Combining in silico and in vitro models to inform cell seeding strategies in tissue engineering

Abstract

The seeding density of therapeutic cells in engineered tissue impacts both cell survival and vascularization. Excessively high seeded cell densities can result in increased death and thus waste of valuable cells, whereas lower seeded cell densities may not provide sufficient support for the tissue in vivo, reducing efficacy. Additionally, the production of growth factors by therapeutic cells in low oxygen environments offers a way of generating growth factor gradients, which are important for vascularization, but hypoxia can also induce unwanted levels of cell death. This is a complex problem that lends itself to a combination of computational modelling and experimentation. Here, we present a spatio-temporal mathematical model parametrized using in vitro data capable of simulating the interactions between a therapeutic cell population, oxygen concentrations and vascular endothelial growth factor (VEGF) concentrations in engineered tissues. Simulations of collagen nerve repair constructs suggest that specific seeded cell densities and non-uniform spatial distributions of seeded cells could enhance cell survival and the generation of VEGF gradients. These predictions can now be tested using targeted experiments.

1. Background

Engineered tissues seeded with therapeutic cells demonstrate improved outcomes, such as rates of regeneration and assimilation, when compared with acellular controls [1]. Therapeutic cells have been implemented successfully in a range of contexts including bone [2], vascular [3,4] and peripheral nervous system repair [5]. However, the death of seeded cells post-implantation due to the often hostile nature of in vivo repair environments remains a problem [6,7]. This has only partly been remedied by the development of scaffolds that act as more amenable microenvironments for the cells in vivo [8,9]. As well as hindering the efficacy of engineered tissue, the death of cells soon after implantation is wasteful and incurs unnecessary expense that will only multiply upon the translation of cell seeding techniques to the mass market clinical setting.

Existing research suggests that specific seeding cell densities that are optimal for promoting cell survival and engineered tissue efficacy in vivo may exist [10–12]. However, experimental studies into the effects of different seeded cell densities are necessarily limited in scope and usually compare only two or three different densities at a time. Furthermore, the impact of different spatial distributions of seeded cells is also generally unknown and unexplored. Varying the distribution of seeded cells throughout an engineered tissue construct could help alleviate excessive cell death in less hospitable regions of a tissue, thereby reducing loss of valuable cells while allowing more cells to flourish in areas of tissue that are more likely to be exposed to the necessary nutrients.

In this paper, we hypothesize that the variation in the seeded cell density throughout engineered tissue could also induce gradients of important growth factors that remain stable over time. The provision of oxygen and nutrients to seeded cells is imperative for their survival, and this can be achieved efficiently only through the growth and remodelling of a highly organized vascular network. Vascular growth factors help to initiate angiogenesis and guide the migration of endothelial cells via chemotactic cues. Vascular endothelial growth factor (VEGF) is generally regarded as the most important of the vascular growth factors. VEGF promotes vascular permeability and basement membrane degradation during the initial stages of angiogenesis [13], as well as acting as a mitogen for endothelial cells. Furthermore, experimental evidence suggests that the concentration and gradient steepness of VEGF in tissue affects the rate and directionality of endothelial cell migration [14–17].

VEGF and other growth factors can be incorporated into tissue-engineered devices [18–22], but the most clinically viable and effective way of doing this remains unknown. Alternatively, VEGF can be produced by a seeded cell population. This allows VEGF to be generated within the graft in a sustained manner over time, while the rate of secretion is regulated by the cells according to local environmental cues. However, little experimental research has been dedicated to determining which densities and distributions of cells within engineered tissue could provide the desired levels of VEGF, in the right regions of the construct.

Overall, the choice of seeded cell distribution and density needs to maintain a careful balance. Cells metabolize oxygen, then respond to a low oxygen environment by releasing VEGF [23–25], which in turn encourages vascularization. However, if the local cell density is too high, then cells will die due to the delay and insufficiency of vascularization. Therefore, there is likely to be an optimal seeding density that both facilitates cell survival and encourages vascularization.

The wide range of possible materials, cells and spatial configurations means that determining experimentally which factors may help or hinder cell survival and vascular growth in engineered tissue is both time-consuming and expensive. Here, we present a multidisciplinary approach (figure 1) to address the connected problems of cell survival and revascularization, demonstrating the potential of using mathematical modelling in combination with experimental work to accelerate and direct tissue engineering research [26]. Integrated mathematical–experimental work has a rich history and a promising future in the field of regenerative medicine [27,28] and general biology [29–31]. Mathematical modelling offers experimental scientists a tool to extract further information from their data. However, this type of multidisciplinary approach has never been used to address the specific problem of cell seeding strategies.

Figure 1.

The multidisciplinary method presented here involves close integration between theoretical and experimental work through the use of in vitro experiments specifically designed for the purpose of parametrization. Model predictions can be used to test and form new hypotheses and thereby direct the course of future experiments.

Here, we formulate a spatio-temporal mathematical model consisting of a set of partial differential equations to describe the interactions between a seeded cell population, oxygen concentration and VEGF concentration within a cell-seeded collagen engineered tissue. This model is developed based on, and parametrized against, in vitro data specifically collected for this purpose. The cell type (adipose-derived stem cells differentiated towards a Schwann cell-like phenotype, dADSCs) and biomaterial (type I collagen) used for the model were chosen to mimic the design of peripheral nerve repair constructs (NRCs) currently under development [32]. The interdisciplinary framework presented here can be used to model other engineered tissues that use alternative materials and cell combinations by substituting parameter values from the literature, or by conducting a new set of dedicated experiments.

