Estimation of Parameter Distributions for Reaction-Diffusion Equations with Competition using Aggregate Spatiotemporal Data

Abstract

Reaction-diffusion equations have been used to model a wide range of biological phenomenon related to population spread and proliferation from ecology to cancer. It is commonly assumed that individuals in a population have homogeneous diffusion and growth rates; however, this assumption can be inaccurate when the population is intrinsically divided into many distinct subpopulations that compete with each other. In previous work, the task of inferring the degree of phenotypic heterogeneity between subpopulations from total population density has been performed within a framework that combines parameter distribution estimation with reaction-diffusion models. Here, we extend this approach so that it is compatible with reaction-diffusion models that include competition between subpopulations. We use a reaction-diffusion model of glioblastoma multiforme, an aggressive type of brain cancer, to test our approach on simulated data that are similar to measurements that could be collected in practice. We use Prokhorov metric framework and convert the reaction-diffusion model to a random differential equation model to estimate joint distributions of diffusion and growth rates among heterogeneous subpopulations. We then compare the new random differential equation model performance against other partial differential equation models’ performance. We find that the random differential equation is more capable at predicting the cell density compared to other models while being more time efficient. Finally, we use k-means clustering to predict the number of subpopulations based on the recovered distributions.

Appendices

Appendix A: Mixture of Two-Gaussian Distribution

1.1 A.1 Elbow plot

Fig. 8 displays the elbow plot showing the sum of squares error for different values of k. The true number of subpopulations (2) is shown in the black solid line. SPSVERBc10 is then used to determine the number of cluster centers, which is shown in the red dashed line.

Fig. 8

Elbow plot showing the sum of squares error for different values of k

Appendix B: Mixture of Three-Gaussian Distribution

1.1 B.1 Fitting and Forecasting Results

In Fig. 9, we plot the simulated results using the estimated parameters from the traditional inverse problem approach on 2-PDE, 4-PDE, and 6-PDE (Figs. 9a-9c) and the simulated aggregated population from the estimated RanDE model using PrMF approach (Fig. 9d). In Fig. 10, we plot the $\textit{SSE}$ comparison between models.

Fig. 9

Aggregated cell density comparison between: a 2-PDE model, b 4-PDE model, c 6-PDE model, and (d) RanDE model. In each figure, we plot the generated data (dashed curves) and model simulation (solid curves) for 4 different time points. For fitting interval, we plot the cell density at $t=0.4$ and $t=0.8$. For the forecasting interval, we plot the cell density at $t=1.2$ and $t=1.4$ (Color figure online)

Fig. 10

Comparison for the fitting and prediction errors between models. The solid red and blue curves show the $\textit{SSE}$ for non-RanDE models with the number of on the x-axis within fitting and prediction intervals, respectively. The red and blue horizontal dashed lines are the $\textit{SSE}$ for the RanDE model within fitting and prediction intervals, respectively (Color figure online)

1.2 B.2 Profile of Traveling Wave Speed

In Fig. 11, we plot the estimated traveling wave speed within the fitting interval (Fig. 11a) and within the forecasting interval (Fig. 11b).

Fig. 11

Wave speed profile comparison between models within: a fitting interval and b prediction interval

1.3 B.3 Recovered Distribution and Cluster Centers

In Fig. 12, we compare the true distribution (Fig. 12a) and the estimated distribution using PrMF (Fig. 12b). In Fig. 13, we plot the cluster centers that were identified by k-means clustering.

Fig. 12

Point-Wise Estimated Parameters of the 2,4,6-PDE models on the true distribution with 30 D-nodes and 60 $\rho$-nodes. b Estimated distribution using Prokhorov metric framework with 10 D-nodes and 5 $\rho$-nodes

Fig. 13

Plotting the predicted cluster centers using k-means clustering

1.4 B.4 Elbow Plot

In Fig. 14, we plot the elbow curve sum of squares error curve against the number of clusters, k. We find that the predicted number of clusters in the elbow plot (red dashed line) is two.

Fig. 14

Elbow Plot showing the sum of squares error for different values of k