A mathematical model of Bacteroides thetaiotaomicron, Methanobrevibacter smithii, and Eubacterium rectale interactions in the human gut.

The human gut microbiota is a complex ecosystem that affects a range of human physiology. In order to explore the dynamics of the human gut microbiota, we used a system of ordinary differential equations to model mathematically the biomass of three microorganism populations: Bacteroides thetaiotaomicron, Eubacterium rectale, and Methanobrevibacter smithii. Additionally, we modeled the concentrations of relevant nutrients necessary to sustain these populations over time. Our model highlights the interactions and the competition among these three species. These three microorganisms were specifically chosen due to the system's end product, butyrate, which is a short chain fatty acid that aids in developing and maintaining the intestinal barrier in the human gut. The basis of our mathematical model assumes the gut is structured such that bacteria and nutrients exit the gut at a rate proportional to its volume, the rate of volumetric flow, and the biomass or concentration of the particular population or nutrient. We performed global sensitivity analyses using Sobol' sensitivities to estimate the relative importance of model parameters on simulation results.

Considerations for Modeling Proteus mirabilis Swarming.

In this chapter we provide some initial guidance to experimentalists on how they might go about creating mathematical representations of their systems under study. Because the interests and goals of different researchers can differ, we try to provide broad instruction on the creation and use of mathematical models. We provide a brief overview of some modeling that has been done with Proteus mirabilis colonies, and discuss the goals of modeling. We suggest ways that collaborative teams may communicate with one another more effectively, and how they can build more confidence in their model results.

Modeling the Effects of Multiple Myeloma on Kidney Function.

Multiple myeloma (MM), a plasma cell cancer, is associated with many health challenges, including damage to the kidney by tubulointerstitial fibrosis. We develop a mathematical model which captures the qualitative behavior of the cell and protein populations involved. Specifically, we model the interaction between cells in the proximal tubule of the kidney, free light chains, renal fibroblasts, and myeloma cells. We analyze the model for steady-state solutions to find a mathematically and biologically relevant stable steady-state solution. This foundational model provides a representation of dynamics between key populations in tubulointerstitial fibrosis that demonstrates how these populations interact to affect patient prognosis in patients with MM and renal impairment.

Modeling the effect of blunt impact on mitochondrial function in cartilage: implications for development of osteoarthritis.

OBJECTIVE: Osteoarthritis (OA) is a disease characterized by degeneration of joint cartilage. It is associated with pain and disability and is the result of either age and activity related joint wear or an injury. Non-invasive treatment options are scarce and prevention and early intervention methods are practically non-existent. The modeling effort presented in this article is constructed based on an emerging biological hypothesis-post-impact oxidative stress leads to cartilage cell apoptosis and hence the degeneration observed with the disease. The objective is to quantitatively describe the loss of cell viability and function in cartilage after an injurious impact and identify the key parameters and variables that contribute to this phenomenon. METHODS: We constructed a system of differential equations that tracks cell viability, mitochondrial function, and concentrations of reactive oxygen species (ROS), adenosine triphosphate (ATP), and glycosaminoglycans (GAG). The system was solved using MATLAB and the equations' parameters were fit to existing data using a particle swarm algorithm. RESULTS: The model fits well the available data for cell viability, ATP production, and GAG content. Local sensitivity analysis shows that the initial amount of ROS is the most important parameter. DISCUSSION: The model we constructed is a viable method for producing in silico studies and with a few modifications, and data calibration and validation, may be a powerful predictive tool in the search for a non-invasive treatment for post-traumatic osteoarthritis.

Complementary models reveal cellular responses to contact stresses that contribute to post-traumatic osteoarthritis.

Two categories of joint overloading cause post-traumatic osteoarthritis (PTOA): single acute traumatic loads/impactions and repetitive overloading due to incongruity/instability. We developed and refined three classes of complementary models to define relationships between joint overloading and progressive cartilage loss across the spectrum of acute injuries and chronic joint abnormalities: explant and whole joint models that allow probing of cellular responses to mechanical injury and contact stresses, animal models that enable study of PTOA pathways in living joints and pre-clinical testing of treatments, and patient-specific computational models that define the overloading that causes OA in humans. We coordinated methodologies across models so that results from each informed the others, maximizing the benefit of this complementary approach. We are incorporating results from these investigations into biomathematical models to provide predictions of PTOA risk and guide treatment. Each approach has limitations, but each provides opportunities to elucidate PTOA pathogenesis. Taken together, they help define levels of joint overloading that cause cartilage destruction, show that both forms of overloading can act through the same biologic pathways, and create a framework for initiating clinical interventions that decrease PTOA risk. Considered collectively, studies extending from explants to humans show that thresholds of joint overloading that cause cartilage loss can be defined, that to at least some extent both forms of joint overloading act through the same biologic pathways, and interventions that interrupt these pathways prevent cartilage damage. These observations suggest that treatments that decrease the risk of all forms of OA progression can be discovered. © 2016 Orthopaedic Research Society. Published by Wiley Periodicals, Inc. J Orthop Res 35:515-523, 2017.

