Background: Multiple myeloma is a hematologic malignancy associated with the development of a des... more Background: Multiple myeloma is a hematologic malignancy associated with the development of a destructive osteolytic bone disease.
Complementary models reveal cellular responses to contact stresses that contribute to post-traumatic osteoarthritis
Journal of orthopaedic research : official publication of the Orthopaedic Research Society, Jan 10, 2016
Two categories of joint overloading cause post-traumatic osteoarthritis (PTOA): single acute trau... more 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 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 ap...
Joint (articular) cartilage is designed to sustain certain level of impact. However, when a large... more Joint (articular) cartilage is designed to sustain certain level of impact. However, when a large enough force is applied, the cells in the cartilage release pro-and anti-inflammatory chemicals that can affect the cell population, the chemical feedback loop, as well as the cartilage itself, causing lesions. Modeling all these processes requires several components: cellular, chemical, and mechanical. The cellular component deals with the population densities of different cells and their interactions and transitions from one type into another. These transitions are induced by, and in turn affect, the chemicals which are released as a result of the impact. The first two components have already been modeled in published papers, using a nonlinear system of age-structured, parabolic PDEs with a radial spacial dimension. The next step is modeling the mechanics of the impact as more than a constant force term, but as a time-dependent effect on the physical and chemical processes in the cartilage. This requires an additional spacial dimension for the depth of the cartilage. Increasing the dimensions, besides adjusting the original equations accordingly, requires the addition of a system of hyperbolic PDEs, to model the effect of the impact using linear elasticity. The current model develops and solves the system numerically, and is validated by experimental results.
Injuries to articular cartilage result in the development of lesions that form on the surface of ... more 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. We present a minimal mathematical model based on known mechanisms to investigate the spread and abatement of such lesions. 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 behaviour 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 behaviour 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. 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.
We present and study an approximation scheme for the mean of a stochastic simulation that models ... more We present and study an approximation scheme for the mean of a stochastic simulation that models a population subject to nonlinear birth and exogenous disturbances. We use the information from the probability distribution for the disturbance times to construct a method that improves upon the mean-®eld approximation. We show through two example systems the effectiveness of the Markov embedding approximation and discuss the contexts in which it is an appropriate method.
Until recently many studies of bone remodeling at the cellular level have focused on the behavior... more 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. 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 theoretically explore many of the most recent experimental results for bone remodeling, and can be utilized to assess the effects of novel bone-targeting agents on the bone remodeling process.
In this paper we present continuous age- and space-structured models and numerical computations o... more 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.
We derive a physiologically structured multiscale model for biofilm development. The model has co... more 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.
Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and m... more 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 \emph{in vitro} and \emph{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 \emph{in vitro} and \emph{in vivo}. This makes it possible to perform \emph{in silica} experiments more closely resembling experimental observations.
It has been known for many years that small fractions of persister cells resist killing in many b... more It has been known for many years that small fractions of persister cells resist killing in many bacterial colony-antimicrobial confrontations. These persisters are not believed to be mutants. Rather it has been hypothesized that they are phenotypic variants. Current models allow cells to switch in and out of the persister phenotype. Here we suggest a different explanation, namely senescence, for persister formation. Using a mathematical model including age structure, we show that senescence provides a natural explanation for persister-related phenomena including the observations that persister fraction depends on growth phase in batch culture and dilution rate in continuous culture.
We present models and computational results which indicate that the spatial and temporal regulari... more 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 sharp age of dedifferentiation in the cell cycle from motile swarmer cells to immotile dividing cells (also called swimmer or vegetative 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 the explicit age structure, and suggest experiments that may help determine the underlying mechanisms empirically. Consequently, we advocate Proteus as a model organism for a multiscale understanding of how and to what extent the life cycle of individual cells affects the macroscopic behavior of a biological system.
Frontiers in bioengineering and biotechnology, 2015
Traumatic injuries of articular cartilage result in the formation of a cartilage lesion and contr... more 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 ...
International journal for numerical methods in biomedical engineering, 2014
We present a model of articular cartilage lesion formation to simulate the effects of cyclic load... more 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.
Modeling and Simulation of Age- and Space-Structured Biological Systems
ABSTRACT This expository paper provides an introduction to computational methods for solving cont... more ABSTRACT This expository paper provides an introduction to computational methods for solving continuous models of certain biological systems. These systems have important behavior in space and within the age distribution of individuals. Very often, the spatial behavior depends critically on the age structure of the population. The different spatial scales induce different time scales in the problem, whereas age and time advance together. Age- and space-structured multiscale systems arise in a wide variety of biological problems ranging from multicellular and tissue-level phenomena to problems in ecology and evolutionary biology. We describe a general modeling framework used to represent such biological systems. We then discuss computational methods used to solve the model equations, beginning with a treatment of the moving-grid Galerkin method used to decouple age and time while they advance together, introduce how this discretization in age works with discretizations in time and space, and then review how more complicated nonlinear problems would be treated. We close by presenting two example systems, swarm-colony development of the bacteria Proteus mirabilis and tumor invasion, which differ in some important respects from the general model, but which are effectively treated by the computational methods presented in this paper.
