed schedules (input streams) for monoclonal antibody production processes are received. The process was computationally modeled according to the workflow illustrated in fig. 1 and 7 and described above. The values of the process variables (arbitrary units) corresponding to the metabolite concentration (CC-GLC, as defined in table 3), as well as the measures of cell viability, Viable Cell Density (VCD) and antibody product (titer) calculated from the model are plotted in fig. 8 as solid lines. Values of metabolite concentration, Viable Cell Density (VCD), cell viability and antibody titer measured during real world execution of the same production process are plotted as points in equal proportion to the corresponding calculated values. The computational model essentially predicts the observed real-world process performance. The drastic changes in the concentrations of certain metabolites are due to discontinuous feeding of the bioreactor — the metabolites are added in batches at specific times of the day. The measured metabolites lack drastic changes because the measurement is limited to the time of day before the addition of the fed-batch metabolites. The measure of Viability (VIAB) represents the fraction of cells in the biomass that can still produce product and depends on the cumulative effect of cell death, as described above. Then, VCD represents the biomass concentration multiplied by the viability.

Fig. 9A and 9B schematically illustrate an example method of modeling spatial heterogeneity in a bioreactor 900. Bioreactor 900 may include an impeller 905 or another mixing device configured to mix the bioreactor contents. FIG. 9A illustrates an example of partitioning a bioreactor 900 into multiple sections 910a-910 d. A homogeneous metabolic model (e.g., 104 model) may represent each of the portions 910a-910 d. Each of the portions 910a-910d may share boundaries with one or more other portions. In the example shown, portion 910c is disposed entirely within portion 910d, while portion 910d borders all three other portions (i.e., 910a-910 c). Inlets and outlets (not shown) into bioreactor 900 may add or remove metabolites from modeled segments 910a-910 c. For each of the sections 910a-910d, the modeled input and output flows (e.g., flows 140, 142) may include flows of virtual contents due to inlets and outlets as well as flows across boundaries with adjacent sections. The flow of virtual contents across the boundaries between bioreactor sections 910a-910d may follow the example velocity field 920 illustrated in FIG. 9B. Computing the input and output streams may include estimating the gradients of the process variables on the boundaries (the gradients being the respective differences in values of the process variables in the regions forming the boundaries), integrating the velocity fields 920 on each boundary (and orthogonal to the boundaries), and multiplying the estimated gradients by the integral values of the velocity fields 920 corresponding to each boundary.

Fig. 9B illustrates velocity field 920 as a set of dashed arrows. The CFD model may calculate the velocity field 920 based on the volumetric discretization of the bioreactor 900 and suitable computational techniques. In some embodiments, the method of computationally modeling the bioreactor 900 (e.g., the method 700) may determine the velocity field 920 based at least in part on measurements (e.g., particle velocimetry techniques) or a combination of pre-calculated or measured velocity fields. The velocity field 920 may depend on the geometry of the bioreactor 900 and the impeller 905, as well as the velocity of the impeller 905. The velocity field 920 may also depend on the input and output flows of any of the inlets and outlets of the bioreactor 900. Importantly, under a set of static conditions, the velocity field 920 may have a steady state from which the model may calculate the flow of virtual contents across the boundary between the bioreactor sections 910a-910 d. Additionally or alternatively, the coupled hydrodynamic and metabolic models may account for dynamic changes in the velocity field 920 and the corresponding effects on the evolution of metabolite concentrations in the bioreactor sections 910a-910 d.

The following list of options reflects the various embodiments explicitly contemplated by the present disclosure.

