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Proliferative Granulation Tissue Formation

The formation of new connective tissue and microscopic blood vessels (angiogenesis). Fibroblasts synthesize collagen, laying down a provisional matrix that supports subsequent remodeling.
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The statement of the theorem

Let M(t,x)M(t, \textbf{x}) be the concentration of the provisional matrix (collagen, fibronectin) at time tt and position x\textbf{x}. The rate of matrix deposition is modeled by a reaction-diffusion equation incorporating cell-mediated synthesis and degradation:\n∂M∂t=DM∇2M+ksynth[Fibroblast]Mmax−MMmax−kdegM[MMP]+Rangi\frac{\partial M}{\partial t} = D_M \nabla^2 M + k_{synth} [\text{Fibroblast}] \frac{M_{max} - M}{M_{max}} - k_{deg} M [\text{MMP}] + R_{angi} \nwhere ksynthk_{synth} is the synthesis rate, MmaxM_{max} is the saturation capacity, and RangiR_{angi} accounts for matrix support provided by newly formed vessels.