Constrained Mixture Model Research Articles

Inverse modeling framework for characterizing patient-specific microstructural changes in the pulmonary arteries

Microstructural changes in the pulmonary arteries associated with pulmonary arterial hypertension (PAH) is not well understood and characterized in humans. To address this issue, we developed and applied a patient-specific inverse finite element (FE) modeling framework to characterize mechanical and structural changes of the micro-constituents in the proximal pulmonary arteries using in-vivo pressure measurements and magnetic resonance images. The framework was applied using data acquired from a pediatric PAH patient and a heart transplant patient with normal pulmonary arterial pressure, which serves as control. Parameters of a constrained mixture model that are associated with the structure and mechanical properties of elastin, collagen fibers and smooth muscle cells were optimized to fit the patient-specific pressure-diameter responses of the main pulmonary artery. Based on the optimized parameters, individual stress and linearized stiffness resultants of the three tissue constituents, as well as their aggregated values, were estimated in the pulmonary artery. Aggregated stress resultant and stiffness are, respectively, 4.6 and 3.4 times higher in the PAH patient than the control subject. Stress and stiffness resultants of each tissue constituent are also higher in the PAH patient. Specifically, the mean stress resultant is highest in elastin (PAH: 69.96, control: 14.42 kPa-mm), followed by those in smooth muscle cell (PAH: 13.95, control: 4.016 kPa-mm) and collagen fibers (PAH: 13.19, control: 2.908 kPa-mm) in both the PAH patient and the control subject. This result implies that elastin may be the key load-bearing constituent in the pulmonary arteries of the PAH patient and the control subject.

Constrained Mixture Models of Soft Tissue Growth and Remodeling - Twenty Years After.

Soft biological tissues compromise diverse cell types and extracellular matrix constituents, each of which can possess individual natural configurations, material properties, and rates of turnover. For this reason, mixture-based models of growth (changes in mass) and remodeling (change in microstructure) are well-suited for studying tissue adaptations, disease progression, and responses to injury or clinical intervention. Such approaches also can be used to design improved tissue engineered constructs to repair, replace, or regenerate tissues. Focusing on blood vessels as archetypes of soft tissues, this paper reviews a constrained mixture theory introduced twenty years ago and explores its usage since by contrasting simulations of diverse vascular conditions. The discussion is framed within the concept of mechanical homeostasis, with consideration of solid-fluid interactions, inflammation, and cell signaling highlighting both past accomplishments and future opportunities as we seek to understand better the evolving composition, geometry, and material behaviors of soft tissues under complex conditions.

From Transcript to Tissue: Multiscale Modeling from Cell Signaling to Matrix Remodeling.

Tissue-level biomechanical properties and function derive from underlying cell signaling, which regulates mass deposition, organization, and removal. Here, we couple two existing modeling frameworks to capture associated multiscale interactions-one for vessel-level growth and remodeling and one for cell-level signaling-and illustrate utility by simulating aortic remodeling. At the vessel level, we employ a constrained mixture model describing turnover of individual wall constituents (elastin, intramural cells, and collagen), which has proven useful in predicting diverse adaptations as well as disease progression using phenomenological constitutive relations. Nevertheless, we now seek an improved mechanistic understanding of these processes; we replace phenomenological relations in the mixture model with a logic-based signaling model, which yields a system of ordinary differential equations predicting changes in collagen synthesis, matrix metalloproteinases, and cell proliferation in response to altered intramural stress, wall shear stress, and exogenous angiotensin II. This coupled approach promises improved understanding of the role of cell signaling in achieving tissue homeostasis and allows us to model feedback between vessel mechanics and cell signaling. We verify our model predictions against data from the hypertensive murine infrarenal abdominal aorta as well as results from validated phenomenological models, and consider effects of noisy signaling and heterogeneous cell populations.