Option 1. a method of computationally modeling a bioreactor, wherein the method comprises:

receiving a plurality of current values of process variables that describe virtual contents in at least a portion of the modeled bioreactor and that include virtual cellular biomass in a virtual extracellular solution;

calculating, by processing hardware of the computing system, new values of the process variables during the simulation time period at least in part by:

generating constraints on flux rates describing metabolic flux of the virtual cellular biomass by modeling one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics,

calculating the flux rates of the metabolic fluxes by performing a flux balance analysis influenced by metabolic targets and generated constraints on the flux rates,

calculating a rate of change of at least some of the process variables based at least in part on the calculated flux rate, an

Updating one or more of the current values of the process variables at least in part by integrating one or more of the calculated rates of change for a virtual time step within the simulation time period;

calculating, by the processing hardware, a metric of the computationally modeled bioreactor based at least in part on the calculated new values of the process variables; and

generating, by the processing hardware and based on the metric of the modeled bioreactor, one or more of: i) information displayed to a user via a user interface, ii) control settings for a real-world bioreactor, or iii) a training set for an artificial intelligence model of a bioreactor.

Option 2. the method of option 1, wherein the process variables comprise at least one of: temperature, acidity, or one or more variables indicative of the total volume of these virtual contents.

Option 3. the method of any of options 1-2, wherein the process variables include one or more variables indicative of the concentration of the virtual cellular biomass and one or more variables indicative of the concentration of extracellular metabolites in the virtual extracellular solution.

Option 4. the method of option 3, wherein the one or more variables indicative of the concentration of extracellular metabolites in the virtual extracellular solution comprise one or more variables indicative of the concentration of at least one of: oxygen, carbon dioxide, or ammonia.

Option 5. the method of any of options 3-4, wherein the one or more variables indicative of the concentration of extracellular metabolites in the virtual extracellular solution comprise variables indicative of the concentration of one or more of: glucose, asparagine, glutamine or glycine.

Option 6. the method of any of options 3-5, wherein the one or more variables indicative of the concentration of the extracellular metabolite in the virtual extracellular solution comprise a variable indicative of the concentration of at least one target metabolite.

Option 7. the method of any of options 3-6, wherein the process variables further comprise one or more variables indicative of one or more intracellular metabolite concentrations in the virtual cellular biomass.

The method of any of options 1-7, wherein the plurality of constraints on the flux rates include an upper limit of at least one of: i) glucose uptake, ii) glutamine uptake, iii) asparagine uptake, or iv) oxygen uptake.

Option 9. the method of any of options 1-8, wherein the plurality of constraints on the rates of change comprise a lower limit on the rate of energy expenditure for cell maintenance.

The method of any of options 1-9, wherein the one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics include an effect of a value of at least one of:

i) temperature, ii) acidity, or iii) osmolarity.

Option 11, the method as described in option 10,

wherein modeling the one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics comprises calculating an effect of a value of at least one of: i) temperature, ii) acidity, or iii) osmolality, the correction factor indicating a decrease in the uptake rate due to non-ideal conditions.

Option 12. the method of any of options 1-11, wherein the one or more effects of the at least some of the current values of the process variables on metabolic reaction kinetics comprises an effect of stress on a lower limit of energy expenditure rate for cell maintenance.

Option 13. the method of option 12, wherein modeling the effect of stress comprises calculating the effect based at least in part on cumulative effects of temperature variations on the virtual cellular biomass.

Option 14. the method of any of options 1-13, wherein the one or more effects of the at least some of the current values of the process variables on metabolic reaction kinetics comprises an effect of a concentration of at least one of metabolic byproducts.

Option 15. the method of any of options 1-14, wherein modeling the one or more effects comprises calibrating the modeled effects with experimental data from real-world cell culture.

Option 16. the method of any of options 1-15, wherein the metabolic goal comprises minimizing a ratio of linear combinations of at least some of the flux rates and the squares of flux rates in the flux balance analysis.

The method of any of options 1-15, wherein the metabolic goal comprises minimizing a linear combination of at least some of the flux rates in the flux balance analysis.

Option 18. the method of option 17, wherein the linear combination is a sum of fluxes in the flux balance analysis.

Option 19. the method of any of options 1-18, wherein the metabolic goal comprises maximizing a rate of energy expenditure for cell maintenance.

Option 20. the method of any of options 1-19, wherein the metabolic goal comprises maximizing the growth rate of the virtual cellular biomass.

