Constrained Mixture Model Research Articles

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.

Computational analysis of heart valve growth and remodeling after the Ross procedure.

During the Ross procedure, an aortic heart valve is replaced by a patient's own pulmonary valve. The pulmonary autograft subsequently undergoes substantial growth and remodeling (G&R) due to its exposure to increased hemodynamic loads. In this study, we developed a homogenized constrained mixture model to understand the observed adaptation of the autograft leaflets in response to the changed hemodynamic environment. This model was based on the hypothesis that tissue G&R aims to preserve mechanical homeostasis for each tissue constituent. To model the Ross procedure, we simulated the exposure of a pulmonary valve to aortic pressure conditions and the subsequent G&R of the valve. Specifically, we investigated the effects of assuming either stress- or stretch-based mechanical homeostasis, the use of blood pressure control, and the effect of root dilation. With this model, we could explain different observations from published clinical studies, such as the increase in thickness, change in collagen organization, and change in tissue composition. In addition, we found that G&R based on stress-based homeostasis could better capture the observed changes in tissue composition than G&R based on stretch-based homeostasis, and that root dilation or blood pressure control can result in more leaflet elongation. Finally, our model demonstrated that successful adaptation can only occur when the mechanically induced tissue deposition is sufficiently larger than tissue degradation, such that leaflet thickening overrules leaflet dilation. In conclusion, our findings demonstrated that G&R based on mechanical homeostasis can capture the observed heart valve adaptation after the Ross procedure. Finally, this study presents a novel homogenized mixture model that can be used to investigate other cases of heart valve G&R as well.

FSGe: A fast and strongly-coupled 3D fluid–solid-growth interaction method

Equilibrated fluid–solid-growth (FSGe) is a fast, open source, three-dimensional (3D) computational platform for simulating interactions between instantaneous hemodynamics and long-term vessel wall adaptation through mechanobiologically equilibrated growth and remodeling (G&R). Such models can capture evolving geometry, composition, and material properties in health and disease and following clinical interventions. In traditional G&R models, this feedback is modeled through highly simplified fluid solutions, neglecting local variations in blood pressure and wall shear stress (WSS). FSGe overcomes these inherent limitations by strongly coupling the 3D Navier–Stokes equations for blood flow with a 3D equilibrated constrained mixture model (CMMe) for vascular tissue G&R. CMMe allows one to predict long-term evolved mechanobiological equilibria from an original homeostatic state at a computational cost equivalent to that of a standard hyperelastic material model. In illustrative computational examples, we focus on the development of a stable aortic aneurysm in a mouse model to highlight key differences in growth patterns between FSGe and solid-only G&R models. We show that FSGe is especially important in blood vessels with asymmetric stimuli. Simulation results reveal greater local variation in fluid-derived WSS than in intramural stress (IMS). Thus, differences between FSGe and G&R models became more pronounced with the growing influence of WSS relative to pressure. Future applications in highly localized disease processes, such as for lesion formation in atherosclerosis, can now include spatial and temporal variations of WSS.

An inverse fitting strategy to determine the constrained mixture model parameters: application in patient-specific aorta.

The Constrained Mixture Model (CMM) is a novel approach to describe arterial wall mechanics, whose formulation is based on a referential physiological state. The CMM considers the arterial wall as a mixture of load-bearing constituents, each of them with characteristic mass fraction, material properties, and deposition stretch levels from its stress-free state to the in-vivo configuration. Although some reports of this model successfully assess its capabilities, they barely explore experimental approaches to model patient-specific scenarios. In this sense, we propose an iterative fitting procedure of numerical-experimental nature to determine material parameters and deposition stretch values. To this end, the model has been implemented in a finite element framework, and it is calibrated using reported experimental data of descending thoracic aorta. The main results obtained from the proposed procedure consist of a set of material parameters for each constituent. Moreover, a relationship between deposition stretches and residual strain measurements (opening angle and axial stretch) has been numerically proved, establishing a strong consistency between the model and experimental data.

Evolution of fiber distributions in homogenized constrained mixture models of soft tissue growth and remodeling: Uniaxial loading

Growth and remodeling in soft tissue changes, in general, the orientation distribution of the collagen fibers in the tissue. So far, the governing principles are still incompletely understood. In this paper, we derive from the homogenized constrained mixture theory of soft tissue growth and remodeling – under some simplifying assumptions – a theoretical framework that enables simple analytical solutions that help to understand the evolution of the collagen fiber distribution in soft tissues during growth and remodeling under uniaxial loading. Our analysis suggests a natural tendency of the collagen fiber distribution to form a peak in the direction of loading and a spread increasing with the ratio of the fiber half-life time and the time constant governing loading-related fiber production. This observation helps to understand why in vivo collagen fiber reinforcement is typically organized in the form of (often von Mises-like) distributions with a finite spread. Our analysis also suggests a considerable history dependence of the fiber distribution evolution, which may explain why, in tissues under the same type of loading, sometimes considerably different fiber orientation distributions are observed.

