Erratum: Mei, Y., et al. Mechanics Based Tomography: A Preliminary Feasibility Study. Sensors 2017, 17, 1075.

The authors wish to correct Figures 12 and 14 in their paper published in Sensors [1], doi:10.3390/s17051075, http://www.mdpi.com/1424-8220/17/5/1075[...].

Mechanics Based Tomography: A Preliminary Feasibility Study.

We present a non-destructive approach to sense inclusion objects embedded in a solid medium remotely from force sensors applied to the medium and boundary displacements that could be measured via a digital image correlation system using a set of cameras. We provide a rationale and strategy to uniquely identify the heterogeneous sample composition based on stiffness (here, shear modulus) maps. The feasibility of this inversion scheme is tested with simulated experiments that could have clinical relevance in diagnostic imaging (e.g., tumor detection) or could be applied to engineering materials. No assumptions are made on the shape or stiffness quantity of the inclusions. We observe that the novel inversion method using solely boundary displacements and force measurements performs well in recovering the heterogeneous material/tissue composition that consists of one and two stiff inclusions embedded in a softer background material. Furthermore, the target shear modulus value for the stiffer inclusion region is underestimated and the inclusion size is overestimated when incomplete boundary displacements on some part of the boundary are utilized. For displacements measured on the entire boundary, the shear modulus reconstruction improves significantly. Additionally, we observe that with increasing number of displacement data sets utilized in solving the inverse problem, the quality of the mapped shear moduli improves. We also analyze the sensitivity of the shear modulus maps on the noise level varied between 0.1% and 5% white Gaussian noise in the boundary displacements, force and corresponding displacement indentation. Finally, a sensitivity analysis of the recovered shear moduli to the depth, stiffness and the shape of the stiff inclusion is performed. We conclude that this approach has potential as a novel imaging modality and refer to it as Mechanics Based Tomography (MBT).

Effect of Outflow Tract Banding on Embryonic Cardiac Hemodynamics.

We analyzed heart wall motion and blood flow dynamics in chicken embryos using in vivo optical coherence tomography (OCT) imaging and computational fluid dynamics (CFD) embryo-specific modeling. We focused on the heart outflow tract (OFT) region of day 3 embryos, and compared normal (control) conditions to conditions after performing an OFT banding intervention, which alters hemodynamics in the embryonic heart and vasculature. We found that hemodynamics and cardiac wall motion in the OFT are affected by banding in ways that might not be intuitive a priori. In addition to the expected increase in ventricular blood pressure, and increase blood flow velocity and, thus, wall shear stress (WSS) at the band site, the characteristic peristaltic-like motion of the OFT was altered, further affecting flow and WSS. Myocardial contractility, however, was affected only close to the band site due to the physical restriction on wall motion imposed by the band. WSS were heterogeneously distributed in both normal and banded OFTs. Our results show how banding affects cardiac mechanics and can lead, in the future, to a better understanding of mechanisms by which altered blood flow conditions affect cardiac development leading to congenital heart disease.

4D subject-specific inverse modeling of the chick embryonic heart outflow tract hemodynamics.

Blood flow plays a critical role in regulating embryonic cardiac growth and development, with altered flow leading to congenital heart disease. Progress in the field, however, is hindered by a lack of quantification of hemodynamic conditions in the developing heart. In this study, we present a methodology to quantify blood flow dynamics in the embryonic heart using subject-specific computational fluid dynamics (CFD) models. While the methodology is general, we focused on a model of the chick embryonic heart outflow tract (OFT), which distally connects the heart to the arterial system, and is the region of origin of many congenital cardiac defects. Using structural and Doppler velocity data collected from optical coherence tomography, we generated 4D ([Formula: see text]) embryo-specific CFD models of the heart OFT. To replicate the blood flow dynamics over time during the cardiac cycle, we developed an iterative inverse-method optimization algorithm, which determines the CFD model boundary conditions such that differences between computed velocities and measured velocities at one point within the OFT lumen are minimized. Results from our developed CFD model agree with previously measured hemodynamics in the OFT. Further, computed velocities and measured velocities differ by [Formula: see text]15 % at locations that were not used in the optimization, validating the model. The presented methodology can be used in quantifications of embryonic cardiac hemodynamics under normal and altered blood flow conditions, enabling an in-depth quantitative study of how blood flow influences cardiac development.

Blood flow dynamics reflect degree of outflow tract banding in Hamburger-Hamilton stage 18 chicken embryos.

