Continuum Modelling of In Vitro Tissue Engineering: A Review

Human Stem Cells for Ophthalmology: Recent Advances in Diagnostic Image Analysis and Computational Modelling

Purpose of Review

To explore the advances and future research directions in image analysis and computational modelling of human stem cells (hSCs) for ophthalmological applications.

Recent Findings

hSCs hold great potential in ocular regenerative medicine due to their application in cell-based therapies and in disease modelling and drug discovery using state-of-the-art 2D and 3D organoid models. However, a deeper characterisation of their complex, multi-scale properties is required to optimise their translation to clinical practice. Image analysis combined with computational modelling is a powerful tool to explore mechanisms of hSC behaviour and aid clinical diagnosis and therapy.

Summary

Many computational models draw on a variety of techniques, often blending continuum and discrete approaches, and have been used to describe cell differentiation and self-organisation. Machine learning tools are having a significant impact in model development and improving image classification processes for clinical diagnosis and treatment and will be the focus of much future research.

Mathematical modelling with Bayesian inference to quantitatively characterize therapeutic cell behaviour in nerve tissue engineering

Cellular engineered neural tissues have significant potential to improve peripheral nerve repair strategies. Traditional approaches depend on quantifying tissue behaviours using experiments in isolation, presenting a challenge for an overarching framework for tissue design. By comparison, mathematical cell-solute models benchmarked against experimental data enable computational experiments to be performed to test the role of biological/biophysical mechanisms, as well as to explore the impact of different design scenarios and thus accelerate the development of new treatment strategies. Such models generally consist of a set of continuous, coupled, partial differential equations relying on a number of parameters and functional forms. They necessitate dedicated in vitro experiments to be informed, which are seldom available and often involve small datasets with limited spatio-temporal resolution, generating uncertainties. We address this issue and propose a pipeline based on Bayesian inference enabling the derivation of experimentally informed cell-solute models describing therapeutic cell behaviour in nerve tissue engineering. We apply our pipeline to three relevant cell types and obtain models that can readily be used to simulate nerve repair scenarios and quantitatively compare therapeutic cells. Beyond parameter estimation, the proposed pipeline enables model selection as well as experiment utility quantification, aimed at improving both model formulation and experimental design.

Perspectives on optimizing local delivery of drugs to peripheral nerves using mathematical models

Drug therapies for treating peripheral nerve injury repair have shown significant promise in preclinical studies. Despite this, drug treatments are not used routinely clinically to treat patients with peripheral nerve injuries. Drugs delivered systemically are often associated with adverse effects to other tissues and organs; it remains challenging to predict the effective concentration needed at an injured nerve and the appropriate delivery strategy. Local drug delivery approaches are being developed to mitigate this, for example via injections or biomaterial-mediated release. We propose the integration of mathematical modeling into the development of local drug delivery protocols for peripheral nerve injury repair. Mathematical models have the potential to inform understanding of the different transport mechanisms at play, as well as quantitative predictions around the efficacy of individual local delivery protocols. We discuss existing approaches in the literature, including drawing from other research fields, and present a process for taking forward an integrated mathematical-experimental approach to accelerate local drug delivery approaches for peripheral nerve injury repair. This article is categorized under: Neurological Diseases > Molecular and Cellular Physiology Neurological Diseases > Computational Models Neurological Diseases > Biomedical Engineering.

A Systematically Reduced Mathematical Model for Organoid Expansion

Organoids are three-dimensional multicellular tissue constructs. When cultured in vitro, they recapitulate the structure, heterogeneity, and function of their in vivo counterparts. As awareness of the multiple uses of organoids has grown, e.g. in drug discovery and personalised medicine, demand has increased for low-cost and efficient methods of producing them in a reproducible manner and at scale. Here we focus on a bioreactor technology for organoid production, which exploits fluid flow to enhance mass transport to and from the organoids. To ensure large numbers of organoids can be grown within the bioreactor in a reproducible manner, nutrient delivery to, and waste product removal from, the organoids must be carefully controlled. We develop a continuum mathematical model to investigate how mass transport within the bioreactor depends on the inlet flow rate and cell seeding density, focusing on the transport of two key metabolites: glucose and lactate. We exploit the thin geometry of the bioreactor to systematically simplify our model. This significantly reduces the computational cost of generating model solutions, and provides insight into the dominant mass transport mechanisms. We test the validity of the reduced models by comparison with simulations of the full model. We then exploit our reduced mathematical model to determine, for a given inlet flow rate and cell seeding density, the evolution of the spatial metabolite distributions throughout the bioreactor. To assess the bioreactor transport characteristics, we introduce metrics quantifying glucose conversion (the ratio between the total amounts of consumed and supplied glucose), the maximum lactate concentration, the proportion of the bioreactor with intolerable lactate concentrations, and the time when intolerable lactate concentrations are first experienced within the bioreactor. We determine the dependence of these metrics on organoid-line characteristics such as proliferation rate and rate of glucose consumption per cell. Finally, for a given organoid line, we determine how the distribution of metabolites and the associated metrics depend on the inlet flow rate. Insights from this study can be used to inform bioreactor operating conditions, ultimately improving the quality and number of bioreactor-expanded organoids.