A Weighted Spatially Constrained Finite Mixture Model for Image Segmentation

Spatially Constrained Mixture Model (SCMM) is an image segmentation model that works over the framework of maximum a-posteriori and Markov Random Field (MAP-MRF). It developed its own maximization step to be used within this framework. This research has proposed an improvement in the SCMM’s maximization step for segmenting simulated brain Magnetic Resonance Images (MRIs). The improved model is named as the Weighted Spatially Constrained Finite Mixture Model (WSCFMM). To compare the performance of SCMM and WSCFMM, simulated T1-Weighted normal MRIs were segmented. A region of interest (ROI) was extracted from segmented images. The similarity level between the extracted ROI and the ground truth (GT) was found by using the Jaccard and Dice similarity measuring method. According to the Jaccard similarity measuring method, WSCFMM showed an overall improvement of 4.72%, whereas the Dice similarity measuring method provided an overall improvement of 2.65% against the SCMM. Besides, WSCFMM significantly stabilized and reduced the execution time by showing an improvement of 83.71%. The study concludes that WSCFMM is a stable model and performs better as compared to the SCMM in noisy and noise-free environments.

Class-Agnostic Weighted Normalization of Staining in Histopathology Images Using a Spatially Constrained Mixture Model.

The colorless biopsied tissue samples are usually stained in order to visualize different microscopic structures for diagnostic purposes. But color variations associated with the process of sample preparation, usage of raw materials, diverse staining protocols, and using different slide scanners may adversely influence both visual inspection and computer-aided image analysis. As a result, many methods are proposed for histopathology image stain normalization in recent years. In this study, we introduce a novel approach for stain normalization based on learning a mixture of multivariate skew-normal distributions for stain clustering and parameter estimation alongside a stain transformation technique. The proposed method, labeled "Class-Agnostic Weighted Normalization" (short CLAW normalization), has the ability to normalize a source image by learning the color distribution of both source and target images within an expectation-maximization framework. The novelty of this approach is its flexibility to quantify the underlying both symmetric and nonsymmetric distributions of the different stain components while it is considering the spatial information. The performance of this new stain normalization scheme is tested on several publicly available digital pathology datasets to compare it against state-of-the-art normalization algorithms in terms of ability to preserve the image structure and information. All in all, our proposed method performed superior more consistently in comparison with existing methods in terms of information preservation, visual quality enhancement, and boosting computer-aided diagnosis algorithm performance.

Efficient obstacle detection based on prior estimation network and spatially constrained mixture model for unmanned surface vehicles

AbstractRecently, spatially constrained mixture model has become the mainstream method for the task of vision‐based obstacle detection in unmanned surface vehicles (USVs), and has shown its potential of modeling the semantic structure of the marine environment. However, the expectation maximization (EM) optimization of this model is quite sensitive to initial values and easily falls into a local optimal solution in the presence of significant rolling and pitching in rough seas. In addition, existing methods based on spatially constrained mixture model are susceptible to false positives in the presence of sun glitter. In this paper, a prior estimation network (PEN) is proposed to improve the mixture model, which together enable reliable monocular obstacle detection for USVs. We develop a weakly supervised E‐step to train the PEN for learning the semantic structure of marine images and estimating initial class priors in obstacle detection. To mitigate the influence of poor initial parameters on the convergence of EM optimization, we use the priors estimated by the PEN to calculate the initial parameters of the mixture model and automatically adjust the hyper priors on the semantic components in the mixture model. The output of the PEN is also applied to set the probability values of the outlier component in the mixture model, aiming to reduce false positives caused by sun glitter. Experimental results show that our approach outperforms the current state‐of‐the‐art monocular method by 15% improvement in sea edge estimation and a 3.3% increase in F‐score on the marine obstacle detection data set, as well as 69.5% improvement in sea edge estimation and a 39.2% increase in F‐score on our data set, while running over 40 fps.

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

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.

Three-dimensional numerical simulation of soft-tissue wound healing using constrained-mixture anisotropic hyperelasticity and gradient-enhanced damage mechanics.

Healing of soft biological tissues is the process of self-recovery or self-repair after injury or damage to the extracellular matrix (ECM). In this work, we assume that healing is a stress-driven process, which works at recovering a homeostatic stress metric in the tissue by replacing the damaged ECM with a new undamaged one. For that, a gradient-enhanced continuum healing model is developed for three-dimensional anisotropic tissues using the modified anisotropic Holzapfel-Gasser-Ogden constitutive model. An adaptive stress-driven approach is proposed for the deposition of new collagen fibres during healing with orientations assigned depending on the principal stress direction. The intrinsic length scales of soft tissues are considered through the gradient-enhanced term, and growth and remodelling are simulated by a constrained-mixture model with temporal homogenization. The proposed model is implemented in the finite-element package Abaqus by means of a user subroutine UEL. Three numerical examples have been achieved to illustrate the performance of the proposed model in simulating the healing process with various damage situations, converging towards stress homeostasis. The orientations of newly deposited collagen fibres and the sensitivity to intrinsic length scales are studied through these examples, showing that both have a significant impact on temporal evolutions of the stress distribution and on the size of the damage region. Applications of the approach to carry out in silico experiments of wound healing are promising and show good agreement with existing experiment results.

