Edited by: Lorenzo Moroni, Maastricht University, Netherlands
Reviewed by: Aurélie Carlier, Maastricht University, Netherlands; Ioannis Papantoniou, KU Leuven, Belgium
This article was submitted to Biomaterials, a section of the journal Frontiers in Bioengineering and Biotechnology
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Organoids are threedimensional multicellular tissue constructs. When cultured
Organoid technology is becoming increasingly prominent as a biomedical tool, with applications in drug discovery and personalised medicine. In biomedical research, brain, kidney, and liver organoids are used to understand the underlying biological mechanisms in tissue development and tissuedrug interactions (Eisenstein,
Organoids are threedimensional, multicellular structures which, when grown
Current methods for organoid expansion are labour intensive, with organoids typically being produced in small numbers at specialist research laboratories. New technologies are required to manufacture large numbers of organoids with uniform and reproducible characteristics, to meet the demands of applications such as highthroughput screening in drug development. One such technology exploits bioreactors, which aim to deliver sufficient nutrients and growth factors to the cells to promote cell proliferation and differentiation, and to prevent the accumulation of toxins, which can lead to cell death. For a more detailed overview of bioreactor technologies used for 3D cell culture see, for example, Martin et al. (
This study is motivated by proprietary organoid expansion bioreactor technology developed by Cellesce (Ellis et al.,
Key priorities in the CXP1 bioreactor design and operation are uniformity of organoid size and system reproducibility, to ensure there is minimal variation in organoid characteristics between and within batches grown under the same operating conditions. The main control parameters for the CXP1 bioreactor are the inlet flow rate (controlled via a peristaltic pump) and the initial cell seeding density (the organoids are grown from single cells). Optimisation of these control parameters requires spatiotemporal information about the flow and metabolite (here glucose and lactate) concentrations throughout the bioreactor (Galban and Locke,
To complement experimental studies, mathematical models of bioreactor systems can be used to predict media flow profiles and the associated metabolite concentrations that cannot easily be measured
Here we review existing mathematical models for metabolite transport in bioreactor systems. A variety of different mathematical modelling approaches have been applied to related problems in tissue engineering, including: ordinary differential equation (ODE) models (Sachs et al.,
We develop a mathematical model of the CXP1 system, with the goal of determining how glucose and lactate levels within the CXP1 bioreactor change as the operating conditions (
The structure of the paper is as follows. In the Methods section, we introduce the full mathematical model, and then systematically derive two reduced models (referred to as the
We derive an unsteady twodimensional model for glucose and lactate transport within the CXP1 bioreactor. Schematics of the CXP1 bioreactor and our model geometry are presented in
We consider organoids grown from single cells seeded in a homogeneous thin layer of hydrogel in the bioreactor (lower yellow layer in
We consider the bioreactor design,
The CXP1 geometry and relevant parameter values (
Definitions of dimensional model parameters, together with typical values.
Diffusivity of glucose in hydrogel  6.0 × 10^{−10}m^{2} s^{−1} (Suhaimi et al., 

Diffusivity of glucose in media  6.0 × 10^{−10}m^{2} s^{−1} (Suhaimi and Das, 

Diffusivity of lactate in hydrogel  1.2 × 10^{−9}m^{2} s^{−1} (Zhou et al., 

Diffusivity of lactate in media  1.4 × 10^{−9}m^{2} s^{−1} (Shipley et al., 

Glucose concentration in upstream reservoir  0.36 mol m^{−2}  
[ 
Maximum velocity of media flow  1 × 10^{−6} m s^{−1} 
Length of bioreactor  9 × 10^{−2}m  
Height of hydrogel layer  1 × 10^{−3}m  
Combined height of hydrogel and media  3 × 10^{−3}m  
Initial cell seeding density  2.7 × 10^{10}cell m^{−2} to 4 × 10^{10}cell m^{−2}  
Proliferation rate  3.9 × 10^{−6}s^{−1}  
ν_{C}  Rate of glucose consumption per unit cell density  9.4 × 10^{−17}m^{2} cell^{−1} s^{−1} 
The diffusivities of glucose and lactate in hydrogel and media used in our model are taken from the literature (see
While the current CXP1 operating conditions have been empirically chosen to be specialised for colorectal cancer organoids, a key advantage of mathematical modelling is that it facilitates consideration of metabolite transport within CXP1 for other cell lines (which is the intent of Cellesce). This knowledge will streamline the adaptation of the CXP1 bioreactor to expanding organoids with significantly different behaviour,
Motivated by the specific bioreactor setup, parameter values, cell densities, and metabolite concentrations, discussed in section 2.1, we neglect stochastic effects and adopt a continuum modelling approach. We consider a twodimensional slice of the bioreactor, and adopt a Cartesian coordinate system
In the hydrogel, the glucose and lactate are transported via diffusion and glucose is consumed by organoids, which subsequently produce lactate. For the organoids (cell aggregates), we model the reaction terms through effective (bulk) sink/source terms over the hydrogel. Such an approach can be mathematically justified through a formal averaging procedure, such as the asymptotic homogenisation carried out for related systems in Dalwadi et al. (
During glycolysis, one glucose molecule produces energy and two lactate molecules (Liberti and Locasale,
In the media, (
Governing equations Equations (2.1)(2.8) require appropriate boundary, initial, and interfacial conditions. The boundaries in the hydrogel are solid walls and we impose zero flux of glucose and lactate at
Schematic of the boundary conditions for the media (blue) and hydrogel (yellow) layers for Equations (2.1), (2.2), (2.6), and (2.7). At the mediahydrogel interface, we impose continuity of concentration and flux. At the airmedia interface and at the impermeable hashed boundaries, we impose no flux. The black arrows indicate the halfPoiseuille flow profile.
