Mathematical modelling of bone adaptation of the metacarpal subchondral bone in racehorses

Abstract

In Thoroughbred racehorses, fractures of the distal limb are commonly catastrophic. Most of these fractures occur due to the accumulation of fatigue damage from repetitive loading, as evidenced by microdamage at the predilection sites for fracture. Adaptation of the bone in response to training loads is important for fatigue resistance. In order to better understand the mechanism of subchondral bone adaptation to its loading environment, we utilised a square root function defining the relationship between bone volume fraction $(f_{BM} )$ and specific surface $(S_v )$ of the subchondral bone of the lateral condyles of the third metacarpal bone (MCIII) of the racehorse, and using this equation, developed a mathematical model of subchondral bone that adapts to loading conditions observed in vivo. The model is expressed as an ordinary differential equation incorporating a formation rate that is dependent on strain energy density. The loading conditions applied to a selected subchondral region, i.e. volume of interest, were estimated based on joint contact forces sustained by racehorses in training. For each of the initial conditions of $f_{BM}$ we found no difference between subsequent homoeostatic $f_{BM}$ at any given loading condition, but the time to reach equilibrium differed by initial $f_{BM}$ and loading condition. We found that the observed values for $f_{BM}$ from the mathematical model output were a good approximation to the existing data for racehorses in training or at rest. This model provides the basis for understanding the effect of changes to training strategies that may reduce the risk of racehorse injury.

Appendix

1.1 Sigmoidal function

Fig. 6

Relationship between strain energy density $(\psi _\mathrm{tissue} )$ and formation rate $(\tau )$, expressed as a hyperbolic function with sigmoidicity $(\gamma =1)$. Minimal and maximum formation rates, respectively, $0.001094 (\tau _\mathrm{min} )$ and $0.0127 (\tau _\mathrm{max} ). \psi _\mathrm{tissue}$ at the half-maximal formation rate $(\delta )$ was set to 7

We approximated the sigmoidal function equation from Peterson and Riggs (2010) to represent the relationship between $\psi _\mathrm{tissue}$ and formation rate $(\tau )$ (Eq. A1; Fig. 6). Formation rates do not differ significantly for racehorses in lower-intensity exercise (i.e. walk to canter), but are higher when unadapted bone is subjected to high-intensity exercise. $H_{\psi _\mathrm{tissue} }^+$ represents the hyperbolic term (H) for the stimulus variable $(\psi _\mathrm{tissue} )$ with an increase (+) from steady-state. Included terms are for sigmoidicity $(\gamma )$ that specifies the curve gradient, maximum estimated formation rate $(\tau _\mathrm{max} )$, minimum estimated formation rate $(\tau _\mathrm{min} )$ and the estimated value of $\psi _\mathrm{tissue}$ that produces the half-maximal formation rate $(\delta =7)$. The latter value was estimated from the maximum value of $\psi _\mathrm{tissue}$ at ultimate failure of the equine metacarpus, which has been reported to be about 14 MPa (Les et al. 1994).

$$\begin{aligned} H_{\psi _\mathrm{tissue} }^+ =\tau _\mathrm{min} +\frac{\left( {\tau _\mathrm{max} -\tau _\mathrm{min} } \right) \cdot \psi _\mathrm{tissue} ^{\gamma }}{\delta ^{\gamma }+\psi _\mathrm{tissue} ^{\gamma }} \end{aligned}$$

(A1)