This work aims to provide a multidisciplinary tool that can inform how the seeded cell densities and distributions in engineered tissue can be controlled to encourage desired levels of vascularization and regeneration through a combination of viable cell survival and targeted VEGF secretion. Model predictions can provide guidance for narrowing the range of designs to be tested in future experiments, refining and reducing the extent of in vitro and in vivo experimentation in the development of engineered tissue therapies.

2. Methods

2.1. Culture of cells

Adipose-derived stem cells were isolated from adult Sprague–Dawley rats as previously described [33]. The animal care and experimental procedures were carried out in accordance with the Directive 2010/63/EU of the European Parliament and of the Council on the protection of animals used for scientific purposes and was also approved by the Northern Swedish Committee for Ethics in Animal Experiments (no. A186-12). The stem cells were differentiated into a Schwann cell-like phenotype (dADSCs) as previously described [33] and were maintained in minimum essential media (MEM with GlutaMAX; Gibco) containing 10% (v/v) fetal bovine serum (FBS) and 1% (v/v) penicillin/streptomycin solution, supplemented with 14 µM forskolin (Sigma), 10 ng ml⁻¹ basic fibroblast growth factor (bFGF; Pepro Tech Ltd, UK) and 252 ng ml⁻¹ neuregulin NR G1 (R&D Systems, UK). The cells were kept at subconfluent levels in a 37°C incubator with 5% CO₂ and passaged with trypsin/EDTA (Invitrogen, UK) approximately every 72 h.

2.2. Fabrication of cell-seeded collagen gels

To prepare the gels, eight volumes of type I rat tail collagen (2 mg ml⁻¹ in 0.6% acetic acid; First Link, UK) were mixed with one volume of 10× MEM (Sigma) and the mixture neutralized using sodium hydroxide before mixing with one volume of cell suspension [5,32,34]. Two hundred and forty microlitres of cellular collagen suspension was pipetted into individual wells of a 96-well plate and then incubated at 37°C for 15 min for the gels to set, before being stabilized by plastic compression for 15 min (RAFT, Lonza). The number of cells in the starting suspensions was varied to give final cell densities of 39, 77, 154, 231 and 385 million cells ml⁻¹ stabilized gel. These values are within the range used within NRCs and other engineered tissues [5,32,35,36]. The resulting gels were immersed in 200 µl media (MEM with GlutaMAX, Gibco), apart from the highest seeded density gels, which were transferred to a 24-well plate with 1 ml media. Samples were incubated at 37°C in a humidified incubator with 5% CO₂ for 24 h. The oxygen level inside the incubator was controlled and maintained (using Biospherix ProOx 110) at each of the following concentrations: 1, 3, 5, 10 and 16%. Incubator oxygen concentration values were chosen to cover a physiological range. The unit % refers to the percentage of volume as a gas, with 1% = 7.6 mmHg = 1.317 mol m⁻³.

2.3. Metabolic viability assay

The medium from each well was aspirated and frozen for further analysis, then the viable cell density of each gel after incubation was determined using the CellTiter-Glo 3D Cell Viability Assay (Promega). Gels were transferred to 100 µl fresh medium (MEM with GlutaMAX) in a white opaque 96-well plate and 100 µl reagent was added, mixed for 3 min at 175 r.p.m. and left at room temperature for a further 25 min before luminescence was measured. Analysis involved subtracting baseline luminescence (from cell-free controls) and determining equivalent viable cell density using a standard curve generated from comparator gels tested immediately after the plastic compression step.

2.4. Proliferation assay (Ki67 staining)

Gels for immunocytochemistry were fixed using 4% paraformaldehyde overnight at 4°C. All storage washes and dilutions were performed using PBS. Cells were permeabilized in 0.5% Triton X-100 (Sigma) for 30 min. Following 3 × 5 min washes, non-specific binding was blocked with 5% normal goat serum (Dako, Ely, UK) for 30 min. After another wash step, a primary antibody used to detect Ki67 (rabbit IgG; 1 : 250 (AB15580 Abcam)) was incubated overnight at 4°C. Following 3 × 10 min washes, secondary antibody (anti-rabbit dylight 488; 1 : 300 (Vector Laboratories)) was added for 90 min. Hoechst 33 258 (1 mg ml⁻¹) was also added in the secondary antibody solution. The mean number of proliferating cells and cells/field was determined using fluorescence microscopy (Zeiss Axio Lab.A1) via the quantification of Ki67 and Hoechst staining in three pre-determined areas per gel.

2.5. Vascular endothelial growth factor A enzyme-linked immunosorbent assay

Secreted VEGF-A protein levels were determined by the enzyme-linked immunosorbent assay (ELISA). The cell culture supernatant from the wells was collected and analysed with a VEGF-A sandwich ELISA kit (RayBiotech, GA, USA) according to the manufacturer's protocols. In brief, samples were diluted 10–200-fold to fit the standard curve (0–80 pg ml⁻¹). All samples were analysed in duplicate, and the end-absorbance was measured at 450 nm (BioTek Synergy microplate reader) with n = 3–6 for each condition (variable % oxygen and initial cell seeding density million ml⁻¹).