Altered blood flow during embryonic development has been shown to cause cardiac defects; however, the mechanisms by which the resulting haemodynamic forces trigger heart malformation are unclear. This study used heart outflow tract banding to alter normal haemodynamics in a chick embryo model at HH18 and characterized the immediate blood flow response versus the degree of band tightness. Optical coherence tomography was used to acquire two-dimensional longitudinal structure and Doppler velocity images from control (n = 16) and banded (n = 25, 6-64% measured band tightness) embryos, from which structural and velocity data were extracted to estimate haemodynamic measures. Peak blood flow velocity and wall shear rate (WSR) initially increased linearly with band tightness (p < 0.01), but then velocity plateaued between 40% and 50% band tightness and started to decrease with constriction greater than 50%, whereas WSR continued to increase up to 60% constriction before it began decreasing with increased band tightness. Time of flow decreased with constriction greater than 20% (p < 0.01), while stroke volume in banded embryos remained comparable to control levels over the entire range of constriction (p > 0.1). The haemodynamic dependence on the degree of banding reveals immediate adaptations of the early embryonic cardiovascular system and could help elucidate a range of cardiac adaptations to gradually increased load.

Algorithms for quantitative quasi-static elasticity imaging using force data.

Quasi-static elasticity imaging can improve diagnosis and detection of diseases that affect the mechanical behavior of tissue. In this methodology, images of the shear modulus of the tissue are reconstructed from the measured displacement field. This is accomplished by seeking the spatial distribution of mechanical properties that minimizes the difference between the predicted and the measured displacement fields, where the former is required to satisfy a finite element approximation to the equations of equilibrium. In the absence of force data, the shear modulus is determined only up to a multiplicative constant. In this manuscript, we address the problem of calibrating quantitative elastic modulus reconstructions created from measurements of quasi-static deformations. We present two methods that utilize the knowledge of the applied force on a portion of the boundary. The first involves rescaling the shear modulus of the original minimization problem to best match the measured force data. This approach is easily implemented but neglects the spatial distribution of tractions. The second involves adding a force-matching term to the original minimization problem and a change of variables wherein we seek the log of the shear modulus. We present numerical results that demonstrate the usefulness of both methods.

Improving the efficiency of abdominal aortic aneurysm wall stress computations.

An abdominal aortic aneurysm is a pathological dilation of the abdominal aorta, which carries a high mortality rate if ruptured. The most commonly used surrogate marker of rupture risk is the maximal transverse diameter of the aneurysm. More recent studies suggest that wall stress from models of patient-specific aneurysm geometries extracted, for instance, from computed tomography images may be a more accurate predictor of rupture risk and an important factor in AAA size progression. However, quantification of wall stress is typically computationally intensive and time-consuming, mainly due to the nonlinear mechanical behavior of the abdominal aortic aneurysm walls. These difficulties have limited the potential of computational models in clinical practice. To facilitate computation of wall stresses, we propose to use a linear approach that ensures equilibrium of wall stresses in the aneurysms. This proposed linear model approach is easy to implement and eliminates the burden of nonlinear computations. To assess the accuracy of our proposed approach to compute wall stresses, results from idealized and patient-specific model simulations were compared to those obtained using conventional approaches and to those of a hypothetical, reference abdominal aortic aneurysm model. For the reference model, wall mechanical properties and the initial unloaded and unstressed configuration were assumed to be known, and the resulting wall stresses were used as reference for comparison. Our proposed linear approach accurately approximates wall stresses for varying model geometries and wall material properties. Our findings suggest that the proposed linear approach could be used as an effective, efficient, easy-to-use clinical tool to estimate patient-specific wall stresses.

Alterations in pulse wave propagation reflect the degree of outflow tract banding in HH18 chicken embryos.

Hemodynamic conditions play a critical role in embryonic cardiovascular development, and altered blood flow leads to congenital heart defects. Chicken embryos are frequently used as models of cardiac development, with abnormal blood flow achieved through surgical interventions such as outflow tract (OFT) banding, in which a suture is tightened around the heart OFT to restrict blood flow. Banding in embryos increases blood pressure and alters blood flow dynamics, leading to cardiac malformations similar to those seen in human congenital heart disease. In studying these hemodynamic changes, synchronization of data to the cardiac cycle is challenging, and alterations in the timing of cardiovascular events after interventions are frequently lost. To overcome this difficulty, we used ECG signals from chicken embryos (Hamburger-Hamilton stage 18, ∼3 days of incubation) to synchronize blood pressure measurements and optical coherence tomography images. Our results revealed that, after 2 h of banding, blood pressure and pulse wave propagation strongly depend on band tightness. In particular, while pulse transit time in the heart OFT of control embryos is ∼10% of the cardiac cycle, after banding (35% to 50% band tightness) it becomes negligible, indicating a faster OFT pulse wave velocity. Pulse wave propagation in the circulation is likewise affected; however, pulse transit time between the ventricle and dorsal aorta (at the level of the heart) is unchanged, suggesting an overall preservation of cardiovascular function. Changes in cardiac pressure wave propagation are likely contributing to the extent of cardiac malformations observed in banded hearts.

