Advanced Chemical Transport Modeling in Dynamic Multicellular Contexts Using CompuCell3D T.J. Sego 1* , Saumya Mehta 2 , James A. Glazier 2 1 Department of Medicine, University of Florida, Gainesville, FL, USA 2 Department of Intelligent Systems Engineering and Biocomplexity Institute, Bloomington, IN, USA * timothy.sego@ufl.edu Abstract Mass transport of soluble signals is a ubiquitous, funda mental physical process in the development and homeostatic functions of living organisms. Cells of complex, multicellular organisms communicate over great distances through release of various soluble biomolecules and rely on the delivery of nutrients into, and removal of waste out of, their local environment by organismal functions at subcellular to organ scales. Tissues like the intestinal epithelium and vascular endothelium organize the molecular complexes of their constituent cell membranes to direct adv ection - and diffusion - based transport of soluble molecules through active transport like ion pumping and transcytosis , resulting in tightly regulated biochemical microenvironments and function al organ units Mathematical modeling has a long history of usin g partial differential equations to describe the relationship between dynamic tissues and their chemical environment, and cell - based computational spatial models describe such systems on the basis of motile, autonomous cells in spatially heterogeneous chem ical distributions. The Glazier - Graner - Hogeweg (GGH) method is the basic mathematical framework for many cell - based spatial models in a large range of biological and biomedical applications in development and disease, and permits developing agent - based mod els of multicellular systems with deformable cell shape, directed cell motility, and a broad range of local cell - chemical interactions ( e.g. , uptake, secretion, induced mitosis). However, the GGH method suffers from a lack of explicit representation of cel l surfaces, neither does it account for the effects of cell shape changes on the distribution of soluble signals. These limitations prevent developing GGH models in a number of important applications in basic biology ( e.g. , emergent tissue transport proper ties) and biomedicine ( e.g. , drug delivery). We addressed such limitations of the GGH method in this work by formalizing a numerical reaction - diffusion solver for modeling transport in GGH models on the basis of surfaces, and by developing an efficient alg orithm for approximating the effects of cellular dynamics on coupled soluble chemical distributions. We constructed our solver formalism using the Finite Volume Method, which describes mass transport on the basis of the surfaces of a discretized space, and implemented robust support for active transport and dynamic transport conditions based on changes in the coupled cellular distribution. Our algorithm for introducing the effects of cell shape changes on coupled chemical distributions, which is inspired by previous GGH methodological work, enables motile GGH model cells to perform basic barrier functions and thus, along with our FVM formalism, allows model 2 development targeting tissues that have historically eluded the GGH method. We implemented our work in CompuCell3D, a publicly and freely available modeling and simulation environment built on the GGH method, and demonstrate new capabilities by providing use - case demonstrations of biological systems that GGH models can now describe as a result of our work Introduction T ra nsport of soluble chemicals occurs at every level of the function and spatial organization of complex organism s Through various modes of transcellular transport, endothelial cells of the vascular and lymphatic systems transport nutrients, fluid and biomolecules between tissues separated over long distances (Yazdani et al. 2019) . In the arterial wall, insufficient efflux of lipoproteins through the endothelium leads to the onset and pathogenesis of atherosclerosis (Jang et al. 2020) , while exchange of nutrients and waste between maternal and fetal vascular systems occurs through the trophoblast layer of the human placenta (Cherubini et al. 2021) Epithelial cells also provide a number of emergent tissue and organ functions through various transport processes. Epithelial cells provide selective, directional transcellular transport through receptor - mediated transcytosis, such as when transporting immunoglobulins in the humoral response, or when controlling the molecules that enter the cerebral spinal fluid at the choroid plexus (Fung et al. 2018) or the retina at the blood - retinal barrier (Yemanyi et al. 2021) Among others, intestinal epithelia provide an absorptive barrier to the vasculature through both passive ( e.g. , transmembrane diffusion, paracellular permeability) and active ( e.g. , receptor - and ion - pump - mediated uptake ) transport of molecules from the intestinal lumen (Gleeson et al. 2021) The biomolecular d