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Abstract Background Multiple myeloma is a hematologic malignancy associated with the development of a destructive osteolytic bone disease. Results Mathematical 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. Conclusions The current model accurately reflects myeloma bone disease and illustrates how treatment approaches may be investigated using such computational approaches. Reviewers This article was reviewed by Ariosto Silva and Mark P. Little.

SUMMARY We present a model of articular cartilage lesion formation to simulate the effects of cyclic loading. This model extends and modifies the reaction-diffusion-delay model by Graham et al., 2012 for the spread of a lesion formed though a single traumatic event. Our model represents 'implicitly' the effects of loading, meaning through a cyclic sink term in the equations for live cells. Our model forms the basis for in silico studies of cartilage damage relevant to questions in osteoarthritis, for example, that may not be easily answered through in vivo or in vitro studies. Computational results are presented that indicate the impact of differing levels of erythropoietin on articular cartilage lesion abatement. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: We present models and computational studies of dormancy in batch cultures to try to understand the relationship between reculturing time and death penalty for low substrate and the relative advantage of fast versus slow reawakening on the part of the bacteria. We find that the advantage goes to the faster waker for shorter reculturing times and lower mortality under low substrate, and moves to the slower waker as reculturing times and death penalty increase. The advantage returns again to the fast waker for very high death penalties. We use an explicit, continuous structure variable to represent dormancy so as to allow for flexibility in substrate usage on the part of dormant cells, and for a more mechanistic representation of the reawakening process. [Copyright &y& Elsevier]

Background: Multiple myeloma is a hematologic malignancy associated with the development of a destructive osteolytic bone disease. Results: Mathematical 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. Conclusions: The current model accurately reflects myeloma bone disease and illustrates how treatment approaches may be investigated using such computational approaches. Reviewers: This article was reviewed by Ariosto Silva and Mark P. Little. [ABSTRACT FROM AUTHOR]

Abstract: We present size distributions of Proteus mirabilis cells obtained from age- and space-structured models and simulations of swarm colonies, and compare them to experimental results from the manuscript of Matsuyama et al. [T. Matsuyama, Y. Takagi, Y. Nakagawa, H. Itoh, J. Wakita, M. Matsushita, Dynamic aspects of the structured cell population in a swarming colony of Proteus mirabilis, J. Bacteriol. 182 (2000) 385–393] that considers the evolution of the cell size structure at fixed points in the colony. Our comparison clarifies the importance of including explicit age structure in models of Proteus mirabilis swarm-colony development, as opposed to reaction–diffusion models that homogenize the system in age. We close with a discussion of our scientific computing methodology and how it enables modelers to move beyond reaction–diffusion models of biological pattern formation and include within their models the distributions of important traits of individuals. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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 computational results from the models, and discuss the scientific-computing methodology used to solve the model equations. The models and methodology presented in this paper form the basis for developing and treating increasingly complex, mechanistic models of tumor invasion that will be more predictive and less phenomenological. Because many of the features of the cancer models, such as taxis, aging, and growth, are seen in other biological systems, the models and methods discussed here also provide a template for handling a broader range of biological problems. [ABSTRACT FROM AUTHOR]

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 methods, where the approximation space in age is the space of piecewise constant functions. In this paper, we allow the use of discontinuous piecewise polynomial subspaces of L² 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. [ABSTRACT FROM AUTHOR]

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 indepen- dent 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. [ABSTRACT FROM AUTHOR]