Tissue engineering

We develop a mathematical model to predict the most effective design of porous tissue engineering scaffolds and analyze the choice of values for input parameters such as channel radius, shear stress, nutrient concentration, and nutrient flow pressure, while assuming a constant inlet flux of suspended nutrients. Our model utilizes a branching structure in which scaffold pores bifurcate at each layer junction, thereby allowing for investigation of the input parameters at a pore level. By assuming all branching structures to be equivalent, we assume homogeneity along two axes, meaning the biological scaffold’s geometry is effectively reduced to one dimension. We employ established fluid dynamics equations such as Darcy’s law, the continuity equation, and the advection–diffusion-reaction equation to model the evolution of parameters such as pore radius, shear stress, pressure, fluid velocity, and nutrient concentration. Further simplification of these equations is achieved via nondimensionalization and asymptotic analysis based on the small aspect ratio of the scaffold. Our results establish quantitative relationships among the input parameters to provide a foundation for effectual construction of engineered tissue in the shortest time possible. Further, we devise a trade-off analysis for differing scaffold geometries by comparing ratios of decreasing layer thickness and decreasing initial pore radius in successive downstream layers; these ratios guide predictions for candidates for ideal scaffold geometries according to priorities of time, cost, or total tissue volume. Particularly, we find that minimal values of both these ratios together yield larger volumes of new tissue, while jointly large ratios produce far less tissue but in much less time. Mixed values (e.g., small layer thickness ratio and large initial radius ratio) and intermediate values (in the middle range for both ratios) of these ratios produce moderate tissue growth, in a moderate amount of time, and yield less waste of material used to construct the scaffolds.

While the effects of external factors like fluid mechanical forces and scaffold geometry on tissue growth have been extensively studied, the influence of cell behavior—particularly nutrient consumption and depletion within the scaffold—has received less attention. Incorporating such factors into mathematical models allows for a more comprehensive understanding of tissue-engineering processes. This work presents a comprehensive continuum model for cell proliferation within two-dimensional tissue-engineering scaffolds. Through mathematical modeling and asymptotic analysis based on the small aspect ratio of the scaffolds, the study aims to reduce computational burdens and solve mathematical models for tissue growth within porous scaffolds. The model incorporates fluid dynamics of nutrient feed flow, nutrient transport, cell concentration, and tissue growth, considering the evolving scaffold porosity due to cell proliferation, with the crux of the work establishing the ideal pore shape for channels within the tissue-engineering scaffold to obtain the maximum tissue growth. We investigate scaffolds with specific two-dimensional initial porosity profiles, and our results show that scaffolds which are uniformly graded in porosity throughout their depth promote more tissue growth.

Tissue-engineering scaffolds contain channels lined by cells that allow nutrient-rich culture medium to pass through to encourage cell proliferation. Several factors have significant impacts on the tissue growth, including the nutrient flow rate, concentration in the feed, scaffold elasticity, and cell properties. Recent studies have investigated these effects separately; however, in this work, we examine all of them simultaneously. Our objectives in this work are as follows: (i) developing a mathematical model describing the nutrient flow dynamics and concentration, scaffold elasticity, and cell proliferation; (ii) solving the model and then simulating the cell proliferation process; and (iii) optimizing the initial configuration of the scaffold channels to maximize the cell growth. The results of our study demonstrate that the rate of nutrient consumption by the cells (cell hunger rate) and the scaffold elastic compliance have an impact on tissue growth, with higher cell hunger rates leading to longer incubation periods, while scaffold elastic compliance slightly affects overall growth. Furthermore, decreasing the scaffold elastic compliance while maintaining a constant nutrient consumption rate results in an optimal funnel-shaped channel geometry, where the upper part of the channel is larger than the downstream, promoting enhanced tissue integration and functionality.

Scaffolds engineered for in vitro tissue engineering consist of multiple pores where cells can migrate along with nutrient-rich culture medium. The presence of the nutrient medium throughout the scaffold pores promotes cell proliferation, and this process depends on several factors such as scaffold geometry, nutrient medium flow rate, shear stress, cell-scaffold focal adhesions and elastic properties of the scaffold mate- rial. While numerous studies have addressed the first four factors, the mathematical approach described herein focuses on cell proliferation rate in elastic scaffolds, under constant flux of nutrients. As cells proliferate, the scaffold pores radius shrinks and thus, in order to sustain the nutrient flux, the inlet applied pressure on the upstream side of the scaffold pore must be increased. This results in expansion of the elastic scaffold pore, which in turn further increases the rate of cell proliferation. Consider- ing the elasticity of the scaffold, the pore deformation allows further cellular growth beyond that of inelastic conditions. In this paper, our objectives are as follows: (i) Develop a mathematical model for describing fluid dynamics, scaffold elasticity and cell proliferation for scaffolds consist of identical nearly cylindrical pores; (ii) Solve the models and then simulate cellular proliferation within an elastic pore. The simula- tion can emulate real life tissue growth in a scaffold and offer a solution which reduces the numerical burdens. Lastly, our results demonstrated are in qualitative agreement with experimental observations reported in the literature.

In a tissue-engineering scaffold pore lined with cells, nutrient-rich culture medium flows through the scaffold and cells proliferate. In this process, both environmental factors such as flow rate, shear stress, as well as cell properties have significant effects on tissue growth. Recent studies focused on effects of scaffold pore geometry on tissue growth, while in this work, we focus on the nutrient depletion and consumption rate by the cells, which cause a change in nutrient concentration of the feed and influence the growth of cells lined downstream. In this paper, our objectives are threefold: (i) design a mathematical model for the cell proliferation describing fluid dynamics, nutrient concentration, and tissue growth; (ii) solve the models and then simulate the tissue proliferation process; (iii) design a ``reverse algorithm" to find the initial configuration of the scaffold with the knowledge of the desired property of the final tissue geometry. Our model reduces the numerical burdens and captures the experimental observations from the literature. In addition, it provides an efficient algorithm to simulate the cell proliferation and determine the design of a tissue engineering scaffold given a desired tissue profile outcome.

Cell proliferation within a fluid-filled porous tissue-engineering scaffold depends on a sensitive choice of pore geometry and flow rates: regions of high curvature encourage cell proliferation, while a critical flow rate is required to promote growth for certain cell types. When the flow rate is too slow the nutrient supply is limited; too fast and cells may be damaged by the high fluid shear stress. As a result, determining appropriate tissue-engineering-construct geometries and operating regimes poses a significant challenge that cannot be addressed by experimentation alone. In this work, we present a mathematical theory for the fluid flow within a pore of a tissue-engineering scaffold, which is coupled to the growth of cells on the pore walls. We exploit the slenderness of a pore that is typical in such a scenario, to derive a reduced model that enables a comprehensive analysis of the system to be performed. We derive analytical solutions in a particular case of a nearly piecewise constant growth law and compare these with numerical solutions of the reduced model. Qualitative comparisons of tissue morphologies predicted by our model, with those observed experimentally, are also made. We demonstrate how the simplified system may be used to make predictions on the design of a tissue-engineering scaffold and the appropriate operating regime that ensures a desired level of tissue growth.