etails of transport mechanisms are of particular interest to a number of research domains, such as targeting delivery through the small intestine (Delon et al. 2022; Padhye et al. 2020) and blood - brain - barrier (Gupta et al. 2019) in drug development. T rafficking of cytoplasmic chemicals across cells and between organelles occurs through a number of physical processes, including diffusion and advection through cytoplasmic flow (S Mogre et al. 2020) and transport within vesicles coordi nated with the cytoskeleton (Burute and Kapitein 2019) Cells can uptake substances from the extracellular environment in vesicles formed f rom the cell membrane, or eject waste and communicate through release of extracellular vesicles, which carry various chemical cargo such as cellular debris, proteins and genetic material (Simeone et al. 2020; Mir and Goettsch 2020) Extracellular vesicles can facilitate mitochondrial transfer between cells (Qin et al. 2021) , and mitochondria can also produce vesicles, which have been recently shown to play critical roles in maintaining mitochondrial quality (Heyn et al. 2023) Understanding the biophysics of v arious microenvironments , including a quantitative description of the distribution of biomolecules, can improve our understanding of the basic properties and functions of homeostatic tissue and mechanisms responsible for the onset of disease, which can inf orm the improvement of drug efficacy (Gładysz et al. 2022) Computational models and methods have been developed to further such understa nding by apply ing domain knowledge from physics and chemistry in computer simulations of mass 3 transport in cellular and multicellular systems , such as those implemented in software like Virtual Cell (Loew and Schaff 2001) , Chaste (Cooper et al. 2020) , Morpheus (Starruß et al. 2014) , and CompuCell3D (Swat et al. 2012) One such method, called the Glazier - Graner - Hogeweg (GGH) method, couples discretized partial differential equations of reaction - diffusion systems with the lattice - based Cell ular Potts Model (CPM) of cellular dynamics (Graner and Glazier 1992) Models using the GGH method have been developed for a number of appli cations in development and disease that include a strong emphasis on transport processes in multicellular systems, including (but not limited to) angiogenesis (Merks et al. 20 08) , cancer (Szabó and Merks 2013; Shirinifard et al. 2009) , somitogenesis (Hester et al. 2011) , viral infection (Aponte - Serrano et al. 2021; Sego et al. 2022) and drug delivery (Sego et al. 2020; Ferrari Gianlupi et al. 2022) However, a significant limitation of the GGH method has historically prevented its application to problems with detailed transport processes. A CPM cell, wh ich is volume excluding and has an explicit, deformable shape, has no explicit representation of surfaces, which are instead implied by adjacent locations in the spatial domain that are occupied by either two different cells, or by a cell and its environme nt. Hence, typically GGH implementations employ central difference schemes for discretization of the partial differential equations (PDEs) used to describe the reaction - diffusion systems of a model, where field values represent the concentration of a chemi cal field at the centroid of a location. This total lack of surface representation , which was observed and partially addressed in (Marée et al. 2012) when studying intracellular organization leading to cell polarity, prevents incorporating two basic processes in GGH models that are fundamentally important to applications like those previously mentioned First, GGH methodological implementation s that employ central difference discretization of PDEs do not present any well - defined representation of surfaces such that active transport on surfaces cannot be considered Second , deformations in the shape of a CPM cell, which occur through fluctuations in the boundary locations of the cell, are not accounted for in the distributions of coupled chemical fields such that deformations in cell shape lead to cells that “leak” mass In this work we addressed these historic limitations by formaliz ing a numerical PDE solver for modeling reaction - diffusion processes in GGH models on the basis of surfaces, and by developing an efficient algorithm for approximating the effects of CPM cell shape changes on coupled reaction - diffusion fields. Our solver f ormalism was constructed from the Finite Volume method (FVM), a well - known numerical method for solving PDEs on the basis of interfaces between the voxels of a discretized spatial domain , and was tailored to support surface transport modeling, heterogeneou s boundary conditions and dynamic changes in numerical solutions that reflect changes in the cellular distribution We developed our algorithm to account for cell shape changes in diffusion field solutions based on the work and assumptions in (Marée et al. 2012) and derived a recursive