Regenerative medicine meets mathematical modelling: developing symbiotic relationships

Successful progression from bench to bedside for regenerative medicine products is challenging and requires a multidisciplinary approach. What has not yet been fully recognised is the potential for quantitative data analysis and mathematical modelling approaches to support this process. In this review, we highlight the wealth of opportunities for embedding mathematical and computational approaches within all stages of the regenerative medicine pipeline. We explore how exploiting quantitative mathematical and computational approaches, alongside state-of-the-art regenerative medicine research, can lead to therapies that potentially can be more rapidly translated into the clinic.

Predicting Bone Formation in Mesenchymal Stromal Cell-Seeded Hydrogels Using Experiment-Based Mathematical Modeling

In vitro bone formation by mesenchymal stromal cells encapsulated in type-1 collagen hydrogels is demonstrated after a 28-day in vitro culture period. Analysis of the hydrogels is carried out by X-ray microcomputed tomography, histology, and immunohistochemistry, which collectively demonstrates that bone formation in the hydrogels was quantifiably proportional to the initial collagen concentration, and subsequently the population density of seeded cells. This was established by varying the initial collagen concentration at a constant cell seeding density (3 × 10⁵ cells/0.3 mL hydrogel), and separately varying cell seeding density at a constant collagen concentration (1 mg/mL). Using these data, a mathematical model is presented for the total hydrogel volume and mineralization volume based on the observed linear contraction dynamics of cell-seeded collagen gels. The model parameters are fitted by comparing the predictions of the mathematical model for the hydrogel and mineralized volumes on day 28 with the experimental data. The model is then used to predict the hydrogel and mineralization volumes for a range of hydrogel collagen concentrations and cell seeding densities, providing comprehensive input/output descriptors for generating mineralized hydrogels for bone tissue engineering. It is proposed that this quantitative approach will be a useful tool for generating in vitro manufactured bone tissue, defining input parameters that yield predictable output measures of tissue maturation. Impact statement This article describes a simple yet powerful quantitative description of in vitro tissue-engineered bone by combining experimental data with mathematical modeling. The overall aim of the article is to examine what is currently known about cell-mediated collagen contraction, and demonstrate that this phenomenon can be exploited to tailor bone formation by choosing a specific set of input parameters in the form of cell seeding density and collagen hydrogel concentration. Our study utilizes a clinically relevant cell source (human mesenchymal stem cells) with a biomaterial that has received regulatory approval for use in humans (collagen type 1), and hence could be useful for clinical applications, as well as furthering our understanding of cell/extracellular matrix interactions in determining in vitro bone tissue formation.

Mathematical modelling of fluid flow and solute transport to define operating parameters for in vitro perfusion cell culture systems

In recent years, there has been a move away from the use of static in vitro two-dimensional cell culture models for testing the chemical safety and efficacy of drugs. Such models are increasingly being replaced by more physiologically relevant cell culture systems featuring dynamic flow and/or three-dimensional structures of cells. While it is acknowledged that such systems provide a more realistic environment within which to test drugs, progress is being hindered by a lack of understanding of the physical and chemical environment that the cells are exposed to. Mathematical and computational modelling may be exploited in this regard to unravel the dependency of the cell response on spatio-temporal differences in chemical and mechanical cues, thereby assisting with the understanding and design of these systems. In this paper, we present a mathematical modelling framework that characterizes the fluid flow and solute transport in perfusion bioreactors featuring an inlet and an outlet. To demonstrate the utility of our model, we simulated the fluid dynamics and solute concentration profiles for a variety of different flow rates, inlet solute concentrations and cell types within a specific commercial bioreactor chamber. Our subsequent analysis has elucidated the basic relationship between inlet flow rate and cell surface flow speed, shear stress and solute concentrations, allowing us to derive simple but useful relationships that enable prediction of the behaviour of the system under a variety of experimental conditions, prior to experimentation. We describe how the model may used by experimentalists to define operating parameters for their particular perfusion cell culture systems and highlight some operating conditions that should be avoided. Finally, we critically comment on the limitations of mathematical and computational modelling in this field, and the challenges associated with the adoption of such methods.

