Stochastic finite element framework for simultaneous estimation of cardiac kinematic functions and material parameters

Analysis of Uncertainty and Variability in Finite Element Computational Models for Biomedical Engineering: Characterization and Propagation

Computational modeling has become a powerful tool in biomedical engineering thanks to its potential to simulate coupled systems. However, real parameters are usually not accurately known, and variability is inherent in living organisms. To cope with this, probabilistic tools, statistical analysis and stochastic approaches have been used. This article aims to review the analysis of uncertainty and variability in the context of finite element modeling in biomedical engineering. Characterization techniques and propagation methods are presented, as well as examples of their applications in biomedical finite element simulations. Uncertainty propagation methods, both non-intrusive and intrusive, are described. Finally, pros and cons of the different approaches and their use in the scientific community are presented. This leads us to identify future directions for research and methodological development of uncertainty modeling in biomedical engineering.

A stochastic filtering approach to recover strain images from quasi-static ultrasound elastography

BACKGROUND

Model-based reconstruction algorithms have shown potentials over conventional strain-based methods in quasi-static elastographic image by using realistic finite element (FE) or bio-mechanical model constraints. However, it is still difficult to properly handle the discrepancies between the model constraint and ultrasound data, and the measurement noise.

METHODS

In this paper, we explore the usage of Kalman filtering algorithm for the estimation of strain imaging in quasi-static ultrasound elastography. The proposed strategy formulates the displacement distribution through biomechanical models, and the ultrasound-derived measurements through observation equations. Through this filtering strategy, the discrepancies are quantitatively modelled as one Gaussian white noise, and the measurement noise of ultrasound data is modelled as another independent Gaussian white noise. The optimal estimation of kinematic functions, i.e. the full displacement and velocity field, are computed through this Kalman filter. Then the strain images can be easily calculated from the estimated displacement field.

RESULTS

The accuracy and robustness of our proposed framework is first evaluated in synthetic data in controlled conditions, and the performance of this framework is then evaluated in the real data collected from elastography phantoms and patients with favourable results.

CONCLUSIONS

The potential of our algorithm is to provide the distribution of mechanically meaningful strain under a proper biomechanical model constraint. We address the model-data discrepancy and measurement noise by introducing process noise and measurement noise in our framework, and then the mechanically meaningful strain is estimated through the Kalman filter in the minimum mean square error (MMSE) sense.

An H∞ strategy for strain estimation in ultrasound elastography using biomechanical modeling constraint

The purpose of ultrasound elastography is to identify lesions by reconstructing the hardness characteristics of tissue reconstructed from ultrasound data. Conventional quasi-static ultrasound elastography is easily applied to obtain axial strain components along the compression direction, with the results inverted to represent the distribution of tissue hardness under the assumption of constant internal stresses. However, previous works of quasi-static ultrasound elastography have found it difficult to obtain the lateral and shear strain components, due to the poor lateral resolution of conventional ultrasound probes. The physical nature of the strain field is a continuous vector field, which should be fully described by the axial, lateral, and shear strain components, and the clinical value of lateral and shear strain components of deformed tissue is gradually being recognized by both engineers and clinicians. Therefore, a biomechanical-model-constrained filtering framework is proposed here for recovering a full displacement field at a high spatial resolution from the noisy ultrasound data. In our implementation, after the biomechanical model constraint is integrated into the state-space equation, both the axial and lateral displacement components can be recovered at a high spatial resolution from the noisy displacement measurements using a robust H∞ filter, which only requires knowledge of the worst-case noise levels in the measurements. All of the strain components can then be calculated by applying a gradient operator to the recovered displacement field. Numerical experiments on synthetic data demonstrated the robustness and effectiveness of our approach, and experiments on phantom data and in-vivo clinical data also produced satisfying results.

Reconstruction of elasticity: a stochastic model-based approach in ultrasound elastography

Background

The convectional strain-based algorithm has been widely utilized in clinical practice. It can only provide the information of relative information of tissue stiffness. However, the exact information of tissue stiffness should be valuable for clinical diagnosis and treatment.

Methods

In this study we propose a reconstruction strategy to recover the mechanical properties of the tissue. After the discrepancies between the biomechanical model and data are modeled as the process noise, and the biomechanical model constraint is transformed into a state space representation the reconstruction of elasticity can be accomplished through one filtering identification process, which is to recursively estimate the material properties and kinematic functions from ultrasound data according to the minimum mean square error (MMSE) criteria. In the implementation of this model-based algorithm, the linear isotropic elasticity is adopted as the biomechanical constraint. The estimation of kinematic functions (i.e., the full displacement and velocity field), and the distribution of Young’s modulus are computed simultaneously through an extended Kalman filter (EKF).

