Modeling biological growth and remodeling: Contrasting methods, contrasting needs

[EN] Biological growth and remodeling processes are necessarily time-dependent due to the finite periods needed for the material to be synthesized, deposited, degraded, and/or reorganized and, hence, so have been predominantly modeled for the past 20+ years. However, a full-spectrum examination of the timescales present in these processes reveals the need to explore a new class of models for which time-dependent effects are negligible. These mechanobiologically (quasi-) equilibrated formulations not only appear to apply well in many cases but also provide the modeler with those additional pieces of information, and intuition, always needed when modeling complex time-dependent responses. Material model determination, optimization involving long-term adaptations, and mechanobiological stability analyses could be leveraged by the simplicity and computational efficiency of time-independent models. Although this concept is general, we address it by means of two particular theories for which we also highlight crucial differences entailed by their diametrically different material memory and heterogeneity descriptions.Latorre, M. (2020). Modeling biological growth and remodeling: Contrasting methods, contrasting needs. Current Opinion in Biomedical Engineering. 15:26-31. https://doi.org/10.1016/j.cobme.2019.11.00526311

On the interpretation of the logarithmic strain tensor in an arbitrary system of representation

[EN] Logarithmic strains are increasingly used in constitutive modelling because of their advantageous properties. In this paper we study the physical interpretation of the components of the logarithmic strain tensor in any arbitrary system of representation, which is crucial in formulating meaningful constitutive models. We use the path-independence property of total logarithmic strains to propose different fictitious paths which can be interpreted as a sum of infinitesimal engineering strain tensors. We show that the angular (engineering) distortion measure is arguably not a good measure of shear and instead we propose area distortions which are an exact interpretation of the shear terms both for engineering and for logarithmic strains. This new interpretation clearly explains the maximum obtained in some constitutive models for the simple shear load case.Partial financial support for this research is given by the Direccion General de Investigacion of the Ministerio de Economia y Competitividad of Spain under Grant DPI2011-26635 of the Plan Nacional de Investigacion.Latorre, M.; Montáns, FJ. (2014). On the interpretation of the logarithmic strain tensor in an arbitrary system of representation. International Journal of Solids and Structures. 51(7-8):1507-1515. https://doi.org/10.1016/j.ijsolstr.2013.12.04115071515517-

Extension of the Sussman-Bathe spline-based hyperelastic model to incompressible transversely isotropic materials

[EN] In this paper we extend the Sussman¿Bathe spline-based hyperelastic isotropic model to predict the behavior of transversely isotropic isochoric materials. The model is based on an uncoupled decomposition of the stored energy function and a generalization of the inversion formula used by Sussman and Bathe. The present extension introduces some approximations that, in all studied cases, do not yield relevant deviations from the experimental data. The isotropic model results in a particular case of the present formulation. Several possibilities of user-prescribed experimental data are addressed. The model is used to predict experiments of calendered rubber and of biological tissues.We acknowledge the anonymous reviewer #2 of the manuscript for providing a more elegant and accurate version of the inversion formula, Eq. (40), than the one given in the original submission. Financial support for this work has been given by the Direccion General de Proyectos de Investigacion of the Ministerio de Ciencia e Innovacion and the Ministerio de Economia y Competitividad of Spain under grants DPI2008-05423 and DPI2011-26635Latorre, M.; Montáns, FJ. (2013). Extension of the Sussman-Bathe spline-based hyperelastic model to incompressible transversely isotropic materials. Computers & Structures. 122:13-26. https://doi.org/10.1016/j.compstruc.2013.01.018132612

A mechanobiologically equilibrated constrained mixture model for growth and remodeling of soft tissues

[EN] Growth and remodeling of soft tissues is a dynamic process and several theoretical frameworks have been developed to analyze the time-dependent, mechanobiological and/or biomechanical responses of these tissues to changes in external loads. Importantly, general processes can often be conveniently separated into truly non-steady contributions and steady-state ones. Depending on characteristic times over which the external loads are applied, time-dependent models can sometimes be specialized to respective time-independent formulations that simplify the mathematical treatment without compromising the goodness of the particularized solutions. Very few studies have analyzed the long-term, steady-state responses of soft tissue growth and remodeling following a direct approach. Here, we derive a mechanobiologically equilibrated formulation that arises from a general constrained mixture model. We see that integral-type evolution equations that characterize these general models can be written in terms of an equivalent set of time-independent, nonlinear algebraic equations that can be solved efficiently to yield long-term outcomes of growth and remodeling processes in response to sustained external stimuli. We discuss the mathematical conditions, in terms of orders of magnitude, that yield the particularized equations and illustrate results numerically for general arterial mechano-adaptations.Universidad Politecnica de Madrid; Ministerio de Educacion, Cultura y Deporte of Spain, Grant/Award Number: CAS17/00068; Ministerio de Economia y Competitividad of Spain, Grant/Award Number: DPI2015-69801-R; National Institutes of Health, Grant/Award Numbers: R01HL086418, R01HL105297, R01HL128602, U01HL116323Latorre, M.; Humphrey, JD. (2018). A mechanobiologically equilibrated constrained mixture model for growth and remodeling of soft tissues. Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. 98(12):2048-2071. https://doi.org/10.1002/zamm.20170030220482071981