PDC-Net+: Enhanced Probabilistic Dense Correspondence Network

Establishing robust and accurate correspondences between a pair of images is a long-standing computer vision problem with numerous applications. While classically dominated by sparse methods, emerging dense approaches offer a compelling alternative paradigm that avoids the keypoint detection step. However, dense flow estimation is often inaccurate in the case of large displacements, occlusions, or homogeneous regions. In order to apply dense methods to real-world applications, such as pose estimation, image manipulation, or 3D reconstruction, it is therefore crucial to estimate the confidence of the predicted matches. We propose the Enhanced Probabilistic Dense Correspondence Network, PDC-Net+, capable of estimating accurate dense correspondences along with a reliable confidence map. We develop a flexible probabilistic approach that jointly learns the flow prediction and its uncertainty. In particular, we parametrize the predictive distribution as a constrained mixture model, ensuring better modelling of both accurate flow predictions and outliers. Moreover, we develop an architecture and an enhanced training strategy tailored for robust and generalizable uncertainty prediction in the context of self-supervised training. Our approach obtains state-of-the-art results on multiple challenging geometric matching and optical flow datasets. We further validate the usefulness of our probabilistic confidence estimation for the tasks of pose estimation, 3D reconstruction, image-based localization, and image retrieval. Code and models are available at https://github.com/PruneTruong/DenseMatching.

An updated Lagrangian constrained mixture model of pathological cardiac growth and remodelling.

Progressive left ventricular (LV) growth and remodelling (G&R) is often induced by volume and pressure overload, characterized by structural and functional adaptation through myocyte hypertrophy and extracellular matrix remodelling, which are dynamically regulated by biomechanical factors, inflammation, neurohormonal pathways, etc. When prolonged, it can eventually lead to irreversible heart failure. In this study, we have developed a new framework for modelling pathological cardiac G&R based on constrained mixture theory using an updated reference configuration, which is triggered by altered biomechanical factors to restore biomechanical homeostasis. Eccentric and concentric growth, and their combination have been explored in a patient-specific human LV model under volume and pressure overload. Eccentric growth is triggered by overstretching of myofibres due to volume overload, i.e. mitral regurgitation, whilst concentric growth is driven by excessive contractile stress due to pressure overload, i.e. aortic stenosis. Different biological constituent's adaptations under pathological conditions are integrated together, which are the ground matrix, myofibres and collagen network. We have shown that this constrained mixture-motivated G&R model can capture different phenotypes of maladaptive LV G&R, such as chamber dilation and wall thinning under volume overload, wall thickening under pressure overload, and more complex patterns under both pressure and volume overload. We have further demonstrated how collagen G&R would affect LV structural and functional adaption by providing mechanistic insight on anti-fibrotic interventions. This updated Lagrangian constrained mixture based myocardial G&R model has the potential to understand the turnover processes of myocytes and collagen due to altered local mechanical stimuli in heart diseases, and in providing mechanistic links between biomechanical factors and biological adaption at both the organ and cellular levels. Once calibrated with patient data, it can be used for assessing heart failure risk and designing optimal treatment therapies. STATEMENT OF SIGNIFICANCE: Computational modelling of cardiac G&R has shown high promise to provide insight into heart disease management when mechanistic understandings are quantified between biomechanical factors and underlying cellular adaptation processes. The kinematic growth theory has been dominantly used to phenomenologically describe the biological G&R process but neglecting underlying cellular mechanisms. We have developed a constrained mixture based G&R model with updated reference by taking into account different mechanobiological processes in the ground matrix, myocytes and collagen fibres. This G&R model can serve as a basis for developing more advanced myocardial G&R models further informed by patient data to assess heart failure risk, predict disease progression, select the optimal treatment by hypothesis testing, and eventually towards a truly precision cardiology using in-silico models.

A biochemomechanical model of collagen turnover in arterial adaptations to hemodynamic loading.

