How to implement constrained mixture growth and remodeling algorithms for soft biological tissues

Biomechanics of soft biological tissues and organs, mechanobiology, homeostasis and modelling

The human body consists of many different soft biological tissues that exhibit diverse microstructures and functions and experience diverse loading conditions. Yet, under many conditions, the mechanical behaviour of these tissues can be described well with similar nonlinearly elastic or inelastic constitutive relations, both in health and some diseases. Such constitutive relations are essential for performing nonlinear stress analyses, which in turn are critical for understanding physiology, pathophysiology and even clinical interventions, including surgery. Indeed, most cells within load-bearing soft tissues are highly sensitive to their local mechanical environment, which can typically be quantified using methods of continuum mechanics only after the constitutive relations are determined from appropriate data, often multi-axial. In this review, we discuss some of the many experimental findings of the structure and the mechanical response, as well as constitutive formulations for 10 representative soft tissues or organs, and present basic concepts of mechanobiology to support continuum biomechanical studies. We conclude by encouraging similar research along these lines, but also the need for models that can describe and predict evolving tissue properties under many conditions, ranging from normal development to disease progression and wound healing. An important foundation for biomechanics and mechanobiology now exists and methods for collecting detailed multi-scale data continue to progress. There is, thus, considerable opportunity for continued advancement of mechanobiology and biomechanics.

Integration of cross-links, discrete fiber distributions and of a non-local theory in the Homogenized Constrained Mixture Model to Simulate Patient-Specific Thoracic Aortic Aneurysm Progression

Thoracic aortic aneurysms (TAA) represent a critical health issue for which computational models can significantly contribute to better understand the physiopathology. Among different computational frameworks, the Homogenized Constrained Mixture Theory has shown to be a computationally efficient option, allowing the inclusion of several mechanically significant constituents into a layer-specific mixture. Different patient-specific Growth and Remodeling (G&R) models correctly predicted TAA progression, although simplifications such as the inclusion of a limited number of collagen fibers and imposed boundary conditions might limit extensive analyses. The current study aims to enhance existing models by incorporating several discrete collagen fibers and to remove restrictive boundary conditions of the previous models. The implementation of discretized fiber dispersion presents a more realistic description of the vessel, while the removal of boundary conditions was addressed by including cross-links in the model to provide a supplemental stiffness against through-thickness shearing, a feature that was previously absent, and by the development of a non-local framework that ensures the stable deposition and degradation of collagen fibers. With these improvements, the current model represents a step forward towards more robust and comprehensive simulations of TAA growth.

Multiscale homogenized constrained mixture model of the bio-chemo-mechanics of soft tissue growth and remodeling

Constrained mixture models have successfully simulated many cases of growth and remodeling in soft biological tissues. So far, extensions of these models have been proposed to include either intracellular signaling or chemo-mechanical coupling on the organ-scale. However, no version of constrained mixture models currently exists that includes both aspects. Here, we propose such a version that resolves cellular signal processing by a set of logic-gated ordinary differential equations and captures chemo-mechanical interactions between cells by coupling a reaction-diffusion equation with the equations of nonlinear continuum mechanics. To demonstrate the potential of the model, we present 2 case studies within vascular solid mechanics: (i) the influence of angiotensin II on aortic growth and remodeling and (ii) the effect of communication between endothelial and intramural arterial cells via nitric oxide and endothelin-1.

Adaptive integration of history variables in constrained mixture models for organ-scale growth and remodeling

In the last decades, many computational models have been developed to predict soft tissue growth and remodeling (G&R). The constrained mixture theory describes fundamental mechanobiological processes in soft tissue G&R and has been widely adopted in cardiovascular models of G&R. However, even after two decades of work, large organ-scale models are rare, mainly due to high computational costs (model evaluation and memory consumption), especially in long-range simulations. We propose two strategies to adaptively integrate history variables in constrained mixture models to enable large organ-scale simulations of G&R. Both strategies exploit that the influence of deposited tissue on the current mixture decreases over time through degradation. One strategy is independent of external loading, allowing the estimation of the computational resources ahead of the simulation. The other adapts the history snapshots based on the local mechanobiological environment so that the additional integration errors can be controlled and kept negligibly small, even in G&R scenarios with severe perturbations. We analyze the adaptively integrated constrained mixture model on a tissue patch for a parameter study and show the performance under different G&R scenarios. To confirm that adaptive strategies enable large organ-scale examples, we show simulations of different hypertension conditions with a real-world example of a biventricular heart discretized with a finite element mesh. In our example, adaptive integrations sped up simulations by a factor of three and reduced memory requirements to one-sixth. The reduction of the computational costs gets even more pronounced for simulations over longer periods. Adaptive integration of the history variables allows studying more finely resolved models and longer G&R periods while computational costs are drastically reduced and largely constant in time.