Lattice and continuum modelling of a bioactive porous tissue scaffold

A contemporary procedure to grow artificial tissue is to seed cells onto a porous biomaterial scaffold and culture it within a perfusion bioreactor to facilitate the transport of nutrients to growing cells. Typical models of cell growth for tissue engineering applications make use of spatially homogeneous or spatially continuous equations to model cell growth, flow of culture medium, nutrient transport and their interactions. The network structure of the physical porous scaffold is often incorporated through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a square grid lattice to demonstrate the importance of explicitly modelling the network structure of the porous scaffold and compare results from this model with those from a modified continuum model from the literature. We capture two-way coupling between cell growth and fluid flow by allowing cells to block pores, and by allowing the shear stress of the fluid to affect cell growth and death. We explore a range of parameters for both models and demonstrate quantitative and qualitative differences between predictions from each of these approaches, including spatial pattern formation and local oscillations in cell density present only in the lattice model. These differences suggest that for some parameter regimes, corresponding to specific cell types and scaffold geometries, the lattice model gives qualitatively different model predictions than typical continuum models. Our results inform model selection for bioactive porous tissue scaffolds, aiding in the development of successful tissue engineering experiments and eventually clinically successful technologies.

A parameterised mathematical model to elucidate osteoblast cell growth in a phosphate-glass microcarrier culture

Tissue engineering has the potential to augment bone grafting. Employing microcarriers as cell-expansion vehicles is a promising bottom-up bone tissue engineering strategy. Here we propose a collaborative approach between experimental work and mathematical modelling to develop protocols for growing microcarrier-based engineered constructs of clinically relevant size. Experiments in 96-well plates characterise cell growth with the model human cell line MG-63 using four phosphate glass microcarrier materials. Three of the materials are doped with 5 mol% TiO₂ and contain 0%, 2% or 5% CoO, and the fourth material is doped only with 7% TiO₂ (0% CoO). A mathematical model of cell growth is parameterised by finding material-specific growth coefficients through data-fitting against these experiments. The parameterised mathematical model offers more insight into the material performance by comparing culture outcome against clinically relevant criteria: maximising final cell number starting with the lowest cell number in the shortest time frame. Based on this analysis, material 7% TiO₂ is identified as the most promising.

Mathematical and computational models for bone tissue engineering in bioreactor systems

Research into cellular engineered bone grafts offers a promising solution to problems associated with the currently used auto- and allografts. Bioreactor systems can facilitate the development of functional cellular bone grafts by augmenting mass transport through media convection and shear flow-induced mechanical stimulation. Developing successful and reproducible protocols for growing bone tissue in vitro is dependent on tuning the bioreactor operating conditions to the specific cell type and graft design. This process, largely reliant on a trial-and-error approach, is challenging, time-consuming and expensive. Modelling can streamline the process by providing further insight into the effect of the bioreactor environment on the cell culture, and by identifying a beneficial range of operational settings to stimulate tissue production. Models can explore the impact of changing flow speeds, scaffold properties, and nutrient and growth factor concentrations. Aiming to act as an introductory reference for bone tissue engineers looking to direct their experimental work, this article presents a comprehensive framework of mathematical models on various aspects of bioreactor bone cultures and overviews modelling case studies from literature.

Curvature- and fluid-stress-driven tissue growth in a tissue-engineering scaffold pore

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; when it is too fast, 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 paper, 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.

Pattern formation in multiphase models of chemotactic cell aggregation

We develop a continuum model for the aggregation of cells cultured in a nutrient-rich medium in a culture well. We consider a 2D geometry, representing a vertical slice through the culture well, and assume that the cell layer depth is small compared with the typical lengthscale of the culture well. We adopt a continuum mechanics approach, treating the cells and culture medium as a two-phase mixture. Specifically, the cells and culture medium are treated as fluids. Additionally, the cell phase can generate forces in response to environmental cues, which include the concentration of a chemoattractant that is produced by the cells within the culture medium. The model leads to a system of coupled nonlinear partial differential equations for the volume fraction and velocity of the cell phase, the culture medium pressure and the chemoattractant concentration, which must be solved subject to appropriate boundary and initial conditions. To gain insight into the system, we consider two model reductions, appropriate when the cell layer depth is thin compared to the typical length scale of the culture well: a (simple) 1D and a (more involved) thin-film extensional flow reduction. By investigating the resulting systems of equations analytically and numerically, we identify conditions under which small amplitude perturbations to a homogeneous steady state (corresponding to a spatially uniform cell distribution) can lead to a spatially varying steady state (pattern formation). Our analysis reveals that the simpler 1D reduction has the same qualitative features as the thin-film extensional flow reduction in the linear and weakly nonlinear regimes, motivating the use of the simpler 1D modelling approach when a qualitative understanding of the system is required. However, the thin-film extensional flow reduction may be more appropriate when detailed quantitative agreement between modelling predictions and experimental data is desired. Furthermore, full numerical simulations of the two model reductions in regions of parameter space when the system is not close to marginal stability reveal significant differences in the evolution of the volume fraction and velocity of the cell phase, and chemoattractant concentration.