The production, removal, and remodeling of fibrillar collagen is fundamental to mechanical homeostasis in arteries, including dynamic morphological and microstructural changes that occur in response to sustained changes in blood flow and pressure under physiological conditions. These dynamic processes involve complex, coupled biological, chemical, and mechanical mechanisms that are not completely understood. Nevertheless, recent simulations using constrained mixture models with phenomenologically motivated constitutive relations have proven able to predict salient features of the progression of certain vascular adaptations as well as disease processes. Collagen turnover is modeled, in part, via stress-dependent changes in collagen half-life, typically within the range of 10-70days. By contrast, in this work we introduce a biochemomechanical approach to model the cellular synthesis of procollagen as well as its transition from an intermediate state of assembled microfibrils to mature cross-linked fibers, with mechano-regulated removal. The resulting model can simulate temporal changes in geometry, composition, and stress during early vascular adaptation (weeks to months) for modest changes in blood flow or pressure. It is shown that these simulations capture salient features from data presented in the literature from different animal models.

A homogenized constrained mixture model of cardiac growth and remodeling: analyzing mechanobiological stability and reversal.

Cardiac growth and remodeling (G&R) patterns change ventricular size, shape, and function both globally and locally. Biomechanical, neurohormonal, and genetic stimuli drive these patterns through changes in myocyte dimension and fibrosis. We propose a novel microstructure-motivated model that predicts organ-scale G&R in the heart based on the homogenized constrained mixture theory. Previous models, based on the kinematic growth theory, reproduced consequences of G&R in bulk myocardial tissue by prescribing the direction and extent of growth but neglected underlying cellular mechanisms. In our model, the direction and extent of G&R emerge naturally from intra- and extracellular turnover processes in myocardial tissue constituents and their preferred homeostatic stretch state. We additionally propose a method to obtain a mechanobiologically equilibrated reference configuration. We test our model on an idealized 3D left ventricular geometry and demonstrate that our model aims to maintain tensional homeostasis in hypertension conditions. In a stability map, we identify regions of stable and unstable G&R from an identical parameter set with varying systolic pressures and growth factors. Furthermore, we show the extent of G&R reversal after returning the systolic pressure to baseline following stage 1 and 2 hypertension. A realistic model of organ-scale cardiac G&R has the potential to identify patients at risk of heart failure, enable personalized cardiac therapies, and facilitate the optimal design of medical devices.

Abstract 359: Multiscale Model Reveals Importance Of Cumulative Vascular Smooth Muscle Cell Responses In Aortic Hypercontractility

Diabetes more than doubles the risk of cardiovascular disease (CVD), and the chronic hyperglycemia leads to vascular maladaptations prior to the diagnostic threshold. While endothelium-independent vascular smooth muscle cell (VSMC) hypercontractility is well-established at the tissue-scale, the mechanical phenotype is inconsistently observed at the cell-scale. Our strategy consisted of experimental characterization at the tissue-scale via an isometric ring contraction assay of WT and OGT KO aortas, the latter an established chronic hyperglycemia model, from male mice and at the cell-scale via published cellular microbiaxial stretching data, coupled with the construction of a multiscale computational model. We hypothesize that the biochemical signaling changes due to increased blood glucose are sufficient to predict VSMC hypercontractility. The multiscale model operates off a biochemical signaling network that determines the activation and contraction of actin stress fibers. The stress fibers are connected in either 2D (cell-scale) or 3D (tissue-scale) cytoskeletal networks where for the tissue-scale, the cytoskeletal mechanics are fit to a macroscale constrained mixture model. The multiscale model correctly predicts the cell-scale euglycemic contractility, however the hyperglycemic condition is too subtle to be resolved at this scale. At the tissue-scale, the model correctly predicts both euglycemic and hyperglycemic active force generation. The model does not currently consider ECM alterations, so the changes in passive tissue mechanics cannot be predicted. The model predictions suggest that while the impetus of hyperglycemia-driven hypercontractility occurs at the cell-scale, these small-scale changes must be amplified across length scales to be experimentally-observable. These results demonstrate the power of the multiscale model to connect cell- and tissue-scale vascular mechanical phenotypes due to chronic hyperglycemia.

A multiscale computational model of arterial growth and remodeling including Notch signaling.

