Constrained Mixture Model Research Articles

Energy Minimization-Based Spatially Constrained Mixture Model and its Application to Image Segmentation

Energy Minimization-Based Spatially Constrained Mixture Model and its Application to Image Segmentation

Prior Distributions of Material Parameters for Bayesian Calibration of Growth and Remodeling Computational Model of Abdominal Aortic Wall.

For the accurate prediction of the vascular disease progression, there is a crucial need for developing a systematic tool aimed toward patient-specific modeling. Considering the interpatient variations, a prior distribution of model parameters has a strong influence on computational results for arterial mechanics. One crucial step toward patient-specific computational modeling is to identify parameters of prior distributions that reflect existing knowledge. In this paper, we present a new systematic method to estimate the prior distribution for the parameters of a constrained mixture model using previous biaxial tests of healthy abdominal aortas (AAs). We investigate the correlation between the estimated parameters for each constituent and the patient's age and gender; however, the results indicate that the parameters are correlated with age only. The parameters are classified into two groups: Group-I in which the parameters ce,ck1,ck2,cm2,Ghc, and ϕe are correlated with age, and Group-II in which the parameters cm1,Ghm,G1e,G2e, and α are not correlated with age. For the parameters in Group-I, we used regression associated with age via linear or inverse relations, in which their prior distributions provide conditional distributions with confidence intervals. For Group-II, the parameter estimated values were subjected to multiple transformations and chosen if the transformed data had a better fit to the normal distribution than the original. This information improves the prior distribution of a subject-specific model by specifying parameters that are correlated with age and their transformed distributions. Therefore, this study is a necessary first step in our group's approach toward a Bayesian calibration of an aortic model. The results from this study will be used as the prior information necessary for the initialization of Bayesian calibration of a computational model for future applications.

A triphasic constrained mixture model of engineered tissue formation under in vitro dynamic mechanical conditioning.

While it has become axiomatic that mechanical signals promote in vitro engineered tissue formation, the underlying mechanisms remain largely unknown. Moreover, efforts to date to determine parameters for optimal extracellular matrix (ECM) development have been largely empirical. In the present work, we propose a two-pronged approach involving novel theoretical developments coupled with key experimental data to develop better mechanistic understanding of growth and development of dense connective tissue under mechanical stimuli. To describe cellular proliferation and ECM synthesis that occur at rates of days to weeks, we employ mixture theory to model the construct constituents as a nutrient-cell-ECM triphasic system, their transport, and their biochemical reactions. Dynamic conditioning protocols with frequencies around 1 Hz are described with multi-scale methods to couple the dissimilar time scales. Enhancement of nutrient transport due to pore fluid advection is upscaled into the growth model, and the spatially dependent ECM distribution describes the evolving poroelastic characteristics of the scaffold-engineered tissue construct. Simulation results compared favorably to the existing experimental data, and most importantly, distinguish between static and dynamic conditioning regimes. The theoretical framework for mechanically conditioned tissue engineering (TE) permits not only the formulation of novel and better-informed mechanistic hypothesis describing the phenomena underlying TE growth and development, but also the exploration/optimization of conditioning protocols in a rational manner.

Coupled Simulation of Hemodynamics and Vascular Growth and Remodeling in a Subject-Specific Geometry.

A computational framework to couple vascular growth and remodeling (G&R) with blood flow simulation in a 3D patient-specific geometry is presented. Hyperelastic and anisotropic properties are considered for the vessel wall material and a constrained mixture model is used to represent multiple constituents in the vessel wall, which was modeled as a membrane. The coupled simulation is divided into two time scales-a longer time scale for G&R and a shorter time scale for fluid dynamics simulation. G&R is simulated to evolve the boundary of the fluid domain, and fluid simulation is in turn used to generate wall shear stress and transmural pressure data that regulates G&R. To minimize required computation cost, the fluid dynamics are only simulated when G&R causes significant vascular geometric change. For demonstration, this coupled model was used to study the influence of stress-mediated growth parameters, and blood flow mechanics, on the behavior of the vascular tissue growth in a model of the infrarenal aorta derived from medical image data.

ODE constrained mixture modelling: a method for unraveling subpopulation structures and dynamics.

