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
10:30
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.
10:55
25 mins
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.
11:20
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 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.
11:45
25 mins
Bridging the single cell with the cell population: opening up for data-driven methodologies
Stefan Engblom
Abstract: In this talk I will discuss a computational framework designed specifically for detailed modeling of populations of living cells [1]. 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. References: [1] 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). [2] S. Engblom, Stochastic simulation of pattern formation in growing tissue: a multilevel approach, Bull. Math. Biol. (2018). [3] S. Engblom, Strong convergence for split-step methods in stochastic jump kinetics, SIAM J. Numer. Anal. 53(6):2655--2676 (2015). [4] 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). [5] 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). [6] 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).