Tag - Mathematical biology

Inverse Reinforcement Learning (IRL) is a compelling technique for revealing the rationale underlying the behaviour of autonomous agents. IRL seeks to estimate the unknown reward function of a Markov decision process (MDP) from observed agent trajectories. However, IRL needs a transition function, and most algorithms assume it is known or can be estimated in advance from data. It therefore becomes even more challenging when such transition dynamics is not known a-priori, since it enters the estimation of the policy in addition to determining the system's evolution. When the dynamics of these agents in the state-action space is described by stochastic differential equations (SDE) in Itô calculus, these transitions can be inferred from the mean-field theory described by the Fokker-Planck (FP) equation. We conjecture there exists an isomorphism between the time-discrete FP and MDP that extends beyond the minimization of free energy (in FP) and maximization of the reward (in MDP). We identify specific manifestations of this isomorphism and use them to create a novel physics-aware IRL algorithm, FP-IRL, which can simultaneously infer the transition and reward functions using only observed trajectories.  We employ variational system identification to infer the potential function in FP, which consequently allows the evaluation of reward, transition, and policy by leveraging the conjecture. We demonstrate the effectiveness of FP-IRL by applying it to a synthetic benchmark and a biological problem of cancer cell dynamics, where the transition function is inaccessible.

White matter changes are a frequent observations in the aging human brain and are considered a reliable indicator for cognitive impairment and long-term functional decline. On T2-weighted fluid attenuated inversion recovery magnetic resonance images, these lesions appear as white matter hyperintensities (WMH) and are commonly associated with vascular degeneration. From a physics perspective, however, the persistent (onset) locations of periventricular WMHs along the edges of the lateral ventricles suggests involvement of mechanical (over)loading of the ependymal cells forming the functional brain-fluid barrier. We use computational modelling to systematically explore the relationship between brain aging, white matter changes, and WMH formation. To that end, we build anatomically accurate brain models and predict the mechanical loading of periventricular tissues. We observe that maximum ependymal cell stretch consistently localizes in the anterior and posterior horns irrespective of ventricular volume or shape. More importantly, these locations coincide with periventricular WMH locations observed in our patient scans. From these results, we pose that further analysis of white matter pathology in the periventricular zone that includes a mechanics-driven deterioration model for the ventricular wall.

Many biological tissues must be structured in such a way as to be able to adapt to two extreme biomechanical scenarios: they have to be strong to resist high pressure and mechanical forces and yet be flexible to allow large expansions and growth. A part of nature’s solution to this intriguing problem are the complex microstructures and microscopic (cellular) processes, that modify tissue’s elastic properties. To analyse the interplay between the mechanics, microstructure, and the chemistry we derive microscopic models for plant biomechanics, assuming that the elastic properties depend on the chemical processes and chemical reactions depend on the mechanical stresses. Multiplicative decomposition of the deformation gradient into elastic and growth parts is used to model the stress or strain based growth. To analyse the properties and behaviour of plant tissues, the macroscopic models are derived using homogenization techniques. Numerical solutions for macroscopic models demonstrate the impact of the microstructure on tissue deformations and growth.

This talk discusses two non-destructive evaluation aims that can be achieved with elastic waves travelling in soft materials.

First, we see how tracking the changes in wave speed with stress or strain gives access to linear and nonlinear material parameters. These can then be used to design biomaterials or to create meaningful Finite Element simulations.

Then, we find that the states of stress and strain existing in a loaded material can be accessed directly from wave speed measurements, without having to determine, or even know, its material properties. These techniques are expected to have important applications in health monitoring of loaded structures. Examples include bulk muscles, and thin membranes such as a stretched rubber sheet, a piece of cling film (~10 μm thick) and the animal skin of a bodhrán, a traditional Irish drum.

Due to their intrinsic complexity, it is not easy to model the mechanical behaviour of biomaterials. Despite the difficulties encountered, some scholars have successfully introduced mathematical models to describe some aspects of the mechanical response of soft tissues. However, none of these models accounts for material dispersion. Since most of materials of biological interest are composite made of different constituents reinforced by collagen and/or elastin fibres, material dispersion and anisotropy are nonnegligible. Within the theory of elasticity, several models for anisotropic materials have been developed. The same cannot be said for anisotropic dispersive materials. The existing models accounting for material dispersion have been introduced only for isotropic materials. The aim of this manuscript is to introduce the most general model for anisotropic dispersion in transversely isotropic materials. Our model has the potentiality to represent a first step toward a better understanding of the mechanical response of fibre-reinforced soft materials.

