Modeling the Growth and Size Distribution of Human Pluripotent Stem Cell Clusters in Culture

Abstract

Human pluripotent stem cells (hPSCs) hold promise for regenerative medicine to replace essential cells that die or become dysfunctional. In some cases, these cells can be used to form clusters whose size distribution affects the growth dynamics. We develop models to predict cluster size distributions of hPSCs based on several plausible hypotheses, including (0) exponential growth, (1) surface growth, (2) Logistic growth, and (3) Gompertz growth. We use experimental data to investigate these models. A partial differential equation for the dynamics of the cluster size distribution is used to fit parameters (rates of growth, mortality, etc.). A comparison of the models using their mean squared error and the Akaike Information criterion suggests that Models 1 (surface growth) or 2 (Logistic growth) best describe the data.

Appendix

1.1 Data availability

All experimental data used in this paper is available from the corresponding author of Iworima et al. (2023).

1.2 Derivation of radial growth rates from the mass growth laws

In each case, we use the following expressions for the spherical cluster mass, volume, and surface area: $M=\rho V=\rho (\frac{4}{3}\pi r^3)$ and $S=4\pi r^2$. We substitute these expressions into the equations that describe the mass of a cluster. In each case, we use the simple calculus step

$$\begin{aligned} \frac{dM}{dt}= \frac{d\rho V}{dt}= \left( \frac{4\pi \rho }{3}\right) \frac{d r^3}{dt}= 4\pi \rho r^2 \frac{dr}{dt}. \end{aligned}$$

1.2.1 Model 1: Linear Growth

In this case, $M'(t)=(\gamma -\alpha ) S - \beta M$. Substituting the above expressions and assuming $r>0$ leads to

$$\begin{aligned} 4\pi \rho r^2 \frac{dr}{dt}= (\gamma -\alpha ) 4 \pi r^2 - \beta \frac{4\pi }{3}r^3 \quad \Rightarrow \quad \frac{dr}{dt}= \frac{(\gamma -\alpha )}{\rho } - \frac{\beta }{3\rho }r\equiv f_1(r). \end{aligned}$$

1.2.2 Model 2: Logistic Growth

Carrying out the same kind of process results in

$$\begin{aligned} \frac{dM}{dt}= \gamma M \left( 1-\frac{M}{K} \right) \quad \Rightarrow \quad 4\rho \pi r^2\frac{dr}{dt}=\frac{4}{3}\gamma \rho \pi r^3\left( 1-\frac{4\rho \pi r^3}{3K}\right) . \end{aligned}$$

(18)

Simplifying (18) yields (12).

1.2.3 Model 3: Gompertz Growth

As above, we obtain

$$\begin{aligned} \frac{dM}{dt}= \gamma M \ln \left( \frac{K}{M} \right) \quad \Rightarrow \quad 4\rho \pi r^2\frac{dr}{dt}=\gamma \frac{4}{3} \rho \pi r^3\ln \left( \left( \frac{\bar{K}}{r}\right) ^3 \right) \end{aligned}$$

where $\bar{K}= (3K/4\pi )^{1/3}$. Simplifying leads to (15).

1.3 Other Model Variants

We briefly summarize a few other models we considered in earlier iterations of this research.

1.3.1 Model2’: Logistic Growth with Shedding

We considered an amalgam of Models 1 and 2 that included both Logistic growth and the surface growth and shedding. This growth law had a total of 5 parameters (including the diffusive parameter $\varepsilon$). We found that this variant was penalized by the AIC and was unreliably fit as we had too few time points to identify the parameters adequately. Hence, we dropped this model in favour of simple Logistic Model 2.