Mathematics as a conduit for translational research in post-traumatic osteoarthritis.

Biomathematical models offer a powerful method of clarifying complex temporal interactions and the relationships among multiple variables in a system. We present a coupled in silico biomathematical model of articular cartilage degeneration in response to impact and/or aberrant loading such as would be associated with injury to an articular joint. The model incorporates fundamental biological and mechanical information obtained from explant and small animal studies to predict post-traumatic osteoarthritis (PTOA) progression, with an eye toward eventual application in human patients. In this sense, we refer to the mathematics as a "conduit of translation." The new in silico framework presented in this paper involves a biomathematical model for the cellular and biochemical response to strains computed using finite element analysis. The model predicts qualitative responses presently, utilizing system parameter values largely taken from the literature. To contribute to accurate predictions, models need to be accurately parameterized with values that are based on solid science. We discuss a parameter identification protocol that will enable us to make increasingly accurate predictions of PTOA progression using additional data from smaller scale explant and small animal assays as they become available. By distilling the data from the explant and animal assays into parameters for biomathematical models, mathematics can translate experimental data to clinically relevant knowledge. © 2016 Orthopaedic Research Society. Published by Wiley Periodicals, Inc. J Orthop Res 35:566-572, 2017.

Linking Cellular and Mechanical Processes in Articular Cartilage Lesion Formation: A Mathematical Model.

Post-traumatic osteoarthritis affects almost 20% of the adult US population. An injurious impact applies a significant amount of physical stress on articular cartilage and can initiate a cascade of biochemical reactions that can lead to the development of osteoarthritis. In our effort to understand the underlying biochemical mechanisms of this debilitating disease, we have constructed a multiscale mathematical model of the process with three components: cellular, chemical, and mechanical. The cellular component describes the different chondrocyte states according to the chemicals these cells release. The chemical component models the change in concentrations of those chemicals. The mechanical component contains a simulation of a blunt impact applied onto a cartilage explant and the resulting strains that initiate the biochemical processes. The scales are modeled through a system of partial-differential equations and solved numerically. The results of the model qualitatively capture the results of laboratory experiments of drop-tower impacts on cartilage explants. The model creates a framework for incorporating explicit mechanics, simulated by finite element analysis, into a theoretical biology framework. The effort is a step toward a complete virtual platform for modeling the development of post-traumatic osteoarthritis, which will be used to inform biomedical researchers on possible non-invasive strategies for mitigating the disease.

Corrigendum: "A Validated Model of the Pro- and Anti-Inflammatory Cytokine Balancing Act in Articular Cartilage Lesion Formation".

[This corrects the article on p. 25 in vol. 3, PMID: 25806365.].

A validated model of the pro- and anti-inflammatory cytokine balancing act in articular cartilage lesion formation.

Traumatic injuries of articular cartilage result in the formation of a cartilage lesion and contribute to cartilage degeneration and the risk of osteoarthritis (OA). A better understanding of the framework for the formation of a cartilage lesion formation would be helpful in therapy development. Toward this end, we present an age and space-structured model of articular cartilage lesion formation after a single blunt impact. This model modifies the reaction-diffusion-delay models in Graham et al. (2012) (single impact) and Wang et al. (2014) (cyclic loading), focusing on the "balancing act" between pro- and anti-inflammatory cytokines. Age structure is introduced to replace the delay terms for cell transitions used in these earlier models; we find age structured models to be more flexible in representing the underlying biological system and more tractable computationally. Numerical results show a successful capture of chondrocyte behavior and chemical activities associated with the cartilage lesion after the initial injury; experimental validation of our computational results is presented. We anticipate that our in silico model of cartilage damage from a single blunt impact can be used to provide information that may not be easily obtained through in in vivo or in vitro studies.

Modeling and simulation of the effects of cyclic loading on articular cartilage lesion formation.

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.

The role of osteocytes in targeted bone remodeling: a mathematical model.