expression for implementing the algorithm with a comparable (but greater) computational cost to the CPM. We th en implemented our FVM formalism as a new PDE solver in CompuCell3D (CC3D) and our algorithm as a CC3D PDE solver extension that is supported by both our new solver and other PDE solvers currently available in CC3D. 4 Models and Methods Finite Volume Formal ism The FVM formalism considers diffusion of an arbitrary set of concentrations through a heterogeneous medium and reactions between them. For a concentration 𝑐 𝑗 of 𝑛 + 1 concentrations with diffusivity field 𝐷 𝑗 and reaction 𝑠 𝑗 , the rate of 𝑐 𝑗 follows the partial differential equation, 𝜕 𝑡 𝑐 𝑗 = 𝜕 𝑖 ( 𝐷 𝑗 𝜕 𝑖 𝑐 𝑗 ) + 𝑠 𝑗 ( 𝑐 0 , 𝑐 1 , ... , 𝑐 𝑛 ) ( 1 ) Generally, the FVM derives discrete transport laws for individual voxels of a discretized space by performing integrals over time and the volume of subdomain of the discretized space . In the case of our formalism, the integrals take form, ∫ ∫∫∫ 𝜕 𝑡 𝑐 𝑗 𝑑𝑉 𝑑 𝑡 = ∫ ∫∫∫ 𝜕 𝑖 ( 𝐷 𝑗 𝜕 𝑖 𝑐 𝑗 ) 𝑑𝑉 𝑑𝑡 + ∫ ∫∫∫ 𝑠 𝑗 ( 𝑐 0 , 𝑐 1 , ... , 𝑐 𝑛 ) 𝑑𝑉 𝑑𝑡 ( 2 ) In this work we consider ( 3 ) applied to a discretized, cuboidal space, where the discretization produces cuboidal subdomains, here referred to as voxels. As such, we refer to a voxel as being centered at a position 𝑥 𝑖 , which is to say that the centroid of the voxel is at 𝑥 𝑖 . Furthermore, we constrain our problems where the discretization of the space is performed such that each voxel surface is either an interface with one neighboring voxel or a surface of the boundar y of the domain. Mathematically, we refer to the set of neighbor voxels of the voxel centered at 𝑥 𝑖 as 𝒩 ( 𝑥 𝑖 ) , and to the centroid of the 𝑘 th neighbor voxel as 𝑥 𝑖 𝑘 ∈ 𝒩 ( 𝑥 𝑖 ) Discretization of the integrals produces a discrete update rule for each concentration in each voxel of a discretized space. For a voxel with centroid at position 𝑥 𝑖 at time 𝑡 , the 𝑗 th concentration 𝑐 𝑗 at time 𝑡 + ∆ 𝑡 is updated according to the updat e rule, 𝑐 𝑗 ( 𝑥 𝑖 , 𝑡 + ∆ 𝑡 ) = 𝑐 𝑗 ( 𝑥 𝑖 , 𝑡 ) + ∆ 𝑡 ( ∑ 𝐹 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) 0 ≤ 𝑘 < | 𝒩 ( 𝑥 𝑖 ) | + ∑ 𝑠 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) 0 ≤ 𝑘 ≤ 𝑛 ) ( 3 ) H ere 𝐹 𝑗 , 𝑘 is a transport law for the 𝑗 th concentration at the interface with the 𝑘 th neighbor voxel with centroid 𝑥 𝑖 𝑘 , 𝒩 ( 𝑥 𝑖 ) is the set of centroids of all voxels that share an interface with the voxel centered at 𝑥 𝑖 , and 𝑠 𝑗 , 𝑘 is the rate of change in 𝑐 𝑗 at 𝑥 𝑖 due to a reaction with the 𝑘 th concentration 𝑐 𝑘 We use the general flux and source term s 𝐹 𝑗 , 𝑘 and 𝑠 𝑗 , 𝑘 , respectively, as interchangeable term s based on the contents of voxels to support models that couple diffusion transport and reactions with cellular dynamics since the contents of voxels, and hence the reactions in them ( e.g. , production in a cell, decay outside of cells) and transport conditions between them, can change. Discretized diffusive transport at the interface between two voxels occurs according to the form, 5 𝐹 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) = 𝐷 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) ‖ 𝑥 𝑖 𝑘 − 𝑥 𝑖 ‖ 2 ( 𝑐 𝑗 ( 𝑥 𝑖 𝑘 , 𝑡 ) − 𝑐 𝑗 ( 𝑥 𝑖 , 𝑡 ) ) , ( 4 ) where 𝐷 𝑗 , 𝑘 is the measur ed diffusivity at the interface. Diffusivity at the interface of two voxels is calculated as the harmonic mean of the diffusivity of each voxel, which accounts for scenarios where the diffusivity of a voxel is zero, 𝐷 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) = 2 𝐷 𝑗 ( 𝑥 𝑖 , 𝑡 ) 𝐷 𝑗 ( 𝑥 𝑖 𝑘 , 𝑡 ) 𝐷 𝑗 ( 𝑥 𝑖 , 𝑡 ) + 𝐷 𝑗 ( 𝑥 𝑖 𝑘 , 𝑡 ) ( 5 ) Transport through a permeable interface occurs, potentially with a bias, according to the form, 𝐹 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) = 𝑃 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) ‖ 𝑥 𝑖 𝑘 − 𝑥 𝑖 ‖ ( 𝑏 𝑗 ( 𝑥 𝑖 𝑘 , 𝑡 ) 𝑐 𝑗 ( 𝑥 𝑖 𝑘 , 𝑡 ) − 𝑎 𝑗 ( 𝑥 𝑖 , 𝑡 ) 𝑐 𝑗 ( 𝑥 𝑖 , 𝑡 ) ) ( 6 ) Here 𝑎 𝑗 and 𝑏 𝑗 are biasing coefficients and 𝑃 𝑗 , 𝑘 is the permeability of the interface. Like interface diffusivity, the interface permeability is calculated as the harmonic mean of the permeability of the two voxels that define the interface, 𝑃 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) = 2 𝑃 𝑗 ( 𝑥 𝑖 , 𝑡 ) 𝑃 𝑗 ( 𝑥 𝑖 𝑘 , 𝑡 ) 𝑃 𝑗 ( 𝑥 𝑖 , 𝑡 ) + 𝑃 𝑗 ( 𝑥 𝑖 𝑘 , 𝑡 ) ( 7 ) A fixed flux 𝑓 𝑖 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) on the interface of the voxel centered at 𝑥 𝑖 with the 𝑘 th neighboring voxel is defined for a 𝑗 th concentration. For outward - facing unit normal 𝑛 𝑖 𝑘 on the interface with respect to the voxel centered at 𝑥 𝑖 , 𝑠 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) can be written to describe the rate into or out of the voxel, 𝑠 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) = − 𝐷 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) 𝑛 𝑝 𝑘 𝑓 𝑝 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) ‖ 𝑥 𝑖 𝑘 − 𝑥 𝑖 ‖ ( 8 ) Likewise, for a fixed concentration 𝑔 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) for