Multiscale modelling of intestinal crypt organization and carcinogenesis

Colorectal cancers are the third most common type of cancer. They originate from intestinal crypts, glands that descend from the intestinal lumen into the underlying connective tissue. Normal crypts are thought to exist in a dynamic equilibrium where the rate of cell production at the base of a crypt is matched by that of loss at the top. Understanding how genetic alterations accumulate and proceed to disrupt this dynamic equilibrium is fundamental to understanding the origins of colorectal cancer. Colorectal cancer emerges from the interaction of biological processes that span several spatial scales, from mutations that cause inappropriate intracellular responses to changes at the cell/tissue level, such as uncontrolled proliferation and altered motility and adhesion. Multiscale mathematical modelling can provide insight into the spatiotemporal organisation of such a complex, highly regulated and dynamic system. Moreover, the aforementioned challenges are inherent to the multiscale modelling of biological tissue more generally. In this review we describe the mathematical approaches that have been applied to investigate multiscale aspects of crypt behavior, highlighting a number of model predictions that have since been validated experimentally. We also discuss some of the key mathematical and computational challenges associated with the multiscale modelling approach. We conclude by discussing recent efforts to derive coarse-grained descriptions of such models, which may offer one way of reducing the computational cost of simulation by leveraging well-established tools of mathematical analysis to address key problems in multiscale modelling

A model of spatially evolving herpesvirus epidemics causing mass mortality in Australian pilchard Sardinops sagax

In 1995 mass mortality of pilchards Sardinops sagax occurred along >5000 km of Australian coast; similar events occurred in 1998/99. This mortality was closely associated with a herpesvirus. The pilchard is an important food source for larger animals and supports commercial fisheries. Both epidemics originated in South Australian waters and spread as waves with velocities of 10 to 40 km d-1. Velocity was constant for a single wave, but varied between the epidemics and between the east- and west-bound waves in each epidemic. The pattern of mortality evolved from recurrent episodes to a single peak with distance from the origin. A 1-dimensional model of these epidemics has been developed. The host population is divided into susceptible, infected and latent, infected and infectious, and removed (recovered and dead) phases; the latent and infectious periods are of fixed duration. This model produces the mortality patterns observed locally and during the spread and evolution of the epidemic. It is consistent with evidence from pathology. The wave velocity is sensitive to diffusion coefficients, viral transmission rates and latent period. These parameters are constrained using the local and large-scale patterns of epidemic spread. The relative roles of these parameters in explaining differences between epidemics and between east- and west-bound waves within epidemics are discussed. The model predicts very high levels of infection, indicating that many surviving pilchards recovered following infection. Control appears impracticable once epidemics are initiated, but impact can be minimised by protecting juvenile stocks

Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities

A class of variable coefficient (1+1)-dimensional nonlinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u^nu_x)_x+h(x)u^m$ is investigated. Different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements. For the class under consideration the complete group classification is performed with respect to convenient equivalence groups (generalized extended and conditional ones) and with respect to the set of all point transformations. Usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results. The corresponding set of admissible transformations is described exhaustively. Then, using the most direct method, we classify local conservation laws. Some exact solutions are constructed by the classical Lie method.Comment: 23 pages, minor misprints are correcte

The Principal Element of a Frobenius Lie Algebra

We introduce the notion of the \textit{principal element} of a Frobenius Lie algebra \f. The principal element corresponds to a choice of F\in \f^* such that $F[-,-]$ non-degenerate. In many natural instances, the principal element is shown to be semisimple, and when associated to \sl_n, its eigenvalues are integers and are independent of $F$. For certain ``small'' functionals $F$, a simple construction is given which readily yields the principal element. When applied to the first maximal parabolic subalgebra of \sl_n, the principal element coincides with semisimple element of the principal three-dimensional subalgebra. We also show that Frobenius algebras are stable under deformation.Comment: 10 page

Information Metric on Instanton Moduli Spaces in Nonlinear Sigma Models

We study the information metric on instanton moduli spaces in two-dimensional nonlinear sigma models. In the CP^1 model, the information metric on the moduli space of one instanton with the topological charge Q=k which is any positive integer is a three-dimensional hyperbolic metric, which corresponds to Euclidean anti--de Sitter space-time metric in three dimensions, and the overall scale factor of the information metric is (4k^2)/3; this means that the sectional curvature is -3/(4k^2). We also calculate the information metric in the CP^2 model.Comment: 9 pages, LaTeX; added references for section 1; typos adde

Projective and Coarse Projective Integration for Problems with Continuous Symmetries

Temporal integration of equations possessing continuous symmetries (e.g. systems with translational invariance associated with traveling solutions and scale invariance associated with self-similar solutions) in a ``co-evolving'' frame (i.e. a frame which is co-traveling, co-collapsing or co-exploding with the evolving solution) leads to improved accuracy because of the smaller time derivative in the new spatial frame. The slower time behavior permits the use of {\it projective} and {\it coarse projective} integration with longer projective steps in the computation of the time evolution of partial differential equations and multiscale systems, respectively. These methods are also demonstrated to be effective for systems which only approximately or asymptotically possess continuous symmetries. The ideas of projective integration in a co-evolving frame are illustrated on the one-dimensional, translationally invariant Nagumo partial differential equation (PDE). A corresponding kinetic Monte Carlo model, motivated from the Nagumo kinetics, is used to illustrate the coarse-grained method. A simple, one-dimensional diffusion problem is used to illustrate the scale invariant case. The efficiency of projective integration in the co-evolving frame for both the macroscopic diffusion PDE and for a random-walker particle based model is again demonstrated

Immunomodulatory Therapy for MIS-C.

Studies comparing initial therapy for multisystem inflammatory syndrome in children (MIS-C) provided conflicting results. To compare outcomes in MIS-C patients treated with intravenous immunoglobulin (IVIG), glucocorticoids, or the combination thereof. Medline, Embase, CENTRAL and WOS, from January 2020 to February 2022. Randomized or observational comparative studies including MIS-C patients <21 years. Two reviewers independently selected studies and obtained individual participant data. The main outcome was cardiovascular dysfunction (CD), defined as left ventricular ejection fraction < 55% or vasopressor requirement ≥ day 2 of initial therapy, analyzed with a propensity score-matched analysis. Of 2635 studies identified, 3 nonrandomized cohorts were included. The meta-analysis included 958 children. IVIG plus glucocorticoids group as compared with IVIG alone had improved CD (odds ratio [OR] 0.62 [0.42-0.91]). Glucocorticoids alone group as compared with IVIG alone did not have improved CD (OR 0.57 [0.31-1.05]). Glucocorticoids alone group as compared with IVIG plus glucocorticoids did not have improved CD (OR 0.67 [0.24-1.86]). Secondary analyses found better outcomes associated with IVIG plus glucocorticoids compared with glucocorticoids alone (fever ≥ day 2, need for secondary therapies) and better outcomes associated with glucocorticoids alone compared with IVIG alone (left ventricular ejection fraction < 55% ≥ day 2). Nonrandomized nature of included studies. In a meta-analysis of MIS-C patients, IVIG plus glucocorticoids was associated with improved CD compared with IVIG alone. Glucocorticoids alone was not associated with improved CD compared with IVIG alone or IVIG plus glucocorticoids

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u_x)_x+h(x)u^m$ ($m\ne0,1$) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with $m=2$ is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case $m\ne2$. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).Comment: 40 pages, this is version published in Acta Applicanda Mathematica