Results

In the following experiments the accuracy and robustness of this filtering framework is first evaluated on synthetic data in controlled conditions, and the performance of this framework is then evaluated in the real data collected from elastography phantom and patients using the ultrasound system. Quantitative analysis verifies that strain fields estimated by our filtering strategy are more closer to the ground truth. The distribution of Young’s modulus is also well estimated. Further, the effects of measurement noise and process noise have been investigated as well.

Conclusions

The advantage of this model-based algorithm over the conventional strain-based algorithm is its potential of providing the distribution of elasticity under a proper biomechanical model constraint. We address the model-data discrepancy and measurement noise by introducing process noise and measurement noise in our framework, and then the absolute values of Young’s modulus are estimated through the EFK in the MMSE sense. However, the initial conditions, and the mesh strategy will affect the performance, i.e., the convergence rate, and computational cost, etc.

Cardiovascular magnetic resonance: deeper insights through bioengineering

Heart disease is the main cause of morbidity and mortality worldwide, with coronary artery disease, diabetes, and obesity being major contributing factors. Cardiovascular magnetic resonance (CMR) can provide a wealth of quantitative information on the performance of the heart, without risk to the patient. Quantitative analyses of these data can substantially augment the diagnostic quality of CMR examinations and can lead to more effective characterization of disease and quantification of treatment benefit. This review provides an overview of the current state of the art in CMR with particular regard to the quantification of motion, both microscopic and macroscopic, and the application of bioengineering analysis for the evaluation of cardiac mechanics. We discuss the current clinical practice and the likely advances in the next 5-10 years, as well as the ways in which clinical examinations can be augmented by bioengineering analysis of strain, compliance, and stress.

Generation of synthetic but visually realistic time series of cardiac images combining a biophysical model and clinical images

We propose a new approach for the generation of synthetic but visually realistic time series of cardiac images based on an electromechanical model of the heart and real clinical 4-D image sequences. This is achieved by combining three steps. The first step is the simulation of a cardiac motion using an electromechanical model of the heart and the segmentation of the end diastolic image of a cardiac sequence. We use biophysical parameters related to the desired condition of the simulated subject. The second step extracts the cardiac motion from the real sequence using nonrigid image registration. Finally, a synthetic time series of cardiac images corresponding to the simulated motion is generated in the third step by combining the motion estimated by image registration and the simulated one. With this approach, image processing algorithms can be evaluated as we know the ground-truth motion underlying the image sequence. Moreover, databases of visually realistic images of controls and patients can be generated for which the underlying cardiac motion and some biophysical parameters are known. Such databases can open new avenues for machine learning approaches.

A meshfree method for simulating myocardial electrical activity

An element-free Galerkin method (EFGM) is proposed to simulate the propagation of myocardial electrical activation without explicit mesh constraints using a monodomain model. In our framework the geometry of myocardium is first defined by a meshfree particle representation that is, a sufficient number of sample nodes without explicit connectivities are placed in and inside the surface of myocardium. Fiber orientations and other material properties of myocardium are then attached to sample nodes according to their geometrical locations, and over the meshfree particle representation spatial variation of these properties is approximated using the shape function of EFGM. After the monodomain equations are converted to their Galerkin weak form and solved using EFGM, the propagation of myocardial activation can be simulated over the meshfree particle representation. The derivation of this solution technique is presented along a series of numerical experiments and a solution of monodomain model using a FitzHugh-Nagumo (FHN) membrane model in a canine ventricular model and a human-heart model which is constructed from digitized virtual Chinese dataset.

Cardiac motion and deformation recovery from MRI: a review

Magnetic resonance imaging (MRI) is a highly advanced and sophisticated imaging modality for cardiac motion tracking and analysis, capable of providing 3D analysis of global and regional cardiac function with great accuracy and reproducibility. In the past few years, numerous efforts have been devoted to cardiac motion recovery and deformation analysis from MR image sequences. Many approaches have been proposed for tracking cardiac motion and for computing deformation parameters and mechanical properties of the heart from a variety of cardiac MR imaging techniques. In this paper, an updated and critical review of cardiac motion tracking methods including major references and those proposed in the past ten years is provided. The MR imaging and analysis techniques surveyed are based on cine MRI, tagged MRI, phase contrast MRI, DENSE, and SENC. This paper can serve as a tutorial for new researchers entering the field.

