A Continuous Intestinal Absorption Model to Predict Drug Enterohepatic Recirculation in Healthy Humans: Nalbuphine as a Model Substrate.

Nalbuphine (NAL) is a κ-agonist/µ-antagonist opioid being developed as an oral extended formulation (ER) for the treatment of chronic cough in idiopathic pulmonary fibrosis and itch in prurigo nodularis. NAL is extensively glucuronidated and likely undergoes enterohepatic recirculation (EHR). The purpose of this work is to develop pharmacokinetic models for NAL absorption and enterohepatic recirculation (EHR). Clinical pharmacokinetic (PK) data sets in healthy subjects from three trials that included IV, oral solution, and ER tablets in fed and fasted state and two published trials were used to parametrize a novel partial differential equation (PDE)-based model, termed "PDE-EHR" model. Experimental inputs included in vitro dissolution and permeability data. The model incorporates a continuous intestinal absorption framework, explicit liver and gall bladder compartments, and compartments for systemic drug disposition. The model was fully PDE-based with well-stirred compartments achieved by rapid diffusion. The PDE-EHR model accurately reproduces NAL concentration-time profiles for all clinical data sets. NAL disposition simulations required inclusion of both parent and glucuronide recirculation. Inclusion of intestinal P-glycoprotein efflux in the simulations suggests that NAL is not expected to be a victim or perpetrator of P-glycoprotein-mediated drug interactions. The PDE-EHR model is a novel tool to predict EHR and food/formulation effects on drug PK. The results strongly suggest that even intravenous dosing studies be conducted in fasted subjects when EHR is suspected. The modeling effort is expected to aid in improved prediction of dosing regimens and drug disposition in patient populations.

The development of drug resistance in metastatic tumours under chemotherapy: An evolutionary perspective.

We present a mathematical model of the evolutionary dynamics of a metastatic tumour under chemotherapy, comprising non-local partial differential equations for the phenotype-structured cell populations in the primary tumour and its metastasis. These equations are coupled with a physiologically-based pharmacokinetic model of drug administration and distribution, implementing a realistic delivery schedule. The model is carefully calibrated from the literature, focusing on BRAF-mutated melanoma treated with Dabrafenib as a case study. By means of long-time asymptotic and global sensitivity analyses, as well as numerical simulations, we explore the impact of cell migration from the primary to the metastatic site, physiological aspects of the tumour tissues and drug dose on the development of chemoresistance and treatment efficacy. Our findings provide a possible explanation for empirical evidence indicating that chemotherapy may foster metastatic spread and that metastases may be less impacted by the chemotherapeutic agent.

A bulk-surface mechanobiochemical modelling approach for single cell migration in two-space dimensions.

In this work, we present a mechanobiochemical model for two-dimensional cell migration which couples mechanical properties of the cell cytosol with biochemical processes taking place near or on the cell plasma membrane. The modelling approach is based on a recently developed mathematical formalism of evolving bulk-surface partial differential equations of reaction-diffusion type. We solve these equations using finite element methods within a moving-mesh framework derived from the weak formulation of the evolving bulk-surface PDEs. In the present work, the cell cytosol interior (bulk) dynamics are coupled to the cell membrane (surface) dynamics through non-homogeneous Dirichlet boundary conditions. The modelling approach exhibits both directed cell migration in response to chemical cues as well as spontaneous migration in the absence of such cues. As a by-product, the approach shows fundamental characteristics associated with single cell migration such as: (i) cytosolic and membrane polarisation, (ii) actin dependent protrusions, and (iii) continuous shape deformation of the cell during migration. Cell migration is an ubiquitous process in life that is mainly triggered by the dynamics of the actin cytoskeleton and therefore is driven by both mechanical and biochemical processes. It is a multistep process essential for mammalian organisms and is closely linked to a vast diversity of processes; from embryonic development to cancer invasion. Experimental, theoretical and computational studies have been key to elucidate the mechanisms underlying cell migration. On one hand, rapid advances in experimental techniques allow for detailed experimental measurements of cell migration pathways, while, on the other, computational approaches allow for the modelling, analysis and understanding of such observations. The bulk-surface mechanobiochemical modelling approach presented in this work, set premises to study single cell migration through complex non-isotropic environments in two- and three-space dimensions.

Bat Motion can be Described by Leap Frogging.

We present models of bat motion derived from radio-tracking data collected over 14 nights. The data presents an initial dispersal period and a return to roost period. Although a simple diffusion model fits the initial dispersal motion we show that simple convection cannot provide a description of the bats returning to their roost. By extending our model to include non-autonomous parameters, or a leap frogging form of motion, where bats on the exterior move back first, we find we are able to accurately capture the bat's motion. We discuss ways of distinguishing between the two movement descriptions and, finally, consider how the different motion descriptions would impact a bat's hunting strategy.