As initial conditions, we assume that the glucose concentration in the media equals the glucose concentration in the upstream reservoir,
The typical parameter values, given in
Timescale groupings of the various physical processes present in the CXP1 bioreactor.
Flow  
Glucose consumption  
Lactate production  
Cell proliferation  
We nondimensionalise the problem to identify the relative importance of each transport mechanism. We introduce the following nondimensional variables, for
Using the scalings Equation (2.15), the governing equations Equations (2.1)(2.7) become,
The dimensionless parameters in Equations (2.16)(2.20) are:
Definitions of nondimensionalised model parameters with their typical values.
ϵ  Ratio of vertical to horizontal lengthscales  1/30 
Ratio of timescale of interest to timescale of diffusion of glucose in hydrogel  6.4 × 10^{−3}  
Ratio of timescale of interest to timescale of diffusion of glucose in media  6.4 × 10^{−3}  
Ratio of timescale of interest to timescale of diffusion of lactate in hydrogel  1.28 × 10^{−2}  
Ratio of timescale of interest to timescale of diffusion of lactate in media  1.49 × 10^{−2}  
μ  Ratio of timescale of interest to timescale of flow  0.96 
ρ  Ratio of timescale of interest to that of glucose consumption per cell  0.220.32 
Ratio of timescale of interest to timescale of cellular proliferation  1/3  
Ratio of hydrogel height to the combined height of hydrogel and media layers  1/3  
Dimensionless maximum tolerated lactate concentration  0.7 
We solve the full twodimensional system, Equations (2.16)(2.19) and (2.22)(2.29), using the parameter values given in
Metabolite concentrations at 1, 3, and 7 days into a typical simulation. The horizontal lines at
As discussed in section 2.2.2, the different transport mechanisms in the system have associated timescales that can be grouped into either hours, days, or months. This is made explicit in the dimensionless system through the presence of the small parameter ϵ. We propose a systematic model reduction, with the key advantage of reducing the complexity of the model while retaining the physical processes which dominate over the timescale of interest.
Motivated by the long, thin geometry of the bioreactor, characterised by ϵ ≪ 1, and the lack of variation in
In the asymptotic analysis that follows, we consider the limit ϵ → 0, and assume all other dimensionless parameters remain
We consider the following asymptotic expansions for the dependent variables:
At leading order, the metabolite transport is given by
Integrating Equation (2.31) subject to the leadingorder versions of the appropriate boundary conditions, Equations (2.25)(2.28), we deduce that
To calculate this dependence, we proceed to
Equations (2.42), (2.43), and (2.45)(2.48) define the longwave approximation model. We will analyse this reduced system in more detail in section 3. First, we derive a further reduction of the longwave approximation, by exploiting the separation in scales between horizontal diffusion and the remaining transport mechanisms, namely advection with the media flow, glucose consumption, and lactate production.
From the typical parameter values given in
A benefit of this sublimit reduction is that we are able to construct analytic solutions for the glucose concentration, using the method of characteristics. The solution is split into two distinct regions: Region 1, given by 0 < β
Using the method of characteristics, we can write the lactate concentration as a single integral of known functions:
We now discuss and compare results obtained from our reduced models and the full system. This will allow us to understand when each reduced model is a useful systematic reduction.
