4. The Inverse Method

4. The Inverse Method - https://urlgoal.com/2tEbcV

Create citation alert 0266-5611/12/4/383 Abstract This paper is concerned with the development of an inversion scheme for two-dimensional inverse scattering problems in the resonance region which does not use nonlinear optimization methods and is relatively independent of the geometry and physical properties of the scatterer. It is assumed that the far field pattern corresponding to observation angle and plane waves incident at angle is known for all . From this information, the support of the scattering obstacle is obtained by solving the integral equation

where k is the wavenumber and is on a rectangular grid containing the scatterer. The support is found by noting that is unbounded as approaches the boundary of the scattering object from inside the scatterer. Numerical examples are given showing the practicality of this method.

We develop a novel algorithm that carries out this inverse method automatically and efficiently for general quantum systems and that can be readily implemented with existing numerical tools. The key step of the algorithm is the evaluation and analysis of the quantum covariance matrix, a quantum mechanical generalization of a statistical covariance matrix. We test our algorithm on numerous wave functions and find a myriad of new models.

The advent of high-resolution in vivo imaging techniques has enabled great progress in determining tissue morphology, enabling the characterization of atherosclerotic tissue in its native state through virtual histology (VH) assessment. The most clinically prevalent of these is VH-intravascular ultrasound (IVUS), which automatically classifies vascular tissue through spectral decomposition of the acoustic acquisition signal10,11,12. Recently, alternative classification schemes have been developed, incorporating machine learning methods to perform similar segmentation using IVUS or optical coherence tomography (OCT) images13,14,15. Such classified images can then be used to build patient-specific FE models16,17.

The determination of physiologically representative material properties remains a comparably challenging task. A number of studies have used ex vivo testing methods to fit observed tissue behavior to material models of differing complexities18,19,20. It is tempting to directly utilize the parameters reported by such studies in a lesion-specific simulation setting. However, such generalizability is precluded by the great variance in atherosclerotic mechanical properties between and within patients19,21, which directly impact the stresses predicted by subsequent FE simulations22. In addition, ex vivo testing can alter the properties of arterial tissue by removing it from its natural environment, rendering the physiological relevance of such material characterization uncertain. Hence, accurate prediction of the patient-specific mechanical response of atherosclerotic tissue mandates subject-specific in vivo tissue material properties.

Single-parameter models were employed to avoid over-parameterization and ensure solution uniqueness when using displacement data derived from images obtained at two distinct intravascular pressures25,26,29. To enable recovery of higher-parameter material models, one could instead acquire data at several intravascular pressures, as previously done in 2D29 and 3D30. However, increasing the number of required image limits clinical translation. An alternative approach is to incorporate a greater degree of information by aiming to match the entire inner and outer surfaces through inverse methods, as done by Liu et al., who recovered a 5-parameter GOH model for homogeneous aortic tissue28. However, their work utilized computed tomography scans lacking heterogeneous morphological information, rendering multi-material, heterogeneous tissue characterization infeasible.

We present a method to recover the material properties of multiple arterial plaque components using two sets of intravascular imaging data acquired at different intraluminal pressures (Fig. 1). A pre-processing step was conducted to first convert in vivo images into 3D FE geometries with heterogeneous material attribution. These geometries were then utilized by the presented inverse method to recover the material properties that reproduce the behavior manifested by the two imaged states.

The FE model generation protocol accepted as input OCT image data acquired in the course of clinical care. Given these data, the inner and outer borders of the vessel wall were identified and fit with a smooth, continuous surface in 3D31. The resulting region of interest was then characterized with a validated deep learning method for classifying tissue micromorphology in OCT images13, which automatically annotated frames with the spatial distribution of non-pathological and diseased (calcified, lipid, fibrotic, or mixed) tissue. A subset of the total acquisition length was taken to exclude low-quality images and reduce computational costs. The final output of these steps was a point cloud set consisting of pixel coordinates and corresponding tissue labels from the selected segment.