Mechanobiological stability of biological soft tissues

[EN] Like all other materials, biological soft tissues are subject to general laws of physics, including those governing mechanical equilibrium and stability. In addition, however, these tissues are able to respond actively to changes in their mechanical and chemical environment. There is, therefore, a pressing need to understand such processes theoretically. In this paper, we present a new rate-based constrained mixture formulation suitable for studying mechanobiological equilibrium and stability of soft tissues exposed to transient or sustained changes in material composition or applied loading. These concepts are illustrated for canonical problems in arterial mechanics, which distinguish possible stable versus unstable mechanobiological responses. Such analyses promise to yield insight into biological processes that govern both health and disease progression.This work was supported, in part, by grants from the US NIH, namely, R01 HL105297 (to C.A. Figueroa and J.D. Humphrey), R01 HL128602 (to J.D. Humphrey, C.K. Breuer, and Y. Wang), P01 HL134605 (to G. Tellides and J.D. Humphrey via a PPG to D. Rifkin), and U01 HL142518 (to J.D. Humphrey and G.E. Karniadakis)Latorre, M.; Humphrey, JD. (2019). Mechanobiological stability of biological soft tissues. Journal of the Mechanics and Physics of Solids. 125:298-325. https://doi.org/10.1016/j.jmps.2018.12.01329832512

Strain-Level Dependent Nonequilibrium Anisotropic Viscoelasticity: Application to the Abdominal Muscle

[EN] Soft connective tissues sustain large strains of viscoelastic nature. The rate-independent component is frequently modeled by means of anisotropic hyperelastic models. The rate-dependent component is usually modeled through linear rheological models or quasilinear viscoelastic (QLV) models. These viscoelastic models are unable, in general, to capture the strain-level dependency of the viscoelastic properties present in many viscoelastic tissues. In linear viscoelastic models, strain-level dependency is frequently accounted for by including the dependence of multipliers of Prony series on strains through additional evolution laws, but the determination of the material parameters is a difficult task and the obtained accuracy is usually not sufficient. In this work, we introduce a model for fully non-linear viscoelasticity in which the instantaneous and quasi-static behaviors are exactly captured and the relaxation curves are predicted to a high accuracy. The model is based on a fully nonlinear standard rheological model and does not necessitate optimization algorithms to obtain material parameters. Furthermore, in contrast to most models used in modeling the viscoelastic behavior of soft tissues, it is valid for the large deviations from thermodynamic equilibrium typically observed in soft tissuesSecretaria de Estado de Investigacion, Desarrollo e Innovacion (DPI2015-69801-R)Latorre, M.; Montáns, FJ. (2017). Strain-Level Dependent Nonequilibrium Anisotropic Viscoelasticity: Application to the Abdominal Muscle. Journal of Biomechanical Engineering. 139(10):1-9. https://doi.org/10.1115/1.4037405191391

On the tension-compression switch of the Gasser-Ogden-Holzapfel model: Analysis and a new pre-integrated proposal

[EN] Many biological soft tissues are structurally composed of a mostly isotropic matrix (elastin) and fibers (collagen). These fibers are not perfectly aligned but dispersed around some referential, preferred directions. In order to account for the dispersion of the fibers, a probability distribution is assumed. The Generalized Structure Tensor (GST) models perform a pre-integration of the distribution in order to achieve improved computational efficiency. The best known model of this kind is the Gasser-Ogden-Holzapfel (GOH) model. However, in these models no singular treatment of fibers is made. Whenever they suffer compression it is usual to consider that fibers should not contribute to the overall stiffness. At this point, a switch criterion is employed. This switch criterion is important because it changes the model predictions and may also result in unphysical stress predictions or strain ranges at which no compatible equilibrium solution is found. We perform an analysis of different tension-compression switch criteria from the literature for the GOH model and show relevant physical and computational drawbacks when using these criteria. In order to overcome these drawbacks, we make a new proposal which yields continuous stress solutions. In our proposal, pre-integrated expressions given in terms of the usual set of invariants take into account an average amount of fibers working either in tension or in compression for a given deformation gradient and fiber family. Two distinct switches naturally emerge from our procedure. Furthermore, we keep the appealing GST pre-integrated approach for any proposed stored energy, including that of the GOH model. (C) 2015 Elsevier Ltd. All rights reserved.Partial financial support for this work has been given by grant DPI2011-26635 from the Direccion General de Proyectos de Investigacion of the Ministerio de Economia y Competitividad of SpainLatorre, M.; Montáns, FJ. (2016). On the tension-compression switch of the Gasser-Ogden-Holzapfel model: Analysis and a new pre-integrated proposal. Journal of the Mechanical Behavior of Biomedical Materials. 57:175-189. https://doi.org/10.1016/j.jmbbm.2015.11.0181751895

