Fig 1.
Schematic summarizing the developed systems-mechanobiological model of wound healing, which aims to capture the temporal evolution of key biochemical, microstructural, and macroscopic mechanical and geometrical variables by representing the cell and tissue regulatory pathways and their interaction across structural scales.
Images used in this figure were downloaded from Labicons.net and openclipart.org.
Fig 2.
Schematic of the hierarchical Bayesian model posed to capture the experimentally-measured mechanical response of a set of Ns inter-related tissue specimens.
A set of 3(m + 1) hyperdistributions, common across all specimens, generates the mNs mechanical model parameters corresponding to each tested specimen, and the Ns parameters representing experimental uncertainty. These parameters yield deterministic predictions for the mechanical behavior of each specimen, which are to be compared to the corresponding experimental evidence in order to establish a best-fitting set of hyperparameters.
Fig 3.
Model preparation before solving the wound healing problem.
(a) Modeled square tissue patch, with symmetric boundary conditions along x = 0 and y = 0 and mechanical constitutive parameters corresponding to unwounded skin. (b) Unwounded skin patch in its in vivo state (i.v.), characterized by an equi-biaxial pre-stretch. (c) Wound infliction in vivo, obtained by setting the mechanical parameters in a circular tissue region to extremely small values; the values of the biochemical and microstructural quantities (α, c, ρ, ϕc) are also adjusted to reflect a freshly-wounded tissue. Note that, immediately after infliction, the wound enlarges due to the corresponding release of tissue pre-stretch, as shown in the inset (white dashed line vs. boundary of the blue region). The reached deformation is made permanent to ensure that the newly-deposited tissue has no initial stress.
Fig 4.
Hierarchical Bayesian calibration of tissue mechanical parameters.
(a) C10, corresponding to the behavior of the non-collagenous ground substance, exhibits marked decrease between 7- and 14-days post-wounding, while mostly remaining within the broad range of values characterizing unwounded skin. (b) k1, corresponding to the behavior of the tissue collagenous matrix, tends to decrease between 7- and 14-days post-wounding, and is typically larger in the wounds than in unwounded skin. (c) k2, relating to the large deformation behavior of the tissue collagenous matrix, can only be inferred for unwounded skin due to the limited deformability of wounded tissues prior to failure. (d–f) Model-based predictions of the specimen tensile behavior accounting for experimental uncertainty. The dots in (a–c) indicate median values of the parameter posteriors for each of the Ns = 8 specimens, cf. S2, S3 and S4 Figs. The boxplots in (a–c) are constructed based on the values indicated by the dots, with orange lines denoting the median and extension of the whiskers denoting the 95% CI. The shadings in (d–f) indicate the 95% CI obtained from 1′000 random tensile curves generated using the calibrated Bayesian model and accounting for experimental uncertainty.
Fig 5.
Results of wound healing simulations over a 21-day period using the wound mechanical parameters directly obtained from the Bayesian calibration procedure (median and 95% CI) and assuming linear variation between known or estimated values.
(a, b) Hard-coded time evolution of the mechanical parameters and
, along with the corresponding values from Bayesian calibration (dots, cf. Fig 4). (c) Decay of the first inflammatory signal, α, in the wound. (d–h) Time and spatial evolution of: second inflammatory signal, c; cell population, ρ; tissue collagen content, ϕc; tissue elastic stretch, θe; tissue plastic stretch, θp. (i) Time evolution and illustration of wound area changes. Dots and error bars in (d,e,f,i): mean ± standard deviation of previously-published experimental data, cf. S1 Appendix and Ref. [10].
Fig 6.
Results of wound healing simulation over a 21-day period for alternative links between the tissue collagen content, ϕc, and the mechanical parameter k1, and alternative values of the parameter that controls collagen production by cells.
(a–d) Temporal evolution of second inflammatory signal, c, cell population, ρ, wound fibrin content, , and mechanical parameter
for either considered constitutive link. These quantities do not depend explicitly on ϕc, hence we report them only once. (e–g) Temporal evolution of tissue collagen content, ϕc, mechanical parameter
, and wound area change resulting from assuming that k1 is proportional to ϕc. (h–j) Temporal evolution of collagen crosslinking, ξc, mechanical parameter
, and wound area change resulting from assuming that k1 depends nonlinearly on ϕc via ξc. Dots and error bars in (a,b,e,g,j): mean ± standard deviation of previously-published experimental data, cf. S1 Appendix and Ref. [10]. Dots in (d,f,i): values of
and
obtained from Bayesian calibration, cf. Fig 4.
Fig 7.
Results of wound healing simulation over a 21-day period for stretch-mediated mechanosensitivity and alternative values of the coupling strength, as controlled by the parameter Ωm.
Temporal evolution of: second inflammatory signal, c, (a); cell population, ρ, (b); tissue elastic stretch, θe (c); tissue collagen content, ϕc, (d); collagen crosslinking, ξc, (e); wound area change (f); mechanical parameters (g) and
(h). The wound healing outcome in terms of tissue mechanical behavior is visualized by evaluating its tensile response at day 21 post-wounding (i).
Fig 8.
Results of wound healing simulation over a 21-day period for stiffness-mediated mechanosensitivity and alternative values of the coupling strength, as controlled by the parameter Ωm.
Temporal evolution of: second inflammatory signal, c, (a); cell population, ρ, (b); tissue elastic stretch, θe (c); tissue collagen content, ϕc, (d); collagen crosslinking, ξc, (e); wound area change (f); mechanical parameters (g) and
(h). The wound healing outcome in terms of tissue mechanical behavior is visualized by evaluating its tensile response at day 21 post-wounding (i).