Bruce P. Ayati

Computational Methods and Results for Structured Multiscale Models of Tumor Invasion

We present multiscale models of cancer tumor invasion with components at the molecular, cellular, and tissue levels. We provide biological justifications for the model components, present computati...

A mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease

BackgroundMultiple myeloma is a hematologic malignancy associated with the development of a destructive osteolytic bone disease.ResultsMathematical models are developed for normal bone remodeling and for the dysregulated bone remodeling that occurs in myeloma bone disease. The models examine the critical signaling between osteoclasts (bone resorption) and osteoblasts (bone formation). The interactions of osteoclasts and osteoblasts are modeled as a system of differential equations for these cell populations, which exhibit stable oscillations in the normal case and unstable oscillations in the myeloma case. In the case of untreated myeloma, osteoclasts increase and osteoblasts decrease, with net bone loss as the tumor grows. The therapeutic effects of targeting both myeloma cells and cells of the bone marrow microenvironment on these dynamics are examined.ConclusionsThe current model accurately reflects myeloma bone disease and illustrates how treatment approaches may be investigated using such computational approaches.ReviewersThis article was reviewed by Ariosto Silva and Mark P. Little.

A Variable Time Step Method for an Age-Dependent Population Model with Nonlinear Diffusion

We propose a method for solving a model of age-dependent population diffusion with random dispersal. This method, unlike previous methods, allows for variable time steps and independent age and time discretizations. We use a moving age discretization that transforms the problem to a coupled system of parabolic equations. The system is then solved by backward differences in time and a Galerkin approximation in space; the equations that need to be solved at each step treat each age group separately. A priori L2 error estimates are obtained by an energy analysis. These estimates are superconvergent in the age variable. We present a postprocessing technique which capitalizes on the superconvergence.

A structured-population model of Proteus mirabilis swarm-colony development

In this paper we present continuous age- and space-structured models and numerical computations of Proteus mirabilis swarm-colony development. We base the mathematical representation of the cell-cycle dynamics of Proteus mirabilis on those developed by Esipov and Shapiro, which are the best understood aspects of the system, and we make minimum assumptions about less-understood mechanisms, such as precise forms of the spatial diffusion. The models in this paper have explicit age-structure and, when solved numerically, display both the temporal and spatial regularity seen in experiments, whereas the Esipov and Shapiro model, when solved accurately, shows only the temporal regularity.The composite hyperbolic-parabolic partial differential equations used to model Proteus mirabilis swarm-colony development are relevant to other biological systems where the spatial dynamics depend on local physiological structure. We use computational methods designed for such systems, with known convergence properties, to obtain the numerical results presented in this paper.

Galerkin Methods in Age and Space for a Population Model with Nonlinear Diffusion

We present Galerkin methods in both the age and space variables for an age-dependent population undergoing nonlinear diffusion. The methods presented are a generalization of the methods presented in {\it A Variable Time Step Method for an Age-dependent Population Model with Nonlinear Diffusion}, where the approximation space in age was taken to be the space of piecewise constant functions. In this paper, we allow the use of discontinuous piecewise polynomial subspaces of

The Role of Osteocytes in Targeted Bone Remodeling: A Mathematical Model

L^2

A MULTISCALE MODEL OF BIOFILM AS A SENESCENCE-STRUCTURED FLUID ∗

as the approximation space in age. As in the piecewise constant case, we move the discretization along characteristic lines. The time variable has been left continuous. The methods are shown to be superconvergent in the age variable.

Modeling the Role of the Cell Cycle in Regulating Proteus mirabilis Swarm-Colony Development

Until recently many studies of bone remodeling at the cellular level have focused on the behavior of mature osteoblasts and osteoclasts, and their respective precursor cells, with the role of osteocytes and bone lining cells left largely unexplored. This is particularly true with respect to the mathematical modeling of bone remodeling. However, there is increasing evidence that osteocytes play important roles in the cycle of targeted bone remodeling, in serving as a significant source of RANKL to support osteoclastogenesis, and in secreting the bone formation inhibitor sclerostin. Moreover, there is also increasing interest in sclerostin, an osteocyte-secreted bone formation inhibitor, and its role in regulating local response to changes in the bone microenvironment. Here we develop a cell population model of bone remodeling that includes the role of osteocytes, sclerostin, and allows for the possibility of RANKL expression by osteocyte cell populations. We have aimed to give a simple, yet still tractable, model that remains faithful to the underlying system based on the known literature. This model extends and complements many of the existing mathematical models for bone remodeling, but can be used to explore aspects of the process of bone remodeling that were previously beyond the scope of prior modeling work. Through numerical simulations we demonstrate that our model can be used to explore theoretically many of the qualitative features of the role of osteocytes in bone biology as presented in recent literature.

Convergence of a Step-doubling Galerkin Method for Parabolic Problems

We derive a physiologically structured multiscale model for biofilm development. The model has components on two spatial scales, which induce different time scales into the problem. The macroscopic behavior of the system is modeled using growth‐induced flow in a domain with a moving boundary. Cell‐level processes are incorporated into the model using a so‐called physiologically structured variable to represent cell senescence, which in turn affects cell division and mortality. We present computational results for our models which shed light on modeling the combined role senescence and the biofilm state play in the defense strategy of bacteria.

Microbial dormancy in batch cultures as a function of substrate-dependent mortality

Abstract We present models and computational results which indicate that the spatial and temporal regularity seen in Proteus mirabilis swarm-colony development is largely an expression of a specific, nearly precise age of dedifferentiation in the cell cycle from motile swarmer cells to immotile dividing cells. This contrasts strongly with reaction–diffusion models of Proteus behavior that ignore or average out the age structure of the cell population and instead use only density-dependent mechanisms. We argue the necessity of retaining this known biological feature using explicit age structure in the model, and suggest that certain experiments may validate this underlying mechanism empirically.