Functional cell-to-cell variability is ubiquitous in multicellular organisms as well as bacterial populations. Even genetically identical cells of the same cell type can respond differently to identical stimuli. Methods have been developed to analyse heterogeneous populations, e.g., mixture models and stochastic population models. The available methods are, however, either incapable of simultaneously analysing different experimental conditions or are computationally demanding and difficult to apply. Furthermore, they do not account for biological information available in the literature. To overcome disadvantages of existing methods, we combine mixture models and ordinary differential equation (ODE) models. The ODE models provide a mechanistic description of the underlying processes while mixture models provide an easy way to capture variability. In a simulation study, we show that the class of ODE constrained mixture models can unravel the subpopulation structure and determine the sources of cell-to-cell variability. In addition, the method provides reliable estimates for kinetic rates and subpopulation characteristics. We use ODE constrained mixture modelling to study NGF-induced Erk1/2 phosphorylation in primary sensory neurones, a process relevant in inflammatory and neuropathic pain. We propose a mechanistic pathway model for this process and reconstructed static and dynamical subpopulation characteristics across experimental conditions. We validate the model predictions experimentally, which verifies the capabilities of ODE constrained mixture models. These results illustrate that ODE constrained mixture models can reveal novel mechanistic insights and possess a high sensitivity.

Sparsity Constrained Mixture Modeling for the Estimation of Kinetic Parameters in Dynamic PET.

The estimation and analysis of kinetic parameters in dynamic positron emission tomography (PET) is frequently confounded by tissue heterogeneity and partial volume effects. We propose a new constrained model of dynamic PET to address these limitations. The proposed formulation incorporates an explicit mixture model in which each image voxel is represented as a mixture of different pure tissue types with distinct temporal dynamics. We use Cramér-Rao lower bounds to demonstrate that the use of prior information is important to stabilize parameter estimation with this model. As a result, we propose a constrained formulation of the estimation problem that we solve using a two-stage algorithm. In the first stage, a sparse signal processing method is applied to estimate the rate parameters for the different tissue compartments from the noisy PET time series. In the second stage, tissue fractions and the linear parameters of different time activity curves are estimated using a combination of spatial-regularity and fractional mixture constraints. A block coordinate descent algorithm is combined with a manifold search to robustly estimate these parameters. The method is evaluated with both simulated and experimental dynamic PET data.

A finite element‐based constrained mixture implementation for arterial growth, remodeling, and adaptation: Theory and numerical verification

We implemented a constrained mixture model of arterial growth and remodeling in a nonlinear finite element framework to facilitate numerical analyses of diverse cases of arterial adaptation and maladaptation, including disease progression, resulting in complex evolving geometries and compositions. This model enables hypothesis testing by predicting consequences of postulated characteristics of cell and matrix turnover, including evolving quantities and orientations of fibrillar constituents and nonhomogenous degradation of elastin or loss of smooth muscle function. The nonlinear finite element formulation is general within the context of arterial mechanics, but we restricted our present numerical verification to cylindrical geometries to allow comparisons with prior results for two special cases: uniform transmural changes in mass and differential growth and remodeling within a two-layered cylindrical model of the human aorta. The present finite element model recovers the results of these simplified semi-inverse analyses with good agreement.

Constrained mixture models as tools for testing competing hypotheses in arterial biomechanics: A brief survey

Hypothesis testing via numerical models has emerged as a powerful tool which permits the verification of theoretical frameworks against canonical experimental and clinical observations. Cleverly designed computational experiments also inspire new methodologies by elucidating important biological processes and restricting parametric spaces. Constrained mixture models of arterial growth and remodeling (G&R) can facilitate the design of computational experiments which can bypass technical limitations in the laboratory, by considering illustrative special cases. The resulting data may then inform the design of focused experimental techniques and the development of improved theories. This work is a survey of computational hypothesis-testing studies, which exploit the unique abilities offered by the constrained mixture theory of arterial G&R. Specifically, we explore the core hypotheses integrated in these models, review their basic mathematical conceptualizations, and recapitulate their most salient and illuminating findings. We then assess how a decade's worth of constrained mixture models have contributed to a lucid, emerging picture of G&R mechanisms.

A Multi-Layered Computational Model of Coupled Elastin Degradation, Vasoactive Dysfunction, and Collagenous Stiffening in Aortic Aging

Arterial responses to diverse pathologies and insults likely occur via similar mechanisms. For example, many studies suggest that the natural process of aging and isolated systolic hypertension share many characteristics in arteries, including loss of functional elastin, decreased smooth muscle tone, and altered rates of deposition, and/or crosslinking of fibrillar collagen. Our aim is to show computationally how these coupled effects can impact evolving aortic geometry and mechanical behavior. Employing a thick-walled, multi-layered constrained mixture model, we suggest that a coupled loss of elastin and vasoactive function are fundamental mechanisms by which aortic aging occurs. Moreover, it is suggested that collagenous stiffening, although itself generally an undesirable process, can play a key role in attenuating excessive dilatation, perhaps including the enlargement of abdominal aortic aneurysms.