the 𝑗 th concentration on the interface of the voxel centered at 𝑥 𝑖 with the 𝑘 th neighboring voxel, 𝐹 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) = 2 𝐷 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) ‖ 𝑥 𝑖 𝑘 − 𝑥 𝑖 ‖ 2 ( 𝑔 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) − 𝑐 𝑗 ( 𝑥 𝑖 , 𝑡 ) ) ( 9 ) Note that for all written expressions for 𝐹 𝑗 , 𝑘 , 𝐹 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) = − 𝐹 𝑗 , 𝑘 ′ ( 𝑥 𝑖 𝑘 , 𝑡 ) for all pairs of 𝑥 𝑖 and 𝑥 𝑖 𝑘 ∈ 𝒩 ( 𝑥 𝑖 ) , where 𝐹 𝑗 , 𝑘 ′ ( 𝑥 𝑖 𝑘 , 𝑡 ) corresponds to the same interface as 𝐹 𝑗 , 𝑘 ( 𝑥 𝑖 , 𝑡 ) but with respect to the voxel centered at 𝑥 𝑖 𝑘 . Note also that, in the case of ( 8 ) , additional reaction term s that describe interactions between multiple concentrations in a voxel may be added to the rate considered in ( 3 ) . We have neglected such reaction terms here to cl early demonstrate the handling of various voxel surface conditions in our implementation. In general, for reaction - 6 diffusion models with non - zero reaction terms, the computations consist of calculating the contribution of the reactions to the rate of chang e in a concentration, and also of adding the contribution as described in ( 8 ) for voxels with a prescribed surface concentration. Accounting for Cell Surface Fluctu ations In the CPM, each cell is described as occupying a set of voxels in space. For a subdomain 𝒳 of the space occupied by a cell, we write 𝜎 ( 𝑥 𝑖 , 𝑡 ) = 𝑠 for each location 𝑥 𝑖 ∈ 𝒳 ( 𝑠 , 𝑡 ) that is occupied by cell 𝑠 at time 𝑡 Our algorithm describes the change in concentration distribution as the shape of each cell changes from one simulation step to the next. In the CPM, such changes occur by through the copy attempt , where the cell at a location overtakes occupancy of a neighboring locatio n that is occupied by a different cell. The CPM also permits locations to be occupied by a generic “medium”, and copy attempts also occur at interfaces between cells and the medium. In general, the CPM algorithm randomly selects a location in the space, ra ndomly selects a neighboring location, and computes the probability of the occurrence of the copy attempt, Pr ( 𝜎 ( 𝑥 𝑖 𝑘 , 𝑡 ) → 𝜎 ( 𝑥 𝑖 , 𝑡 ) ) = 𝑒 − max { 0 , ∆ ℋ ℋ ∗ } ( 10 ) Here 𝜎 ( 𝑥 𝑖 𝑘 , 𝑡 ) → 𝜎 ( 𝑥 𝑖 , 𝑡 ) denotes the copying of 𝜎 ( 𝑥 𝑖 , 𝑡 ) to 𝑥 𝑖 𝑘 , ℋ ∗ is a scaling factor and ∆ ℋ is the change in effective energy due to the copy . We refer to 𝑥 𝑖 in this notation as the source location of the copy attempt, and to 𝑥 𝑖 𝑘 as the target location of the copy attempt . In general, the effective energy is the m athematical description of a CPM - based model, and the algorithm evolves the cellular distribution to minimize the effective energy. Here w e develop a recursive algorithm for efficiently approximating the effect of fluctuations in the surface of a cell on t he concentration fields coupled with the distribution of cells . Models that employ this coupling of the CPM to describe cellular distributions and reaction - diffusion models to describe soluble chemicals are often referred to as Glazier - Graner - Hogeweg (GGH) models The CPM does not define the number of copy attempts that constitute a simulation step , called the Monte Carlo step (MCS) , though typically a simulation step consists of more than copy attempt, and often of a number of copy attempts equal to the number of voxels in a discretized domain (hence, each voxel is likely to be selected once as both a source and target location of a copy attempt). We derive our algorithm for an arbitrary number of copy attempts in one MCS , and denote time 𝑡 𝑘 ∈ [ 𝑡 , 𝑡 + ∆ 𝑡 ] = [ 𝑡 0 , 𝑡 𝐾 ] to describe the time at the 𝑘 th copy attempt of 𝐾 + 1 copy attempts in one MCS beginning at time 𝑡 We construct our alg orithm according to the following three rules, Rule 1 (mass conservation) : In each subdomain occupied by a cell or the medium, the total amount of each concentration is unchanged over the copy attempts of a MCS. Rule 2 (uniform correction) : Corrections a pplied to concentrations in each subdomain so to account for changes in the subdomain are uniformly applied. Rule 3 (Neumann conditions) : The amount of species in the source location of a copy attempt are exactly copied to the target location of the copy attempt. 