OCT Image Denoising Based on Asymmetric Normal Laplace Mixture Model.

Optical Coherence Tomography (OCT) is one of the well-known imaging systems in ophthalmology that provides images with high resolution from retinal tissue. However, like other coherent imaging systems, OCT images suffer from speckle noise which decreases the image quality. Denoising can be considered as an estimation problem in a Bayesian framework. So, finding a suitable distribution for noiseless data is an important issue. We propose a statistical model for OCT data, namely Asymmetric Normal Laplace Mixture Model (ANLMM), and then convert its distribution to normal by Gaussianization Transform (GT). Finally, by applying the Spatially Constrained Gaussian Mixture Model (SC-GMM), a new OCT denoising algorithm is introduced, which significantly outperforms the other methods in terms of Contrast-to-Noise Ratio (CNR).

Patient-specific predictions of aneurysm growth and remodeling in the ascending thoracic aorta using the homogenized constrained mixture model.

In its permanent quest of mechanobiological homeostasis, our vasculature significantly adapts across multiple length and timescales in various physiological and pathological conditions. Computational modeling of vascular growth and remodeling (G&R) has significantly improved our insights into the mechanobiological processes of diseases such as hypertension or aneurysms. However, patient-specific computational modeling of ascending thoracic aortic aneurysm (ATAA) evolution, based on finite element models (FEM), remains a challenging scientific problem with rare contributions, despite the major significance of this topic of research. Challenges are related to complex boundary conditions and geometries combined with layer-specific G&R responses. To address these challenges, in the current paper, we employed the constrained mixture model (CMM) to model the arterial wall as a mixture of different constituents such as elastin, collagen fiber families and smooth muscle cells. Implemented in Abaqus as a UMAT, this first patient-specific CMM-based FEM of G&R in human ATAA was first validated for canonical problems such as single-layer thick-wall cylindrical and bilayer thick-wall toric arterial geometries. Then it was used to predict ATAA evolution for a patient-specific aortic geometry, showing that the typical shape of an ATAA can be simply produced by elastin proteolysis localized in regions of deranged hemodymanics. The results indicate a transfer of stress to the adventitia by elastin loss and continuous adaptation of the stress distribution due to change in ATAA shape. Moreover, stress redistribution leads to collagen deposition where the maximum elastin mass is lost, which in turn leads to stiffening of the arterial wall. As future work, the predictions of this G&R framework will be validated on datasets of patient-specific ATAA geometries followed up over a significant number of years.

Constrained mixture modeling affects material parameter identification from planar biaxial tests

The constrained mixture theory is an elegant way to incorporate the phenomenon of residual stresses in patient-specific finite element models of arteries. This theory assumes an in vivo reference geometry, obtained from medical imaging, and constituent-specific deposition stretches in the assumed reference state. It allows to model residual stresses and prestretches in arteries without the need for a stress-free reference configuration, most often unknown in patient-specific modeling.A finite element (FE) model requires material parameters, which are classically obtained by fitting the constitutive model to experimental data. The characterization of arterial tissue is often based on planar biaxial test data, to which nonlinear elastic fiber-reinforced material parameters are fitted. However, the introduction of the constrained mixture theory requires an adapted approach to parameter fitting. Therefore, we introduce an iterative fitting method, alternating between nonlinear least squares parameter optimization and an FE prestressing algorithm to obtain the correct constrained mixture material state during the mechanical test.We verify the method based on numerically constructed planar biaxial test data sets, containing ground truth sets of material parameters. The results show that the method converges to the correct parameter sets in just a few iterations. Next, the iterative fitting approach is applied to planar biaxial test data of ovine pulmonary artery tissue. The obtained results demonstrate a convergence towards constrained mixture compatible parameters, which differ significantly from classically obtained parameters.We show that this new modeling approach yields in vivo wall stresses similar to when using classically obtained parameters. However, due to the numerous advantages of constrained mixture modeling, our fitting method is relevant to obtain compatible material parameters, that may not be confused with parameters obtained in a classical way.