The longwave approximation model, Equations (2.42), (2.43), and (2.45)(2.48), is solved numerically using the Chebfun toolbox in MATLAB. For the sublimit approaximation model, Equations (2.49)(2.52), we obtain an analytical expression for the glucose concentration, and the lactate concentration is numerically computed from Equation (2.50) subject to Equation (2.51) with a RungeKutta method using the inbuilt ODE solver
Computationally, there is a significant difference between the models: on a standard desktop, the full problem is solved in
To present the model solutions over space and time, we average solutions of the full 2D model over
Results showing how the glucose
At this stage, we conclude that when information close to the dividing characteristic is of interest, the longwave approximation should be used instead of the sublimit approximation. If this information is not important, the sublimit approximation should be used since it is faster to solve than the full model and the longwave approximation, and it admits analytic solutions for glucose concentration.
We emphasise that our analytic solutions in the sublimit approximation allow us to understand observations from the full numerical solutions. That is, we can use our analytic solutions from the sublimit model to physically interpret our results and provide insight into the underlying physical system. For example, the dividing characteristic (α
Additionally, the analytic solution of our sublimit approximation provides insight into why the glucose and lactate concentration appear to be spatiallyindependent in the lowerright regions (
To quantitatively compare the model predictions, we consider the following timedependent variables:
In
Comparison of outputs from the different mathematical models and their evolution in time:
Similar plots showing how the maximum lactate concentration,
We compare the position at which the maximum lactate concentration occurs,
It is infeasible to obtain experimental data for maximum lactate concentrations, which we would need to validate our model. Therefore, we consider the lactate concentration at the media outlet,
In this section, we start by exploiting our reduced modelling approach to
We investigate and quantify the metabolite behaviour by introducing the following timedependent metrics. We previously defined the
There is a tradeoff between high glucose conversion and minimising the fraction of the domain which is
In addition to the metrics we have introduced to assess metabolite distribution, an important cellspecific metric is the
Organoid lines differ in many ways including, but not limited to, proliferation rate, glucose consumption rate, the maximum lactate concentration cells can tolerate without affecting cell properties, and minimum glucose level needed for cellular proliferation. To understand the metabolic environment experienced by different organoid lines within the bioreactor, we perform a discrete parameter sensitivity analysis in which we vary the rates of proliferation,
Evolution of glucose
In
We consider organoid line (ii), with low proliferation and high glucose uptake rates, in
For rapidly proliferating cells with a low rate of glucose uptake [organoid line (iii)]
Finally, we consider cells with high proliferation and high uptake [organoid line (iv)], in
Using the metrics we introduced above, we now quantify the behaviour of the bioreactor environment during cell culture for each of the five organoid lines. In
Comparison of
The glucose conversion generically increases over time, as the cells grow. However, the shape of this increase over time varies significantly between the different organoid lines. While solely considering the standard case [organoid line (v), given by parameters in
We show the maximum lactate concentration in
We examine the time at which the lactate concentration equals the tolerated lactate concentration in
There is a tradeoff between promoting: (1) high glucose conversion, to ensure resources are not wasted; (2) high glucose consumption rate per cell, to ensure cells absorb sufficient glucose to proliferate; and (3) increasing the turnoff time, to ensure the lactate concentrations within the bioreactor remain tolerable everywhere throughout the experiment. Our model framework allows for efficient quantification of all these metrics. By determining how these metrics vary with bioreactor operating parameters, we can then identify operating conditions that enhance cell growth. We illustrate this in the next section.
In this subsection, we focus on the standard organoid line (
We now determine how the metrics depend on the inlet flow rate for this organoid line, and show how this leads to the identification of flow rates that enhance cell growth. We focus on flow rate as this is an experimental parameter that is easily varied. We investigate flow rates over two order of magnitudes, [
In
Results for a specific organoid line within the CXP1 bioreactor showing the evolution of:
While the timedependent maximum lactate concentration within the domain monotonically increases for a given flow rate, the effect of varying the flow rate is nonmonotonic (
In
We now consider a more finely refined investigation of the effect of flow rate of the system metrics. In
We see that the relationship between glucose conversion at 7 days and media flow velocity is monotonically decreasing, and the rate of decrease is larger for flows faster than [
An advantage of our mathematical modelling framework is that we have been able to easily explore a wide range of parameter values, in this case the flow rate, and explore the nonlinear effects of varying experimental parameters. For example, an experimentalist may start with a slow flow rate of 10^{−7}m s^{−1} and conduct a set of experiments over which they increased the flow. Over an order of magnitude increase in flow, they would see no improvement in turnoff time, and therefore might be discouraged from increasing the flow any further. In such a scenario, they would miss finding the flow rate values required for turnoff times greater than 4 days.
