A Damage-Driven Model for Duchenne Muscular Dystrophy: Early-Stage Dynamics and Invasion Thresholds

Abstract

We introduce a spatially extended mathematical model for Duchenne muscular dystrophy based on a damage-driven paradigm, in which immune recruitment is triggered by tissue injury. The model is formulated as a reaction--diffusion--chemotaxis system describing the interaction between healthy tissue, damaged fibers, immune cells and inflammatory signals. We establish the global well-posedness of the system and investigate the early-stage dynamics through linearization around the healthy equilibrium. Our analysis shows that diffusion does not induce Turing instabilities, so that spatial heterogeneity cannot arise from diffusion-driven mechanisms. Instead, disease progression occurs through invasion processes. We derive explicit conditions for the onset of invasion, interpreted as an effective damage reproduction threshold and characterize the minimal propagation speed of pathological fronts, showing that the dynamics is governed by a pulled-front mechanism. Numerical simulations support the analytical results and confirm the transition between decay and invasion. These results provide a mathematical framework for early-stage disease progression and indicate that spatial patterns arise from the expansion of localized damage rather than from intrinsic pattern-forming mechanisms.

Figures (4)

• Figure 1: Numerically computed bifurcation diagram in the $(\delta, \alpha)$ parameter plane. The solid black line corresponds to the threshold condition $\Theta = 0$, with $\Theta$ given in \ref{['Theta']}, separating regions with different stability properties of the healthy equilibrium. In the white region, the healthy equilibrium is stable and no biologically admissible pathological equilibrium exists. In the red region, the healthy equilibrium is unstable and a stable pathological equilibrium exists. The green region corresponds to a bistable regime, where the healthy equilibrium coexists with two pathological equilibria, one stable and one unstable. The parameters are chosen as $\sigma=1$, $\rho=0.9$, $c_\epsilon=0.1$, $\nu=7$, $k=r=1$ and $\mu=168.$
• Figure 2: Temporal and spatial dynamics of the system for different values of $\alpha$, illustrating the transition across the invasion threshold and the emergence of traveling-wave propagation. The three rows correspond, respectively, to the below-threshold ($\alpha=30$), near-threshold ($\alpha=32$) and strong invasion ($\alpha=50$) regimes. In the first row, the four panels (from left to right) represent the temporal evolution at $x=0$ of healthy tissue ($h$), damaged tissue ($d$), macrophages ($m$) and chemokines ($c$). In this regime, the perturbation decays and the system returns to the healthy equilibrium. In the second and third rows, the four panels show the spatial profiles of $h$, $d$, $m$, and $c$ at successive times, highlighting the formation and propagation of traveling waves connecting the healthy state to the diseased state. Near the threshold ($\alpha=32$), invasion is initiated but progresses slowly, while for larger values of $\alpha$ ($\alpha=50$) the invasion front propagates faster and becomes more pronounced. Simulations are performed with parameters $\sigma=1$, $\rho=0.9$, $c_\epsilon=0.1$, $\delta=1.1$, $\nu=7$, $k=r=1$, $\mu=168$, $D_d=1$, $D_m=10$, $D_c=1000$, and $\chi_0=5$. For this choice of parameters, the analytical threshold predicted by the linearized system is $\alpha_c \approx 31$. Initial conditions consist of a small localized perturbation of the healthy equilibrium.
• Figure 3: Left: Comparison between the analytical dispersion relation $s(\gamma)$ and numerical estimates of the front speed obtained from exponential initial conditions of the form $e^{-\gamma x}$. The minimal speed $s^*$ corresponds to the minimum of the dispersion curve and is selected for $\gamma \geq \gamma^*$. Right: Comparison between analytical and numerical front speeds for Gaussian initial conditions, for different domain sizes $L$. The agreement improves as $L$ increases, indicating that discrepancies observed for small domains are due to finite-size effects. The remaining parameters are as in Fig. \ref{['1Dinvasion']}, with $\alpha=32$ in the left panel.
• Figure 4: Comparison between the analytical wave speed predicted by the linearized system and the numerical front velocity. Left: propagation speed as a function of $\alpha$, with $\rho=0.9, \delta=1.1$. Increasing $\alpha$ leads to faster invasion. Middle: propagation speed as a function of $\rho$, with $\alpha=40, \delta=1.1$. Increasing $\rho$ slows down the propagation. Right: propagation speed as a function of $\delta$, with $\alpha=40, \rho=0.9$. Increasing $\delta$ slows down the propagation. The remaining parameters are set to $c_\epsilon=0.1$, $r=\kappa=\sigma=1$, $\nu=7$, $\mu=168$, $D_d=1$, $D_m=10$, $D_c=100$, and $\chi_0=5$, with domain size $L=800$.

Theorems & Definitions (23)

• Theorem 1
• Corollary 2
• Lemma 3: Total tissue density is strictly below saturation at equilibrium
• Proposition 4
• Theorem 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10