A Mechanobiologically Equilibrated Constrained Mixture Model for Growth and Remodeling of Soft Tissues.

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.

Model-Based segmentation of image data using spatially constrained mixture models

Abstract In this paper, a novel Bayesian statistical approach is proposed to tackle the problem of natural image segmentation. The proposed approach is based on finite Dirichlet mixture models in which contextual proportions (i.e., the probabilities of class labels) are modeled with spatial smoothness constraints. The major merits of our approach are summarized as follows: Firstly, it exploits the Dirichlet mixture model which can obtain a better statistical performance than commonly used mixture models (such as the Gaussian mixture model), especially for proportional data (i.e, normalized histogram). Secondly, it explicitly models the mixing contextual proportions as probability vectors and simultaneously integrate spatial relationship between pixels into the Dirichlet mixture model, which results in a more robust framework for image segmentation. Finally, we develop a variational Bayes learning method to update the parameters in a closed-form expression. The effectiveness of the proposed approach is compared with other mixture modeling-based image segmentation approaches through extensive experiments that involve both simulated and natural color images.

Reactive Constrained Mixtures for Modeling the Solid Matrix of Biological Tissues.

This article illustrates our approach for modeling the solid matrix of biological tissues using reactive constrained mixtures. Several examples are presented to highlight the potential benefits of this approach, showing that seemingly disparate fields of mechanics and chemical kinetics are actually closely interrelated and may be elegantly expressed in a unified framework. Thus, constrained mixture models recover classical theories for fibrous materials with bundles oriented in different directions or having different reference configurations, that produce characteristic fiber recruitment patterns under loading. Reactions that exchange mass among various constituents of a mixture may be used to describe tissue growth and remodeling, which may also alter the material's anisotropy. Similarly, reactions that describe the breaking and reforming of bonds may be used to model free energy dissipation in a viscoelastic material. Therefore, this framework is particularly well suited for modeling biological tissues.

Homogenized constrained mixture models for anisotropic volumetric growth and remodeling.

Constrained mixture models for soft tissue growth and remodeling have attracted increasing attention over the last decade. They can capture the effects of the simultaneous presence of multiple constituents that are continuously deposited and degraded at in general different rates, which is important to understand essential features of living soft tissues that cannot be captured by simple kinematic growth models. Recently the novel concept of homogenized constrained mixture models was introduced. It was shown that these models produce results which are very similar (and in certain limit cases even identical) to the ones of constrained mixture models based on multi-network theory. At the same time, the computational cost and complexity of homogenized constrained mixture models are much lower. This paper discusses the theory and implementation of homogenized constrained mixture models for anisotropic volumetric growth and remodeling in three dimensions. Previous constrained mixture models of volumetric growth in three dimensions were limited to the special case of isotropic growth. By numerical examples, comparison with experimental data and a theoretical discussion, we demonstrate that there is some evidence raising doubts whether isotropic growth models are appropriate to represent growth and remodeling of soft tissue in the vasculature. Anisotropic constrained mixture models, as introduced in this paper for the first time, may be required to avoid unphysiological results in simulations of vascular growth and remodeling.

Spatially Constrained Mixture Model and Image Segmentation: A Review

The mixture model is a commonly used approach for image segmentation. However, it doesn’t consider the spatial information. In order to overcome this disadvantage, several spatially constrained mixture models have been proposed. In this paper, these spatially constrained mixture models and their experimental results on synthetic and real world images are presented. These experimental results demonstrate that the spatially constrained mixture models can achieve competitive performance compared to the standard mixture model.

Stability analysis of a continuum-based constrained mixture model for vascular growth and remodeling.