Hemodynamics and wall shear metrics in a pulmonary autograft: Comparing a fluid-structure interaction and computational fluid dynamics approach

OBJECTIVE

In young patients, aortic valve disease is often treated by placement of a pulmonary autograft (PA) which adapts to its new environment through growth and remodeling. To better understand the hemodynamic forces acting on the highly distensible PA in the acute phase after surgery, we developed a fluid-structure interaction (FSI) framework and comprehensively compared hemodynamics and wall shear-stress (WSS) metrics with a computational fluid dynamic (CFD) simulation.

METHODS

The FSI framework couples a prestressed non-linear hyperelastic arterial tissue model with a fluid model using the in-house coupling code CoCoNuT. Geometry, material parameters and boundary conditions are based on in-vivo measurements. Hemodynamics, time-averaged WSS (TAWSS), oscillatory shear index (OSI) and topological shear variation index (TSVI) are evaluated qualitatively and quantitatively for 3 different sheeps.

RESULTS

Despite systolic-to-diastolic volumetric changes of the PA in the order of 20 %, the point-by-point correlation of TAWSS and OSI obtained through CFD and FSI remains high (r > 0.9, p < 0.01) for TAWSS and (r > 0.8, p < 0.01) for OSI). Instantaneous WSS divergence patterns qualitatively preserve similarities, but large deformations of the PA leads to a decrease of the correlation between FSI and CFD resolved TSVI (r < 0.7, p < 0.01). Moderate co-localization between FSI and CFD is observed for low thresholds of TAWSS and high thresholds of OSI and TSVI.

CONCLUSION

FSI might be warranted if we were to use the TSVI as a mechano-biological driver for growth and remodeling of PA due to varying intra-vascular flow structures and near wall hemodynamics because of the large expansion of the PA.

Growth and remodeling of the dissected membrane in an idealized dissected aorta model

While transitioning from the acute to chronic phase, the wall of a dissected aorta often expands in diameter and adaptations in thickness and microstructure take place in the dissected membrane. Including the mechanisms, leading to these changes, in a computational model is expected to improve the accuracy of predictions of the long-term complications and optimal treatment timing of dissection patients. An idealized dissected wall was modeled to represent the elastin and collagen production and/or degradation imposed by stress- and inflammation-mediated growth and remodeling, using the homogenized constrained mixture theory. As no optimal growth and remodeling parameters have been defined for aortic dissections, a Latin hypercube sampling with 1000 parameter combinations was assessed for four inflammation patterns, with a varying spatial extent (full/local) and temporal evolution (permanent/transient). The dissected membrane thickening and microstructure was considered together with the diameter expansion over a period of 90 days. The highest success rate was found for the transient inflammation patterns, with about 15% of the samples leading to converged solutions after 90 days. Clinically observed thickening rates were found for 2-4% of the transient inflammation samples, which represented median total diameter expansion rates of about 5 mm/year. The dissected membrane microstructure showed an elastin decrease and, in most cases, a collagen increase. In conclusion, the model with the transient inflammation pattern allowed the reproduction of clinically observed dissected membrane thickening rates, diameter expansion rates and adaptations in microstructure, thus providing guidance in reducing the parameter space in growth and remodeling models of aortic dissections.

A Fluid-Solid-Growth Solver for Cardiovascular Modeling

We implement full, three-dimensional constrained mixture theory for vascular growth and remodeling into a finite element fluid-structure interaction (FSI) solver. The resulting "fluid-solid-growth" (FSG) solver allows long term, patient-specific predictions of changing hemodynamics, vessel wall morphology, tissue composition, and material properties. This extension from short term (FSI) to long term (FSG) simulations increases clinical relevance by enabling mechanobioloigcally-dependent studies of disease progression in complex domains.