Adult Stem Cell Therapies for Wound Healing: Biomaterials and Computational Models

The increased incidence of diabetes and tumors, associated with global demographic issues (aging and life styles), has pointed out the importance to develop new strategies for the effective management of skin wounds. Individuals affected by these diseases are in fact highly exposed to the risk of delayed healing of the injured tissue that typically leads to a pathological inflammatory state and consequently to chronic wounds. Therapies based on stem cells (SCs) have been proposed for the treatment of these wounds, thanks to the ability of SCs to self-renew and specifically differentiate in response to the target bimolecular environment. Here, we discuss how advanced biomedical devices can be developed by combining SCs with properly engineered biomaterials and computational models. Examples include composite skin substitutes and bioactive dressings with controlled porosity and surface topography for controlling the infiltration and differentiation of the cells. In this scenario, mathematical frameworks for the simulation of cell population growth can provide support for the design of bioconstructs, reducing the need of expensive, time-consuming, and ethically controversial animal experimentation.

Evaluation of the growth environment of a hydrostatic force bioreactor for preconditioning of tissue-engineered constructs

Bioreactors have been widely acknowledged as valuable tools to provide a growth environment for engineering tissues and to investigate the effect of physical forces on cells and cell-scaffold constructs. However, evaluation of the bioreactor environment during culture is critical to defining outcomes. In this study, the performance of a hydrostatic force bioreactor was examined by experimental measurements of changes in dissolved oxygen (O2), carbon dioxide (CO2), and pH after mechanical stimulation and the determination of physical forces (pressure and stress) in the bioreactor through mathematical modeling and numerical simulation. To determine the effect of hydrostatic pressure on bone formation, chick femur skeletal cell-seeded hydrogels were subjected to cyclic hydrostatic pressure at 0-270 kPa and 1 Hz for 1 h daily (5 days per week) over a period of 14 days. At the start of mechanical stimulation, dissolved O2 and CO2 in the medium increased and the pH of the medium decreased, but remained within human physiological ranges. Changes in physiological parameters (O2, CO2, and pH) were reversible when medium samples were placed in a standard cell culture incubator. In addition, computational modeling showed that the distribution and magnitude of physical forces depends on the shape and position of the cell-hydrogel constructs in the tissue culture format. Finally, hydrostatic pressure was seen to enhance mineralization of chick femur skeletal cell-seeded hydrogels.

Multiphase modelling of the effect of fluid shear stress on cell yield and distribution in a hollow fibre membrane bioreactor

We present a simplified two-dimensional model of fluid flow, nutrient transport and cell distribution in a hollow fibre membrane bioreactor, with the aim of exploring how fluid flow can be used to control the distribution and yield of a cell population which is sensitive to both fluid shear stress and nutrient concentration. The cells are seeded in a scaffold in a layer on top of the hollow fibre, only partially occupying the extracapillary space. Above this layer is a region of free-flowing fluid which we refer to as the upper fluid layer. The flow in the lumen and upper fluid layer is described by the Stokes equations, whilst the flow in the porous fibre membrane is assumed to follow Darcy’s law. Porous mixture theory is used to model the dynamics of and interactions between the cells, scaffold and fluid in the cell–scaffold construct. The concentration of a limiting nutrient (e.g. oxygen) is governed by an advection–reaction–diffusion equation in each region. Through exploitation of the small aspect ratio of each region and asymptotic analysis, we derive a coupled system of partial differential equations for the cell volume fraction and nutrient concentration. We use this model to investigate the effect of mechanotransduction on the distribution and yield of the cell population, by considering cases in which cell proliferation is either enhanced or limited by fluid shear stress and by varying experimentally controllable parameters such as flow rate and cell–scaffold construct thickness.