Mathematical modeling of the bioprocesses
In a very general term Model is the description of the realty. The realty could be described in terms of maps, graphs or pictures. In a way these are the miniaturized versions of realty (or model) of the actual realty. If above analogy is further extended to bioprocess then one can describe the bio reactions either verbally (Verbal model) or in terms of mathematical equations (Mathematical model). The mathematical description of the bioprocess is developed to describe the complex intracellular reactions occurring during the metabolism of the cell which is converting the substrate to products in the fermentation reactions. While developing mathematical equations, one has to propose such equations which are simple, biologically relevant (not fitting polynomials) and easily verifiable by the experiments. In order to cater to above objectives generally unstructured mathematical models are used to describe the metabolic activities of the cells. The basic assumption of these models is the similar activity of all homogenously distributed cells in the bioreactor. Lumped parameter biomass (living+dead) is assumed to describe the activities of the bio-catalyst, knowing fully well that complex intracellular reactions are occurring in the cell and the cell population in the bioreactor consists of all kinds of cells (young, old, dying and dead).
Once the mathematical description of the bioprocess is ready, it can help to understand the system in a better way as to what is going to happen if the substrate is changed, how the Temperature, pH changes are changing the outcome of the process and so on. A good mathematical model of a culture system has to be sensitive not only to the environment but also to the physiology of the culture. Suitable physiological state markers are identified which are incorporated in the mathematical model to describe the biological activity in a more realistic manner (structured mathematical model). These models describe the reality of the biological process in a more realistic manner.
In order to attempt the representation the mathematical description of a biological process one has to initiate with the batch kinetics of the process. Multiple repeat experiments are done of the batch cultivation process so as to get the average values and standard deviations of the process variables.
The mathematical model development consist of following steps -
1. Elimination of the experimental errors, smoothing of data points and calculation of rates and specific rates
The experimental values of the process variables invariably contain some experimental errors. In order to eliminate these errors and get suitable smooth curves, smoothing of the data points is done. For this a nth degree polynomial equation is passed through the observed experimental values of process variables such that the fitting curve is as smooth as possible and it is as close as possible to the observed experimental data points. A minimization function containing above two (contradicting) objectives is minimized by the theory of splines. The volumetric and specific rates of the key process variables (Biomass, Substrate and Products) are also figured out to attempt mathematical correlation between these specific rates and observed process variables.
2. Proposal of the mathematical model
Using the specific rates of biomass, substrate and product formation and the observed experimental batch kinetics, the key mathematical correlations are developed, which form the approximate representation (description) of the Bioprocess. Approximate values of the model parameters are calculated from the graphical observations using above correlations.
3. Parameter optimization
Once the model structure and the approximate values of the model parameters are known, the model differential equations are solved (numerically) to figure out the capability of the model to describe the experimental observations. The error (discrepancy) between the model simulation and experimental data points is squared and normalized with respect to the maximum value of the process variable(s). The overall sum of normalized squared error at all data points and all process variables (Biomass, Substrate and Product(s)) has to be as low as possible, for the model to adequately describe the observed experimental kinetics. In order to do this, initial guess of the model parameters is used to numerically solve the model equations. The objective function described above is then calculated. The model parameters are then changed according to well defined non linear regression algorithms (Such Rosenbrock Optimization algorithm) simulation followed by calculation of the objective function is done again and again (iterated) to optimize the model parameters. The new objective function is compared from the previous value to figure out if the error between the model simulation and experimental data points is reducing or not. This is continued for several iterations till the model accurately describes the experimental kinetics.
4. Parameter Sensitivity Analysis
The model may contain so many parameters to accurately describe the observed kinetics of microbial cultivation. All the parameters may not be so important for the description of the kinetics of cultivation. Some parameters may be very important and some may not be so important in the overall description of the cultivation process. In order to attach significance to each model parameter, Parameter sensitivity analysis is done. In this analysis the value of a particular parameter is increased slightly (by 1 %) and its effect in the overall simulation figured out. If it is observed that the model simulation has significantly gone away from the data points (more error) then that parameter is more sensitive. Efforts are made to determine that parameter accurately by independent experiment. If such a analysis shows that (out of so many parameters of the model) some parameter is least sensitive then it might as well be dropped from the model structure in order to make the model more robust.
5. Statistical Validity
Once the model is ready, the capability of the mathematical model to describe particular batch kinetics is tested statistically. F– tests are done to find out the confidence level for the developed mathematical model. Statistical tests are also done to test the suitability of different model structure to describe same batch kinetics of fermentation.