2.6. Mathematical model of cell–solute interactions in cell-seeded collagen gels

Here, we present the mathematical model for cell–solute interactions in cell-seeded collagen gels. The mathematical model consists of three_­­­ coupled differential equations describing the interactions between the cell population density (n), oxygen concentration (c) and VEGF concentration (v) within engineered tissue. These variables are important factors that affect the survival of seeded cells and vascularization. The parameters used in the equations are defined in table 1. The generalized form of the oxygen concentration governing equation incorporates oxygen diffusion and metabolism,

$$\frac{\mathrm{\partial }c}{\mathrm{\partial }t}=D_c{\mathrm{\nabla }}^{2}c-Mf(n,c).$$ 2.1

Here D_c is the diffusion coefficient of oxygen and M is the cell metabolism constant. The function f determines how the oxygen consumption rate of the cells depends on the local oxygen concentration and cell density. The cell density governing equation includes cell proliferation and death,

$$\frac{\mathrm{\partial }n}{\mathrm{\partial }t}=\beta cn\left(1-\frac{n}{n_{\max }}\right)-\delta d(n,c),$$ 2.2

with β denoting the cell proliferation constant, δ the cell death constant and n_max the maximum cell density. The function d determines how cell death depends on the local oxygen concentration and cell density. Lastly, the VEGF governing equation combines VEGF diffusion, secretion by the cells and decay:

$$\frac{\mathrm{\partial }v}{\mathrm{\partial }t}=D_v{\mathrm{\nabla }}^{2}v+\alpha g(n,c)-d_{v}v.$$ 2.3

Here, D_v is the VEGF diffusion coefficient, α is the VEGF secretion constant and d_v the VEGF degradation rate.

Table 1.

Model parameters, including those identified from the literature or previous work, and those derived via fitting against the experimental data (*). The concentration at which oxygen is half-maximal $\overline{c}$ was provided by R. J. Shipley based on previous unpublished research using data published elsewhere [37]. Sensitivity analysis showed that in comparison to the other parameters in the oxygen concentration governing equation, the model predictions are not highly sensitive to $\overline{c}$; therefore, this parameter was not prioritized for fitting against the experimental data.

	approximated bounds used for parametrization	parameter value
cell density parameters
maximal cell density		nmax = 4 × 108 cell ml−1
proliferation rate constant*	unbounded	β = 2.2 × 10−4 m3 mol−1 s−1
cell death rate constant*	unbounded	δ(n0) = 1.1334 × 10−5 + 9.1256 × 10−14 n0 s−1 (where n0 is in cells ml−1)
oxygen concentration parameters
diffusion coefficient for oxygen in media		$D_{c_m}=\hspace{0.17em}2.624\hspace{0.17em}\times \hspace{0.17em}{10}^{-5}\hspace{0.17em}\hspace{0.17em}\text{c}{\text{m}}^2\hspace{0.17em}{\text{s}}^{-1}$[38]
diffusion coefficient for oxygen in gel		$D_{c_g}=\hspace{0.17em}4.50\hspace{0.17em}\times \hspace{0.17em}{10}^{-6}\hspace{0.17em}\hspace{0.17em}\text{c}{\text{m}}^2\hspace{0.17em}{\text{s}}^{-1}$ [39]
concentration at which oxygen consumption is ½ maximal		$\overline{c}=6.66\hspace{0.17em}\times \hspace{0.17em}{10}^{-9}\hspace{0.17em}\text{mol}\hspace{0.17em}\text{m}{\text{l}}^{-1}=6.66\hspace{0.17em}\text{nmol}\hspace{0.17em}\text{m}{\text{l}}^{-1}$
maximal rate of oxygen consumption*	[2 × 10−19, 1 × 10−15] mol cell−1 s−1 [40,41]	M = 2 × 10−19 mol cell−1 s−1
VEGF concentration parameters
diffusion coefficient for VEGF in media		$D_{v_m}=\hspace{0.17em}1.37\hspace{0.17em}\times {10}^{-6}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{c}{\text{m}}^2\hspace{0.17em}{\text{s}}^{-1}$ [42]
diffusion coefficient for VEGF in gel*	[3.77 × 10−7, 1.13 × 10−6] cm2 s−1 [42,43]	$D_{v_g}=1.13\hspace{0.17em}\times {10}^{-6}\hspace{0.17em}\hspace{0.17em}\text{c}{\text{m}}^2\hspace{0.17em}{\text{s}}^{-1}$
VEGF degradation rate*	[2.67 × 10−6, 1.28 × 10−4] s−1 [43,44]	dv = 29.874 × 10−6 s−1
hypoxia threshold for VEGF secretion*	unbounded	ch = 6.281 × 10−8 mol ml−1 = 4.768%
VEGF secretion gradient constant*	unbounded	Bv = 90 ml mol−1
baseline VEGF secretion rate*	unbounded	$\alpha (n_0)=0.04596\times {10}^{-23}+6.7225\times {10}^{-34}{n}_0+5.4325\times {10}^{-42}{n}_0^2\hspace{0.17em}\text{mol}\hspace{0.17em}{\text{cell}}^{-1}\hspace{0.17em}{\text{s}}^{-1}$ (where n0 is in cell ml−1)
VEGF secretion multiplier*	[0,10] [42,45,46]	Vm(n0) = 5.217 − 9.0375 × 10−9n0, where n0 is in cell ml−1

Similar previously developed cell–solute models [41,42,47] and statistical analysis of the in vitro data were used to inform the functional forms of the model; these functional forms (Mf(n, c), δd(n, c), αg(n, c)) and contained parameters are presented in Results, alongside the in vitro data that motivate them. Equations (2.1)–(2.3) represent an explicit hypothesis based upon data and previous research about the relationships between the three variables (figure 2).