The “optimal” operating conditions for the bioreactor will determine glucose and lactate concentrations which (1) yield a specified value for glucose conversion; (2) maintain a glucose consumption rate per cell which is sufficient for cellular proliferation; and (3) predict a turnoff time which is greater than the run time of the experiment. The specific values and relative importance of each of these requirements will depend on the user. Our model reduction facilitates rapid calculation of each metric. Hence, our work could be combined with an optimisation algorithm, with userspecified cost functions, to produce an efficient framework that can identify the bioreactor operating conditions that optimise for growth of organoids.
We have presented an unsteady, twodimensional model of metabolite transport that predicts metabolite concentrations within the CXP1 bioreactor system. We used an asymptotic analysis to systematically derive two reduced models which exploit the extreme spatial and temporal parameter ratios in the system. Our model predicts the spatiotemporal distribution of the metabolic environment within the bioreactor, information which is challenging to obtain experimentally. Both reduced models are onedimensional in space; the
Although the above may appear to suggest that the sublimit approximation is not useful, it does have additional benefits over the longwave approximation. A notable benefit is that it admits analytic solutions in the entire domain. Interpreting these analytic results, and understanding why they are discontinuous across the specific line in spacetime, provides insight into the underlying physical system. We find that the specific line in spacetime is a dividing characteristic in the (hyperbolic) sublimit approximation we derive. We are able to infer that this line divides the domain into two regions, depending on whether or not the effect of replenishment from the inlet has been experienced.
The flow of media through the bioreactor has the dual function of delivering nutrients to, and removing waste from, the growing organoids. As such, the inlet flow rate needs to be chosen carefully. The systematic reduction we have performed yields models that are easier to solve numerically than the full model. More importantly, they provide insight into the behaviour of the full model, particularly the dominant transport mechanisms. This systematic reduction has enabled us to efficiently characterise the experimental parameter space for given cell characteristics. One key outcome from this analysis is our prediction of a “worstcase” flow rate that minimises the turnoff time (the time when intolerable lactate concentrations first occur), Equation (3.5). Our model reduction has allowed us to understand why this minimum arises: for higher flow rates, the lactate is washed away more quickly (the bioreactor is in a proliferationlimited regime), for lower flow rates the lactate is produced more slowly since glucose is not delivered quickly enough (the bioreactor is in a transportlimited regime).
To understand how outcomes change as the control parameters are varied, we introduced the following timedependent metrics which characterise bioreactor performance:
Different bioreactor operating conditions will yield different values of these metrics. The relative importance of each metric will depend on the particular organoid line being investigated and the specific user requirements. Our work provides a framework for efficiently determining desirable bioreactor operating conditions for given cell properties.
In this study, we performed a systematic model reduction to study metabolite transport within the CXP1 bioreactor, whose geometry differs significantly from other bioreactors, such as hollow fibre or perfusion bioreactors. An important insight gained from our model reduction is the identification of the transport mechanisms that are dominant on our timescale of interest. We performed model reductions in two ways: (1) we exploited the slender geometry of the system, to obtain the
There are a number of interesting possible extensions to this work. For example, the optimal operating conditions are likely to change during the course of organoid growth. Future modelling work could predict how, and when, operating conditions should change to account for this growth. While we have considered steady flows, it would be straightforward to extend our framework to examine more complex flow behaviours, such as oscillating flows, or threedimensional effects. The potential use of unsteady flows will be of particular interest when minimisation of spatial variation in metabolite concentrations across the bioreactor is important, as we have seen that steady flows with little spatial variation in metabolite concentration also have very low conversion (see
In this work, we considered a spatially constant cell density, with growth rates independent of the local biochemical environment. Future modelling work will represent individual organoids as small, localised regions within the hydrogel where glucose consumption and lactate production occur, and regulate organoid growth. We will use a mathematical homogenisation approach (see
The mathematical modelling approach developed in this paper provides a framework for establishing how organoid viability can be improved by varying bioreactor operating conditions. The framework has the flexibility to consider different organoid lines, via characterisation of their proliferation and nutrient consumption rates and their tolerance to the presence of waste metabolite. Our work has the potential to improve the quality and reproducibility of bioreactorexpanded organoid output. We intend our theoretical framework to be used to scaleup the production of viable organoids, contributing to overall organoid technology development, and enabling organoids to be exploited as a powerful tool for accelerating drug discovery and testing.
The datasets generated for this study can be found in the GitHub repository
All authors designed the research and wrote the paper. MAE performed the research.
MJE is a cofounder of Cellesce. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The authors wish to thank the Cellesce technical team for fruitful and informative discussions on organoids, bioprocessing, and bioreactor design and operation.
The Supplementary Material for this article can be found online at: