Reaction-Diffusion-Delay Model for EPO/TNF-α Interaction in articular cartilage lesion abatement

BackgroundInjuries 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 discussionWe 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.Table 1Model Parameter Values for Results in Figure 5Table of Parameter Values Corresponding to Simulations in Figure 5ParameterValueUnitsReasonDR0.1cm2dayDetermined from [13]DM0.05cm2dayDetermined from [13]DF0.05cm2dayDetermined from [13]DP0.005cm2dayDetermined from [13]δR0.011dayApproximatedδM0.61dayApproximatedδF0.61dayApproximatedδP0.00871dayApproximatedδU0.00011dayApproximatedσR0.0001micromolar⋅cm2day⋅cellsApproximatedσM0.00001micromolar⋅cm2day⋅cellsApproximatedσF0.0001micromolar⋅cm2day⋅cellsApproximatedσP0micromolar⋅cm2day⋅cellsCase with no anti-inflammatory responseΛ10micromolarApproximatedλR10micromolarApproximatedλM10micromolarApproximatedλF10micromolarApproximatedλP10micromolarApproximatedα01dayCase with no anti-inflammatory responseβ11001dayApproximatedΒ2501dayApproximatedγ101dayApproximatedν0.51dayApproximatedμSA11dayApproximatedμDN0.51dayApproximatedτ10.5daysTaken from [5]τ21daysTaken from [5]Table 2Model Parameter Values for Results in Figure 6Table of Parameter Values Corresponding to Simulations in Figure 6ParameterValueUnitsReasonDR0.1cm2dayDetermined from [13]DM0.05cm2dayDetermined from [13]DF0.05cm2dayDetermined from [13]DP0.005cm2dayDetermined from [13]δR0.011dayApproximatedδM0.61dayApproximatedδF0.61dayApproximatedδP0.00871dayApproximatedδU0.00011dayApproximatedσR0.0001micromolar⋅cm2day⋅cellsApproximatedσM0.00001micromolar⋅cm2day⋅cellsApproximatedσF0.0001micromolar⋅cm2day⋅cellsApproximatedσP0.001micromolar⋅cm2day⋅cellsApproximatedΛ10micromolarApproximatedλR10micromolarApproximatedλM10micromolarApproximatedλF10micromolarApproximatedλP10micromolarApproximatedα101dayApproximatedβ11001dayApproximatedβ2501dayApproximatedγ101dayApproximatedν0.51dayApproximatedμSA11dayApproximatedμDN0.51dayApproximatedτ10.5daysTaken from [5]τ21daysTaken from [5]ConclusionsThe mathematical model presented herein suggests that not only are anti-inflammatory cy-tokines, 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.ReviewersThis paper has been reviewed by Yang Kuang, James Faeder and Anna Marciniak-Czochra.