High Resolution Finite Volume Schemes for Solving Multivariable Biological Cell Population Balance Models

Abstract

Multivariable cell population balance models are commonly used to explain complicated biological phenomena associated with product formation and cell growth in microbial populations. Such models typically consist of a partial integrodifferential equation for describing cell growth and an ordinary integrodifferential equation for representing substrate consumption. Due to their mathematical complexities, the numerical solutions of such models are hard tasks for numerical schemes. In this article, semidiscrete high resolution flux-limiting finite volume schemes are applied to solve single-variate and bivariate cell population balance models. The schemes have the abilities to achieve narrow peaks and resolve sharp discontinuities in the solutions on coarse meshes. These schemes are cheaper due to their short and reliable computational coding for complex problems. Several case studies are carried out. The numerical results of the schemes are compared with each other in terms of CPU time and accuracy. After consideration of different growth rate functions and incorporating equal and unequal partitioning, the suggested schemes were found to be more reliable and effective.

Population balances in case of crossing characteristic curves: Application to T‐cells immune response

“…For this reason, the concentration of CD25 protein starts decreasing depending on the rate of interaction between CD25 and IL-2 proteins. The (10) In this article, the IL-2 protein concentration 4 () ct is increased linearly by injection: (11) A set of values is considered for the kinetic parameters ki (i = 1...5) describing the single T-cell dynamics. The numerical solutions for the system of ODEs (see Equations (8-10)) are solved using the RK-4 method as shown in Figure 7.…”

“…[5] It has diverse applications including polymerization, crystallization, precipitation, and several other chemical industrial processes. [6,7] In the biological sciences, PBM has been employed heavily due to vast applications of population dynamics in the modelling of continuous variation in the microbial populations, [2,8,9] phenomena such as cell growth, [10] intracellular reactions and division with stochastic partitioning, [11][12][13] and gene regulatory processes by population mediated signalling. [14] Mathematical modelling for the concentrations of multi proteins in a given T-cell has been investigated by several researchers.…”

Mathematical modelling of T‐cells activation dynamics for CD3 molecules recycling process

“…PBMs are well established in the field of chemical engineering where the particulate processes require accurate predictions to understand the overall behaviour of the underlying physical system . In cell biology, this technique is used frequently as a cell population balance model (CPBM) where single and multi‐variable cell population dynamics are studied in deterministic and stochastic settings by assimilating the chemical processes of particulate systems defined in PBM into biologically plausible expressions represented by CPBM …”

“…[26][27][28] In cell biology, this technique is used frequently as a cell population balance model (CPBM) where single and multivariable cell population dynamics are studied in deterministic and stochastic settings by assimilating the chemical processes of particulate systems defined in PBM into biologically plausible expressions represented by CPBM. [25,[29][30][31][32]. The variables represent characteristics such as cell contents, size, age, or biomass that are distributed over the whole population.…”

Central Upwind Scheme for Solving Multivariate Cell Population Balance Models

“…Thus, researchers started to find its numerical solution instead of the exact solution. A large number of numerical techniques are present for approximating population balance equations (PBEs) in chemical and biological engineering, such as the weighted residual method [7], the Monte Carto simulation [8], the method of moments [9] [10], the finite difference method [11]- [13], the spectral methods [14], the finite element method [15] [16], and the high resolution finite volume scheme [17] [18], etc.…”