Figure 2.

Flow diagram demonstrating the relationships between the variables in the model.

2.7. Mathematical model parametrization based on in vitro data

The in vitro data were split into a training subset and a validation subset. The training subset consists of the mean viable cell densities in the gel and the mean VEGF concentrations in the media for initial cell densities n₀ = 39, 154 and 385 million cells ml⁻¹. The model was parametrized using the training dataset. The validation subset consists of the corresponding data with n₀ = 77 and 231 million cells ml⁻¹, and this was used to validate the model fit against the training data.

The mathematical model (2.1–2.3) was parametrized against the training dataset and solved for subsequent simulations using the finite-element modelling software COMSOL Multiphysics. To achieve this, an axisymmetric 2D geometry representative of the in vitro well was created, with domains representing the cell-seeded collagen gel at the base of the well and the volume of culture media above it (figure 3). The equations (2.1–2.3) were adapted to reflect the material properties of each domain of the geometry via the provision of different parameters and constitutive relationships where appropriate. Functional forms and embedded parameters are summarized next for completeness in presenting the mathematical model, but these functional relationships were informed through analysis of the in vitro data, as discussed in Results.

Figure 3.

In vitro 96-well plate schematic showing a single well containing a cell-seeded collagen gel and media (left) and corresponding geometry created in COMSOL (right). Here, the sample oxygen distribution ranges from 10% at the top of the well, according to the applied boundary condition, to 9.6% at the bottom.

2.7.1. Oxygen concentration governing equation

Equation (2.1) is applied across both domains shown in figure 3, consisting of a diffusion term only in the media, $\mathrm{\partial }c/\mathrm{\partial }t=D_{c_m}{\mathrm{\nabla }}^{2}c$, and a diffusion term and oxygen uptake term in the gel to represent the consumption of oxygen by the cells:

$$\frac{\mathrm{\partial }c}{\mathrm{\partial }t}=D_{c_g}{\mathrm{\nabla }}^{2}c-Mn\frac{c}{c+\overline{c}}.$$ 2.4

The diffusion constants used ($D_{c_m}$ and $D_{c_g}$ in table 1) were based upon the aqueous diffusion rate of oxygen [38] and an experimentally derived value for diffusion of oxygen in collagen gel [39]. In the gel domain, the oxygen uptake term is dictated by Michaelis–Menten kinetics. Here, $\overline{c}$ is the concentration at which the oxygen consumption is half-maximal (this was assigned a value, presented in table 1, based on unpublished data by the authors). The metabolism constant, M, was derived by fitting to experimental data, as described below.

2.7.2. Cell density governing equation

Equation (2.2) applies only to the gel domain since there is no experimental evidence to suggest that the seeded cells would migrate into the media (figure 3). This equation consists of a cell proliferation term, dependent upon oxygen and mediated by a term in n with cell proliferation coefficient β, and a cell death term dependent upon cell density to incorporate competition for space and nutrients, giving the following:

$$\frac{\mathrm{\partial }n}{\mathrm{\partial }t}=\beta cn\left(1-\frac{n}{n_{\max }}\right)-\delta n.$$ 2.5

The maximum cell density n_max was set at around the maximum cell density observed during the experiments. This maximum cell density is lower than the density that could physically fit into the gel (calculated to be approx. 2500–3000 million cells ml⁻¹ based on cellular dimensions of cylinder-shaped dADSCs [33]), but represents the carrying capacity of the gel taking into account environmental factors other than volumetric space.

A logistic growth model with a linear dependence on oxygen concentration was chosen to describe cell proliferation. This functional form captures the impact of cellular competition for space and resources by limiting cell proliferation according to the maximum cell density, n_max. Other growth models were trialled during the parametrization of the model, including a model that incorporated a Michaelis–Menten function of oxygen concentration, but the form presented here achieved the best fit to the experimental data.

The cell death coefficient δ was determined as a function of the initial cell density n₀ via parametrization against the in vitro data. A death term was incorporated into the model based on the in vitro data; under some experimental conditions, the measured viable cell density after 24 h was lower than the initial cell density, demonstrating that some cells died over this time period (figure 4). A death term dependent on the oxygen concentration was also trialled during the parametrization process, but this did not result in a better fit to the data.

Figure 4.

Comparison of simulated mean viable cell density over collagen gel after 24 h and experimental means, plotted against ambient oxygen concentration. Error bars denote the standard deviations of the experimental data subsets used to calculate each of the means. R² values indicate the accuracy of the model fit to data: (a) training dataset and (b) validation dataset.

