Multiscale modelling of vascular tumour growth in 3D: the roles of domain size & boundary condition

We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour

Cahn-Hilliard-Brinkman models for tumour growth: Modelling, analysis and optimal control

Phase field models recently gained a lot of interest in the context of tumour growth models. In this work we study several diffuse interface models for tumour growth in a bounded domain with sufficiently smooth boundary. The basic model consists of a Cahn–Hilliard type equation for the concentration of tumour cells coupled to a convection-reaction-diffusion-type equation for an unknown species acting as a nutrient and a Brinkman-type equation for the velocity. The system is equipped with Neumann boundary conditions for the phase field and the chemical potential, a Robin-type boundary condition for the nutrient and a “no-friction” boundary condition for the velocity which allows us to consider solution dependent source terms. We derive the model from basic thermodynamic principles, conservation laws for mass and momentum and constitutive assumptions. Using the method of formal matched asymptotics, we relate our diffuse interface model with free boundary problems for tumour growth that have been studied earlier. For the basic model, we show the existence of weak solutions under suitable assumptions on the source terms and the potential by using a Galerkin method, energy estimates and compactness arguments. If the velocity satisfies a no-slip boundary condition and is divergence free, we can establish the existence of weak solutions for degenerate mobilities and singular potentials. From a modelling point of view, it seems to be more appropriate to describe the nutrient evolution by a so-called quasi-static equation of reaction-diffusion type. For this model we establish existence of both weak and strong solutions for regular potentials and a continuous dependence result yields the uniqueness of weak solutions and thus the model is well-posed. These results build the basis to study an optimal control problem where the control acts as a cytotoxic drug. Moreover, we rigorously prove the zero viscosity limit in two and three space dimensions which allows us to relate the Cahn–Hilliard–Brinkman model with Cahn–Hilliard–Darcy models which have been studied earlier. Finally, we also analyse the model with quasi-static nutrients and classical singular potentials like the logarithmic and double-obstacle potential which enforce the phase field to stay in the physical relevant range. Under suitable assumptions on the source terms, we can establish the existence of weak solutions for these kinds of potentials

Programmable models of growth and mutation of cancer-cell populations

In this paper we propose a systematic approach to construct mathematical models describing populations of cancer-cells at different stages of disease development. The methodology we propose is based on stochastic Concurrent Constraint Programming, a flexible stochastic modelling language. The methodology is tested on (and partially motivated by) the study of prostate cancer. In particular, we prove how our method is suitable to systematically reconstruct different mathematical models of prostate cancer growth - together with interactions with different kinds of hormone therapy - at different levels of refinement.Comment: In Proceedings CompMod 2011, arXiv:1109.104

Fibre tract segmentation for intraoperative diffusion MRI in neurosurgical patients using tract-specific orientation atlas and tumour deformation modelling

Purpose:: Intraoperative diffusion MRI could provide a means of visualising brain fibre tracts near a neurosurgical target after preoperative images have been invalidated by brain shift. We propose an atlas-based intraoperative tract segmentation method, as the standard preoperative method, streamline tractography, is unsuitable for intraoperative implementation. Methods:: A tract-specific voxel-wise fibre orientation atlas is constructed from healthy training data. After registration with a target image, a radial tumour deformation model is applied to the orientation atlas to account for displacement caused by lesions. The final tract map is obtained from the inner product of the atlas and target image fibre orientation data derived from intraoperative diffusion MRI. Results:: The simple tumour model takes only seconds to effectively deform the atlas into alignment with the target image. With minimal processing time and operator effort, maps of surgically relevant tracts can be achieved that are visually and qualitatively comparable with results obtained from streamline tractography. Conclusion:: Preliminary results demonstrate feasibility of intraoperative streamline-free tract segmentation in challenging neurosurgical cases. Demonstrated results in a small number of representative sample subjects are realistic despite the simplicity of the tumour deformation model employed. Following this proof of concept, future studies will focus on achieving robustness in a wide range of tumour types and clinical scenarios, as well as quantitative validation of segmentations

Model Pertumbuhan Logistik Dengan Waktu Tunda

The logistic growth model with time delay has developed from the classical logistic model, where as in the growth logistic model with time delay, the growth process delay from a population is calculated. This delay cause population decrease then increase so oscillation appears in population growth. So, the solution is not a monotonous function. The result of analysis indicate that the logistic growth model with time delay have two equilibrium points. Each equilibrium points is analyzed for their stability based on time delay variation on the population growth. The longer time delay in the population growth can cause the unstable growth, hence the population decrease and become extinct

Modelling The Cancer Growth Process By Stochastic Delay Diffferential Equations Under Verhults And Gompertz's Law

In this paper, the uncontrolled environmental factors are perturbed into the intrinsic growth rate factor of deterministic equations of the growth process. The growth process under two different laws which are Verhults and Gompertz’s law are considered, thus leading to stochastic delay differential equations (SDDEs) of logistic and Gompertzian, respectively. Gompertzian deterministic model has been proved to fit well the clinical data of cancerous growth, however the performance of stochastic model towards clinical data is yet to be confirmed. The prediction quality of logistic and Gompertzian SDDEs are evaluating by comparing the simulated results with the clinical data of cervical cancer growth. The parameter estimation of stochastic models is computed by using simulated maximum likelihood method. We adopt 4-stage stochastic Runge-Kutta to simulate the solution of stochastic models

Modified Fractional Logistic Equation

In the article [B.J.West, Exact solution to fractional logistic equation, Physica A: Statistical Mechanics and its Applications 429 (2015) 103-108], the author has obtained a function as the solution to fractional logistic equation (FLE). As demonstrated later in [I. Area, J. Losada, J. J. Nieto, A note on the fractional logistic equation, Physica A: Statistical Mechanics and its Applications 444 (2016) 182-187], this function (West function) is not the solution to FLE, but nevertheless as shown by West, it is in good agreement with the numerical solution to FLE. The West function indicates a compelling feature, in which the exponentials are substituted by Mittag-Leffler functions. In this paper, a modified fractional logistic equation (MFLE) is introduced, to which the West function is a solution. The proposed fractional integro-differential equation possesses a nonlinear additive term related to the solution of the logistic equation (LE). The method utilized in this article, may be applied to the analysis of solutions to nonlinear fractional differential equations of mathematical physics.Comment: 16 pages, 3 figure