Parameter Identifiability in PDE Models of Fluorescence Recovery After Photobleaching.

Identifying unique parameters for mathematical models describing biological data can be challenging and often impossible. Parameter identifiability for partial differential equations models in cell biology is especially difficult given that many established in vivo measurements of protein dynamics average out the spatial dimensions. Here, we are motivated by recent experiments on the binding dynamics of the RNA-binding protein PTBP3 in RNP granules of frog oocytes based on fluorescence recovery after photobleaching (FRAP) measurements. FRAP is a widely-used experimental technique for probing protein dynamics in living cells, and is often modeled using simple reaction-diffusion models of the protein dynamics. We show that current methods of structural and practical parameter identifiability provide limited insights into identifiability of kinetic parameters for these PDE models and spatially-averaged FRAP data. We thus propose a pipeline for assessing parameter identifiability and for learning parameter combinations based on re-parametrization and profile likelihoods analysis. We show that this method is able to recover parameter combinations for synthetic FRAP datasets and investigate its application to real experimental data.

A Genuinely Hybrid, Multiscale 3D Cancer Invasion and Metastasis Modelling Framework.

We introduce in this paper substantial enhancements to a previously proposed hybrid multiscale cancer invasion modelling framework to better reflect the biological reality and dynamics of cancer. These model updates contribute to a more accurate representation of cancer dynamics, they provide deeper insights and enhance our predictive capabilities. Key updates include the integration of porous medium-like diffusion for the evolution of Epithelial-like Cancer Cells and other essential cellular constituents of the system, more realistic modelling of Epithelial-Mesenchymal Transition and Mesenchymal-Epithelial Transition models with the inclusion of Transforming Growth Factor beta within the tumour microenvironment, and the introduction of Compound Poisson Process in the Stochastic Differential Equations that describe the migration behaviour of the Mesenchymal-like Cancer Cells. Another innovative feature of the model is its extension into a multi-organ metastatic framework. This framework connects various organs through a circulatory network, enabling the study of how cancer cells spread to secondary sites.

A Lifshitz-Slyozov type model for adipocyte size dynamics: limit from Becker-Döring system and numerical simulation.

Biological data show that the size distribution of adipose cells follows a bimodal distribution. In this work, we introduce a Lifshitz-Slyozov type model, based on a transport partial differential equation, for the dynamics of the size distribution of adipose cells. We prove a new convergence result from the related Becker-Döring model, a system composed of several ordinary differential equations, toward mild solutions of the Lifshitz-Slyozov model using distribution tail techniques. Then, this result allows us to propose a new advective-diffusive model, the second-order diffusive Lifshitz-Slyozov model, which is expected to better fit the experimental data. Numerical simulations of the solutions to the diffusive Lifshitz-Slyozov model are performed using a well-balanced scheme and compared to solutions to the transport model. Those simulations show that both bimodal and unimodal profiles can be reached asymptotically depending on several parameters. We put in evidence that the asymptotic profile for the second-order system does not depend on initial conditions, unlike for the transport Lifshitz-Slyozov model.

Machine learning-accelerated computational fluid dynamics.

Numerical simulation of fluids plays an essential role in modeling many physical phenomena, such as weather, climate, aerodynamics, and plasma physics. Fluids are well described by the Navier-Stokes equations, but solving these equations at scale remains daunting, limited by the computational cost of resolving the smallest spatiotemporal features. This leads to unfavorable trade-offs between accuracy and tractability. Here we use end-to-end deep learning to improve approximations inside computational fluid dynamics for modeling two-dimensional turbulent flows. For both direct numerical simulation of turbulence and large-eddy simulation, our results are as accurate as baseline solvers with 8 to 10× finer resolution in each spatial dimension, resulting in 40- to 80-fold computational speedups. Our method remains stable during long simulations and generalizes to forcing functions and Reynolds numbers outside of the flows where it is trained, in contrast to black-box machine-learning approaches. Our approach exemplifies how scientific computing can leverage machine learning and hardware accelerators to improve simulations without sacrificing accuracy or generalization.

Efficient Estimates of Surface Diffusion Parameters for Spatio-Temporally Resolved Virus Replication Dynamics.