Relevant subtask learning by constrained mixture models

We introduce relevant subtask learning, a new learning problem which is a variant of multi-task learning. The goal is to build a classifier for a task-of-interest for which we have too few training samples. We additionally have supplementary data'' collected from other tasks, but it is uncertain which of these other samples are relevant, that is, which samples are classified in the same way as in the task-of-interest. The research problem is how to use the supplementary data from the other tasks to improve the classifier in the task-of-interest. We show how to solve the problem, and demonstrate the solution with logistic regression classifiers. The key idea is to model all tasks as mixtures of relevant and irrelevant samples, and model the irrelevant part with a sufficiently flexible model such that it does not distort the model of relevant data. We give two learning algorithms for the method - a simple maximum likelihood optimization algorithm and a more advanced variational Bayes inference algorithm; in both cases we show that the method works better than a comparable multi-task learning model and naive methods.

A constrained mixture model for developing mouse aorta

Mechanical stresses influence the structure and function of adult and developing blood vessels. When these stresses are perturbed, the vessel wall remodels to return the stresses to homeostatic levels. Constrained mixture models have been used to predict remodeling of adult vessels in response to step changes in blood pressure, axial length and blood flow, but have not yet been applied to developing vessels. Models of developing blood vessels are complicated by continuous and simultaneous changes in the mechanical forces. Understanding developmental growth and remodeling is important for treating human diseases and designing tissue-engineered blood vessels. This study presents a constrained mixture model for postnatal development of mouse aorta with multiple step increases in pressure, length and flow. The baseline model assumes that smooth muscle cells (SMCs) in the vessel wall immediately constrict or dilate the inner radius after a perturbation to maintain the shear stress and then remodel the wall thickness to maintain the circumferential stress. The elastin, collagen and SMCs have homeostatic stretch ratios and passive material constants that do not change with developmental age. The baseline model does not predict previously published experimental data. To approximate the experimental data, it must be assumed that the SMCs dilate a constant amount, regardless of the step change in mechanical forces. It must also be assumed that the homeostatic stretch ratios and passive material constants change with age. With these alterations, the model approximates experimental data on the mechanical properties and dimensions of aorta from 3- to 30-day-old mice.

A 3-D framework for arterial growth and remodeling in response to altered hemodynamics

We present a three-dimensional mathematical framework for modeling the evolving geometry, structure, and mechanical properties of a representative straight cylindrical artery subjected to changes in mean blood pressure and flow. We show that numerical predictions recover prior findings from a validated two-dimensional framework, but extend those findings by allowing effects of transmural gradients in wall constituents and vasoactive molecules to be simulated directly. Of particular note, we show that the predicted evolution of the residual stress related opening angle in response to an abrupt, sustained increase in blood pressure is qualitatively similar to measured changes when one accounts for a nonlinear transmural distribution of pre-stretched elastin. We submit that continuum-based constrained mixture models of arterial adaptation hold significant promise for deepening our basic understanding of arterial mechanobiology and thus for designing improved clinical interventions to treat many different types of arterial disease and injury.

Modelling carotid artery adaptations to dynamic alterations in pressure and flow over the cardiac cycle.

Motivated by recent clinical and laboratory findings of important effects of pulsatile pressure and flow on arterial adaptations, we employ and extend an established constrained mixture framework of growth (change in mass) and remodelling (change in structure) to include such dynamical effects. New descriptors of cell and tissue behavior (constitutive relations) are postulated and refined based on new experimental data from a transverse aortic arch banding model in the mouse that increases pulsatile pressure and flow in one carotid artery. In particular, it is shown that there was a need to refine constitutive relations for the active stress generated by smooth muscle, to include both stress- and stress rate-mediated control of the turnover of cells and matrix and to account for a cyclic stress-mediated loss of elastic fibre integrity and decrease in collagen stiffness in order to capture the reported evolution, over 8 weeks, of luminal radius, wall thickness, axial force and in vivo axial stretch of the hypertensive mouse carotid artery. We submit, therefore, that complex aspects of adaptation by elastic arteries can be predicted by constrained mixture models wherein individual constituents are produced or removed at individual rates and to individual extents depending on changes in both stress and stress rate from normal values.