Blood vessels grow and remodel in response to mechanical stimuli. Many computational models capture this process phenomenologically, by assuming stress homeostasis, but this approach cannot unravel the underlying cellular mechanisms. Mechano-sensitive Notch signaling is well-known to be key in vascular development and homeostasis. Here, we present a multiscale framework coupling a constrained mixture model, capturing the mechanics and turnover of arterial constituents, to a cell-cell signaling model, describing Notch signaling dynamics among vascular smooth muscle cells (SMCs) as influenced by mechanical stimuli. Tissue turnover was regulated by both Notch activity, informed by in vitro data, and a phenomenological contribution, accounting for mechanisms other than Notch. This novel framework predicted changes in wall thickness and arterial composition in response to hypertension similar to previous in vivo data. The simulations suggested that Notch contributes to arterial growth in hypertension mainly by promoting SMC proliferation, while other mechanisms are needed to fully capture remodeling. The results also indicated that interventions to Notch, such as external Jagged ligands, can alter both the geometry and composition of hypertensive vessels, especially in the short term. Overall, our model enables a deeper analysis of the role of Notch and Notch interventions in arterial growth and remodeling and could be adopted to investigate therapeutic strategies and optimize vascular regeneration protocols.

An Updated Lagrangian Constrained Mixture Model of Pathological Cardiac Growth and Remodelling

An Updated Lagrangian Constrained Mixture Model of Pathological Cardiac Growth and Remodelling

Remodeling of the uterine artery during and early after pregnancy in the mouse.

Pregnancy associates with dramatic changes in maternal cardiovascular physiology that ensure that the utero-placental circulation can support the developing fetus. Particularly striking is the marked flow-induced remodeling of uterine arteries during pregnancy and their recovery following birth. Whereas details are available in the literature on alterations in hemodynamics within and changes in the dimensions of uterine arteries during and following pregnancy in mice, we report here the first biaxial biomechanical phenotyping of these arteries during this dynamic period of growth and remodeling (G&R). To gain additional insight into the measured G&R, we also use a computational constrained mixture model to describe and predict findings, including simulations related to complications that may arise during pregnancy. It is found that dramatic pregnancy-induced remodeling of the uterine artery is largely, but not completely, reversed in the postpartum period, which appears to be driven by increases in collagen turnover among other intramural changes. By contrast, data on the remodeling of the ascending aorta, an elastic artery, reveal modest changes that are fully recovered postpartum. There is strong motivation to continue biomechanical studies on this critical aspect of women's health, which has heretofore not received appropriate consideration from the biomechanics community.

Region-dependent mechanical characterization of porcine thoracic aorta with a one-to-many correspondence method to create virtual datasets using uniaxial tensile tests.

The simulation of the cardiovascular system and in silico clinical trials have garnered attention in the biomedical engineering field. Physics-based modeling is essential to associate with physical and clinical features. In physics-based constitutive modeling, the identification of the parameters and estimation of their ranges based on appropriate experiments are required. Uniaxial tests are commonly used in the field of vascular mechanics, but they have limitations in fully characterizing the regional mechanical behavior of the aorta. Therefore, this study is aimed at identifying a method to integrate constitutive models with experimental data to elucidate regional aortic behavior. To create a virtual two-dimensional dataset, a pair of uniaxial experimental datasets in the longitudinal and circumferential directions was combined using a one-to-many correspondence method such as bootstrap aggregation. The proposed approach is subsequently applied to three constitutive models, i.e., the Fung model, Holzapfel model, and constrained mixture model, to estimate the material parameters based on the four test regions of the porcine thoracic aorta. Finally, the regional difference in the mechanical behavior of the aorta, the correlation between the experimental characteristics and model parameters, and the inter-correlation of the material parameters are confirmed. This integrative approach will enhance the prediction capability of the model with respect to the regions of the aorta.

Neural operator learning of heterogeneous mechanobiological insults contributing to aortic aneurysms.

Thoracic aortic aneurysm (TAA) is a localized dilatation of the aorta that can lead to life-threatening dissection or rupture. In vivo assessments of TAA progression are largely limited to measurements of aneurysm size and growth rate. There is promise, however, that computational modelling of the evolving biomechanics of the aorta could predict future geometry and properties from initiating mechanobiological insults. We present an integrated framework to train a deep operator network (DeepONet)-based surrogate model to identify TAA contributing factors using synthetic finite-element-based datasets. For training, we employ a constrained mixture model of aortic growth and remodelling to generate maps of local aortic dilatation and distensibility for multiple TAA risk factors. We evaluate the performance of the surrogate model for insult distributions varying from fusiform (analytically defined) to complex (randomly generated). We propose two frameworks, one trained on sparse information and one on full-field greyscale images, to gain insight into a preferred neural operator-based approach. We show that this continuous learning approach can predict the patient-specific insult profile associated with any given dilatation and distensibility map with high accuracy, particularly when based on full-field images. Our findings demonstrate the feasibility of applying DeepONet to support transfer learning of patient-specific inputs to predict TAA progression.