Biomechanics of early cardiac development.

Biomechanics affect early cardiac development, from looping to the development of chambers and valves. Hemodynamic forces are essential for proper cardiac development, and their disruption leads to congenital heart defects. A wealth of information already exists on early cardiac adaptations to hemodynamic loading, and new technologies, including high-resolution imaging modalities and computational modeling, are enabling a more thorough understanding of relationships between hemodynamics and cardiac development. Imaging and modeling approaches, used in combination with biological data on cell behavior and adaptation, are paving the road for new discoveries on links between biomechanics and biology and their effect on cardiac development and fetal programming.

Linear and nonlinear elastic modulus imaging: an application to breast cancer diagnosis.

We reconstruct the in vivo spatial distribution of linear and nonlinear elastic parameters in ten patients with benign (five) and malignant (five) tumors. The mechanical behavior of breast tissue is represented by a modified Veronda-Westmann model with one linear and one nonlinear elastic parameter. The spatial distribution of these elastic parameters is determined by solving an inverse problem within the region of interest (ROI). This inverse problem solution requires the knowledge of the displacement fields at small and large strains. The displacement fields are measured using a free-hand ultrasound strain imaging technique wherein, a linear array ultrasound transducer is positioned on the breast and radio frequency echo signals are recorded within the ROI while the tissue is slowly deformed with the transducer. Incremental displacement fields are determined from successive radio-frequency frames by employing cross-correlation techniques. The rectangular regions of interest were subjectively selected to obtain low noise displacement estimates and therefore were variables that ranged from 346 to 849.6 mm2 . It is observed that malignant tumors stiffen at a faster rate than benign tumors and based on this criterion nine out of ten tumors were correctly classified as being either benign or malignant.

Solution of the nonlinear elasticity imaging inverse problem: The incompressible case.

We have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material. The data for this problem are the quasi-static deformation fields within the solid measured at two distinct overall strain levels. The main ingredients of our algorithm are a gradient based quasi-Newton minimization strategy, the use of adjoint equations and a novel strategy for continuation in the material parameters. In this paper we present several extensions to this algorithm. First, we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation. We achieve this by solving the forward problem using a residual-based, stabilized, mixed finite element formulation which circumvents the Ladyzenskaya-Babuska-Brezzi condition. Second, we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters, or by splitting the problem into two distinct steps. Finally, we present a new strain energy density function with an exponential stress-strain behavior that yields a deviatoric stress tensor, thereby simplifying the interpretation of pressure when compared with other exponential functions. We test the overall approach by solving for the spatial distribution of material parameters from noisy, synthetic deformation fields.

Recent results in nonlinear strain and modulus imaging.

We report a summary of recent developments and current status of our team's efforts to image and quantify in vivo nonlinear strain and tissue mechanical properties. Our work is guided by a focus on applications to cancer diagnosis and treatment using clinical ultrasound imaging and quasi-static tissue deformations. We review our recent developments in displacement estimation from ultrasound image sequences. We discuss cross correlation approaches, regularized optimization approaches, guided search methods, multiscale methods, and hybrid methods. Current implementations can return results of high accuracy in both axial and lateral directions at several frames per second.We compare several strain estimators. Again we see a benefit from a regularized optimization approach. We then discuss both direct and iterative methods to reconstruct tissue mechanical property distributions from measured strain and displacement fields. We review the formulation, discretization, and algorithmic considerations that come into play when attempting to infer linear and nonlinear elastic properties from strain and displacement measurements. Finally we illustrate our progress with example applications in breast disease diagnosis and tumor ablation monitoring. Our current status shows that we have demonstrated quantitative determination of nonlinear parameters in phantoms and in vivo, in the context of 2D models and data. We look forward to incorporating 3D data from 2D transducer arrays to noninvasively create calibrated 3D quantitative maps of nonlinear elastic properties of breast tissues in vivo.

Linear and nonlinear elasticity imaging of soft tissue in vivo: demonstration of feasibility.

We establish the feasibility of imaging the linear and nonlinear elastic properties of soft tissue using ultrasound. We report results for breast tissue where it is conjectured that these properties may be used to discern malignant tumors from benign tumors. We consider and compare three different quantities that describe nonlinear behavior, including the variation of strain distribution with overall strain, the variation of the secant modulus with overall applied strain and finally the distribution of the nonlinear parameter in a fully nonlinear hyperelastic model of the breast tissue.