7 Here Rule 1 accomplishes the objective of the algorithm Rule 2 provides sufficient simplification to allow the development of a computationally efficient approximation Rule 3 provides the necessary information to approximate local advection of concentration distributions. Consider the concentration 𝑐 ( 𝑥 𝑖 , 𝑡 𝑘 ) of a field at location 𝑥 𝑖 and time 𝑡 𝑘 Let ∆ 𝑐 ( 𝑥 𝑖 , 𝑡 𝑘 ) be the change in concentration at time 𝑡 𝑘 due to changes in the shape of the subdomain 𝒳 ( 𝑠 , 𝑡 ) of cell 𝑠 , where 𝑥 𝑖 ∈ 𝒳 ( 𝑠 , 𝑡 ) . We seek a scalar scaling factor to relate the concentration at all locations in 𝒳 after the copy attempts of a MCS to the concentration at those locations before the copy attempts according to Rules 1 and 2. To accomplish deriving this scaling factor, we define a scaling factor 𝑓 𝑘 ( 𝑠 ) that describes the change in concentration from time 𝑡 𝑘 − 1 to time 𝑡 𝑘 in cell 𝑠 , 𝑐 ( 𝑥 𝑖 , 𝑡 𝑘 ) = 𝑐 ( 𝑥 𝑖 , 𝑡 𝑘 − 1 ) + ∆ 𝑐 ( 𝑥 𝑖 , 𝑡 𝑘 ) = 𝑓 𝑘 ( 𝜎 ( 𝑥 𝑖 , 𝑡 𝑘 ) ) 𝑐 ( 𝑥 𝑖 , 𝑡 𝑘 − 1 ) ( 11 ) Let 𝐹 𝑘 ( 𝑠 ) be the scaling factor for cell 𝑠 over 𝑘 + 1 copy attempts , 𝐹 𝑘 ( 𝑠 ) = ∏ 𝑓 𝑛 ( 𝑠 ) 1 ≤ 𝑛 ≤ 𝑘 ( 12 ) The scaling factor to relate the concentration distribution in cell 𝑠 over 𝐾 + 1 copy attempts in a MCS is then 𝐹 𝐾 ( 𝑠 ) Rule 1 prescribes that the total concentration in the domain of a cell remains unchanged over the copy attempts of a MCS. The discrete volume integral of the concentration 𝐶 ( 𝑠 , 𝑡 𝑘 ) over the domain of cell 𝑠 at time 𝑡 𝑘 can be written in terms of the cu rrent concentration, as well as the concentration before the copy attempts and current scaling factor using ( 11 ) and ( 12 ) , 𝐶 ( 𝑠 , 𝑡 𝑘 ) = ∑ 𝑐 ( 𝑥 𝑖 , 𝑡 𝑘 ) 𝑥 𝑖 ∈ 𝒳 ( 𝑠 , 𝑡 𝑘 ) = 𝐹 𝑘 ( 𝑠 ) ∑ 𝑐 ( 𝑥 𝑖 , 𝑡 0 ) 𝑥 𝑖 ∈ 𝒳 ( 𝑠 , 𝑡 𝑘 ) ( 13 ) Rule 1 dictates that ( 13 ) is constant over all times of an MCS, 𝐶 ( 𝑠 , 𝑡 𝑘 ) = 𝐶 ( 𝑠 , 𝑡 0 ) , ∀ 𝑘 ∈ [ 0 , 𝐾 ] ( 14 ) Consider the case of adding a location 𝑦 𝑖 to the domain of cell 𝑠 at time 𝑡 𝑘 ( 13 ) can b e written for such a case as 𝐶 ( 𝑠 , 𝑡 𝑘 ) = 𝐹 𝑘 ( 𝑠 ) ( ∑ 𝑐 ( 𝑥 𝑖 , 𝑡 0 ) 𝑥 𝑖 ∈ 𝒳 ( 𝑡 𝑘 − 1 ) + 𝑐 ( 𝑦 𝑖 , 𝑡 0 ) ) ( 15 ) Substituting ( 13 ) and ( 14 ) into ( 15 ) presents a solution for 𝐹 𝑘 in terms of previous solutions, 8 the total amount of the concentration, and the concentration of the location that is added to the domain, 𝐹 𝑘 ( 𝑠 ) = 1 1 𝐹 𝑘 − 1 ( 𝑠 ) + 𝑐 ( 𝑦 𝑖 , 𝑡 0 ) 𝐶 ( 𝑠 , 𝑡 0 ) ( 16 ) The same derivations hold for removing a point 𝑧 𝑖 from 𝒳 . Removing 𝑧 𝑖 from 𝒳 at time 𝑡 𝑘 produces the expression for the volume integral, 𝐶 ( 𝑠 , 𝑡 𝑘 ) = 𝐹 𝑘 ( 𝑠 ) ( ∑ 𝑐 ( 𝑥 𝑖 , 𝑡 0 ) 𝑥 ∈ 𝒳 ( 𝑡 𝑘 − 1 ) − 𝑐 ( 𝑧 𝑖 , 𝑡 0 ) ) ( 17 ) Using the same substitutions as those used for ( 16 ) also presents a solution for 𝐹 𝑘 in terms of previous solutions, the total amount of the concentration, and the concentration of the location that is removed from the domain, 𝐹 𝑘 ( 𝑠 ) = 1 1 𝐹 𝑘 − 1 ( 𝑠 ) − 𝑐 ( 𝑧 𝑖 , 𝑡 0 ) 𝐶 ( 𝑠 , 𝑡 0 ) ( 18 ) We then find from ( 16 ) and ( 18 ) that, in general, a change in total amount ∆ 𝑐 ( 𝑥 𝑖 , 𝑡 𝑘 ) due to a change in cell shape associated with location 𝑥 𝑖 can be corrected to accomplish Rule 1 subject to Rule 2 with the general form, 𝐹 𝑘 ( 𝑠 ) = { 1 𝑘 = 0 1 1 𝐹 𝑘 − 1 ( 𝑠 ) + ∆ 𝑐 ( 𝑥 𝑖 , 𝑡 𝑘 ) 𝐶 0 ( 𝑠 , 𝑡 0 ) 𝑘 > 0 ( 19 ) ∆ 𝑐 ( 𝑥 𝑖 , 𝑡 𝑘 ) is determined for the c ell that occupies the source location during a copy attempt according to Rule 3, in which case the concentration at the source location is copied to the target location ( i.e. , no gradient normal to the motion of the cell surface). For the cell that occupie s the target location of the copy attempt, we assume the simple case and calculate ∆ 𝑐 ( 𝑥 𝑖 , 𝑡 𝑘 ) as the negative of the concentration at the target location , thereby removing the need for further modifications in the neighborhood of the copy Applying thes e local modifications to a concentration distribution while maintaining ( 19 ) during the copy attempts of a MCS and applying it after a MCS provides concentration distributions that respect the motion of cell boundaries and cross cell boundaries only according to the transport laws of a GGH model ( Figure 1 ). 9 Figure 1 A multicellular aggregate in two chemical gradients while accounting ( i.e. , “Compensator”) or not accounting ( i.e. , “No compensator”) for cell surface fluctuations. A first species “C1” has zero diffusion coefficient inside each cell, and a second species “C2” has a nonzero, but significantly less, diffusion coefficient inside each cell compared to the medium around the cells. A: Without ac counting for surface fluctuations (top two rows), cell motility in the CPM allows chemical species to cross the cell membrane, as demonstrated by the presence of nonzero values of species C1 inside cells and smooth gradients of species C2 inside cells. Whe n accounting for surface fluctuations (bottom two rows), species C1 diffuse around, but not into, the cells, and the cell boundaries create discontinuities in the distribution of species C2. B: Measurements of average concentration of spe cies C1 (left) and C2 (right) inside each cell. Nonzero measurements of species C1 occur only when not accounting for cell surface fluctuations. Species C2 measurements show similar values when accounting or not accounting for cell surface fluctuations, though concentration measurements are smoother when accounting for cell surface fluctuations. In application, the algorithm works in tandem with an implementation of the CPM. Assuming that the total concentration for each cell is known at the beginning of a MCS, 10 implementing the algorithm consists of updating a scaling factor of initial value 1 for each cell when modifications to the cellular distribution