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.

Towards a new spatial representation of bone remodeling.

Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and multiple myeloma. Computational and mathematical modeling can aid in therapy and treatment as well as understanding fundamental biology. Different approaches to modeling give insight into different aspects of a phenomena so it is useful to have an arsenal of various computational and mathematical models. Here we develop a mathematical representation of bone remodeling that can effectively describe many aspects of the complicated geometries and spatial behavior observed. There is a sharp interface between bone and marrow regions. Also the surface of bone moves in and out, i.e. in the normal direction, due to remodeling. Based on these observations we employ the use of a level-set function to represent the spatial behavior of remodeling. We elaborate on a temporal model for osteoclast and osteoblast population dynamics to determine the change in bone mass which influences how the interface between bone and marrow changes. We exhibit simulations based on our computational model that show the motion of the interface between bone and marrow as a consequence of bone remodeling. The simulations show that it is possible to capture spatial behavior of bone remodeling in complicated geometries as they occur in vitro and in vivo. By employing the level set approach it is possible to develop computational and mathematical representations of the spatial behavior of bone remodeling. By including in this formalism further details, such as more complex cytokine interactions and accurate parameter values, it is possible to obtain simulations of phenomena related to bone remodeling with spatial behavior much as in vitro and in vivo. This makes it possible to perform in silica experiments more closely resembling experimental observations.

Reaction-diffusion-delay model for EPO/TNF-α interaction in articular cartilage lesion abatement.

BACKGROUND: Injuries to articular cartilage result in the development of lesions that form on the surface of the cartilage. Such lesions are associated with articular cartilage degeneration and osteoarthritis. The typical injury response often causes collateral damage, primarily an effect of inflammation, which results in the spread of lesions beyond the region where the initial injury occurs. RESULTS AND DISCUSSION: We present a minimal mathematical model based on known mechanisms to investigate the spread and abatement of such lesions. The first case corresponds to the parameter values listed in Table 1, while the second case has parameter values as in Table 2. In particular we represent the "balancing act" between pro-inflammatory and anti-inflammatory cytokines that is hypothesized to be a principal mechanism in the expansion properties of cartilage damage during the typical injury response. We present preliminary results of in vitro studies that confirm the anti-inflammatory activities of the cytokine erythropoietin (EPO). We assume that the diffusion of cytokines determine the spatial behavior of injury response and lesion expansion so that a reaction diffusion system involving chemical species and chondrocyte cell state population densities is a natural way to represent cartilage injury response. We present computational results using the mathematical model showing that our representation is successful in capturing much of the interesting spatial behavior of injury associated lesion development and abatement in articular cartilage. Further, we discuss the use of this model to study the possibility of using EPO as a therapy for reducing the amount of inflammation induced collateral damage to cartilage during the typical injury response. CONCLUSIONS: The mathematical model presented herein suggests that not only are anti-inflammatory cytokines, such as EPO necessary to prevent chondrocytes signaled by pro-inflammatory cytokines from entering apoptosis, they may also influence how chondrocytes respond to signaling by pro-inflammatory cytokines.

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

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.

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

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.

A spatial model of tumor-host interaction: application of chemotherapy.

In this paper we consider chemotherapy in a spatial model of tumor growth. The model, which is of reaction-diffusion type, takes into account the complex interactions between the tumor and surrounding stromal cells by including densities of endothelial cells and the extra-cellular matrix. When no treatment is applied the model reproduces the typical dynamics of early tumor growth. The initially avascular tumor reaches a diffusion limited size of the order of millimeters and initiates angiogenesis through the release of vascular endothelial growth factor (VEGF) secreted by hypoxic cells in the core of the tumor. This stimulates endothelial cells to migrate towards the tumor and establishes a nutrient supply sufficient for sustained invasion. To this model we apply cytostatic treatment in the form of a VEGF-inhibitor, which reduces the proliferation and chemotaxis of endothelial cells. This treatment has the capability to reduce tumor mass, but more importantly, we were able to determine that inhibition of endothelial cell proliferation is the more important of the two cellular functions targeted by the drug. Further, we considered the application of a cytotoxic drug that targets proliferating tumor cells. The drug was treated as a diffusible substance entering the tissue from the blood vessels. Our results show that depending on the characteristics of the drug it can either reduce the tumor mass significantly or in fact accelerate the growth rate of the tumor. This result seems to be due to complicated interplay between the stromal and tumor cell types and highlights the importance of considering chemotherapy in a spatial context.

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.