European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:30   Mathematical Biology
Chair: Fred Vermolen
25 mins
Point Forces and Their Alternatives in Cell-Based Models for Skin Contraction
Qiyao Peng, Fred Vermolen
Abstract: During wound healing, contractions occur due to the pulling forces released by (myo)fibroblasts. We consider a cell-based approach in which the balance of momentum is used to predict the cellular impact on the mechanics of the tissue. To this extent, the elasticity equation and Dirac Delta distributions are combined. However, Dirac Delta distributions cause a singular solution. Hence, alternative approaches are developed and a Gaussian distribution is often used as a smoothed approach. Based on the application that the pulling force is pointing inward the cell, the smoothed particle approach is probed as well. In one dimension, it turns out that the aforementioned three approaches are consistent. For two dimensions, the ratio of the force magnitude is only worked out in a special case, but for the general case, the numerical results show consistency between the direct approach and the smoothed particle approach.
25 mins
Several agent-based and cellular automata mathematical frameworks for modeling pancreatic cancer
Jiao Chen, Daphne Weihs, Fred Vermolen
Abstract: Cancer is a fatal disease with a rising global mortality rate. Typically, cancer initiates with multiple gene mutations and subsequently progresses in a rapid way. Since there are almost no obvious symptoms, cancer hardly can be detected at early stages. However, if cancer develops to advanced stages, it is difficult to be cured. Mathematical modeling of cancer, known as an efficient and cheap approach, has shown prospects for the further understanding of cancer pathology and has proven to be an alternative to some animal-based experiments. We have developed mathematical modeling of cancer progression and treatment by using a lattice-based method (cellular automata model) and cell-based models (an off-lattice method), respectively. Our cellular automata model is able to phenomenologically show pancreatic cancer initiation and its recession under oncolytic virotherapy, where each single lattice site is occupied by a cluster of cells. In addition, a couple of biological processes, i.e. cell division, cell death, cell mutation, etc., are modeled by stochastic processes. Moreover, the time-dependent reaction-diffusion equation is used to simulate the spread of oncolytic viruses. In contrast, we also develop a cell-based model to mimic pancreatic cancer at a smaller cell number level, where cell migration is considered. The mechanotaxis or/and chemotaxis migrations of cells are modeled by solving a large system of ordinary stochastic differential equations. Furthermore, the impact of the isotropic desmoplastic stroma of pancreatic cancer on the migration of T-lymphocytes has been incorporated. Targeting on the degradation of the desmoplastic stroma, a drug-oriented therapy is proposed and modeled, where the protocols of administration are compared and guidelines towards successful treatment are given based on the computational results. Regarding the simulation at a single cell level, we set up a cell-based model to describe an extensive deformation of cell morphology during cancer metastasis. The migrating cell is attracted by a generic emitting source, which is dealt with by utilizing Green's Fundamental solutions. Moreover, a microvascular flow is taken into account by using Poisseuille flow. To investigate the uncertainties in the input variables and their potential influences, Monte Carlo simulations are performed. The likelihood of cancer cell metastasis is estimated.
25 mins
A multi-scale flow model for studying blood circulation in vascular system
Ulin Nuha Abdul Qohar, Antonella Zanna Munthe-Kaas, Jan Martin Nordbotten, Erik Andreas Hanson
Abstract: In the last few decades, numerical models have become an essential tool for medical science. In this paper, we develop a multi-scale model for studying blood flow in the existing vascular structure of an organ. The model simulates tracer concentration flow that replicates Dynamic Contrast-Enhanced Magnetic Resonance Imaging (DCE-MRI) data. We coupled a 1D vascular graph model to represent blood flow through vascular vessel network and flow in the porous medium based on Darcy's law for the capillary bed. Numerical experiments show the blood circulation in the system closely related to the structure and parameter of the vascular system, that gives realistic tracer concentration flow. The proposed model is complex enough so that it captures reliably the physiology of the system, and at the same time, it is simple enough so that simulations can be performed on a regular PC.\\~\\ Keywords: blood circulation, Darcy flow, multi-scale flow, vascular graph model.
25 mins
Towards confident Bayesian parameter estimation in stochastic chemical kinetics
Stefan Engblom, Robin Eriksson, Pedro Vilanova
Abstract: We investigate the feasibility of Bayesian parameter inference for chemical reaction networks described in the low copy number regime. Here stochastic models are often favorable implying that the Bayesian approach becomes natural. Our discussion circles around a concrete oscillating system describing a circadian rhythm, and we ask if its parameters can be inferred from observational data. The main challenge is the lack of analytic likelihood and we circumvent this through the use of a synthetic likelihood based on summarizing statistics. We are particularly interested in the robustness and confidence of the inference procedure and therefore estimates a priori as well as a posteriori the information content available in the data. Our all-synthetic experiments are successful but also point out several challenges when it comes to real data sets.