Fast, rate-independent, finite element implementation of a 3D constrained mixture model of soft tissue growth and remodeling

Abstract

Constrained mixture models of soft tissue growth and remodeling can simulate many evolving conditions in health as well as in disease and its treatment, but they can be computationally expensive. In this paper, we derive a new fast, robust finite element implementation based on a concept of mechanobiological equilibrium that yields fully resolved solutions and allows computation of quasi-equilibrated evolutions when imposed perturbations are slow relative to the adaptive process. We demonstrate quadratic convergence and verify the model via comparisons with semi-analytical solutions for arterial mechanics. We further examine the enlargement of aortic aneurysms for which we identify new mechanobiological insights into factors that affect the nearby non-aneurysmal segment as it responds to the changing mechanics within the diseased segment. Because this new 3D approach can be implemented within many existing finite element solvers, constrained mixture models of growth and remodeling can now be used more widely.

Numerical knockouts–In silico assessment of factors predisposing to thoracic aortic aneurysms

“…We have previously presented finite element implementations of our constrained mixture model (e.g., [ 34 , 35 ]), but herein we use a fast, efficient 3-D implementation [ 36 ] that is based on the underlying assumption that each G&R state is mechanobiologically equilibrated [ 27 ], which holds for cases wherein the characteristic timescale of the remodeling process is shorter than the timescale for the driving stimulus, that is, for fully quasi-static G&R [ 37 , 38 ].…”

“…Consistently, the Cauchy stress in Eq ( 4 ) also adopts a rule-of-mixtures expression in terms of evolved constituent-specific passive and active stresses with the active contribution in Eq ( 8 ), and λ θ , act , h = 1, relaxed as [ 27 ] where both and Δ τ wh are deformation-dependent, and the equilibrated (also deformation-dependent) Lagrange multiplier p h is determined from the scalar stress-like constraint (Eq 11 ) during the quasi-static G&R evolution. An exact linearization of the formulation consistent with these constraints enables implementation within a finite element framework, where simultaneous solution of mechanical and mechanobiological equilibrium can be ensured efficiently at load steps that capture evolving geometries, compositions, and properties of interest for complex boundary value problems; see [ 36 ] for specific details.…”

“…Moreover, note that the deviation in shear stress from its homeostatic value Δ τ wh = (1− ξ ) τ wh / τ wo −1 in Eqs ( 11 ) and ( 13 ) must be assessed pointwise. We assume the analytical expression , with a luminal radius and Q blood flow rate, whose value and derivatives may be approximated and computed at Gauss points in terms of local circumferential and axial stretches λ θh and λ zh from reference to current homeostatic configurations (for cylindrical segments; see Appendix C in [ 36 ]), which, in turn, enables additional straightforward computations of Eq ( 13 ) and its consistent tangent from the equilibrated second Piola-Kirchhoff active stresses (cf. Eq (40) in [ 36 ]).…”

“…We assume the analytical expression , with a luminal radius and Q blood flow rate, whose value and derivatives may be approximated and computed at Gauss points in terms of local circumferential and axial stretches λ θh and λ zh from reference to current homeostatic configurations (for cylindrical segments; see Appendix C in [ 36 ]), which, in turn, enables additional straightforward computations of Eq ( 13 ) and its consistent tangent from the equilibrated second Piola-Kirchhoff active stresses (cf. Eq (40) in [ 36 ]). For the intramural stress deviation in Eq ( 11 ), we consider as a stimulus, which has repeatedly yielded numerical responses that reflect experimental observations well [ 25 , 36 , 40 ].…”

“…For the intramural stress deviation in Eq ( 11 ), we consider as a stimulus, which has repeatedly yielded numerical responses that reflect experimental observations well [ 25 , 36 , 40 ]. Moreover, for the sake of clarity in our prior presentation of this rate-independent computational framework [ 36 ], we did not posit specific evolution laws for the magnitude or orientation of deposition pre-stretches or -stresses; rather, the rotated (i.e., “right”) deposition stretch tensors , and hence associated pre-stresses at the constituent level , were prescribed during the evolution, which were particularly critical to control the anisotropic enlargement of the simulated aneurysms. Following previous studies on aneurysms (e.g., [ 34 , 35 ]), here we let the (referential) deposition angle α 0 h of newly synthesized collagen fibers for the two symmetric diagonal families to evolve during G&R according to with α 0 o given in Table 1 .…”