1.3.2 Model 4: Nutrient Depletion

A model in which each cluster (of mass M(t)) grows in its own nutrient bath (concentration C(t)), with nutrient depletion as it grows is

$$\begin{aligned} \frac{dM}{dt}=\alpha _1 CS, \quad \frac{dC}{dt}=-\alpha _2 CS, \end{aligned}$$

(19)

where $\alpha _1$ is the nutrient consumption rate per unit surface area of a cluster and $\alpha _2$ is the corresponding nutrient depletion rate. Then mass conservation allows elimination of C, since $\alpha _2M'(t)+\alpha _1C'(t)=0$. Integrating once and using initial conditions for mass and nutrient, $M_0=M(0), C_0=C(0)$ results in $C=(\alpha _2/\alpha _1)(M_0-M)+C_0$. Substituting into (19) then results in the single ODE for the mass (and for the radius)

$$\begin{aligned} \frac{dM}{dt}=(\alpha _2(M_0-M)+\alpha _1C_0)S, \quad \Rightarrow \quad \frac{dr}{dt}=\frac{4}{3}\alpha _2 \pi (r_0^3-r^3)+\frac{\alpha _1 C_0}{\rho } \equiv f_4(r). \end{aligned}$$

(20)

where $r_0= [(3/4\pi ) M_0]^{1/3}$ is a parameter depicting the initial cluster size. This model has a stable cluster size $r_\infty$. We rejected this model for several reasons: (1) In the experiments, nutrient is renewed daily and does not get significantly deplete. (2) The independence of clusters, while convenient mathematically, is not consistent with the experiments in which many clusters share a common nutrient bath. (The model could be modified to take this into account.)

1.3.3 Model 5: Necrotic Core

We considered a “necrotic core” model, where nutrient is not able to penetrate beyond some diffusion-limited depth $r_1\approx \sqrt{D/k_d}$. Here D is rate of nutrient diffusion, and $k_d$ a rate of nutrient consumption, and $r_1>0$ is assumed to be constant. We assumed that the mass of viable cells occupies a spherical shell (radius R(t)), Volume $V_1(t)$ of depth $r_1\ge 0$, with a spherical necrotic core (radius $R(t)-r_1$, volume $V_2(t)$). The mass of viable cells in the cluster is then $M=\rho (V_1-V_2)$. Assuming surface growth and shedding analogous to Model 1 (7) results in

$$\begin{aligned} \frac{dM}{dt}=(\gamma -\alpha )S - \beta M \quad \Rightarrow \quad \frac{d}{dt}(\rho (V_1-V_2)) = (\gamma - \alpha )(4\pi R^2)-\beta \rho (V_1-V_2) . \end{aligned}$$

The corresponding cluster radius, R, assuming $R>r_1$, satisfies

$$\begin{aligned} \frac{dR}{dt}=\left( \frac{\gamma -\alpha }{\rho }-\beta r_1\right) \frac{R^2}{2Rr_1-r_1^2}+\beta r_1^2\frac{R}{2Rr_1-r_1^2} -\beta r_1^3\frac{1}{6Rr_1-3r_1^3}. \end{aligned}$$

(21)

We rejected this model for two reasons: (1) The experimental data was not consistent with significant necrosis of the cluster interior on the timescale of the experiments (Iworima et al. 2023, 2024; Iworima 2023). (2) This model also proves to be more challenging to fit. See also Komatsu et al. (2017) for the effect of oxygen on necrosis in islets and cell clusters.

1.4 Steady State Solution of PDE for Model 1

We can solve for the steady state profile of (22) in Model 1 as follows

$$\begin{aligned} 0 = -\frac{\partial }{\partial r} [p(r) f(r)] + \varepsilon \frac{\partial ^2 p(r) }{\partial r^2}. \end{aligned}$$

(22)

Integrate once and use the BCs to conclude that

$$\begin{aligned} 0 = -p f(r)+\varepsilon \frac{dp}{dr} \quad \Rightarrow \quad \frac{\varepsilon }{p}\frac{dp}{dr}=f(r), \quad \Rightarrow \quad \frac{d \ln (p)}{dr}=\frac{1}{\varepsilon }f(r). \end{aligned}$$