Experimental data reduction for hyperelasticity

[EN] WYPiWYG hyperelasticity is a data-driven, model-free computational procedure for finite element analysis of soft materials. The procedure does not assume the shape of the stored energy function and does not employ material parameters, predicting accurately any smooth prescribed behavior from a complete set of experimental tests. However, fuzzy experimental data may yield useless highly oscillatory, unstable stored energy functions, and classical curvature smoothing frequently gives unsatisfactory results. Aside, the possibility of having experimental data from different specimens for the same test was not considered in previous procedures. In this work we present a novel technique based on spline regression and smoothing penalization using stability conditions. In general, this procedure reduces noisy experimental data or data from multiple specimens for ulterior determination of the stored energy. The procedure only needs the solution of a linear system of equations. Instead of classical curvature-based smoothing, we employ a novel stability-based smoothing, determining for each branch of the uniaxial stress-strain curve the most restrictive stability condition during uniaxial and equibiaxial tests. The resulting stored energy functions are smooth and stable. The procedure has little sensitivity to the number of spline segments or to the choice of the penalization parameter, which are computed automatically.Partial financial support for this work has been given by grant DPI2015-69801-R from the Direccion General de Proyectos de Investigacion of the Ministerio de Economia y Competitividad of SpainLatorre, M.; Montáns, FJ. (2020). Experimental data reduction for hyperelasticity. Computers & Structures. 232:1-16. https://doi.org/10.1016/j.compstruc.2018.02.01111623

Modeling mechano-driven and immuno-mediated aortic maladaptation in hypertension

[EN] Uncontrolled hypertension is a primary risk factor for diverse cardiovascular diseases and thus remains responsible for significant morbidity and mortality. Hypertension leads to marked changes in the composition, structure, properties, and function of central arteries; hence, there has long been interest in quantifying the associated wall mechanics. Indeed, over the past 20 years there has been increasing interest in formulating mathematical models of the evolving geometry and biomechanical behavior of central arteries that occur during hypertension. In this paper, we introduce a new mathematical model of growth (changes in mass) and remodeling (changes in microstructure) of the aortic wall for an animal model of induced hypertension that exhibits both mechano-driven and immuno-mediated matrix turnover. In particular, we present a bilayered model of the aortic wall to account for differences in medial versus adventitial growth and remodeling and we include mechanical stress and inflammatory cell density as determinants of matrix turnover. Using this approach, we can capture results from a recent report of adventitial fibrosis that resulted in marked aortic maladaptation in hypertension. We submit that this model can also be used to identify novel hypotheses to guide future experimentation.This work was supported, in part, by grants from the US NIH: R01 HL105297 (to C.A. Figueroa and J.D. Humphrey), U01 HL116323 (to J.D. Humphrey and G.E. Karniadakis), R01 HL128602 (to J.D. Humphrey, C.K. Breuer, and Y. Wang), P01 HL134605 (to G. Tellides and J.D. Humphrey via a PPG Award to D. Rifkin, NYU), and R03 EB021430 (to J.D. Humphrey); from the Ministerio de Educacion, Cultura y Deporte of Spain: CAS17/00068 (to M. Latorre); and from Universidad Politecnica de Madrid: 'Ayudas al personal docente e investigador para estancias breves en el extranjero 2017' (to M. Latorre). Additional support was given to M. Latorre by grant DPI2015-69801-R from the Direccion General de Proyectos de Investigacion, Ministerio de Economia y Competitividad of Spain (to F.J. Montans and J.M. Benitez). ML gratefully acknowledges the support given by the Department of Biomedical Engineering, Yale University, during his postdoctoral stayLatorre, M.; Humphrey, JD. (2018). Modeling mechano-driven and immuno-mediated aortic maladaptation in hypertension. Biomechanics and Modeling in Mechanobiology. 17(5):1497-1511. https://doi.org/10.1007/s10237-018-1041-81497151117

Anisotropic finite strain viscoelasticity based on the Sidoroff multiplicative decomposition and logarithmic strains

[EN] In this paper a purely phenomenological formulation and finite element numerical implementation for quasi-incompressible transversely isotropic and orthotropic materials is presented. The stored energy is composed of distinct anisotropic equilibrated and non-equilibrated parts. The nonequilibrated strains are obtained from the multiplicative decomposition of the deformation gradient. The procedure can be considered as an extension of the Reese and Govindjee framework to anisotropic materials and reduces to such formulation for isotropic materials. The stress-point algorithmic implementation is based on an elastic-predictor viscous-corrector algorithm similar to that employed in plasticity. The consistent tangent moduli for the general anisotropic case are also derived. Numerical examples explain the procedure to obtain the material parameters, show the quadratic convergence of the algorithm and usefulness in multiaxial loading. One example also highlights the importance of prescribing a complete set of stress-strain curves in orthotropic materials.Partial financial support for this work has been given by grant DPI2011-26635 from the Direccion General de Proyectos de Investigacion of the Ministerio de Economia y Competitividad of SpainLatorre, M.; Montáns, FJ. (2015). Anisotropic finite strain viscoelasticity based on the Sidoroff multiplicative decomposition and logarithmic strains. Computational Mechanics. 56(3):506-531. https://doi.org/10.1007/s00466-015-1184-850653156