A 3-D constrained mixture model for mechanically mediated vascular growth and remodeling

In contrast to the widely applied approach to model soft tissue remodeling employing the concept of volumetric growth, microstructurally motivated models are capable of capturing many of the underlying mechanisms of growth and remodeling; i.e., the production, removal, and remodeling of individual constituents at different rates and to different extents. A 3-dimensional constrained mixture computational framework has been developed for vascular growth and remodeling, considering new, microstructurally motivated kinematics and constitutive equations and new stress and muscle activation mediated evolution equations. Our computational results for alterations in flow and pressure, using reasonable physiological values for rates of constituent growth and turnover, concur with findings in the literature. For example, for flow-induced remodeling, our simulations predict that, although the wall shear stress is restored completely, the circumferential stress is not restored employing realistic physiological rate parameters. Also, our simulations predict different levels of thickening on inner versus outer wall locations, as shown in numerous reports of pressure-induced remodeling. Whereas the simulations are meant to be illustrative, they serve to highlight the experimental data currently lacking to fully quantify mechanically mediated adaptations in the vasculature.

Evaluation of fundamental hypotheses underlying constrained mixture models of arterial growth and remodelling

Evolving constituent composition and organization are important determinants of the biomechanical behaviour of soft tissues. In arteries, vascular smooth muscle cells and fibroblasts continually produce and degrade matrix constituents in preferred modes and at altered rates in response to changing mechanical stimuli. Smooth muscle cells similarly exhibit vasoactive changes that contribute to the control of overall structure, function and mechanical behaviour. Constrained mixture models provide a useful framework in which to quantify arterial growth and remodelling for they can account for cell-mediated changes in individual structurally significant constituents. Our simulations show that the combined effects of changing mass density turnover and vasoactivity, as well as the prestretch at which constituents are incorporated within extant matrix, are essential to capture salient features of bounded arterial growth and remodelling. These findings emphasize the importance of formulating biologically motivated constitutive relations in any theory of growth and remodelling and distinct advantages of the constrained mixture approach, in particular.

Parameter Sensitivity Study of a Constrained Mixture Model of Arterial Growth and Remodeling

Computational models of arterial growth and remodeling promise to increase our understanding of basic biological processes, such as development, tissue maintenance, and aging, the biomechanics of functional adaptation, the progression and treatment of disease, responses to injuries, and even the design of improved replacement vessels and implanted medical devices. Ensuring reliability of and confidence in such models requires appropriate attention to verification and validation, including parameter sensitivity studies. In this paper, we classify different types of parameters within a constrained mixture model of arterial growth and remodeling; we then evaluate the sensitivity of model predictions to parameter values that are not known directly from experiments for cases of modest sustained alterations in blood flow and pressure as well as increased axial extension. Particular attention is directed toward complementary roles of smooth muscle vasoactivity and matrix turnover, with an emphasis on mechanosensitive changes in the rates of turnover of intramural fibrillar collagen and smooth muscle in maturity. It is shown that vasoactive changes influence the rapid change in caliber that is needed to maintain wall shear stress near its homeostatic level and the longer term changes in wall thickness that are needed to maintain circumferential wall stress near its homeostatic target. Moreover, it is shown that competing effects of intramural and wall shear stress-regulated rates of turnover can develop complex coupled responses. Finally, results demonstrate that the sensitivity to parameter values depends upon the type of perturbation from normalcy, with changes in axial stretch being most sensitive consistent with empirical reports.

Modeling effects of axial extension on arterial growth and remodeling

Diverse mechanical perturbations elicit arterial growth and remodeling responses that appear to optimize structure and function so as to promote mechanical homeostasis. For example, it is well known that functional adaptations to sustained changes in transmural pressure and blood flow primarily affect wall thickness and caliber to restore circumferential and wall shear stresses toward normal. More recently, however, it has been shown that changes in axial extension similarly prompt dramatic cell and matrix reorganization and turnover, resulting in marked changes in unloaded geometry and mechanical behavior that presumably restore axial stress toward normal. Because of the inability to infer axial stress from in vivo measurements, simulations are needed to examine this hypothesis and to guide the design of future experiments. In this paper, we show that a constrained mixture model predicts salient features of observed responses to step increases in axial extension, including marked increases in fibrous constituent production, leading to a compensatory lengthening that restores original mechanical behavior. Because axial extension can be modified via diverse surgical procedures, including bypass operations, and exploited in tissue regeneration research, there is a need for increased attention to this important aspect of arterial biomechanics and mechanobiology.

