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Efficient Methods to Calculate Partial Sphere Surface Areas for A Mathematical and Computational Applications Article Efficient Methods to Calculate Partial Sphere Surface Areas for a Higher Resolution Finite Volume Method for Diffusion-Reaction Systems in Biological Modeling Abigail Bowers, Jared Bunn and Myles Kim * Department of Applied Mathematics, Florida Polytechnic University, Lakeland, FL 33805, USA; abowers@floridapoly.edu (A.B.); jbunn@floridapoly.edu (J.B.) * Correspondence: mkim@floridapoly.edu Received: 14 November 2019; Accepted: 20 December 2019; Published: 23 December 2019 Abstract: Computational models for multicellular biological systems, in both in vitro and in vivo environments, require solving systems of differential equations to incorporate molecular transport and their reactions such as release, uptake, or decay. Examples can be found from drugs, growth nutrients, and signaling factors. The systems of differential equations frequently fall into the category of the diffusion-reaction system due to the nature of the spatial and temporal change. Due to the complexity of equations and complexity of the modeled systems, an analytical solution for the systems of the differential equations is not possible. Therefore, numerical calculation schemes are required and have been used for multicellular biological systems such as bacterial population dynamics or cancer cell dynamics. Finite volume methods in conjunction with agent-based models have been popular choices to simulate such reaction-diffusion systems. In such implementations, the reaction occurs within each finite volume and finite volumes interact with one another following the law of diffusion. The characteristic of the reaction can be determined by the agents in the finite volume. In the case of cancer cell growth dynamics, it is observed that cell behavior can be different by a matter of a few cell size distances because of the chemical gradient. Therefore, in the modeling of such systems, the spatial resolution must be comparable to the cell size. Such spatial resolution poses an extra challenge in the development and execution of the computational model due to the agents sitting over multiple finite volumes. In this article, a few computational methods for cell surface-based reaction for the finite volume method will be introduced and tested for their performance in terms of accuracy and computation speed. Keywords: partial sphere surface areas; finite volume method; diffusion reaction equation; mathematical model for biology 1. Introduction Agent-based models (also called single-cell based models or individual-based models) have been widely popular in various fields as tools to build computational models. They have been adopted in biological system modeling [1–3], fluid dynamics simulation [4,5], ecosystem modeling [6,7], human organizational simulation [8,9], business simulation [10], etc. In agent-based modeling, a system is modeled as a collection of autonomous self-decision-making entities called agents. Each agent individually assesses its situation and makes decisions on the basis of a set of rules. These rules can be designed to make internal decisions altering the fate of an agent or can be written to define interactions among spatially neighboring agents. The rule set can also be identical to each agent or can be designed differently for different groups of agents. Agent-based models were proven to be an effective modeling Math. Comput. Appl. 2020, 25, 2; doi:10.3390/mca25010002 www.mdpi.com/journal/mca Math. Comput. Appl. 2020, 25, 2 2 of 13 approach for complex heterogeneous systems. Though all rules are usually written for small scale, agent level or local interaction level, agent-based models are capable of showing population level emerging characteristics that are often unpredictable from the local rule sets. Due to these reasons, cancerMath. biology Comput. Appl. is one 2020 of, 25 those, x FOR PEER areas REVIEW where agent-based models have been successful [3,52, 11of 13]. In a biological system, cells interact with one another in various ways, through physical interaction and/or throughcan be chemical designed signaling. differently Cellsfor different uptake groups molecules of agents. from Agent the surrounding-based models medium,were proven and to moleculesbe an may comeeffective from modeling nearby cellsapproach because for theycomplex synthesize heterogeneous and secrete systems. molecules. Though Due all torules the are diff usiveusually nature of thewritten molecules for small in media, scale, theagent mathematical level or local model interaction for the level, biological agent-base cellsd models that consume are capable and of secrete moleculesshowing in mediapopulation would level be emerging a diffusion-reaction characteristics equation that are andoften finite unpredictable volume methodsfrom the local area rule popular sets. Due to these reasons, cancer biology is one of those areas where agent-based models have been choice for the numerical implementation [11,12]. successful [3,5,11. In a biological system, cells interact with one another in various ways, through It has been