2.7.3. Vascular endothelial growth factor concentration governing equation

The VEGF concentration governing equation applied to the region representing the media consists of a diffusion term and a degradation term:

$$\frac{\mathrm{\partial }v}{\mathrm{\partial }t}=D_{v_m}{\mathrm{\nabla }}^{2}v-d_{v}v.$$ 2.6

In the gel, an additional production term was incorporated to represent the release of VEGF by the seeded cells:

$$\begin{array}{c}\frac{\partial v}{\partial t}=D_{v_g}{\nabla }^{2}v+\alpha n\left(\frac{(V_m+1)}{2}-\frac{V_m-1}{2}\tan h(B_v(c-c_h)\right)\\ -d_{v}v.\end{array}$$ 2.7

The form of the VEGF production term was informed by research, suggesting that VEGF mRNA and protein production by cells is upregulated at low oxygen levels [22–24], as well as previous modelling attempts of similar processes [42]. Here, α denotes the baseline VEGF secretion rate and is determined as a function of the initial cell density n₀ (table 1); c_h is the hypoxia threshold value at which VEGF secretion is upregulated; V_m determines the factor by which VEGF secretion is upregulated at oxygen values below this threshold and B_v determines the gradient of this oxygen-dependent upregulation.

2.7.4. Parametrization

Key unknown parameter values were derived by fitting the model against the experimental training dataset using COMSOL Multiphysics software (table 1). Where possible, existing literature values were used to approximate bounds (displayed in table 1) for the unknown parameter values prior to fitting. It was necessary to let the cell death coefficient δ, VEGF secretion coefficient α and VEGF secretion multiplier V_m be functions of n₀ to achieve a closer fit to the training dataset. Figure 4 presents the final fit to the cell viability data, and figure 5 displays the final fit to the VEGF concentration data.

Figure 5.

Comparison of simulated mean VEGF concentration over media after 24 h and experimental means, plotted against ambient oxygen concentration. Error bars denote the standard deviations of the experimental data subsets used to calculate each of the means. R² values indicate the accuracy of the model fit to data: (a) training dataset and (b) validation dataset.

Boundary conditions were set to match the geometry and relevant transport characteristics. The gel–media boundary condition enforces the continuity of concentration for VEGF and oxygen, and zero flux for the cells. Zero flux boundary conditions were used for all variables around the edges of the well, with only oxygen permitted to diffuse from the external environment (the air surrounding the well) into the media volume via a continuity of concentration boundary condition (c = c_a, the ambient oxygen as prescribed during the experiments) at the exposed surface of the well. The initial oxygen concentration in the gel and the media was set uniformly at c(0) = 18% based on oxygen probe measurements of EngNT in ambient oxygen conditions prior to incubation. The uniform initial VEGF concentration was set at v(0) = 0 pg ml⁻¹, and the initial cell density was varied to match the data.

A mesh was generated to cover this geometry, containing a total of 8032 triangular elements automatically generated with a maximum element size of 0.4 mm, with a more refined mesh at the geometry boundaries and in the region representing the gel. Convergence was verified by comparing the results of simulations run using progressively more fine meshes; criteria for convergence were set at 0.05% of the converged values for viable cell density over the gel and VEGF concentration over the media. The equations were solved simultaneously to obtain spatial distributions of the variables across the well geometry over time. An example oxygen distribution is given in figure 3.

Parametrization was conducted in two stages: the first focused upon the interdependent oxygen concentration and cell density governing equations, and the second upon the VEGF governing equation (figure 1). Prior to parametrization, parameter sweeps were carried out to identify parameter regimes in which simulation predictions were comparable in order of magnitude to the experimental data.

The parameters identified for fitting were the oxygen metabolism constant, cell proliferation and death constants. Other parameters were estimated according to existing research or knowledge. Parametrization of the cell-oxygen equations was carried out in COMSOL using the least-squares method in combination with the SNOPT (sparse nonlinear optimizer) algorithm, which is often implemented to solve constrained nonlinear least-squares problems.

Initially, global values for the cell death and proliferation coefficients, β and δ, were sought, but this did not produce any satisfactory results, indicating that further functionality in the constitutive relationships would be required. Parametrization was then carried out separately for each initial cell density of the training dataset, with each computation using five data points consisting of the mean viable cell density across five different ambient oxygen concentrations. This resulted in a pair of parameter values (β, δ) that differed for each initial cell density. The cell proliferation coefficient then remained set as a constant, whereas the death parameters were then expressed as functions of the initial cell density within the mathematical model, and parametrization was conducted again to find the exact form of this equation. This equation (table 1) can be used to estimate the corresponding parameter value for alternative initial cell densities.

Similarly, parametrization of the VEGF governing equation was carried out using the SNOPT algorithm. In this case, it was necessary to derive the baseline VEGF secretion constant and the VEGF upregulation constant as functions of n₀. This was carried out through iterative runs of the SNOPT algorithm combined with parameter sweeps to identify the correct parameter space.

2.8. Simulation of nerve repair construct geometry

Simulations of different cell densities and distributions over an NRC geometry were also carried using COMSOL Multiphysics. An axisymmetric 2D geometry, with dimensions length 15 mm and diameter 0.5 mm, was used to mimic NRCs used in our ongoing programme of nerve tissue engineering work [5,32,34]. The same mathematical model as for the gel phase of the in vitro model (equations (2.1–2.5), together with (2.7)), combined with the parameter values found via data fitting, was applied to the NRC geometry to model a possible in vivo scenario. For all three species, zero flux boundary conditions were applied along the curved edges of the cylindrical geometry to represent an impermeable wrap encasing the NRC. Concentration boundary conditions for both oxygen concentration (5%) and VEGF concentration (0 ng ml⁻¹) were applied at either end of the geometry to represent an open-ended NRC. The 5% oxygen boundary condition was based on measurements of rat skeletal muscle oxygen concentrations, which fall in the range of 5–7% [48].