A stabilizing criterion is derived for equations governing vascular growth and remodeling. We start from the integral state equations of the continuum-based constrained mixture theory of vascular growth and remodeling and obtain a system of time-delayed differential equations describing vascular growth. By employing an exponential form of the constituent survival function, the delayed differential equations can be reduced to a nonlinear ODE system. We demonstrate the degeneracy of the linearized system about the homeostatic state, which is a fundamental cause of the neutral stability observations reported in prior studies. Due to this degeneracy, stability conclusions for the original nonlinear system cannot be directly inferred. To resolve this problem, a sub-system is constructed by recognizing a linear relation between two states. Subsequently, Lyapunov's indirect method is used to connect stability properties between the linearized system and the original nonlinear system, to rigorously establish the neutral stability properties of the original system. In particular, this analysis leads to a stability criterion for vascular expansion in terms of growth and remodeling kinetic parameters, geometric quantities and material properties. Numerical simulations were conducted to evaluate the theoretical stability criterion under broader conditions, as well as study the influence of key parameters and physical factors on growth properties. The theoretical results are also compared with prior numerical and experimental findings in the literature.

A homogenized constrained mixture (and mechanical analog) model for growth and remodeling of soft tissue.

Most mathematical models of the growth and remodeling of load-bearing soft tissues are based on one of two major approaches: a kinematic theory that specifies an evolution equation for the stress-free configuration of the tissue as a whole or a constrained mixture theory that specifies rates of mass production and removal of individual constituents within stressed configurations. The former is popular because of its conceptual simplicity, but relies largely on heuristic definitions of growth; the latter is based on biologically motivated micromechanical models, but suffers from higher computational costs due to the need to track all past configurations. In this paper, we present a temporally homogenized constrained mixture model that combines advantages of both classical approaches, namely a biologically motivated micromechanical foundation, a simple computational implementation, and low computational cost. As illustrative examples, we show that this approach describes well both cell-mediated remodeling of tissue equivalents in vitro and the growth and remodeling of aneurysms in vivo. We also show that this homogenized constrained mixture model suggests an intimate relationship between models of growth and remodeling and viscoelasticity. That is, important aspects of tissue adaptation can be understood in terms of a simple mechanical analog model, a Maxwell fluid (i.e., spring and dashpot in series) in parallel with a "motor element" that represents cell-mediated mechanoregulation of extracellular matrix. This analogy allows a simple implementation of homogenized constrained mixture models within commercially available simulation codes by exploiting available models of viscoelasticity.

Spatially constrained mixture model via energy minimization and its application to image segmentation

A mixture model with spatial constraint is proposed for image segmentation. This model assumes that the pixel label priors are similar if the pixels are close in geometry. An energy function is defined on the spatial space for measuring the spatial information. We also derive an energy function on the observed data space from the log-likelihood function of the standard mixture model. We estimate the model parameters by minimizing the combination of the two energy functions, using the gradient descent algorithm. Then we use the parameters to compute the posterior probability. Finally, each pixel can be assigned to a class using the maximum a posterior decision rule. Numerical experiments are presented where the proposed method and other mixture model-based methods are tested on synthetic and real-world images. These experimental results demonstrate that the proposed method achieves competitive performance compared with other spatially constrained mixture model-based methods.

Image Retrieval Based on Multiview Constrained Nonnegative Matrix Factorization and Gaussian Mixture Model Spectral Clustering Method

Content-based image retrieval has recently become an important research topic and has been widely used for managing images from repertories. In this article, we address an efficient technique, called MNGS, which integrates multiview constrained nonnegative matrix factorization (NMF) and Gaussian mixture model- (GMM-) based spectral clustering for image retrieval. In the proposed methodology, the multiview NMF scheme provides competitive sparse representations of underlying images through decomposition of a similarity-preserving matrix that is formed by fusing multiple features from different visual aspects. In particular, the proposed method merges manifold constraints into the standard NMF objective function to impose an orthogonality constraint on the basis matrix and satisfy the structure preservation requirement of the coefficient matrix. To manipulate the clustering method on sparse representations, this paper has developed a GMM-based spectral clustering method in which the Gaussian components are regrouped in spectral space, which significantly improves the retrieval effectiveness. In this way, image retrieval of the whole database translates to a nearest-neighbour search in the cluster containing the query image. Simultaneously, this study investigates the proof of convergence of the objective function and the analysis of the computational complexity. Experimental results on three standard image datasets reveal the advantages that can be achieved with the proposed retrieval scheme.