Computationally efficient mechanism discovery for cell invasion with uncertainty quantification

Abstract

Parameter estimation for mathematical models of biological processes is often difficult and depends significantly on the quality and quantity of available data. We introduce an efficient framework using Gaussian processes to discover mechanisms underlying delay, migration, and proliferation in a cell invasion experiment. Gaussian processes are leveraged with bootstrapping to provide uncertainty quantification for the mechanisms that drive the invasion process. Our framework is efficient, parallelisable, and can be applied to other biological problems. We illustrate our methods using a canonical scratch assay experiment, demonstrating how simply we can explore different functional forms and develop and test hypotheses about underlying mechanisms, such as whether delay is present. All code and data to reproduce this work are available at https://github.com/DanielVandH/EquationLearning.jl.

Pushing coarse-grained models beyond the continuum limit using equation learning

“…Our procedure was able to recover terms that conserved mass, despite not enforcing conservation of mass explicitly. The procedure as we have described does have some limitations, such as assuming that the mechanisms are linear combinations of basis functions, which could be handled more generally by instead using nonlinear least squares [42]. The procedure may also be sensitive to the quality of the data points included in the matrices, and thus to the parameters used for the procedure.…”

“…Working with heterogeneous populations of cells, where parameters in the discrete model can vary between individuals in the population, is also another interesting option for future exploration [14]. Uncertainty quantification could also be considered using bootstrapping [42] or Bayesian inference [62]. Allowing for uncertainty quantification would also allow for noisy datasets to be modelled, unlike the idealized, noise-free data used in this work.…”

“…, θ d ) T in (4.1). We use ideas similar to the basis function approach from VandenHeuvel et al [42], using (4.1) to construct a matrix problem for θ. In particular, let us take the PDE (2.6), with R(q) = 0 and H(q) = 0, and expand the spatial derivative term so that we can isolate the θ k terms,…”

“…Lagergren et al [41] use this framework to discover a model describing data from simple in vitro experiments that describe the invasion of populations of motile and proliferative prostate cancer cells. VandenHeuvel et al [42] extend the work of Lagergren et al [41], incorporating uncertainty quantification into the equation learning procedure through a bootstrapping approach. Nardini et al [32] use discrete data from agent-based models to learn associated continuum ODE models, combining a user-provided library of functions together with sparse regression methods to give simple ODE models describing population densities.…”

“…These previous approaches to equation learning consider various methods to estimate the parameters bold-italicθ, such as sparse regression or nonlinear optimization [32,37–39,41,42], representing N as a library of functions [37–39], neural networks [41], or in the form of a conservation law with individual components to be learned [41,42]. In this work, we introduce a stepwise equation learning framework, inspired from stepwise regression [44], for estimating bold-italicθ from averaged discrete data with a given N representing a proposed form for the continuum model description.…”