For Model 1, with $f(r)=a-br$, this ODE can be integrated to obtain

$$\begin{aligned} \ln (p)= \frac{1}{\varepsilon } (a r -(b/2)r^2 + C_0)= -\frac{b}{2\varepsilon }([r-(a/b)]^2+C_1) \end{aligned}$$

for arbitrary constants $C_0, C_1$. After some algebraic manipulation, the solution can be expressed as a Gaussian

$$\begin{aligned} p(r)=C_2 \exp \left( -\frac{ \left( r-\frac{a}{b}\right) ^2}{2\varepsilon /b}\right) , \end{aligned}$$

whose mean is $r=a/b$ and whose width is proportional to $\varepsilon /b$. The constant $C_2$ can be found from the total mass of the clusters, which is constant in our models.

1.5 Numerical Solutions

We use finite differences to approximate solutions of (16). Note that for the right hand side, we use the standard upwind scheme that preserves the direction of flow. For simplicity, we write $f(r)=g(r)+h(r)$, where g(r) is the negative terms and h(r) is the positive terms of f(r).

Let $p_i^n$ represent a distribution of clusters with size $r_i=i\varDelta r$ at time $t_n=n\varDelta t$, where $i\in \{1,2,3,...,M\ |\ M$ is number of spatial nodes$\}$ (cluster radius size “bins”), and $n\in \{1,2,3,...,N\ |\ N \text { is number of time steps}\}$. Here, $\varDelta r$ and $\varDelta t$ are the radius and time step sizes, respectively. We used $\varDelta t = 1/1440 \approx 0.0007$ days and $\varDelta r = 0.01$ for the scaled radius $0\le r \le 1$. Our discretization is:

$$\begin{aligned} \frac{p_i^{n+1} - p_i^n}{\varDelta t}&=-\left( \frac{p^n_{i+1}g(r_{i+1}) - p^n_{i}g(r_i)}{\varDelta r}+ \frac{p^n_{i}h(r_{i}) - p^n_{i-1}h(r_{i-1})}{\varDelta r} \right) ,\nonumber \\ p_i^{n+1}&= p_i^n - \frac{\varDelta t}{\varDelta r}\left( p^n_{i+1}g(r_{i+1}) - p^n_{i}g(r_i)+ p^n_{i}h(r_{i}) - p^n_{i-1}h(r_{i-1}) \right) . \end{aligned}$$

(23)

For the PDE with small diffusive term, (16), we superimposed a centered difference approximation of the diffusion, to obtain

$$\begin{aligned} p_i^{n+1}&= p_i^n - \frac{\varDelta t}{\varDelta r}\left( p^n_{i+1}\left[ g(r_{i+1})+\frac{\varepsilon }{\varDelta r}\right] + p^n_{i}\left[ h(r_{i})- g(r_i) -\frac{2\varepsilon }{\varDelta r}\right] \right. \\&\quad \left. + p^n_{i-1}\left[ \frac{\varepsilon }{\varDelta r} -h(r_{i-1}) \right] \right) . \end{aligned}$$

To simulate the initial condition, we first obtain the mean $\mu$ and the variance $\sigma ^2$ from the data on the first day by using Python function +numpy.mean+ and +numpy.var+. Then we generate the initial condition from the Gaussian distribution:

$$\begin{aligned} p_n^0 =\frac{1}{\sigma \sqrt{2\pi }} e^{-\frac{1}{2}\left( \frac{r_n-\mu }{\sigma }\right) ^2}. \end{aligned}$$

The numerical method shown here has first-order derivative approximations. Future applications of these methods where higher accuracy is essential would benefit from using a higher order scheme.

1.6 Parameter Fitting

We implement parameter estimates using Python Optimization (+scipy.optimize.minimize+). The objective function Z for our problem is given by (17). For each time point, the numerical solution $p_\text {PDE}(r_i,t_n)$ of the discretized PDE is obtained from (23) with the added small diffusion, using the current values of the parameters $\alpha ,\beta ,\gamma , K, \varepsilon$. The method is illustrated in Fig. 8.