The solution space for the inverse problem can be highly nonlinear and non-convex. As such, there can exist multiple local minima to which an optimizer may converge, resulting in erroneous recovery of material parameters. To overcome this inherent challenge and increase the probability of attaining a global minimum, a multi-objective genetic algorithm called the Non-dominated Sorting Genetic Algorithm (NSGA-II)32 was first used for identifying the global region within which the optimal material parameter set was likely to exist. In NSGA-II, an initial population of parameters was generated using the space-filling Latin Hypercube Sampling method33. This population was then propagated over several generations, with the fittest individuals being chosen to stochastically exchange parameters with each other. As it is a multi-objective optimization algorithm, the fitness was measured using the vector of all interface errors \(\delta_{MO}^i\) (Eq. 2) and the algorithm converged towards a Pareto optimal set of solutions, defined as a set within which one cannot improve in one dimension of the vector \(\delta_{MO}^i\) without sacrificing progress in another dimension.

The inverse FE method was verified in silico by recovering mechanical constitutive properties of a patient-specific model generated from clinical images. Noise sensitivity analysis was also carried out on the same patient-specific model to ascertain robustness of the inverse FE framework. Finally, the method was applied to different patient models to assess its generalizability across a range of clinically-relevant diseased vessel phenotypes. All procedures were performed on a CPU with 6 cores running at 2.8 GHz and with 24 GB RAM.

Using the image-based FE model generation methodology described earlier, the characterized OCT images from a single patient were transformed into a 3D FE mesh with five material regions: fibrous, lipid, calcium, mixed, and healthy wall tissue. As indicated earlier, the presented inverse method assumes the availability of two sets of pullback data with known acquisition pressures. However, common clinical practice and guidelines directs acquisition of only a single image sequence with no corresponding lumen pressure recording. Furthermore, excised tissue was not available from the imaged patients for mechanical testing. Hence, for the sake of verification, the geometries corresponding to the target pullback datasets were generated in silico for assumed material properties (Fig. 3).

The simulated target shapes used in the earlier section provided an effective testbed for evaluating the accuracy of the proposed inverse FE method. However, such synthetic data represents an idealized version of the clinical equivalent, where acquisition noise, cardiopulmonary variations and motion, spatiotemporal inaccuracies, and meshing discrepancies may all influence data quality and subsequently influence the performance of the proposed method. Therefore, to study the effect of spurious noise and mimic real-world conditions, 5% and 20% random Gaussian noise were added to the displacement data used to generate \({\Omega_{{\text{target}}}}\) using linear elastic parameters:

As a validation of versatility beyond the model employed in the aforementioned studies, the presented inverse method was used to recover nonlinear (Yeoh) material parameters in two additional patient-specific models. As evinced by the differing volume percentage of each tissue class (Supplementary Table S2), each model had varying degrees of material heterogeneity and phenotypes. Since the calcium volume percentages in these lesion models were greater than 1%, the C10 parameter for calcium was included in the recovery process for these cases, resulting in 9 recovered parameters.

Errors in linear elastic (top row) and Yeoh (middle and bottom rows) material parameter recovery over 8 runs. The performance of the current interface matching method (left column) is compared to inner and outer surface matching alone (middle column), and diameter matching (right column). Full interface matching outperforms surface matching and diameter matching in all cases, but the distinction is more pronounced for the higher-order Yeoh model and for the predominantly intramural materials (mixed and lipid tissue). The median errors (dashed horizontal lines) were greater for material parameters with lower assigned value magnitudes. Boxes extend from 25th to 75th percentiles; range bars extend to extrema.

The stand-out feature of the presented method is its ability to recover multiple parameters for multiple materials through the incorporation of micro-morphological information in the form of intra-plaque tissue interfaces into the objective function, as evinced by the 8-parameter recovery (Fig. 4). To demonstrate the distinction of the method, its performance was directly compared to two approaches that do not take into account the micromorphology of the vessel wall, but instead use only macro-morphological information. In one approach, only the inner and outer vessel wall surfaces, instead of all intra-plaque interfaces, were considered in the objective function, in a manner inspired by Liu et al.28. In the other approach, the minimum and maximum diameters at a number of slices were compared, in a method similar to that used by Noble et al.28. 781b155fdc