Zero flux conditions were also applied at the construct ends for the cell population. The initial VEGF concentration in the NRC geometry was set at 0, and the initial oxygen concentration across the geometry was set at 18% to mimic the likely oxygen concentration in a collagen NRC post-manufacture. The oxygen concentration initial condition is based on the NRC existing in an ambient oxygen concentration environment prior to implantation. Initial seeding cell densities and distributions were varied, and simulations of all three species were run over a time period of 24 h to investigate the effect of different seeding strategies upon VEGF and cell distributions over time.

3. Results

3.1. In vitro data and model parametrization

In vitro experiments using cells seeded within hydrogel biomaterials were designed and carried out specifically for parametrization of the mathematical model according to the combined experimental–theoretical method (figure 1). This necessitated quantification of the relationship between therapeutic cell density, oxygen and VEGF concentration. Collagen gels seeded with Schwann cell-like dADSCs were cast and stabilized within 96-well plates (figure 3) at five densities and cultured at five ambient oxygen concentrations for 24 h, then cell viability, VEGF-A concentration and cell proliferation were measured (Methods). First of all, the results were analysed to inform the functional forms of the governing equation terms.

The in vitro data exhibited a positive trend between the number of viable cells in the gel after 24 h and the ambient oxygen concentration, and a negative trend between the former and the initial seeded cell density (figure 4). Multiple regression analysis of the in vitro 24 h cell viability data found that the ambient oxygen concentration and initial cell density statistically significantly predicted the final viable cell density, F_3,123 = 46.999, p = 2.62 × 10⁻²⁰ < 0.0005, overall model fit R² = 0.534. The ambient oxygen concentration (standardized β = 0.658, p = 2.95 × 10⁻²⁰ < 0.0005) and initial cell density (standardized β = −0.300, p = 3.38 × 10⁻⁶ < 0.0005) added statistically significantly to the prediction. The interaction term (standardized β = −0.081, p = 0.190), included to test the hypothesis that the initial cell density moderates the relationship between the ambient oxygen concentration and the final viable cell density, was not a significant predictor. Therefore, the moderation hypothesis was rejected. This suggested that the cell governing equation should incorporate a dependency upon the local oxygen concentration, as this is largely determined by diffusion of the ambient oxygen into the gel.

To assess the impact of different initial conditions upon cell proliferation, a multiple regression analysis was run on the Ki-67 proliferation dataset to predict % Ki-67+ from initial cell density, n₀, and ambient oxygen concentration, c_a. The variables did not statistically significantly predict the % Ki-67+, F_3,56 = 0.972, p = 0.412, R² = 0.049. However, the increase in viable cell density with ambient oxygen concentration pointed towards increased proliferation with increasing oxygen concentration, and therefore, a linear dependency on oxygen was incorporated into the cell proliferation term.

The in vitro data suggested a nonlinear relationship between the VEGF concentration after 24 h and the ambient oxygen concentration (figure 5), which we hypothesize is caused by a combination of a nonlinear increase in the production of VEGF per cell in low oxygen environments and a proportional increase in the total number of viable cells in higher oxygen environments. This is in accordance with the cell viability results presented here and evidence that VEGF upregulation occurs in hypoxic environments [23–25]. Other mathematical frameworks have also implemented this or similar hypotheses [47]. Multiple regression analysis was run on the VEGF data to predict VEGF concentration at 24 h from initial cell density n₀ and ambient oxygen concentration c_a. An interaction term was included to test the moderation hypothesis that ambient oxygen concentration moderates the relationship between initial cell density and the final VEGF concentration. The variables statistically significantly predicted the final VEGF concentration, F_3,105 = 4.641, p = 0.004, overall model fit R² = 0.117. The initial cell density (standardized β = 0.268, p = 0.004) added statistically significantly to the prediction, but the ambient oxygen concentration (standardized β = 0.059, p = 0.520) did not. The interaction term was not found to add statistically significantly to the prediction (standardized β = 0.192, p = 0.39). Therefore, the moderation hypothesis was rejected.

3.2. Specific uniform-seeded cell densities result in higher mean viable cell densities and vascular endothelial growth factor concentrations within a simulated engineered tissue after 24 h

The model and accompanying parameters derived from the in vitro data fitting exercise were used to make predictions to inform future in vivo experimentation, according to figure 1. To achieve this, the mathematical model was applied to a cylindrical geometry representing a NRC, with length 15 mm and diameter 0.5 mm to match the dimensions of engineered tissues tested in a rat sciatic nerve repair model [5,32]. These constructs can be made using the same materials and cells as in the in vitro experiments, justifying the use of parameters derived from the in vitro data.

Simulations were then run to predict spatio-temporal changes in oxygen concentration, VEGF concentration and cell density within an NRC during the first 24 h after implantation in vivo. Oxygen and VEGF boundary conditions were applied to reflect the presence of an impermeable sheath covering the curved sides of the NRC, thus permitting solute transport only through the construct ends which interface with the surrounding tissue in vivo.

First of all, simulations were run using uniform initial distributions of seeded cells with a range of magnitudes. The simulated range of initial cell densities was chosen to cover the range currently used in our standard experimental models (100–400 million cells ml⁻¹) and in previously published in vivo studies [49], as well as densities below this which have the potential to reduce cell waste and decrease costs.

