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Biol Direct. 2012; 7: 9.
Published online Feb 21, 2012. doi:  10.1186/1745-6150-7-9
PMCID: PMC3356234
Reaction-Diffusion-Delay Model for EPO/TNF-α Interaction in articular cartilage lesion abatement
Jason M Graham,corresponding author1 Bruce P Ayati,1 Lei Ding,2 Prem S Ramakrishnan,2 and James A Martin2
1Department of Mathematics/Program in Applied Mathematical and Computational Sciences, University of Iowa, Iowa City, Iowa, USA
2Department of Orthopaedics and Rehabilitation, University of Iowa Hospitals and Clinics, Iowa City, Iowa, USA
corresponding authorCorresponding author.
Jason M Graham: jason-graham/at/uiowa.edu; Bruce P Ayati: bruce-ayati/at/uiowa.edu; Lei Ding: lei-ding/at/uiowa.edu; Prem S Ramakrishnan: prem-ramakrishnan/at/uiowa.edu; James A Martin: james-martin/at/uiowa.edu
Received September 27, 2011; Accepted February 21, 2012.
Abstract
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 Table1,1, while the second case has parameter values as in Table Table2.2. 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 1
Model Parameter Values for Results in Figure 5
Table of Parameter Values Corresponding to Simulations in Figure 5
ParameterValueUnitsReason

DR0.1equation M1Determined from []

DM0.05equation M2Determined from []

DF0.05equation M3Determined from []

DP0.005equation M4Determined from []

δR0.01equation M5Approximated

δM0.6equation M6Approximated

δF0.6equation M7Approximated

δP0.0087equation M8Approximated

δU0.0001equation M9Approximated

σR0.0001equation M10Approximated

σM0.00001equation M11Approximated

σF0.0001equation M12Approximated

σP0equation M13Case with no anti-inflammatory response

Λ10micromolarApproximated

λR10micromolarApproximated

λM10micromolarApproximated

λF10micromolarApproximated

λP10micromolarApproximated

α0equation M14Case with no anti-inflammatory response

β1100equation M15Approximated

Β250equation M16Approximated

γ10equation M17Approximated

ν0.5equation M18Approximated

equation M191equation M20Approximated

equation M210.5equation M22Approximated

τ10.5daysTaken from []

τ21daysTaken from []
Table 2
Model Parameter Values for Results in Figure 6
Table of Parameter Values Corresponding to Simulations in Figure 6
ParameterValueUnitsReason

DR0.1equation M23Determined from []

DM0.05equation M24Determined from []

DF0.05equation M25Determined from []

DP0.005equation M26Determined from []

δR0.01equation M27Approximated

δM0.6equation M28Approximated

δF0.6equation M29Approximated

δP0.0087equation M30Approximated

δU0.0001equation M31Approximated

σR0.0001equation M32Approximated

σM0.00001equation M33Approximated

σF0.0001equation M34Approximated

σP0.001equation M35Approximated

Λ10micromolarApproximated

λR10micromolarApproximated

λM10micromolarApproximated

λF10micromolarApproximated

λP10micromolarApproximated

α10equation M36Approximated

β1100equation M37Approximated

β250equation M38Approximated

γ10equation M39Approximated

ν0.5equation M40Approximated

equation M411equation M42Approximated

equation M430.5equation M44Approximated

τ10.5daysTaken from []

τ21daysTaken from []
Conclusions
The 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.
Reviewers
This paper has been reviewed by Yang Kuang, James Faeder and Anna Marciniak-Czochra.
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