experimentally observed that cancer cells’ fates can be different by a matter of one or physical interaction and/or through chemical signaling. Cells uptake molecules from the surrounding two cellmedium, lengths and due molecules to the spatial may come nutrient from nearby concentration cells because change they [13 synthesize]. Thus, aand computational secrete molecules. model for cancerDue cells to the with diffusive physiology nature includingof the molecules consumption in media, andthe mathematical secretion of moleculesmodel for the will biological require cells a spatial resolutionthat consume of single and cell secrete length. molecules Uptake ofin moleculesmedia would by cellsbe a maydiffusion involve-reaction receptors equation on theand cell finite surface and endocytosisvolume methods process. are a Ifpopular a 3D finite choice volume for the numeric implementational implementation is designed [11,12]. with a cell surface reaction, one cell lengthIt has been spatial experimentally resolution, and observed cellsrepresented that cancer cells by’ spheres fates can (agents), be different then by a a snapshot matter of aroundone or one cell wouldtwo cell be lengths like Figure due to1 .the The spatial cell innutrient Figure concentration1 overlaps with change eight [13 finite]. Thus, volumes. a computational The contribution model of thisfor cell cancer to the cells whole with diphysiolffusion-reactionogy including system consumption needs toand be secretion divided of into molecules eight parts will require proportional a spatial resolution of single cell length. Uptake of molecules by cells may involve receptors on the cell to the cell surface area in each finite volume that the cell overlaps; S , S , , S in Figure1. In a surface and endocytosis process. If a 3D finite volume implementation is designed1 2 ··· with8 a cell surface diffusion-reaction equation like Equation (1), the reaction coefficient, λ, would depend, as a function of reaction, one cell length spatial resolution, and cells represented by spheres (agents), then a snapshot spatialaround variables, one cell on would the values be like of FigureSi’s from 1. The all cell cells in inFigure the system. 1 overlaps Thus, with it eight is important finite volumes. to calculate The the precisecontribution values of ofSi ’sthis to cell capture to the the whole correct diffusion rate- ofreaction reaction. system needs to be divided into eight parts proportional to the cell surface area in each finite volume that the cell overlaps; 푆1, 푆2, ⋯, 푆8 in @c Figure 1. In a diffusion-reaction equation like =EquationD 2c + (1),λ cthe reaction coefficient, 휆, would depend, (1) @t r as a function of spatial variables, on the values of 푆푖’s from all cells in the system. Thus, it is important to calculate the precise values of 푆 ’s to capture the correctr rate of reaction. S =푖 x q dA (2) R 휕푐2 ( )2 ( )2 r = x퐷∇2a푐 + 휆푐 y b (1) 휕푡− − − − Figure 1. Schematic diagram of an agent-based model for 3D cell proliferation (left) and a spherical Figure 1. Schematic diagram of an agent-based model for 3D cell proliferation (left) and a spherical right agentagent representing representing a cell a cell overlapping overlapping with with surroundingsurrounding finite finite volumes volumes (right ( ). There). There are more are more than than 46004600 agents agents in the in the snapshot snapshot of of a simulationa simulation ((leftleft).). A A small small zoomed zoomed-in-in view view of a of cell a cellwith with surrounding surrounding finitefinite volumes volumes (rectangular (rectangular boxes) boxes) is is shown shown inin the middle. AA further further zoomed zoomed-in-in view view of the of cell the is cell is displayeddisplayed on theon the right. right. When When thethe agent’s diameter diameter is smaller is smaller than thanthe edge the of edge the rectangular of the rectangular box, the box, the agentagent cancan overlapoverlap with up up to to eight eight finite finite volumes. volumes. 푆1,S 푆12,,S …2,, :::and, and 푆8 areS8 aresurface surface areas areas that overlap that overlap eighteight finite finite volumes. volumes. Math. Comput. Appl. 2020, 25, 2 3 of 13 Analytic formulas are not readily available for the calculation of these areas even when all geometric specifications are given. An attempt to calculate these surface areas, S ’s, can be made by q i using the function f (x, y) = r2 (x a)2 (y b)2, which results in a double integral as in Equation − − − − (2). This integral, however, is possible analytically only when the domain R is extremely simple in polar coordinates. Therefore, numerical calculation methods need to be deployed. Diffusions in biological contexts are usually fast. The stability of numerical schemes for a fast diffusion requires small time stepping. Furthermore, typical cancer cell simulations consider thousands of cells and use durations of days and weeks of their lives [3,5]. Considering all these, we need to find an efficient numerical method to calculate the cell surface area in finite volumes for thousands of cells repeatedly for the duration of simulation not to burden the overall numerical simulation.
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