Tag - Mathematical biology

Biological tissues typically lack the regular spatial organisation or homogeneity of engineered materials. To understand the impact of the inherent granularity and stochastic heterogeneity of multicellular tissues on their macroscopic mechanical properties, it can be fruitful to resort to discrete models that capture key physical features of individual tissue components. One popular approach in developmental biology is the vertex model, which assigns a mechanical energy to individual cells and predicts distributions of stress across growing tissues, capturing certain features that risk being 'washed away' using traditional homogenisation approaches, while exploiting geometric features that are readily measured experimentally. Complementing numerical simulations, discrete calculus methods provide a powerful set of tools with which to analyse the vertex model. I will show how this approach has revealed features such as couple stresses at tricellular junctions in spatially disordered cellular monolayers.

From the beating heart to tissue assembly and repair, it is well accepted that mechanics plays an important role in the behaviour of biological systems. Mechanical forces are not only fundamentally important to biological materials (e.g., the mechanics of growth), but are also fundamental drivers of cellular behaviour change. However, it is often difficult to determine mechanical state both in vitro and in vivo, and it is often difficult to determine how mechanical perturbations (e.g., changes to boundary conditions) will change the mechanical state throughout the domain. Over the past several decades, mathematical modelling has emerged as an important tool to bridge this gap. And, more recently, there has been a surge in interest towards using data-driven statistical techniques to create predictive models of biological system behaviour. As experimental techniques and data-driven methods simultaneously advance, there is an unprecedented opportunity to gain biological insight. In this talk, we will describe our preliminary and ongoing work in data driven modelling of in vitro biological systems with applications focused on both cardiac tissue engineering and wound healing. In brief, we envision a methodological framework with three essential components: (1) open access datasets of time-lapse movies of cells and tissue, (2) open source software to extract interpretable quantities of interest from these time-lapse movies, and (3) combined mechanistic and statistical models of biological behaviour informed by these data. We are presently working on creating these datasets, software, and models in partnership with experimental collaborators, and releasing them to the community under permissive licenses. Looking forward, we anticipate that these large open access curated datasets combined with open source tools to extract information from them will enable significant advances in our understanding of, and ability to control, living systems. Through this talk, we hope to foster further discussion and collaborations at the interface of mechanics, biology, and open science.

A comprehensive image-based computational modelling pipeline is required for high-fidelity patient-specific cardiac simulations. However, traditional simulation methods are a limitation in these approaches due to their prohibitively slow speeds. We developed a physics-based training scheme using differentiable finite elements to compute the residual force vector of the governing PDE, which is then minimized to find the optimal network parameters. We used neural networks for their representation power, and finite elements for defining the problem domain, specifying the boundary conditions, and performing numerical integrations. We incorporated spatially varying fibre structures into a prolate spheroidal model of the left ventricle. A Fung-type material model including active contraction was used. We developed two versions of our model, one was trained on a reduced basis of the solution space, and one was trained on the full solution space. The models were trained against two pressure-volume loops and validated on a third loop (Fig. 1). We validated our implementation against conventional FEM simulation using FEniCS. While the reduced order model was trained faster than the full-order model, we achieved mean and standard deviation of the nodal error between the NNFE solution and the FE solution with 10^-3 cm, with both models, where the characteristic length was 1 cm. The NNFE model predicted each solution within 0.6 ms whereas the FE models took up to 500 ms for each state. The NNFE method can be simultaneously trained over the entire range of physiological boundary conditions. The trained NNFE can predict stress–strain responses for any physiological boundary condition without retraining.

Tendons are the connective tissues that attach muscles to bones. Ligaments have a similar function but connect two bones. They both have complex microstructures consisting of a hierarchical arrangement of collagen fibres embedded in a compliant matrix and can have varied macroscale geometries ranging from cylinders to tapered bands to hourglass shapes. The interplay between their micro- and macro-structure makes it challenging to model their mechanical behaviour. In this seminar, I will present a microscale constitutive model that can be used to predict how the structural arrangement of the collagen fibres impacts tendon and ligament stress-strain behaviour. I will explain how it can be used to infer microscale parameter values from macroscale mechanical data. Then, I will discuss its implementation in a finite element model to investigate the effects of macroscale geometry, highlighting the importance of realistic fibre alignment and how this has been neglected in the literature.

This study presents a continuum multiphysics approach to mathematically and physically describe the behaviour of living systems, capturing the interaction between cells and extracellular matrix (ECM) to simulate tissue-level responses such as wound healing and tissue reconfiguration. As most processes in living systems involve highly nonlinear multiphysical phenomena, this study aims to provide insight into the locally-averaged details of these processes. To achieve this, experimental observations and physical concepts are utilized to motivate theoretical formulations in a nonlinear solid mechanics framework. This work utilizes an array of mixed finite element formulations, where special discretization techniques and numerical analysis algorithms are discussed to simulate these complex systems accurately in a robust manner. The proposed models are validated through experimental results that capture the changes in tissue construct shape and cell concentration for wounded and intact microtissues, enabling the interpretation of experimental data. One of the main points of this study is to understand the collective response of cells during remodelling in the context of cell-ECM interactions and its effect on tissue morphology. This continuum multiphysics framework provides valuable insights into cell-ECM interactions that can be extended to development and cancer, tissue engineering, and regenerative medicine, and can aid in developing novel regenerative therapies.

