A mathematical model is developed for fluid flow, nutrient transport, and cell growth inside a bioreactor with a scaffold at the periphery and lumen at the centerline. The scaffold material is assumed to be deformable. In order to deal with the deformation of the solid phase and fluid phase inside the scaffold, biphasic mixture theory equations are adopted, which are derived from the theory of mixtures. The flow inside the lumen is governed by Stokes equation. Advection–diffusion–reaction equation is used for the mass balance of the nutrient within the scaffold region. Cell growth depends on the nutrient concentration and is expressed by the Contois equation that accounts for contact inhibition. We use lubrication approximation to reduce the system of hydrodynamic and nutrient transport equations. This leads to a coupled system of partial differential equations (PDEs) with time‐dependent variables. Laplace transformation is used to deal with time‐dependent terms, and Durbin's algorithm is used to retrieve the time dependency. We investigate the outreach of nutrients inside the scaffold region, which regulates the growth of cells at a particular time. Based on the available experimental data, we consider relevant reaction kinematics of the cells and observe the corresponding nutrient distribution inside the bioreactor. The factors that affect the nutrient concentration are lumen radius, porosity, and permeability of the scaffold, Thiele modulus, pressure gradient, and so forth. The total mass transfer rate is computed to understand the nutrient distribution across various regions of the bioreactor.

中文翻译：

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装配在生物反应器外围的可变形支架内部的弹性动力学和细胞生长的数学模型

建立了一个数学模型，用于生物反应器内部的流体流动，营养物运输和细胞生长，其中支架位于外围，内腔位于中心线。假定支架材料是可变形的。为了处理支架内部固相和液相的变形，采用了双相混合理论方程式，该方程式是从混合理论推导而来的。管腔内部的流动由斯托克斯方程控制。对流-扩散-反应方程用于支架区域内营养物的质量平衡。细胞生长取决于营养物的浓度，并由解释接触抑制的Contois方程表示。我们使用润滑近似来简化流体动力和养分运移方程式的系统。这导致具有时间相关变量的偏微分方程（PDE）耦合系统。拉普拉斯变换用于处理与时间有关的项，而杜宾算法用于检索与时间有关的项。我们研究支架区域内营养的扩展，从而调节特定时间的细胞生长。基于可用的实验数据，我们考虑细胞的相关反应运动学，并观察生物反应器内部相应的营养分布。影响营养物浓度的因素是管腔半径，孔隙率和支架的渗透性，Thiele模量，压力梯度等。计算总传质速率以了解生物反应器各个区域的养分分布。