Advanced methods of treatment are needed to fight the threats of virus-transmitted diseases and pandemics. Often, they are based on an improved biophysical understanding of virus replication strategies and processes in their host cells. For instance, an essential component of the replication of the hepatitis C virus (HCV) proceeds under the influence of nonstructural HCV proteins (NSPs) that are anchored to the endoplasmatic reticulum (ER), such as the NS5A protein. The diffusion of NSPs has been studied by in vitro fluorescence recovery after photobleaching (FRAP) experiments. The diffusive evolution of the concentration field of NSPs on the ER can be described by means of surface partial differential equations (sufPDEs). Previous work estimated the diffusion coefficient of the NS5A protein by minimizing the discrepancy between an extended set of sufPDE simulations and experimental FRAP time-series data. Here, we provide a scaling analysis of the sufPDEs that describe the diffusive evolution of the concentration field of NSPs on the ER. This analysis provides an estimate of the diffusion coefficient that is based only on the ratio of the membrane surface area in the FRAP region to its contour length. The quality of this estimate is explored by a comparison to numerical solutions of the sufPDE for a flat geometry and for ten different 3D embedded 2D ER grids that are derived from fluorescence z-stack data of the ER. Finally, we apply the new data analysis to the experimental FRAP time-series data analyzed in our previous paper, and we discuss the opportunities of the new approach.

An Adaptive Sampling Algorithm with Dynamic Iterative Probability Adjustment Incorporating Positional Information.

Physics-informed neural networks (PINNs) have garnered widespread use for solving a variety of complex partial differential equations (PDEs). Nevertheless, when addressing certain specific problem types, traditional sampling algorithms still reveal deficiencies in efficiency and precision. In response, this paper builds upon the progress of adaptive sampling techniques, addressing the inadequacy of existing algorithms to fully leverage the spatial location information of sample points, and introduces an innovative adaptive sampling method. This approach incorporates the Dual Inverse Distance Weighting (DIDW) algorithm, embedding the spatial characteristics of sampling points within the probability sampling process. Furthermore, it introduces reward factors derived from reinforcement learning principles to dynamically refine the probability sampling formula. This strategy more effectively captures the essential characteristics of PDEs with each iteration. We utilize sparsely connected networks and have adjusted the sampling process, which has proven to effectively reduce the training time. In numerical experiments on fluid mechanics problems, such as the two-dimensional Burgers' equation with sharp solutions, pipe flow, flow around a circular cylinder, lid-driven cavity flow, and Kovasznay flow, our proposed adaptive sampling algorithm markedly enhances accuracy over conventional PINN methods, validating the algorithm's efficacy.

Effect of permeability on the initiation of Atherosclerosis modeled as an inflammatory process.

This work presents a mathematical model, based on partial differential equations, that analyzes the inflammatory stage of atherosclerosis. Four leading players are taken into consideration: Low Density Lipoproteins, oxidized Low Density Lipoproteins, immune cells and the inflammatory cytokines. In addition to this, the permeability of the endothelial layer is taken into account in the model. A stability analysis of the fixed points of the kinetic system is presented in details followed by the proof of existence of traveling wave solutions of the system of partial differential equations. The mathematical analysis leads to a biological interpretation. We distinguish three main cases of the disease state that correlate with the permeability of the endothelial layer. In fact, having a low permeability indicates the disease free state since no chronic inflammatory reaction occurs due to the non initiation of the inflammation. With intermediate permeability, a wave propagation corresponding to a chronic inflammatory reaction might occur whether the initial perturbation overcomes a threshold or not. With high permeability, even a small perturbation of the disease free state leads to a chronic inflammatory reaction represented by a wave propagation. We perform numerical simulations of the solutions to illustrate the biological results.

How to scale up from animal movement decisions to spatiotemporal patterns: An approach via step selection.

Uncovering the mechanisms behind animal space use patterns is of vital importance for predictive ecology, thus conservation and management of ecosystems. Movement is a core driver of those patterns so understanding how movement mechanisms give rise to space use patterns has become an increasingly active area of research. This study focuses on a particular strand of research in this area, based around step selection analysis (SSA). SSA is a popular way of inferring drivers of movement decisions, but, perhaps less well appreciated, it also parametrises a model of animal movement. Of key interest is that this model can be propagated forwards in time to predict the space use patterns over broader spatial and temporal scales than those that pertain to the proximate movement decisions of animals. Here, we provide a guide for understanding and using the various existing techniques for scaling up step selection models to predict broad-scale space use patterns. We give practical guidance on when to use which technique, as well as specific examples together with code in R and Python. By pulling together various disparate techniques into one place, and providing code and instructions in simple examples, we hope to highlight the importance of these techniques and make them accessible to a wider range of ecologists, ultimately helping expand the usefulness of SSA.

