Frontiers in growth and remodeling

Abstract

Unlike common engineering materials, living matter can autonomously respond to environmental changes. Living structures can grow stronger, weaker, larger, or smaller within months, weeks, or days as a result of a continuous microstructural turnover and renewal. Hard tissues can adapt by increasing their density and grow strong. Soft tissues can adapt by increasing their volume and grow large. For more than three decades, the mechanics community has actively contributed to understand the phenomena of growth and remodeling from a mechanistic point of view. However, to date, there is no single, unified characterization of growth, which is equally accepted by all scientists in the field. Here we shed light on the continuum modeling of growth and remodeling of living matter, and give a comprehensive overview of historical developments and trends. We provide a state-of-the-art review of current research highlights, and discuss challenges and potential future directions. Using the example of volumetric growth, we illustrate how we can establish and utilize growth theories to characterize the functional adaptation of soft living matter. We anticipate this review to be the starting point for critical discussions and future research in growth and remodeling, with a potential impact on life science and medicine.

Figure 1

Graphical illustration of the direct motion problem φ(X, t) mapping referential particles X at time t onto their spatial position x = φ(X, t) and of the inverse motion problem Φ(x, t) mapping spatial particles x at time t onto their referential position X = Φ(x, t). The local material transformation F^{g − 1} maps infinitesimal line elements dX_g at time t onto referential line elements dX = F^{g − 1} · dX.

Figure 2

Graphical illustration of the multiplicative decomposition of the deformation gradient F = ∇_Xφ = F^e · F^g into a reversible elastic part F^e and an irreversible growth part F^g. The local mass density ρ_t = j ρ₀ and ρ_g = j ρ₀ are transformations of the referential density ρ₀ in terms of the Jacobians j = det(F⁻¹) and j^g = det(F^g−1).

Figure 3

Surface growth of a narwhal tusk with an upward pointing material velocity, V = V^Г + V^g, according to equation (16). Photograph of a narwhal tusk, left, demonstrates the characteristic helical growth pattern. Computational simulation of surface growth, right, with an outward pointing velocity V^Г of the growth surface Г, here characterized through the bottom ring, and a helically upward pointing velocity V^g of material grown at the surface Г.

Figure 4

Density growth of the proximal femur for an energy-driven mass source, ρ̇₀ = R₀ with $R_0=k_{\rho }[{[{\rho }_0/{\rho }_0^{*}]}^{-m}{\psi }_0-{\psi }_0^{*}]$, according to equations (17) and (39). Photograph of a thin section, left, demonstrates microstructural arrangement of trabeculae in the femur head aligned with the axis of maximum principal stress [112]. Computational simulation of density growth, right, predicts higher bone densities in regions of large mechanical stress and lower bone densities in unloaded regions [64, 65].

Figure 5

Density growth of the proximal tibia for an energy-driven mass source, ρ̇₀ = R₀ with $R_0=k_{\rho }[{[{\rho }_0/{\rho }_0^{*}]}^{-m}{\psi }_0-{\psi }_0^{*}]$, according to equations (17) and (39). Photograph of a thin section, left, displays microstructural arrangement of trabeculae in the tibia head aligned with the axis of maximum principal stress [112]. Computational simulation of density growth, right, predicts a higher bone density in regions of large mechanical stress and lower bone density in unloaded regions [91, 110].

Figure 6

Volume growth of an artery for stress-driven isotropic growth, F^g = ϑ^g I with ϑ̇^g = k_ϑ ${\overline{I}}_1^{M^{\mathrm{e}}}/{\vartheta }^{\mathrm{g}}$, according to equations (62) and (66.1). Photograph of restenosis following balloon angioplasty, left, demonstrating residual atherosclerotic plaque and a new proliferative lesion caused by intimal thickening [70]. Computational simulation of isotropic volume growth, right, predicts wall thickening and re-narrowing of the lumen in response to stent-induced changes in the mechanical environment [43, 68].