Option 21. the method of any of options 1-20, further comprising:

receiving into the modeled bioreactor one or more variables indicative of flow rate and composition of one or more virtual input streams, and wherein

Updating the at least some of the current values of the process variables includes interpreting a flow rate and a composition of the one or more virtual input streams.

Option 22. the method of option 21, further comprising:

receiving one or more variables indicative of a property of a virtual output filter;

calculating a composition of one or more virtual output streams from the modeled bioreactor using properties of the virtual output filter, and wherein

Updating the at least some of the current values of the process variables includes interpreting a composition of the one or more virtual output streams.

Option 23. the method of option 21, wherein updating at least some of the current values of the process variables comprises solving mass balance equations.

Option 24. the method of any of options 1-23, further comprising:

receiving one or more control variables, an

The affected set of the current values of the process variables is updated based at least in part on the received control variables.

Option 25. the method of any of options 1-24, wherein

The metric for the modeled bioreactor includes a value of at least one of the process variables at different virtual times within the simulation time period.

Option 26. the method of any of options 1-25, wherein

The measure of the computationally modeled bioreactor is indicative of the effectiveness or efficiency of the computationally modeled bioreactor in producing the target product.

Option 27. the method of any of options 1-26, wherein

The method includes generating the control setting, and wherein

The method further includes controlling an input to the real-world bioreactor based on the generated control settings.

Option 28. the method of any of options 1-27, wherein

The method includes generating a training set of the artificial intelligence model for the bioreactor.

Option 29. the method of any of options 1-28, wherein

The virtual cellular biomass comprises virtual Chinese Hamster Ovary (CHO) cells.

Option 30. the method of any of options 1-29, wherein receiving the plurality of current values comprises at least one of

i) Receiving a value from a user via a graphical user interface;

ii) receiving values based on measurements of the real bioreactor;

iii) loading a value from the computer memory and based on a previously calculated value; or

iv) loading predetermined default values from the computer memory.

Option 31. the method of any of options 1-30, wherein the modeled bioreactor is a spatially homogeneous bioreactor and the at least part of the modeled bioreactor is the entirety of the spatially homogeneous bioreactor.

Option 32. the method of any of options 1-30, wherein:

the modeled bioreactor is a spatially heterogeneous bioreactor;

the at least part of the modeled bioreactor is a first part of the modeled bioreactor;

the process variables are first process variables;

the method further includes receiving a plurality of current values of second process variables describing a second virtual content in a second portion of the modeled bioreactor; and

the calculation of the new values of the first process variables during the simulation time period is based in part on the received current values of the second process variables.

Option 33. the method of option 32, wherein calculating new values for the first process variables during the simulation time period includes calculating a gradient of at least one of the first process variables based on values for corresponding ones of the second process variables.

Option 34. the method of any of options 32-33, further comprising:

one or more velocities associated with the virtual contents of the first portion of the modeled bioreactor are determined.

Option 35. the method of option 34, wherein:

determining the one or more velocities is based at least in part on computational fluid dynamics.

Option 36. a non-transitory computer-readable medium storing instructions for computationally modeling a bioreactor, wherein the instructions, when executed by one or more processors, cause the one or more processors to:

receiving a plurality of current values of process variables that describe virtual contents of the modeled bioreactor and that include virtual cellular biomass in a virtual extracellular solution;

calculating, by processing hardware of the computing system, new values of the process variables during the simulation time period at least in part by:

generating constraints on flux rates describing metabolic flux of the virtual cellular biomass by modeling one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics,

calculating the flux rates of the metabolic fluxes by performing a flux balance analysis influenced by metabolic targets and generated constraints on the flux rates,

calculating a rate of change of at least some of the process variables based at least in part on the calculated flux rate, an

Updating one or more of the current values of the process variables at least in part by integrating one or more of the calculated rates of change for a virtual time step within the simulation time period;

calculating, by the processing hardware, metrics for the modeled bioreactor based on the calculated new values of the process variables; and

generating, by the processing hardware and based on the metric of the modeled bioreactor, one or more of: i) information displayed to a user via a user interface, ii) control settings for a real-world bioreactor, or iii) a training set for an artificial intelligence model of a bioreactor.