Modelling of surface reactions and diffusion mediated by bulk diffusion.

We develop a continuum framework applicable to solid-state hydrogen storage, cell biology and other scenarios where the diffusion of a single constituent within a bulk region is coupled via adsorption/desorption to reactions and diffusion on the boundary of the region. We formulate content balances for all relevant constituents and develop thermodynamically consistent constitutive equations. The latter encompass two classes of kinetics for adsorption/desorption and chemical reactions-fast and Marcelin-De Donder, and the second class includes mass action kinetics as a special case. We apply the framework to derive a system consisting of the standard diffusion equation in bulk and FitzHugh-Nagumo type surface reaction-diffusion system of equations on the boundary. We also study the linear stability of a homogeneous steady state in a spherical region and establish sufficient conditions for the occurrence of instabilities driven by surface diffusion. These findings are verified through numerical simulations which reveal that instabilities driven by diffusion lead to the emergence of steady-state spatial patterns from random initial conditions and that bulk diffusion can suppress spatial patterns, in which case temporal oscillations can ensue. We include an extension of our framework that accounts for mechanochemical coupling when the bulk region is occupied by a deformable solid. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

Forecasting Pathogen Dynamics with Bayesian Model-Averaging: Application to Xylella fastidiosa.

Forecasting invasive-pathogen dynamics is paramount to anticipate eradication and containment strategies. Such predictions can be obtained using a model grounded on partial differential equations (PDE; often exploited to model invasions) and fitted to surveillance data. This framework allows the construction of phenomenological but concise models relying on mechanistic hypotheses and real observations. However, it may lead to models with overly rigid behavior and possible data-model mismatches. Hence, to avoid drawing a forecast grounded on a single PDE-based model that would be prone to errors, we propose to apply Bayesian model averaging (BMA), which allows us to account for both parameter and model uncertainties. Thus, we propose a set of different competing PDE-based models for representing the pathogen dynamics, we use an adaptive multiple importance sampling algorithm (AMIS) to estimate parameters of each competing model from surveillance data in a mechanistic-statistical framework, we evaluate the posterior probabilities of models by comparing different approaches proposed in the literature, and we apply BMA to draw posterior distributions of parameters and a posterior forecast of the pathogen dynamics. This approach is applied to predict the extent of Xylella fastidiosa in South Corsica, France, a phytopathogenic bacterium detected in situ in Europe less than 10 years ago (Italy 2013, France 2015). Separating data into training and validation sets, we show that the BMA forecast outperforms competing forecast approaches.

VisualPDE: Rapid Interactive Simulations of Partial Differential Equations.

Computing has revolutionised the study of complex nonlinear systems, both by allowing us to solve previously intractable models and through the ability to visualise solutions in different ways. Using ubiquitous computing infrastructure, we provide a means to go one step further in using computers to understand complex models through instantaneous and interactive exploration. This ubiquitous infrastructure has enormous potential in education, outreach and research. Here, we present VisualPDE, an online, interactive solver for a broad class of 1D and 2D partial differential equation (PDE) systems. Abstract dynamical systems concepts such as symmetry-breaking instabilities, subcritical bifurcations and the role of initial data in multistable nonlinear models become much more intuitive when you can play with these models yourself, and immediately answer questions about how the system responds to changes in parameters, initial conditions, boundary conditions or even spatiotemporal forcing. Importantly, VisualPDE is freely available, open source and highly customisable. We give several examples in teaching, research and knowledge exchange, providing high-level discussions of how it may be employed in different settings. This includes designing web-based course materials structured around interactive simulations, or easily crafting specific simulations that can be shared with students or collaborators via a simple URL. We envisage VisualPDE becoming an invaluable resource for teaching and research in mathematical biology and beyond. We also hope that it inspires other efforts to make mathematics more interactive and accessible.

Stochastic differential equation modelling of cancer cell migration and tissue invasion.

Invasion of the surrounding tissue is a key aspect of cancer growth and spread involving a coordinated effort between cell migration and matrix degradation, and has been the subject of mathematical modelling for almost 30 years. In this current paper we address a long-standing question in the field of cancer cell migration modelling. Namely, identify the migratory pattern and spread of individual cancer cells, or small clusters of cancer cells, when the macroscopic evolution of the cancer cell colony is dictated by a specific partial differential equation (PDE). We show that the usual heuristic understanding of the diffusion and advection terms of the PDE being one-to-one responsible for the random and biased motion of the solitary cancer cells, respectively, is not precise. On the contrary, we show that the drift term of the correct stochastic differential equation scheme that dictates the individual cancer cell migration, should account also for the divergence of the diffusion of the PDE. We support our claims with a number of numerical experiments and computational simulations.

Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data.

We propose a machine learning framework for the data-driven discovery of macroscopic chemotactic Partial Differential Equations (PDEs)-and the closures that lead to them- from high-fidelity, individual-based stochastic simulations of Escherichia coli bacterial motility. The fine scale, chemomechanical, hybrid (continuum-Monte Carlo) simulation model embodies the underlying biophysics, and its parameters are informed from experimental observations of individual cells. Using a parsimonious set of collective observables, we learn effective, coarse-grained "Keller-Segel class" chemotactic PDEs using machine learning regressors: (a) (shallow) feedforward neural networks and (b) Gaussian Processes. The learned laws can be black-box (when no prior knowledge about the PDE law structure is assumed) or gray-box when parts of the equation (e.g. the pure diffusion part) is known and "hardwired" in the regression process. More importantly, we discuss data-driven corrections (both additive and functional), to analytically known, approximate closures.

Solid Boundary Output Feedback Control of the Stefan Problem: The Enthalpy Approach.

By taking enthalpy-an internal energy of a diffusion-type system-as the system state and expressing it in terms of the temperature profile and the phase-change interface position, the output feedback boundary control laws for a fundamentally nonlinear single-phase one-dimensional (1-D) PDE process model with moving boundaries, referred to as the Stefan problem, are developed. The control objective is tracking of the spatiotemporal temperature and temporal interface (solidification front) trajectory generated by the reference model. The external boundaries through which temperature sensing and heat flux actuation are performed are assumed to be solid. First, a full-state single-sided tracking feedback controller is presented. Then, an observer is proposed and proven to provide a stable full-state reconstruction. Finally, by combining a full-state controller with an observer, the output feedback trajectory tracking control laws are presented and the closed-loop convergence of the temperature and the interface errors proven for the single-sided and the two-sided Stefan problems. Simulation shows the exponential-like trajectory convergence attained by the implementable smooth bounded control signals.

Soft Sensor Modeling for 3D Transient Temperature Field of Large-Scale Aluminum Alloy Workpieces Based on Multi-Loss Consistency Optimization PINN.

Uniform temperature distribution during quenching thermal treatment is crucial for achieving exceptional mechanical and physical properties of alloy materials. Accurate and rapid prediction of the 3D transient temperature field model of large-scale aluminum alloy workpieces is key to realizing effective thermal treatment. This paper establishes a 3D transient temperature field model of large aluminum alloy workpieces and proposes a multi-loss consistency optimization-based physics-informed neural network (MCO-PINN) to realize soft sensing of the 3D temperature field model. The method is based on a MLP structure and adopts Gaussian activation functions. A surrogate model of the partial differential equation (PDE) is first constructed, and the residuals of the PDE, initial and boundary conditions, and observed data are encoded into the loss functions of the network. By establishing a Gaussian probability distribution model of each loss function and combining it with maximum likelihood estimation, the weight consistency optimization method of each loss function is then proposed to further improve the approximation ability of the model. To optimize the training speed of the network, an adaptive initial-value-eigenvector coding clustering (AIV-ECC) algorithm is finally proposed, which quickly determines the parameters of the Gaussian activation function, reduces the dependence on the initial value and improves the generalization performance of the network. Simulation and industrial experiments demonstrate that the proposed MCO-PINN can solve the 3D transient temperature field model with high precision and high time efficiency based on sparse measurements.

Optimization of hyperthermia process applied to cancer treatment using multi-objective optimization differential evolution.

Due to the number of cancer cases diagnosed each year and the fatality rate resulting from some more severe types, the improvement of less invasive and more efficient treatment techniques is of great importance. In this context, hyperthermia is a medical procedure in which the tumor region is heated by using an applicator for a certain period, aiming to destroy pathological cells. Computational models can be used to simulate the heating effect of tumors and adjacent cells. In general, the solution to an optimization problem considering factors such as heating temperature, applicator position, and the time in which the region will be subjected to heating can provide important information about the procedure. Traditionally, this type of problem has been addressed in a single objective context, focusing on minimizing the destruction of adjacent healthy tissue considering the area of the applicator constant. Our fundamental objective is to propose a multi-objective design problem considering the minimization of the area subject to the procedure and the time required for the process of hyperthermia in a breast cancer treatment. The problem is constrained by the degree of tissue destruction and by a partial differential equation that describes the phenomenon of heat transfer in both healthy and tumor tissues. The results obtained demonstrate that a point with a good compromise between the objectives can be chosen in such a way that a particular strategy can be defined for each patient.