The simulation results showed a maximum in the mean viable cell density across the simulated construct after 24 h at a seeding cell density of 95 million cells ml⁻¹ (figure 6a), suggesting that the use of seeding densities within a certain range of values could help to increase viable cell density at later time points. Figure 6b demonstrates that the highest initial cell densities result in higher viable cell densities after 12 h, but this is reversed by 24 h.

Figure 6.

Model simulation results demonstrate the potential impact of different seeded cell densities. (a) Mean viable cell density over the construct has a maximum at n₀ = 95 million cells ml⁻¹ (white point). (b) Median viable cell densities decrease with time. (c) Visual demonstration of the increase in viable cell density across the construct at 95 million cells ml⁻¹. Viable cell densities remain predominantly uniform across the construct geometry independent of the initial cell density, with the exception of an increase in viable cell density at the proximal and distal ends, corresponding to higher oxygen concentrations at these locations. (d) Mean VEGF concentration over the construct has a maximum at n₀ = 245 million cells ml⁻¹ (white point). (e) An initial cell density of 170 million cells ml⁻¹ provides a balance of high viable cell density after 24 h, and high VEGF concentrations with a relatively large range after 24 h, suggesting elevated gradients of VEGF. (f) Visual demonstration of the increase in VEGF quantity and range at n₀ = 245 million cells ml⁻¹ in comparison to both lower initial cell densities. This initial seeding cell density results in greater gradients of VEGF, with higher concentrations in the centre, as well as higher overall concentrations over the length of the construct when compared to other seeding cell densities.

VEGF concentrations throughout the simulated construct were generally higher and slightly wider in range after 24 h for initial seeded cell densities greater than 95 million ml⁻¹, as shown in figure 6e. A peak in the mean is achieved at 245 million cells ml⁻¹ initial seeding density (figure 6d), and this design also achieves a comparatively large range of VEGF concentration values in silico, indicating the presence of more pronounced VEGF gradients (figure 6c).

These results suggest that the optimal uniform-seeded cell density for inducing a sizeable VEGF gradient as well as relatively high densities of viable cells over time could lie in the range of 95–245 million cells ml⁻¹, demonstrated by the simulation results for a seeded cell density of 170 million cells ml⁻¹ (figure 6b,e).

3.3. Non-uniform-seeded cell distributions could help to generate steeper gradients of vascular endothelial growth factor across engineered tissue

The initial spatial distribution of the seeded cells was varied to understand whether moving away from the current uniformly distributed approach could reduce waste of cells and improve the vascularization of the engineered tissue via the generation of VEGF gradients. A step-like function was tested which splits the length of an NRC into thirds (figure 7a), with the middle third having a seeded cell density that is a specified multiple (Sc) of the seeded cell densities of the end thirds (which are equal at n₀). This specific non-uniform distribution was investigated because it is relatively simple and therefore feasible to manufacture.

Figure 7.

(a) Example of a range of non-uniform initial distributions of seeded cells, where the total number of seeded cells is 50 000. The case where Sc = 1 corresponds to a uniform distribution. (b) The total number of viable cells in a construct after 24 h depends upon the distribution of the initial seeded cells. The optimal distribution for the maximization of the total number of viable cells after 24 h is approximately uniform (Sc = 1) regardless of the total number of cells used. (c) Simulations suggest that the mean VEGF concentration across the construct after 24 h is lower when there are fewer cells seeded in the centre than in the proximal and distal thirds (Sc < 1). For Sc > 1, the maximum VEGF concentration in the construct increases slightly with Sc, but the mean VEGF concentration remains similar to the uniform case, Sc = 1.

A range of designs with different (Sc, n₀) pairs were simulated. Figure 7b demonstrates that different configurations of the same number of cells can result in different viable cell totals after 24 h. The simulation results suggest that distributing a specific number of cells non-uniformly instead of uniformly generally results in a decreased number of viable cells after 24 h, particularly when the total number of cells seeded is less than 500 000. Figure 7b also suggests that a total number of seeded cells in the range of 200 000–400 000 produces the highest viable cell number after 24 h regardless of the distribution of the cells, with the exception of distributions corresponding to Sc values close to 0.

Furthermore, the simulation results suggest that implementing non-uniform-seeded cell distributions could have an effect upon subsequent VEGF distributions across the construct (figures 7c and 8). In particular, figure 8 demonstrates that a non-uniform seeding distribution could be used to increase gradients of VEGF along the length of a construct after 24 h when the total number of cells seeded is relatively low (100 000); whereas when more cells are seeded (500 000), a uniform distribution achieves the highest gradients of VEGF at the same time point. Thus, the distribution that induces the greatest range of VEGF concentrations, and thus the largest gradient, appears to vary according to the total number of cells seeded.

Figure 8.

Different seeding cell strategies result in varying (a) cell density, (b) VEGF concentration and (c) oxygen concentration profiles along the length of the NRC geometry at different time points. In particular, non-uniform-seeded cell distributions (Sc = 3) result in more pronounced VEGF gradients in the centre of the constructs than uniform distributions (Sc = 1), although this varies over time and according the total number of cells seeded. Simulations suggest that the impact of the initial seeded cell distribution upon the VEGF and viable cell density distributions after 24 h varies according to the total number of cells seeded. Seeding 500 000 cells uniformly (Sc = 1) generates a steeper VEGF concentration gradient after 24 h than seeding more cells in the centre (Sc = 3); whereas the converse is true, when seeding 100 000 cells.