Option 37. the non-transitory computer readable medium of option 36, wherein the process variables include at least one of: temperature, acidity, or one or more variables indicative of the total volume of these virtual contents.

Option 38 the non-transitory computer-readable medium of any one of options 36-37, wherein the process variables include one or more variables indicative of a concentration of the virtual cellular biomass and one or more variables indicative of a concentration of an extracellular metabolite in the virtual extracellular solution.

Option 39. the non-transitory computer-readable medium of option 38, wherein the one or more variables indicative of the concentration of the extracellular metabolite in the virtual extracellular solution comprise one or more variables indicative of a concentration of at least one of: oxygen, carbon dioxide, or ammonia.

Option 40 the non-transitory computer-readable medium of any one of options 38-39, wherein the one or more variables indicative of the concentration of the extracellular metabolite in the virtual extracellular solution comprise variables indicative of the concentration of one or more of: glucose, asparagine, glutamine or glycine.

Option 41. the non-transitory computer-readable medium of any one of options 38-40, wherein the one or more variables indicative of the concentration of the extracellular metabolite in the virtual extracellular solution comprise a variable indicative of the concentration of at least one target metabolite.

Option 42. the non-transitory computer-readable medium of any of options 38-41, wherein the process variables further comprise a variable indicative of one or more intracellular metabolite concentrations in the virtual cellular biomass.

The non-transitory computer-readable medium of any of options 36-42, wherein the plurality of constraints on the rate of change include an upper limit of at least one of: i) glucose uptake, ii) glutamine uptake, iii) asparagine uptake, or iv) oxygen uptake.

Option 44 the non-transitory computer-readable medium of any one of options 36-43, wherein the plurality of constraints on the rate of change includes a lower limit on the rate of energy expenditure for cell maintenance.

Option 45 the non-transitory computer-readable medium of any one of options 36-44, wherein the one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics include an effect of a value of at least one of:

i) temperature, ii) acidity, or iii) osmolarity.

Option 46. the non-transitory computer readable medium of option 45, wherein

Modeling the one or more effects of at least some of the current values of the process variables on metabolic reaction kinetics includes calculating an effect of a value of at least one of: i) temperature, ii) acidity, or iii) osmolality, the correction factor indicating a decrease in the uptake rate due to non-ideal conditions.

Option 47. the non-transitory computer-readable medium of any one of options 36-46, wherein the one or more effects of the at least some of the current values of the process variables on metabolic reaction kinetics include an effect of stress on a lower limit of an energy expenditure rate for cell maintenance.

Option 48. the non-transitory computer-readable medium of option 47, wherein modeling the effect of stress comprises calculating the effect based at least in part on a cumulative effect of temperature variations on the virtual cellular biomass.

Option 49 the non-transitory computer-readable medium of any one of options 36-48, wherein the one or more effects of the at least some of the current values of the process variables on metabolic reaction kinetics includes an effect of a concentration of at least one of metabolic byproducts.

Option 50. the non-transitory computer-readable medium of any of options 36-49, wherein modeling the one or more effects comprises calibrating the modeled effects with experimental data from real-world cell culture.

Option 51. the non-transitory computer-readable medium of any of options 36-50, wherein the metabolic goal comprises minimizing a linear combination of at least some of the rates of change.

Option 52. the non-transitory computer-readable medium of any of options 36-51, wherein the metabolic goal comprises minimizing a linear combination of at least some of the fluxes in the flux balance analysis.

Option 53. the non-transitory computer-readable medium of option 52, wherein the linear combination is a sum of fluxes in the flux balance analysis.

Option 54. the non-transitory computer-readable medium of any of options 36-53, wherein the metabolic goal comprises maximizing a rate of energy expenditure for cell maintenance.