Automatic generation of boundary conditions using demons nonrigid image registration for use in 3-D modality-independent elastography

Modality-independent elastography (MIE) is a method of elastography that reconstructs the elastic properties of tissue using images acquired under different loading conditions and a biomechanical model. Boundary conditions are a critical input to the algorithm and are often determined by time-consuming point correspondence methods requiring manual user input. This study presents a novel method of automatically generating boundary conditions by nonrigidly registering two image sets with a demons diffusion-based registration algorithm. The use of this method was successfully performed in silico using magnetic resonance and X-ray-computed tomography image data with known boundary conditions. These preliminary results produced boundary conditions with an accuracy of up to 80% compared to the known conditions. Demons-based boundary conditions were utilized within a 3-D MIE reconstruction to determine an elasticity contrast ratio between tumor and normal tissue. Two phantom experiments were then conducted to further test the accuracy of the demons boundary conditions and the MIE reconstruction arising from the use of these conditions. Preliminary results show a reasonable characterization of the material properties on this first attempt and a significant improvement in the automation level and viability of the method.

A comparative study of physiological models on cardiac deformation recovery: effects of biomechanical constraints

Cardiac deformation recovery is to recover quantitative subject-specific myocardial deformation from imaging data. In the last decade, cardiac physiological models derived from anatomy, biomechanics, and cardiac electrophysiology have become increasingly popular in constraining the recovery problems because of their physiological meaningfulness. Although physiological models with various electrical and biomechanical components have been adopted by different frameworks and have exhibited promising results, these models have not been systematically compared under the same recovery framework, input data, and experimental setups. As different models comprise varying physiological plausibilities and complexities, comparisons under the same settings can aid choosing the proper models for specific goals and available resources. In this paper, under a state-space filtering framework for statistically optimal couplings between models and image data, we compare the performances of six different cardiac physiological models with different biomechanical constraints. Experiments were performed on synthetic data for quantitative comparisons, and on clinical data for their capabilities in identifying pathological situations.

Maximum a posteriori strategy for the simultaneous motion and material property estimation of the heart

In addition to its technical merits as a challenging nonrigid motion and structural integrity analysis problem, quantitative estimation of cardiac regional functions and material characteristics has significant physiological and clinical value. We developed a stochastic finite-element framework for the simultaneous recovery of myocardial motion and material parameters from medical image sequences with an extended Kalman filter approach, and we have shown that this simultaneous estimation strategy achieves more accurate and robust results than separated motion and material estimation efforts. In this paper, we present a new computational strategy for the framework based upon the maximum a posteriori estimation principles, realized through the extended Kalman smoother, that produces a sequence of kinematics state and material parameter estimation of the entire myocardium, including the endocardial, epicardial, and midwall tissues. The system dynamics equations of the heart are constructed using a biomechanical model with stochastic parameters, and the tissue material and deformation parameters are jointly estimated from the periodic imaging data. Noise-corrupted synthetic image sequences with known kinematics and material parameters are used to assess the accuracy and robustness of the framework. Experiments with canine magnetic resonance tagging and phase-contrast image sequences have been conducted with very promising results, as validated through comparison to the histological staining of postmortem myocardium.

Temporal DeltaB0 and relaxation in the rat heart

Field maps of the induced main magnetic field offset (DeltaB(0)) were measured in the rat heart at various points in the cardiac cycle for the purpose of identifying their effects on relaxation measurements. The mean DeltaB(0) of the left ventricle averaged across rats was found to be 0.11 +/- 0.35 ppm and 0.19 +/- 0.39 ppm at the onset of systole and diastole, respectively. The root mean square (RMS) variation in resonant frequency of the left ventricle averaged across rats was found to be 0.09 +/- 0.05 ppm and 0.06 +/- 0.04 ppm during systole and diastole, respectively. Temporal variations in DeltaB(0) could substantially affect quantitative MRI measurements. To assess this, transverse relaxation rates (R(2) and R(2)(*)) were measured at different points in the cardiac cycle, and the effects of DeltaB(0) were estimated using measured field map data. For a given region of the left ventricle, DeltaB(0) induced a mean error across rats of < or =3.9% for R(2) and < or =9.6% for R(2)(*). For R(2)(*) measurements, the static component of the field inhomogeneity was found to be responsible for most of the error induced.