Fig. 8

Schematic diagram of the fitting method. The scheme minimizes the deviation between the solution of the model PDEs p(r, t) for the given model, and the experimental size-distribution data. $t_n$ is the time point (Day 1, 2, 3), $r_i$ is the i’th radial bin of the size distribution, and m is the experimental replicate

1.6.1 Validation of Fitting Scheme Using Synthetic Data

To test the parameter fitting routine, we created “synthetic data” where the ground-truth was known, and investigated how close the method results came to that ground-truth. To do so, we set the values of the (‘input’) parameters for each of the models and then solved the PDEs to obtain “artificial data” both with and without additive normally distributed random noise (mean $\mu =0$ and $\sigma = 1$), (that is, SynData with noise = max( 0, SynData+np.random.normal(0,1,1)*0.1)).

We then ran the fitting routine and compared values it found (‘return’) to the true model parameter values.

In each case, we set a Gaussian distribution with variance $=0.004$ and mean $= 0.2$ as synthetic Day 1 data (initial conditions). Then we generate Day 2 and Day 3 synthetic data from the predicted PDE solutions for the given model. For Model 1, the optimization returns fitted parameters that are within 10% of our ground-truth (“input”) parameters, and within 30% for Model 2 and 3 (see Table 3). These results indicate that our optimization scheme is reasonable for the given models and numbers of parameters to be fit.

Table 3 Fitting routine validation: We tested the routine on synthetic data produced using PDE predictions with the Input parameter values

1.7 Model Selection

We use the Akaike Information Criterion (AIC) to help select the most parsimonious model that fits the data. (Motulsky and Christopoulos 2004). Briefly, the AIC selects the model with least sum of squared errors that has fewest parameters.

$$\begin{aligned} \text {AIC}=N \ln \left( \frac{SS}{N} \right) + 2n_p. \end{aligned}$$

(24)

Here N is number of residuals. For example, when there are 3 experimental replicates, then $N=100$ radial size bins $\times 3$ replicates $\times 3$ days = 900. $n_p$ is number of model parameters to be fit, and SS is the minimized sum of square errors

$$\begin{aligned} SS = \min \sum _{i=1}^{N}(y_i-f_i)^2, \end{aligned}$$

(25)

where $y_i$ is the observed value of a data point and $f_i$ is its predicted value. In a given comparison, the models are tested using the same number of residuals N. Models 1, 2, and 3 have the same number of parameters. Hence, we are mainly concerned with $\varDelta$AIC between Models i and Model 0, where $i=1,\dots 3$.

1.8 Spinner Flask and PBS-Mini Tables

Table 4 Spinner flask experiment: Summary of parameters obtained by fitting Models 0, 1, and 2 to the spinner flask data

Fig. 9

Behaviour of replicates with Model 0 fits for the Aggrewell plates, as in Fig. 5. For each row, the parameter $\bar{c}$ is the same (fit to combined three replicates). The initial conditions for the model PDE are determined from Day 1 for each replicate. (See also Figs. 10 and 5 for Models 1 and 2 replicates.)

Fig. 10

Behaviour of replicates with Model 1 fits for the Aggrewell plates, as in Fig. 5. The parameters were fit for each row by pooling the replicates. The initial conditions for the PDE model were determined from Day 1. See also Figs. 5 and 9 for Models 2 and 0 fit to the same replicates

1.9 Replicate Fits

In addition to Fig. 5 (Model 0 fits to the Aggrewell plates experimental data), similar figures for Model 0, and 1 are shown in Figs. 9 and 10.

Table 5 Spinner flask model comparison: Mean squared error (SS/N) and AIC values

Table 6 PBS-mini experiment: Summary of parameters obtained by fitting Model 0, 1 and 2 to the PBS min data

Table 7 PBS-mini model comparison: Mean squared error (SS/N) and AIC values