Bayesian uncertainty quantification for data-driven equation learning

How modulation of the tumor microenvironment drives cancer immune escape dynamics

Metastatic disease is the leading cause of cancer-related death, despite recent advances in therapeutic interventions. Prior modeling approaches have accounted for the adaptive immune system's role in combating tumors, which has led to the development of stochastic models that explain cancer immunoediting and tumor-immune co-evolution. However, cancer immune-mediated dormancy, wherein the adaptive immune system maintains a micrometastatic population by keeping its growth in check, remains poorly understood. Immune-mediated dormancy can significantly delay the emergence (and therefore detection) of metastasis. An improved quantitative understanding of this process will thereby improve our ability to identify and treat cancer during the micrometastatic period. Here, we introduce a generalized stochastic model that incorporates the dynamic effects of immunomodulation within the tumor microenvironment on T cell-mediated cancer killing. This broad class of nonlinear birth-death model can account for a variety of cytotoxic T cell immunosuppressive effects, including regulatory T cells, cancer-associated fibroblasts, and myeloid-derived suppressor cells. We develop analytic expressions for the likelihood and mean time of immune escape. We also develop a method for identifying a corresponding diffusion approximation applicable to estimating population dynamics across a wide range of nonlinear birth-death processes. Lastly, we apply our model to estimate the nature and extent of immunomodulation that best explains the timing of disease recurrence in bladder and breast cancer patients. Our findings quantify the effects that stochastic tumor-immune interaction dynamics can play in the timing and likelihood of disease progression. Our analytical approximations provide a method of studying population escape in other ecological contexts involving nonlinear transition rates.

Master-slave coupling scheme for synchronization and parameter estimation in the generalized Kuramoto-Sivashinsky equation

The problem of estimating the constant parameters of the Kuramoto-Sivashinsky (KS) equation from observed data has received attention from researchers in physics, applied mathematics, and statistics. This is motivated by the various physical applications of the equation and also because it often serves as a test model for the study of space-time pattern formation. Remarkably, most existing inference techniques rely on statistical tools, which are computationally very costly yet do not exploit the dynamical features of the system. In this paper, we introduce a simple, online parameter estimation method that relies on the synchronization properties of the KS equation. In particular, we describe a master-slave setup where the slave model is driven by observations from the master system. The slave dynamics are data-driven and designed to continuously adapt the model parameters until identical synchronization with the master system is achieved. We provide a simple analysis that supports the proposed approach and also present and discuss the results of an extensive set of computer simulations. Our numerical study shows that the proposed method is computationally fast and also robust to initialization errors, observational noise, and variations in the spatial resolution of the numerical scheme used to integrate the KS equation.

Direct Estimation of Parameters in ODE Models Using WENDy: Weak-Form Estimation of Nonlinear Dynamics

We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of [Formula: see text] bump functions of varying support sizes.We demonstrate the high robustness and computational efficiency by applying WENDy to estimate parameters in some common models from population biology, neuroscience, and biochemistry, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available at https://github.com/MathBioCU/WENDy .

Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics

We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of C-infinity bump functions of varying support sizes. We demonstrate the high robustness and computational efficiency by applying WENDy to estimate parameters in some common models from population biology, neuroscience, and biochemistry, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available at (https://github.com/MathBioCU/WENDy).

Computationally efficient mechanism discovery for cell invasion with uncertainty quantification

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.