Mathematical Biology (Part 1)
Agent-based modelling and likelihood estimation of a finite-element model for wound contraction
Qiyao Peng, Fred Vermolen
Abstract: Deep tissue injury, such as severe burns, often go with the contraction of skin. This contraction can become problematic if it causes loss of mobility of the joint of the patient. Our final objective is to predict the likelihood that such a contracture occurs and to analyze the impact of several treatments on the likelihood of the development of a serious contraction. To do so, we develop a mathematical framework based on tracking the migration and proliferation of individual cells, in particular macrophages, fibroblasts and myofibroblasts. Furthermore, we take into account the forces that are exerted by the (myo)fibroblasts on their environment, which is the reason behind the contraction of the wound and surrounding tissue. To this end, we use a stochastic finite element framework that eventually will be used to estimate the probability of an occurring contraction. In the presentation, we will outline the model in terms of assumptions and mathematical formulation, as well as its clinical implications.
Mathematical modeling of cancer progression and treatment using different simulation frameworks
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.
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 decade, numerical models have become an important tool in medical science for understanding human bodies and diagnosing diseases. In this paper, we develop a multi-scale model for studying blood flow, and blood regulation in the existing vascular structure of organ. We coupled 1D vascular graph model to represent blood flow through vessel network and two-compartments porous medium based on Darcy's law for the capillary bed. The vascular model is based on Poiseuille's law with pressure correction by incorporating elasticity and pressure drop estimation on junctions. Transfer fluid between two compartments in capillary bed is defined as blood perfusion. Numerical experiments show the blood circulation in the system closely related to the structure and parameter of both vascular vessel and capillary bed, that gives realistic blood circulation in the vascular system. 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.
Bridging the single cell with the cell population: opening up for data-driven methodologies
Abstract: In this talk I will discuss a computational framework designed
specifically for detailed modeling of populations of living cells
. Selected motivating examples include wound healing processes,
gradient sensing migrating cells, development of colon crypts, and
quorum sensing mechanisms. I will next consider some of the
intricacies of capturing cell-to-cell communication and being able to
do so efficiently within a non-static population of cells [1,2].
I will next review some results towards multiscale convergence in the
above setting of spatial stochastic models [3,4,5,6]. The
computational challenge here is to bridge the vast scale separation
inherent with these types of applications, and to provide for
computational efficiency enough that the model can either be
effectively parameterized given data, or homogenized using multiscale
techniques. Some final results evolve around Bayesian data-driven
modeling in the above framework.
 S. Engblom, D. B. Wilson, and R. E. Baker, Scalable
population-level modeling of biological cells incorporating
mechanics and kinetics in continuous time, Roy. Soc. Open
Sci. 5(8) (2018).
 S. Engblom, Stochastic simulation of pattern formation in growing
tissue: a multilevel approach, Bull. Math. Biol. (2018).
 S. Engblom, Strong convergence for split-step methods in
stochastic jump kinetics, SIAM J. Numer. Anal. 53(6):2655--2676
 S. Engblom, Stability and Strong Convergence for Spatial
Stochastic Kinetics, Chap. 3.3 in D. Holcman (editor), Stochastic
Processes, Multiscale Modeling, and Numerical Methods for
Computational Cellular Biology, Springer (2017).
 A. Chevallier, and S. Engblom, Pathwise error bounds in Multiscale
variable splitting methods for spatial stochastic kinetics, SIAM
J. Numer. Anal. 56(1):469--498 (2018).
 P. Bauer, S. Engblom, S. Mikulovic, and A. Senek, Multiscale
modeling via split-step methods in neural firing,
Math. Comput. Model. Dyn. Syst. 24(4):409--425 (2018).