Figure 7

Area growth of skin for stretch-driven transversely isotropic growth, ${\mathit{F}}^{\mathrm{g}}=\sqrt{{\vartheta }^{\mathrm{g}}}\mathit{I}+[1-\sqrt{{\vartheta }^{\mathrm{g}}}]\mathit{n}\otimes \mathit{n}$ with ${\dot{\vartheta }}^{\mathrm{g}}=k_{\vartheta }[{\overline{I}}_1^{C^{\mathrm{e}}}-{\overline{I}}_4^{C^{\mathrm{e}}}]/[2{\vartheta }^{\mathrm{g}}]$, according to equations (69) and (73.2). Photograph of a tissue expander to induce controlled in situ skin growth for defect correction in reconstructive surgery, left, reprinted with permission, Mentor Worldwide LLC [17]. Computational simulation of transversely isotropic area growth, right, predicts area growth in response to controlled mechanical overstretch during tissue expansion [18].

Figure 8

Area growth of skin for stretch-driven transversely isotropic growth, ${\mathit{F}}^{\mathrm{g}}=\sqrt{{\vartheta }^{\mathrm{g}}}\mathit{I}+[1-\sqrt{{\vartheta }^{\mathrm{g}}}]\mathit{n}\otimes \mathit{n}$ with ${\dot{\vartheta }}^{\mathrm{g}}=k_{\vartheta }[{\overline{I}}_1^{C^{\mathrm{e}}}-{\overline{I}}_4^{C^{\mathrm{e}}}]/[2{\vartheta }^{\mathrm{g}}]$, according to equations (69) and (73.2). Photograph of tissue expansion in pediatric forehead reconstruction, left, shows forehead, anterior and posterior scalp expansion to trigger skin growth in situ [17, 39]. Computational simulation of transversely isotropic area growth, right, predicts area growth in response to controlled mechanical overstretch during tissue expansion [115, 116].

Figure 9

Fiber growth of the heart for strain-driven transversely isotropic growth, F^g = I + [ϑ^g − 1] n ⊗ n with ${\dot{\vartheta }}^{\mathrm{g}}=k_{\vartheta }{\overline{I}}_1^{C^{\mathrm{e}}}/{\vartheta }^{\mathrm{g}}$, according to equations (76) and (80.2). Photograph of a heart in dilated cardiomyopathy, left, illustrates an increase in cavity size at a constant wall thickness, typically associated with volume-overload induced eccentric growth [70]. Computational simulation of transversely isotropic fiber growth, right, predicts an enlargement of the left ventricular cavity in response to mechanical overstretch [34, 35].

Figure 10

Cross-fiber growth of the heart for stress-driven wall thickening, F^g = I + [ϑ^g − 1] n ⊗ n with ${\dot{\vartheta }}^{\mathrm{g}}=k_{\vartheta }{\overline{I}}_1^{M^{\mathrm{e}}}/{\vartheta }^{\mathrm{g}}$, according to equations (76) and (66.1). Photograph of a heart in hypertrophic cardiomyopathy, left, illustrates an increase in wall thickness at a constant cardiac size, typically associated with pressure-overload induced concentric growth [70]. Computational simulation of transversely isotropic cross-fiber growth, right, predicts a significant wall thickening of the left ventricular wall in response to hypertension [34, 94].

Figure 11

Evolution of structural tensor in layered artery for stress- and strain-driven evolution of microstructural direction, ṅ = [I − n ⊗ n] · M^e · n / t_* and ṅ = [I − n ⊗ n] · C^e · n / t_* according to equations (87.1) and (87.2). Polarized light micrograph of tangentially sectioned brain artery, left, showing variation of collagen fiber orientation from circumferential inner to helical outer layer [30]. Computational simulation of stress- and strain-driven fiber distribution, left and right, predicts a smooth variation of collagen fiber orientation from circumferential inner to helical outer layer [68].