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 EPO on articular cartilage lesion abatement.
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 develop mathematical models for collective cell motility. Initially we develop a model using a linear diffusion-advection type equation and fit the parameters to data from cell motility assays. This approach is helpful in classifying
the results of cell motility assay experiments. In particular, this model can determine degrees of directed versus undirected collective cell motility. Next we develop a model using a nonlinear diffusion term that is able capture in a unified way directed and undirected collective cell motility. Finally we apply the nonlinear diffusion approach to a problem in tumor cell invasion, noting that neither chemotaxis or haptotaxis are present in the system under consideration in this article.
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.
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 appro
aches 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.
We present models of dormancy in a planktonic culture and in biofilm, and examine the relative advantage of short dormancy versus long dormancy times in each case. Simulations and analyses indicate that in planktonic batch cultures and in chemostats,
live biomass is maximized by the fastest possible exit from dormancy. The lower limit of time to reawakening is thus perhaps governed by physiological, biochemical or other constraints within the cells. In biofilm we see that the slower waker has a defensive advantage over the fast waker due to a larger amount of dormant biomass, without an appreciable difference in total live biomass. Thus it would seem that typical laboratory culture conditions can be unrepresentative of the natural state. We discuss the computational methods developed for this work.