Information field theory (IFT) is Bayesian statistics for fields (a.k.a. functions of space and time). It uses the same mathematics found in statistical field theory and quantum field theory, namely functional integration or Feynman path integrals. IFT starts by imposing a prior probability measure over the space of physical fields (e.g., temperature, pressure, strain, stress). Then, one constructs a likelihood function that models the measurement process to connect the fields to the available data. Finally, one uses Bayes’ rule to build a posterior over the space of physical fields, which they proceed to characterize either analytically (see Feynman diagrams) or numerically. IFT has been used successfully in various field reconstruction problems, primarily astrophysical applications. In this talk, we will discuss how IFT can be used to perform uncertainty quantification tasks in physical problems governed by ordinary and partial differential equations. We will show how one can 1) use knowledge of the governing equations to construct suitable prior measures over the space of fields; 2) sample from the fields’ posterior numerically via advanced Markov chain Monte Carlo and variational inference without the need to call a numerical solver; and 3) sample from the posterior of any physical parameters, initial conditions, boundary conditions, and source terms without the need to evaluate a normalization constant. The method offers several potential advantages compared to traditional uncertainty quantification techniques. The approach has a mechanism for quantifying the model-form uncertainty. It naturally fuses data from multiple modalities and elegantly deals with ill-posed problems (e.g., missing boundary conditions).

The Ascona B-DNA Consortium (or ABC) has generated very large data sets of molecular dynamics simulations of double stranded nucleic acids (or dsNA). Predictive Gaussian coarse-grain sequence-dependent models of dsNA can then be parametrised by fitting to statistics drawn from this training set data using Kullback-Leibler divergence (or relative entropy) as objective function. However the precision, or stiffness, matrices in these Gaussian models have very particular structures arising from the specifics of the application, e.g. block bandedness. I will focus on describing special features of the parameter fitting process for such structured Gaussians, which seem to be of interest beyond the specific application.

Complex materials such as soft tissues exhibit nonlinear anisotropic response and heterogeneous mechanical properties. Data-driven methods have been recently developed to capture the rich mechanical behaviour of these materials under extreme deformations. In particular, we have contributed to the field by leveraging neural ordinary differential equations (NODEs) as the building blocks of strain energy density functions that automatically satisfy polyconvexity, objectivity, material symmetry and positive energy dissipation requirements for realistic and physically plausible material models. However, these data-driven models have lacked consideration of uncertainty. This is particularly problematic for soft tissues which exhibit a large variation in mechanical properties from one individual to another. Here we establish a generative modelling framework based on stable diffusion to model distributions of materials while satisfying physics constraints. We use NODEs to describe the material response. Because the NODE framework automatically satisfies the desired physics, any samples of parameters of the NODE produces feasible  materials. For a given material of interest e.g. skin, we assume that stress-strain curves from the population are available. Fitting a subset of the NODE parameters to the stress-strain data yields samples over the parameter space of the NODEs. Diffusion probabilistic models are then employed to learn that distribution over these NODE parameters and, implicitly, over the constitutive models. We showcase the ability of the framework to learn the distribution of material behaviour for both synthetic examples and murine skin data, outperforming standard density estimation techniques. We anticipate that this work will further establish the use of data-driven methods for materials that exhibit large variation across a population for which uncertainty quantification is essential.

The embryonic elongation of C. elegans  represents an attractive model of matter reorganization occurring without mass increase in an initial egg. Within four hours, the embryo elongates along the anterior/posterior axis by four times and its circumference reduces one-third. Unlike the embryonic development of Drosophila and Zebrafish, there is neither cell migration, cell division nor a notable change but only a noticeable epidermis elongation which  drives the whole morphogenetic before hatching. The identified driving forces are  the actomyosin contractility  of  the epidermis and the muscle activity that starts after the 1.8-fold stage. The late elongation shows cyclic periods of bending, rotation and torsion depending on the activation of the muscles. In this talk I will present experimental results of the Labouesse's group and models which explain the role of the different active filaments in the embryo.