are made according to the distribution of a concentration field where the copy occurs. Modifications to a concentration di stribution during a copy occur in the same way as the modifications that occur in the cellular distribution, where the value at the source location is copied to the target location (according to Rule 3). After the completion of all copy attempts of a MCS, the concentration distribution is updated according to the final scaling factor of the cell that occupies each location. Hence, implementation of the algorithm is of comparable (but greater) computational complexity to the CPM, and significantly less compl ex than solving the d iffusion equation, which involves neighborhood calculations. Implementation We implemented our FVM formalism as an available diffusion solver in CC3D, henceforth referred to as the FVM solver , with robust support for model specificati on that reflects the features available in currently available CC3D diffusion solvers. The FVM solver supports an arbitrary number of concentration fields, explicit declaration of reactions as algebraic expressions, declaration of diffusivity on the basis of cell type, and locally modifiable diffusivity fields that can be prescribed during simulation in CC3D Python code. The FVM solver supports fixed concentration, fixed flux and periodic boundary conditions, and heterogeneous fixed concentration and flux c onditions can be specified along each voxel surface of a boundary and also modified during simulation. The FVM solver provides adaptive time stepping for maintaining numerical stability using the Scarborough criterion Computations for transport , local reactions and adaptive time stepping are parallelized using OpenMP according to a CC3D simulation specification Results Here we present two examples demonstrating select GGH modeling and simulation capabilities enabled by the methods and implementation developed in this work. These examples were selected to demonstrate modeling critical biophysical processes that historically have eluded G GH models of multicellular systems , and to motivate future work that rigorously develops such models in relevant biological scenarios As such, we limited development of these models to the scope of the present work and leave development of detailed models to future investigations that focus on particular biological applications. Intracellular Transport with Subcellular Resolution Of the possible processes of chemical transport in multicellular systems, transcellular vesicular transport of large molecules is perhaps the least feasible to describe using GGH models. Generally speaking, vesicular transport involves forming a vesicle from a section of the cell membrane ( i.e. , endocytosis) that carries materials from the extracellular environment (van der Pol et al. 2012) , which is then transported elsewhere within the cell and sometimes recombined with the cell membrane to release the contents of the vesicle back into the extracellular environment ( i.e. , exocytosis) This transport process is especially difficult to describe with GGH models because of the intimate relations hip between transport of vesicular contents and motion of the vesicular interface with its environment , which GGH models historically do not respect. 11 We developed a model of transcellular transport through vesicular trafficking to demonstrate how our work provides the capability to model vesicular trafficking in GGH models. Our model describes the cross - section of a two - dimensional monolayer of cell s, which separates two chemically different environments and actively transports a chemical species between t he two environments through vesicular trafficking . Vesicles initially form at the interface of the monolayer and one environment, where the vesicles are initially static and fill with the chemical species ( i.e. , endocytosis) . A vesicle separate s from the i nterface and travel s across the monolayer un til it reach es the interface of the monolayer and other environment, at which point the vesicle recombine s with the cell surface and expel s its carried chemical species into the environment ( i.e. , exocytosis) We model vesicles as compartments within cells, and the shape of cells and vesicles evolve according to the effective energy, ℋ = ∑ 𝜆 𝑣 ( 𝜏 ( 𝑠 ) ) ( 𝑉 ( 𝑠 ) − 𝑉 𝑜 ( 𝜏 ( 𝑠 ) ) ) 2 𝑠 + ∑ ∑ ( 1 − 𝛿 𝜎 ( 𝑥 ) , 𝜎 ( 𝑥 ′ ) ) 𝐽 ( 𝜏 ( 𝜎 ( 𝑥 ) ) , 𝜏 ( 𝜎 ( 𝑥 ′ ) ) ) 𝑥 ′ ∈ 𝒩 ( 𝑥 ) 𝑥 ∈ 𝒳 + ℋ 𝑑𝑖𝑟𝑒𝑐𝑡 ( 20 ) H ere 𝜆 𝑣 ( 𝜏 ( 𝑠 ) ) is the volume constraint model parameter of the type 𝜏 ( 𝑠 ) of cell or vesicle 𝑠 , 𝑉 ( 𝑠 ) is the volume of cell or vesicle 𝑠 , 𝑉 𝑜 ( 𝜏 ) is the volume constraint of type 𝜏 , 𝑥 is a position in the spatial domain 𝒳 , 𝑥 ′ is a position in the neighborhood 𝒩 ( 𝑥 ) of position 𝑥 , 𝐽 ( 𝜏 , 𝜏 ′ ) is the adhesion parameter between types 𝜏 and 𝜏 ′ , 𝜎 ( 𝑥 ) is the cell at 𝑥 , 𝛿 𝑥 , 𝑥 ′ is the Kronecker delta , and ℋ 𝑑𝑖𝑟𝑒𝑐𝑡 is an expression that models a force acting on