About prestretch in homogenized constrained mixture models simulating growth and remodeling in patient-specific aortic geometries

Evolution of mechanical and structural properties in the Ascending Thoracic Aorta (ATA) is the results of complex mechanobiological processes. In this work, we address some numerical challenges in order to elaborate computational models of these processes. For that, we extend the state of the art of homogenized constrained mixture (hCM) models. In these models, prestretches are assigned to the mixed constituents in order to ensure local mechanical equilibrium macroscopically, and to maintain a homeostatic level of tension in collagen fibers microscopically. Although the initial prestretches were assumed as homogeneous in idealized straight tubes, more elaborate prestretch distributions need to be considered for curved geometrical models such as patient-specific ATA. Therefore, we introduce prestretches having a three-dimensional gradient across the ATA geometry in the homeostatic reference state. We test different schemes with the objective to ensure stable growth and remodeling (G&R) simulations on patient-specific curved vessels. In these simulations, aneurysm progression is triggered by tissue changes in the constituents such as mass degradation of intramural elastin. The results show that the initial prestretches are not only critical for the stability of numerical simulations, but they also affect the G&R response. Eventually, we submit that initial conditions required for G&R simulations need to be identified regionally for ensuring realistic patient-specific predictions of aneurysm progression.

Random Walks with Local Memory

A solution manifold is the collection of points in a $d$-dimensional space satisfying a system of $s$ equations with $s<d$. Solution manifolds occur in several statistical problems including hypothesis testing, curved-exponential families, constrained mixture models, partial identifications, and nonparametric set estimation. We analyze solution manifolds both theoretically and algorithmically. In terms of theory, we derive five useful results: the smoothness theorem, the stability theorem (which implies the consistency of a plug-in estimator), the convergence of a gradient flow, the local center manifold theorem and the convergence of the gradient descent algorithm. To numerically approximate a solution manifold, we propose a Monte Carlo gradient descent algorithm. In the case of likelihood inference, we design a manifold constraint maximization procedure to find the maximum likelihood estimator on the manifold. We also develop a method to approximate a posterior distribution defined on a solution manifold.

Global Sensitivity Analysis of a Homogenized Constrained Mixture Model of Arterial Growth and Remodeling

Growth and remodeling in arterial tissue have attracted considerable attention over the last decade. Mathematical models have been proposed, and computational studies with these have helped to understand the role of the different model parameters. So far it remains, however, poorly understood how much of the model output variability can be attributed to the individual input parameters and their interactions. To clarify this, we propose herein a global sensitivity analysis, based on Sobol indices, for a homogenized constrained mixture model of aortic growth and remodeling. In two representative examples, we found that 54–80% of the long term output variability resulted from only three model parameters. In our study, the two most influential parameters were the one characterizing the ability of the tissue to increase collagen production under increased stress and the one characterizing the collagen half-life time. The third most influential parameter was the one characterizing the strain-stiffening of collagen under large deformation. Our results suggest that in future computational studies it may - at least in scenarios similar to the ones studied herein - suffice to use population average values for the other parameters. Moreover, our results suggest that developing methods to measure the said three most influential parameters may be an important step towards reliable patient-specific predictions of the enlargement of abdominal aortic aneurysms in clinical practice.

A homogenized constrained mixture model of restenosis and vascular remodelling after balloon angioplasty.

Restenosis is one of the main adverse effects of the treatment of atherosclerosis through balloon angioplasty or stenting. During the intervention, the arterial wall is overstretched, causing a cascade of cellular events and subsequent neointima formation. This mechanical stimulus and its mechanobiological effects can be reproduced in biomechanical simulations. The aim of these models is to predict the long-term outcome of these procedures, to help increase the understanding of restenosis formation and to allow for in silico optimization of the treatment. We propose a predictive finite-element model of restenosis, using the homogenized constrained mixture modelling framework designed to model growth and remodelling in soft tissues. We compare the results with clinical observations in human coronary arteries and experimental findings in non-human primate models. We also explore the model's clinical relevance by testing its response to different balloon loads and to the use of drug-eluting balloons. The comparison of the results with experimental data shows the relevance of the model. We show its ability to predict both inward and outward remodelling as observed in vivo and we show the importance of an improved understanding of restenosis formation from a biomechanical point of view.