Origin of axial prestretch and residual stress in arteries.

The structural protein elastin endows large arteries with unique biological functionality and mechanical integrity, hence its disorganization, fragmentation, or degradation can have important consequences on the progression and treatment of vascular diseases. There is, therefore, a need in arterial mechanics to move from materially uniform, phenomenological, constitutive relations for the wall to those that account for separate contributions of the primary structural constituents: elastin, fibrillar collagens, smooth muscle, and amorphous matrix. In this paper, we employ a recently proposed constrained mixture model of the arterial wall and show that prestretched elastin contributes significantly to both the retraction of arteries that is observed upon transection and the opening angle that follows the introduction of a radial cut in an unloaded segment. We also show that the transmural distributions of elastin and collagen, compressive stiffness of collagen, and smooth muscle tone play complementary roles. Axial prestresses and residual stresses in arteries contribute to the homeostatic state of stress in vivo as well as adaptations to perturbed loads, disease, or injury. Understanding better the development of and changes in wall stress due to individual extracellular matrix constituents thus promises to provide considerable clinically important insight into arterial health and disease.

A mathematical model of evolving mechanical properties of intraluminal thrombus

Quantifying mechanical properties of blood clots is fundamental to understanding many aspects of cardiovascular disease and its treatment. Nevertheless, there has been little attention to quantifying the evolving composition, structure and properties when a clot transforms from an initial fibrin-based mesh to a predominantly collagenous mass. Although more data are needed to formulate a complete mathematical model of the evolution of clot properties, we propose a general constrained mixture model based on diverse data available from in vitro tests on fibrinogenesis, the stiffness of fibrin gels, and fibrinolysis as well as histological and mechanical data from clots retrieved from patients at surgery or autopsy. In particular, albeit resulting from complex kinetics involving many clotting factors, we show that the rapid (minutes) in vitro production of fibrin from fibrinogen can be modeled well by an Avrami-type relation and similarly that the fast (tens of minutes) in vitro degradation of fibrin in response to different concentrations of plasmin can be captured via a single "master function" parameterized by appropriate half-times that can be inferred from laboratory or clinical data. Accounting simultaneously for the production and removal of fibrin as well as chemo-mechano-stimulated production of fibrillar collagens yields predictions of changing mass fractions and bulk mechanical properties that correspond well to experimentally available data. Constrained mixture models thus hold considerable promise for modeling the biomechanics of clot evolution and can guide the design and interpretation of needed experiments and stress analyses.

A 3-D constrained mixture model for vascular growth and remodeling

Although the cardiac biomarker troponin T (cTnT) is related strongly to mortality in patients with end-stage renal disease, the independent association of N-terminal pro–B-type natriuretic peptide (NT–pro-BNP) and cTnT levels in predicting outcomes is unknown. The objective of this study is to determine factors associated with NT–pro-BNP and cTnT and determine whether these levels are associated with mortality.Cohort study.Asymptomatic hemodialysis patients (n = 150) in 4 university-affiliated hemodialysis units.For cross-sectional analysis, echocardiographic variables as exposures and NT–pro-BNP and cTnT levels as outcomes; for longitudinal analysis, association of NT–pro-BNP and cTnT levels as exposures to all-cause and cardiovascular disease mortality as outcomes.In a multivariate regression analysis, low midwall fractional shortening, a measure of poor systolic function, was an independent correlate of log NT–pro-BNP level (P < 0.01), whereas left ventricular mass index was an independent correlate of cTnT level (P < 0.01). During a median follow-up of 24 months, 46 patients died, 26 of cardiovascular causes. NT–pro-BNP levels had a strong graded relationship with all-cause (hazard ratios [HRs], 1.54, 4.78, and 4.03 for increasing quartiles; P < 0.001) and cardiovascular mortality (HRs, 2.99, 10.95, and 8.54; P < 0.01), whereas cTnT level had a weaker relationship with all-cause (HRs, 1.57, 2.32, and 3.39; P < 0.01) and cardiovascular mortality (HRs, 0.81, 2.12, and 2.14; P = 0.1). The combination of the 2 biomarker levels did not improve the association with all-cause or cardiovascular mortality compared with NT–pro-BNP level alone. NT–pro-BNP level was a marker of mortality even after adjusting for left ventricular mass index and midwall fractional shortening.Our cohort was predominantly black and of limited sample size.NT–pro-BNP level strongly correlates with left ventricular systolic dysfunction and is associated more strongly with mortality than cTnT level in asymptomatic hemodialysis patients.