Sampled-data filtering framework for cardiac motion recovery: optimal estimation of continuous dynamics from discrete measurements

Quantitative and noninvasive estimation of cardiac kinematics has significant physiological and clinical implications. In this paper, a sampled-data filtering framework is presented for the recovery of cardiac motion and deformation functions from periodic medical image sequences. Cardiac dynamics is a continuously evolving physical/physiological process, whereas the imaging data can provide only sampled measurements at discrete time instants. Given such a hybrid paradigm, stochastic multiframe filtering frameworks are constructed to couple the continuous dynamics with the discrete measurements, and to coordinately deal with the parameter uncertainty of the biomechanical constraining model and the noisy nature of the imaging data. The state estimates are predicted according to the continuous-time biomechanically constructed state equation between observation time points, and then updated with the new imaging-derived measurements at discrete time instants, yielding physically more meaningful and more accurate estimation results. Both continuous-discrete Kalman filter and sampled-data Hinfinity filter are applied for motion recovery. While Kalman filter is the optimal estimator under Gaussian noises, the Hinfinity scheme can give robust estimation results when the types and levels of model uncertainties and data disturbances are not available a priori. The strategies are validated through synthetic data experiments to illustrate their advantages and on canine MR phase contrast images and human MR tagging data to show their clinical potential.

Physiome-model-based state-space framework for cardiac deformation recovery

RATIONALE AND OBJECTIVES

To more reliably recover cardiac information from noise-corrupted, patient-specific measurements, it is essential to employ meaningful constraining models and adopt appropriate optimization criteria to couple the models with the measurements. Although biomechanical models have been extensively used for myocardial motion recovery with encouraging results, the passive nature of such constraints limits their ability to fully count for the deformation caused by active forces of the myocytes. To overcome such limitations, we propose to adopt a cardiac physiome model as the prior constraint for cardiac motion analysis.

MATERIALS AND METHODS

The cardiac physiome model comprises an electric wave propagation model, an electromechanical coupling model, and a biomechanical model, which are connected through a cardiac system dynamics for a more complete description of the macroscopic cardiac physiology. Embedded within a multiframe state-space framework, the uncertainties of the model and the patient's measurements are systematically dealt with to arrive at optimal cardiac kinematic estimates and possibly beyond.

RESULTS

Experiments have been conducted to compare our proposed cardiac-physiome-model-based framework with the solely biomechanical model-based framework. The results show that our proposed framework recovers more accurate cardiac deformation from synthetic data and obtains more sensible estimates from real magnetic resonance image sequences.

CONCLUSION

With the active components introduced by the cardiac physiome model, cardiac deformations recovered from patient's medical images are more physiologically plausible.

Cardiac Motion Recovery: Continuous Dynamics, Discrete Measurements, and Optimal Estimation

A sampled-data filtering framework is presented for cardiac motion recovery from periodic medical image sequences. Cardiac dynamics is a continuously evolving physiological process, whereas the imaging data can provide only sampled measurements at discrete time instants. Stochastic multi-frame filtering frameworks are constructed to couple the continuous cardiac dynamics with the discrete measurements, and to deal with the parameter uncertainty of the biomechanical constraining model and the noisy nature of the imaging data in a coordinated fashion. The state estimates are predicted according to the continuous-time state equation between observation time points, and then updated with the new measurements obtained at discrete time instants, yielding physically more meaningful and more accurate estimation results. Both continuous-discrete Kalman filter and sampled-data Hinfinity filter are applied, and the Hinfinity scheme can give robust estimation results when the noise statistics is not available a priori. The sampled-data estimation strategies are validated through synthetic data experiments to illustrate their advantages and on canine MR phase contrast images to show their clinical relevance.

Finite element modelling and simulations in cardiovascular mechanics and cardiology: A bibliography 1993–2004

The paper gives a bibliographical review of the finite element modelling and simulations in cardiovascular mechanics and cardiology from the theoretical as well as practical points of views. The bibliography lists references to papers, conference proceedings and theses/dissertations that were published between 1993 and 2004. At the end of this paper, more than 890 references are given dealing with subjects as: Cardiovascular soft tissue modelling; material properties; mechanisms of cardiovascular components; blood flow; artificial components; cardiac diseases examination; surgery; and other topics.