cells and/or vesicles (Li and Lowengrub 2014) . Adhesion parameters between vesicles are defined on the basis of whether two vesicles are in the same cell ( i.e. , intracellular) or different cells ( i.e. , intercellular). ℋ 𝑑𝑖𝑟𝑒𝑐𝑡 is defined in terms of the change in the expression ∆ ℋ 𝑑𝑖𝑟𝑒𝑐𝑡 for a copy attempt, ∆ ℋ 𝑑𝑖𝑟𝑒𝑐𝑡 = ∑ 𝑃 𝑖 ( 𝑠 ) 𝑆 𝑖 ( 𝑠 ) 𝑠 ( 21 ) Here 𝑃 𝑖 ( 𝑠 ) is a directional bias vector on cell or vesicle 𝑠 and 𝑆 𝑖 is the direction of a copy attempt. The directional bias vector acts like a constant force that alters the probability of the occurrence of a copy attempt based on the orientation of the copy att empt with respect to the directional bias vector. We represent the action of vesicular transport by applying ( 21 ) to all vesicles such that they tend to travel acros s the epithelium ( Table 1 ) Table 1 . CPM model parameters for the GGH model of transcellular transport through vesicular trafficking. Name Symbol Value Type label – medium 𝜏 0 Type label – cell 1 Type label – endocytosing vesicle 2 Type label – transporting vesicle 3 Volume constraint – cell 𝑉 𝑜 ( 1 ) 800 voxels Volume constraint – transporting vesicle 𝑉 𝑜 ( 2 ) 5 voxels 12 Volume constraint model parameter 𝜆 𝑣 ( 1 ) , 𝜆 𝑣 ( 2 ) 2.0 Directional bias vector - transporting vesicle 𝑃 𝑖 {0, 100, 0} Intrinsic random motility ℋ ∗ 10 Adhesion parameters M edium – cell 𝐽 ( 0 , 1 ) 20 M edium – endocytosing vesicle 𝐽 ( 0 , 2 ) 10 M edium – transporting vesicle 𝐽 ( 0 , 3 ) 30 C ell – cell 𝐽 ( 1 , 1 ) 10 C ell – endocytosing vesicle 𝐽 ( 1 , 2 ) 10 C ell – transporting vesicle (intercellular) 𝐽 ( 1 , 3 ) 30 C ell – transporting vesicle (intracellular) 10 E ndocytosing vesicle – endocytosing vesicle 𝐽 ( 2 , 2 ) 40 E ndocytosing vesicle – transporting vesicle 𝐽 ( 2 , 3 ) 20 T ransporting vesicle – transporting vesicle (intercellular) 𝐽 ( 3 , 3 ) 10 T ransporting vesicle – transporting vesicle (intracellular) 40 W e consider two types of vesicles in our simplified model of vesicular transport . One type of vesicle, called an endocytosing vesicle , represents a vesicle during endocytosis. The other type of vesicle, called a transporting vesicle , represents a vesicle as the cell transports across the monolayer We neglected the subcellular dynamics of endocytosis and assumed that endocytosing vesicle s form with a constant probability , called the endocytosis probability , at cellular interfaces with the extracellular environment from which cells transport the soluble chemical ( Table 2 ) After creation, and endocytosing vesicle remains at the cell interface for a fixed period, called the endocytosis period , during which time the soluble chemical is actively pumped into the vesicle according to ( 6 ) . After the endocytosis period, the endocytosing vesicle is converted into a transporting vesicle and is trafficked across the monolayer . When a transporting vesicle contacts the medium on t he opposite interface of the monolayer, exocytosis occurs by instantaneously converting all sites of the vesicle into site s occupied by the medium , thus releasing the contents of the vesicle into the environment We assumed perfect barrier function of the monolayer and so assigned zero diffusive transport through cells. Thus, all transport across the monolayer occurs through vesicular trafficking. Table 2 . Diffusive and transcellular transport model parameters for the GGH model of transcellular transport through vesicular trafficking. Permeable interface bias coefficients are listed for an outward - oriented flux from a voxel occupied by the medium. Name Symbol Value Endocytosis period N/A 10 MCSs Endocytosis probabil ity N/A 0.01 Diffusion coefficients Medium 𝐷 0.1 Cells 0 Endocytosing vesicle 0.1 Transporting vesicle 0.1 Permeable interface, medium – endocytosing vesicle Permeability coefficient 𝑃 0.5 Bias coefficient, vesicle side 𝑎 0.0 Bias coefficient, medium side 𝑏 1.0 13 To develop as a sense of the resulting transcellular transport from our simplified model and chosen parameters, we simulated a two - dimensional cross - section of fifteen cells of a monolayer for ten thousand MCSs ( Figure 2 ). We assigned a fixed concentration boundary value of one to the boundaries of the environment from which the monolayer transports the chemical, a no - flux boundary condition to the boundaries of the en vironment to which the monolayer transports the chemical , and initial values of one and zero, respectively Notable accumulation of transported chemical was visible by five thousand MCSs, and after ten thousand MCSs transcellular transport reduced the chem ical gradient across the monolayer from initially one to approximately 0.7. Figure 2 . Simulati on of transcellular transport of a chemical species across a two - dimensional monolayer with explicit modeling of vesicular trafficking Cell and vesicle borders are shown as yellow A: Representative snapshot of the simulation with boundary conditions indicated . Domain boundaries in t he