4. Discussion and conclusion

The simulation results for uniform-seeded cell densities in the NRC geometry reiterate the need to achieve a balance between cell survival and the production of sufficient quantities and gradients of VEGF, and confirm our hypothesis that such an optimum exists. The optimal initial seeded cell density to achieve cell survival was approximated at 95 million cells ml⁻¹, whereas higher initial seeded cell densities of approximately 245 million cells ml⁻¹ resulted in lower viable cell density values after 24 h but greater concentrations of VEGF. These predictions will enable future in vivo experiments to focus on testing these cell seeding densities, accelerating and refining the development process and reducing animal usage.

The use of non-uniform initial seeded cell distributions is currently uncommon in tissue engineering, but this work confirms that spatial effects are important and could be exploited. The simulations highlight the potential to use non-uniform distributions to increase the severity of VEGF gradients, which could improve the vascularization of such constructs. Various tissue engineering techniques and advances in additive manufacturing technologies such as bioprinting provide promising ways to manufacture complex cell distributions in a precise and scalable manner [50,51].

The model simulations using the multiwell plate geometry appear to capture the main trends of both the training and the validation dataset. A notable exception is the difference between the viable cell density after 24 h under 3% oxygen concentration, in contrast to both 1 and 5%, for all but the lowest initial cell density. This pattern could be interpreted in two ways. The first is to assume that increases in oxygen should promote an increase in viable cell density, in which case the 5% values appear to be lower than expected. The second, and perhaps more likely, cause could be that the dADSCs are particularly well adapted to the 3% ambient oxygen niche, due perhaps to culturing conditions or source cell environment, and therefore survive more readily under these conditions. Further experiments could help to test these theories.

Forms of the cell death and baseline VEGF secretion coefficients and VEGF secretion multiplier that explicitly depend upon the value of the initial cell density were incorporated into the model to achieve an adequate fit to the training dataset. This indicates that the constitutive relationships that form the cell density and VEGF governing equations do not fully capture all of the mechanisms involved in the proliferation and death of cells, and the secretion and decay of VEGF. It is possible that variables other than oxygen concentration and the cell density could have an effect upon these processes, such as glucose concentration [52,53]. However, over the span of 24 h, it was expected that oxygen, rather than any other nutrient, would be the limiting factor for maintaining a population of cells.

The model thus far has been parametrized using data collected after 24 h. In terms of repair and cell survival, this is a short but vital time period. The extension of the model to longer timescales is a possibility, although, at later time points, the impact of growing vasculature and other processes would have to be taken into account, increasing the complexity of the model. Any further experimental data could be used to improve the model further as part of a theoretical–experimental iterative feedback loop [26].

Our work suggests that specific densities and distributions of therapeutic cells could be used to reduce the total number of cells required and to help generate growth factor gradients crucial for encouraging vascularization. These predictions require validation, but they should at least encourage more careful thought about the placement of cells by tissue engineers in the future, and at best could offer significant reductions in cost and improvements in efficacy.

Simulations of the cell well scenario were validated using the validation dataset. However, additional in vitro validation of the NRC construct geometry simulations would be of limited value: any such in vitro model would likely provide a poor representation of the in vivo environment due to the biological complexity of the peripheral nerve repair scenario. Therefore, in this case, in vivo validation would be the most useful next step.

A possible in vivo validation study could involve implanting EngNT NRCs into a rat sciatic nerve model (the standard in vivo peripheral nerve injury model) seeded using three different uniform cell densities, chosen according to the results of this study and harvesting at the 5-day point. Histological staining and imaging could be used to assess cell survival, and VEGF measurements could be taken at various longitudinal positions along the NRC. Other important information, such as assessments of endothelial cell migration, vascular growth and in vivo oxygen levels, could also be taken and used to inform future iterations of the model. This is a significant body of work, which lies outside the scope of the current study.

In the future, the framework and method of parametrization presented in this paper could be applied to predict outcomes for engineered tissues composed of a broader range of different combinations of biomaterials and therapeutic cells. This would likely require the collection of additional in vitro data corresponding to these alternative cell and material types. This model could also be expanded to incorporate the growth of blood vessels and neurites into the engineered tissue. The iterative theoretical–experimental approach demonstrated in this paper has the potential to be applied to other aspects of tissue engineering and could become a useful tool when implemented alongside experimental work.

Data accessibility

The data and code are available as part of the electronic supplementary material.

Authors' contributions

R.C. conducted the theoretical work, wrote the main manuscript text and prepared figures. G.A.-B, C.K., C.O. and P.J.K. conducted the experimental work. R.J.S. and J.B.P. designed the study, oversaw the experimental and modelling work, and contributed to the writing and editing of the manuscript. All authors reviewed the final version of the manuscript.

Competing interests

We declare we have no competing interests

Funding

This work was supported by EPSRC (grant nos. EP/N033493/1 and EP/R004463/1) and a doctoral training grant no. SP/08/004 from the British Heart Foundation (BHF) to R.C. R.C. was also supported through the UCL CoMPLEX Doctoral Training Programme. P.K. was supported by Umeå University Insamlingsstiftelsen.