The development of computational models in the cardiovascular field is a challenging research area, where the need for accurate responses in short timeframes conflicts with the complexity of the underlying physical processes and the great anatomical and functional variability among patients. In this context, physics-based models require long times and computational resources for the numerical discretization of multi-scale and multi-physics systems of differential equations, while data-driven methods rarely achieve high accuracy and generalization capabilities. In this talk, we present scientific machine learning methods that integrate physical knowledge with data-driven techniques to accelerate the evaluation of differential models and address many-query problems - such as sensitivity analysis, robust parameter estimation, and uncertainty quantification - in cardiovascular applications. To speed up input-output evaluations, we develop emulators of time-dependent processes capable of predicting spatial outputs and accounting for geometric variability from patient to patient. Our methods also enable data-driven learning of mathematical models for the slow-scale remodelling associated with processes whose fast scale is well characterized by physics-based models. Numerical results demonstrate that these scientific machine learning methods enhance efficiency and accuracy in approximating quantities of interest, as well as in solving parameter estimation and uncertainty quantification problems.

We present new approaches for a direct deduction of the macroscopic properties of biological and rubberlike materials, starting from a detailed discussion of the behaviour at the molecular and cellular scale. By adapting classical methods of equilibrium and non equilibrium statistical mechanics, we describe different types of instabilities observed in polypeptide chains. From one side conformational transitions corresponding to the unfolding of crystal hard domains, such as beta-sheets and alpha helices secondary structures are modelled by considering multiwells energy functions. From the other side bonds breaking as in the case of RNA/DNA denaturation or molecules and cell decohesion are modelled by considering a ‘degenerate’ second energy well, corresponding to constant (debonding) energy and zero force. In this way the purely mechanical approaches for phase transition and hysteresis that we previously proposed  in discrete models based on the introduction of internal (spin type) variables, are extended to the cases when thermal, rate, and entropic effects cannot be neglected. Different phenomena will be described such as protein unfolding taking care of intermolecular or external devices interaction, non-local interactions and interfaces energy effects, intermediate ‘damaged’ configurations anticipating decohesion. Based then on classical multiscale modelling and continuum mechanics we deduce effective macroscale constitutive models with damage, residual strains, hysteresis, healing, growth and rate effects.

For many decades, physics-based PDEs have been commonly employed for modelling the mechanical responses of biological tissues, then traditional numerical methods were employed to solve the PDEs and provide predictions. However, when governing laws are unknown or when high degrees of heterogeneity present, these classical models may become inaccurate. In this talk we propose to use data-driven modelling which directly utilizes experimental measurements to learn the hidden physics and provide further predictions. In particular, we develop PDE-inspired neural operator architectures, to learn the mapping between loading conditions and the corresponding mechanical responses. By parameterizing the increment between layers as an integral operator, our neural operator can be seen as the analogue of a time-dependent nonlocal equation, which captures the long-range dependencies in the feature space and is guaranteed to be resolution-independent. Moreover, when applying to (hidden) PDE solving tasks, our neural operator provides a universal approximator to a fixed point iterative procedure, and partial physical knowledge can be incorporated to further improve the model’s generalizability and transferability. As a real-world application, we learn the material models directly from digital image correlation (DIC) displacement tracking measurements on a porcine tricuspid valve leaflet tissue, and show that the learnt model substantially outperforms conventional constitutive models.

The motor-driven intracellular transport plays a crucial role in supporting a neuron cell’s survival and function, with motor proteins and microtubule (MT) structures collaborating to promptly deliver the essential materials to the right location in neuron. The disruption of transport may lead to the onset of various neurodegenerative diseases. To study how neurons regulate the material transport process and have a better understanding of the traffic jam formation, we develop a PDE-constrained optimization model and an isogeometric analysis (IGA) solver to simulate traffic jams induced by MT reduction and swirl. We also develop a novel IGA-based physics-informed graph neural network (PGNN) to quickly predict normal and abnormal transport phenomena in different neuron geometries. The IGA-based PGNN model contains simulators to handle local prediction of both normal and two MT-induced traffic jams in pipes, as well as another simulator to predict normal transport in bifurcations. Bézier extraction is adopted to incorporate the geometry information into the simulators to accurately compute the physics informed loss function with PDE residuals. Moreover, a GNN assembly model is adopted to tackle different neuron morphologies by assembling local prediction into the entire geometry. The well-trained model effectively predicts the distribution of transport velocity and material concentration during traffic jam and normal transport with an average error less than 10% compared to IGA simulations.

To model neuron growth, we develop a new computational framework and an open-source software package "NeuronGrowth_IGAcollocation” based on the phase field method. Neurons consist of a cell body, dendrites, and axons. Axons and dendrites are long processes extending from the cell body and enabling information transfer to and from other neurons. There is high variation in neuron morphology based on their location and function, thus increasing the complexity in mathematical modelling of neuron growth. We propose a novel phase field model with isogeometric collocation to simulate different stages of neuron growth by considering the effect of tubulin. The stages modelled include lamellipodia formation, initial neurite outgrowth, axon differentiation, and dendrite formation considering the effect of intracellular transport of tubulin on neurite outgrowth. By incorporating neurite features from experiments, we can demonstrate similar reproduction of neuron morphologies at different stages of growth and allow extension towards the formation of neurite networks. Based on the IGA simulation data, a CNN model is also built to efficiently predict the growth process.