medium above the monolayer have a constant concentration of one, and domain boundaries below the monolayer have a constant normal flux of zero . Transport in the edge cells is not considered . Detail views show examples of endocytosis (bottom left) and ve sicular transport (bottom right). B: Concentration distribution during simulation, as vesicular transport moves the chemical concentration from the upper side of the monolayer to the lower side. Spatial g radients in the distribution due to exocytosis were most apparent at around 7k simulation steps. Signaling During Repair in the Alveolar Microenvironment Various insults like hyperoxia, bacteria and viral pathogens, and inflammation can injure the alveolar epithelium, resulting in several processes that lead to epithelial proliferation (of Type II airway epithelial cells), transdifferentiation (into Type II airway epithelial cells) and overall re turn to homeostasis. Of the growth factors released to induce proliferation in airway epithelial cells, endothelial cells of the local vasculature secrete matrix metalloproteinase s (MMPs) that cleave heparin - bound epidermal growth factor ( HB - EGF) , allowing epidermal growth factor (EGF) to diffuse and ligate epithelial cell EGF receptors (Matthay et al. 2019) Under normal, 14 homeostatic con ditions, EGFR is limited to the basolateral membrane and isolated from apically situated large molecules by intercellular adhesion molecules (Brune et al. 2015) In the case of alveolar insult and repair, the spatial distribution of soluble EGF is then especially interesting since the source of soluble EGF ( i.e. , heparin) is apically situated , and since insult leads to compromised barrier function , which for EGF is known to cause hyperproliferation and epithelial - mesenchymal transition (Tanos and Rodriguez - Boulan 2008) We d eveloped a simplified, two - dimensional simulation of EGF signaling during wound healing in the alveolus. Our model neglects the intricate arrangements of airways epithelial cells in the alveolus and endothelial cells in the local vasculature, and instead considers a cross - section of a planar epithelium and perfectly straight nearby capillary endothelium , where the epithelium is pla ced at a variable distance from the endothelium and wounded at the beginning of simulation This arrangement divides our simulation domain into three subdomain s: the alveolar space , which represents inside the alveolus; the interstitial space , which is the space between the epithelium and the endothelium ; and the capillary space , which represents the space inside the capillary ( Figure 3 A ). The epithelium and endothelium are both motile and each adheres to its own basement membrane , where each basement membrane is static. The epithelium is wounded in our simulations by removing a section of the epithelium and its basement membrane along the vertical centerline of the simu lation domain. 15 Figure 3 . T wo - dimensional simulation of signaling during wound healing in a simplified alveolar microenvironment . All fields solutions are shown with contours scaled from zero (blue) to a maximum value specified by an inset value 𝐶 (red) . A: Signaling by a wounded epithelium (pink) leads to release of MMPs by endothelial cells (red) in the local vasculature, which degrades heparin - bound EGF (HB - EGF) . Detailed views show the characteristic short diffusion length of MMPs diffusing across the endothelial basement membrane (green) and degrading non - diffusive HB - EGF near the endothelium B: Degradation of HB - EGF produces diffusive EGF, which diffuses across the interstitial space between the vasculature and alveolus. Level of exposure of epithelial cells to soluble EGF decreases with increasing size of the interstitial space (top row: 25 μ m distance from endothelium to ep ithelium; middle row: 50 μ m distance; bottom row: 100 μ m distance) , though all epithelial cells experience similar levels of exposure within three thousand simulated steps ( five minutes of simulated time ) 16 We modeled epithelial and endothelial cells as mo tile along epithelial and endothelial basement membranes, respectively. Epithelial and endothelial cells were modeled with a volume constraint and intercellular adhesion similarly to ( 20 ) using the effective energy, ℋ = ∑ 𝜆 𝑣 ( 𝜏 ( 𝑠 ) ) ( 𝑉 ( 𝑠 ) − 𝑉 𝑜 ( 𝜏 ( 𝑠 ) ) ) 2 𝑠 + ∑ ∑ ( 1 − 𝛿 𝜎 ( 𝑥 ) , 𝜎 ( 𝑥 ′ ) ) 𝐽 ( 𝜏 ( 𝜎 ( 𝑥 ) ) , 𝜏 ( 𝜎 ( 𝑥 ′ ) ) ) 𝑥 ′ ∈ 𝒩 ( 𝑥 ) 𝑥 ∈ 𝒳 ( 22 ) Each basement membrane was modeled as a layer of cell objects that are prohibited from changing shape. Epithelial and endothelial cells were assumed to occupy approximately the same amount of volume, and we assumed that the effects of type - specific cell sh ape were negligible for our simulation. CPM model parameters used in the simulation of wound healing in the alveolus are available in Table 3 Table 3 CPM model parameters for the GGH model of wound healing in the alveolus. Name Symbol Value