Background: Multiple myeloma is a hematologic malignancy associated with the development of a des... more Background: Multiple myeloma is a hematologic malignancy associated with the development of a destructive osteolytic bone disease.
Complementary models reveal cellular responses to contact stresses that contribute to post-traumatic osteoarthritis
Journal of orthopaedic research : official publication of the Orthopaedic Research Society, Jan 10, 2016
Two categories of joint overloading cause post-traumatic osteoarthritis (PTOA): single acute trau... more 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 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 ap...
Joint (articular) cartilage is designed to sustain certain level of impact. However, when a large... more Joint (articular) cartilage is designed to sustain certain level of impact. However, when a large enough force is applied, the cells in the cartilage release pro-and anti-inflammatory chemicals that can affect the cell population, the chemical feedback loop, as well as the cartilage itself, causing lesions. Modeling all these processes requires several components: cellular, chemical, and mechanical. The cellular component deals with the population densities of different cells and their interactions and transitions from one type into another. These transitions are induced by, and in turn affect, the chemicals which are released as a result of the impact. The first two components have already been modeled in published papers, using a nonlinear system of age-structured, parabolic PDEs with a radial spacial dimension. The next step is modeling the mechanics of the impact as more than a constant force term, but as a time-dependent effect on the physical and chemical processes in the cartilage. This requires an additional spacial dimension for the depth of the cartilage. Increasing the dimensions, besides adjusting the original equations accordingly, requires the addition of a system of hyperbolic PDEs, to model the effect of the impact using linear elasticity. The current model develops and solves the system numerically, and is validated by experimental results.
Injuries to articular cartilage result in the development of lesions that form on the surface of ... more 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. We present a minimal mathematical model based on known mechanisms to investigate the spread and abatement of such lesions. 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 behaviour 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 behaviour 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. 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.
We present and study an approximation scheme for the mean of a stochastic simulation that models ... more We present and study an approximation scheme for the mean of a stochastic simulation that models a population subject to nonlinear birth and exogenous disturbances. We use the information from the probability distribution for the disturbance times to construct a method that improves upon the mean-®eld approximation. We show through two example systems the effectiveness of the Markov embedding approximation and discuss the contexts in which it is an appropriate method.
Until recently many studies of bone remodeling at the cellular level have focused on the behavior... more 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. 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 theoretically explore many of the most recent experimental results for bone remodeling, and can be utilized to assess the effects of novel bone-targeting agents on the bone remodeling process.
In this paper we present continuous age- and space-structured models and numerical computations o... more 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.
We derive a physiologically structured multiscale model for biofilm development. The model has co... more 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.
Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and m... more 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 \emph{in vitro} and \emph{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 \emph{in vitro} and \emph{in vivo}. This makes it possible to perform \emph{in silica} experiments more closely resembling experimental observations.
It has been known for many years that small fractions of persister cells resist killing in many b... more It has been known for many years that small fractions of persister cells resist killing in many bacterial colony-antimicrobial confrontations. These persisters are not believed to be mutants. Rather it has been hypothesized that they are phenotypic variants. Current models allow cells to switch in and out of the persister phenotype. Here we suggest a different explanation, namely senescence, for persister formation. Using a mathematical model including age structure, we show that senescence provides a natural explanation for persister-related phenomena including the observations that persister fraction depends on growth phase in batch culture and dilution rate in continuous culture.
We present models and computational results which indicate that the spatial and temporal regulari... more 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 sharp age of dedifferentiation in the cell cycle from motile swarmer cells to immotile dividing cells (also called swimmer or vegetative 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 the explicit age structure, and suggest experiments that may help determine the underlying mechanisms empirically. Consequently, we advocate Proteus as a model organism for a multiscale understanding of how and to what extent the life cycle of individual cells affects the macroscopic behavior of a biological system.
Frontiers in bioengineering and biotechnology, 2015
Traumatic injuries of articular cartilage result in the formation of a cartilage lesion and contr... more 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 ...
International journal for numerical methods in biomedical engineering, 2014
We present a model of articular cartilage lesion formation to simulate the effects of cyclic load... more 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.
Modeling and Simulation of Age- and Space-Structured Biological Systems
ABSTRACT This expository paper provides an introduction to computational methods for solving cont... more ABSTRACT This expository paper provides an introduction to computational methods for solving continuous models of certain biological systems. These systems have important behavior in space and within the age distribution of individuals. Very often, the spatial behavior depends critically on the age structure of the population. The different spatial scales induce different time scales in the problem, whereas age and time advance together. Age- and space-structured multiscale systems arise in a wide variety of biological problems ranging from multicellular and tissue-level phenomena to problems in ecology and evolutionary biology. We describe a general modeling framework used to represent such biological systems. We then discuss computational methods used to solve the model equations, beginning with a treatment of the moving-grid Galerkin method used to decouple age and time while they advance together, introduce how this discretization in age works with discretizations in time and space, and then review how more complicated nonlinear problems would be treated. We close by presenting two example systems, swarm-colony development of the bacteria Proteus mirabilis and tumor invasion, which differ in some important respects from the general model, but which are effectively treated by the computational methods presented in this paper.
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Papers by Bruce Ayati