Option 55 the non-transitory computer-readable medium of any one of options 36-54, wherein the metabolic goal comprises maximizing a growth rate of the virtual cellular biomass.

Option 56 the non-transitory computer-readable medium of any one of options 36-55, wherein the instructions further cause the one or more processors to:

receiving into the modeled bioreactor one or more variables indicative of flow rate and composition of one or more virtual input streams, and wherein

Updating the at least some of the current values of the process variables includes interpreting a flow rate and a composition of the one or more virtual input streams.

Option 57. the non-transitory computer-readable medium of option 56, wherein the instructions further cause the one or more processors to:

receiving one or more variables indicative of a property of a virtual output filter;

calculating a composition of one or more virtual output streams from the modeled bioreactor using properties of the virtual output filter, and wherein

Updating the at least some of the current values of the process variables includes interpreting a composition of the one or more virtual output streams.

Option 58. the non-transitory computer-readable medium of options 56-57, wherein updating the at least some of the current values of the process variables includes solving mass balance equations.

Option 59. the non-transitory computer-readable medium of any of options 36-58, wherein the instructions further cause the one or more processors to:

receiving one or more control variables, an

The affected set of the current values of the process variables is updated based at least in part on the received control variables.

Option 60 the non-transitory computer-readable medium of any one of options 36-59, wherein

The metric for the modeled bioreactor includes a value of at least one of the process variables at different virtual times within the simulation time period.

Option 61. the non-transitory computer-readable medium of any of options 36-60, wherein

This measure of the modeled bioreactor is indicative of the effectiveness or efficiency of the modeled bioreactor in producing the target product.

Option 62. the non-transitory computer readable medium of any of options 36-61, wherein the instructions

Cause the one or more processors to generate the control setting, and

the one or more processors are further caused to control the input to the real-world bioreactor based on the generated control settings.

Option 63 the non-transitory computer readable medium of any of options 36-62, wherein the instructions

Causing the one or more processors to generate a training set of the artificial intelligence model for the bioreactor.

Option 64 the non-transitory computer-readable medium of any one of options 36-63, wherein

The virtual cellular biomass comprises virtual Chinese Hamster Ovary (CHO) cells.

Option 65 the non-transitory computer-readable medium of any one of options 36-64, wherein receiving the plurality of current values comprises at least one of:

i) receiving a value from a user via a graphical user interface;

ii) receiving values based on measurements of the real bioreactor;

iii) loading a value from the computer memory and based on a previously calculated value; or

iv) loading predetermined default values from the computer memory.

Option 66. the non-transitory computer-readable medium of any of options 36-65, wherein the modeled bioreactor is a spatially homogeneous bioreactor and the at least a portion of the modeled bioreactor is all of the spatially homogeneous bioreactor.

Option 67. the non-transitory computer-readable medium of any one of options 36-65, wherein:

the modeled bioreactor is a spatially heterogeneous bioreactor;

the at least part of the modeled bioreactor is a first part of the modeled bioreactor;

the process variables are first process variables;

the instructions further cause the one or more processors to receive current values of second process variables describing a second virtual content in a second portion of the modeled bioreactor; and

the calculation of the new values of the first process variables during the simulation time period is based in part on the received current values of the second process variables.

Option 68. the non-transitory computer-readable medium of option 67, wherein calculating new values for the first process variables during the simulation time period includes calculating a gradient of at least one of the first process variables based on values of corresponding ones of the second process variables.

Option 69. the non-transitory computer-readable medium of any of options 67-68, wherein the instructions further cause the one or more processors to determine one or more velocities associated with the virtual contents of the first portion of the modeled bioreactor.

Option 70. the non-transitory computer readable medium of any one of options 67-69, wherein:

determining the one or more velocities is based at least in part on computational fluid dynamics.

Those skilled in the art will recognize that a wide variety of modifications, alterations, and combinations can be made with respect to the above described embodiments without departing from the scope of the invention, and that such modifications, alterations, and combinations are to be viewed as being within the ambit of the inventive concept.