An inverse approach to determining spatially varying arterial compliance using ultrasound imaging

Abstract

The mechanical properties of arteries are implicated in a wide variety of cardiovascular diseases, many of which are expected to involve a strong spatial variation in properties that can be depicted by diagnostic imaging. A pulse wave inverse problem (PWIP) is presented, which can produce spatially resolved estimates of vessel compliance from ultrasound measurements of the vessel wall displacements. The 1D equations governing pulse wave propagation in a flexible tube are parameterized by the spatially varying properties, discrete cosine transform components of the inlet pressure boundary conditions, viscous loss constant and a resistance outlet boundary condition. Gradient descent optimization is used to fit displacements from the model to the measured data by updating the model parameters. Inversion of simulated data showed that the PWIP can accurately recover the correct compliance distribution and inlet pressure under realistic conditions, even under high simulated measurement noise conditions. Silicone phantoms with known compliance contrast were imaged with a clinical ultrasound system. The PWIP produced spatially and quantitatively accurate maps of the phantom compliance compared to independent static property estimates, and the known locations of stiff inclusions (which were as small as 7 mm). The PWIP is necessary for these phantom experiments as the spatiotemporal resolution, measurement noise and compliance contrast does not allow accurate tracking of the pulse wave velocity using traditional approaches (e.g. 50% upstroke markers). Results from simulations indicate reflections generated from material interfaces may negatively affect wave velocity estimates, whereas these reflections are accounted for in the PWIP and do not cause problems.

Figure 1

Schematic of the pulse wave inverse problem (PWIP) process, where a discretized set of unknown properties, θ, is fitted to spatiotemporal vessel wall displacement and/or blood velocity data. These data can be generated through simulation to investigate performance under ideal conditions, from phantom experiments to validate accuracy in a controllable system, or from in-vivo examinations of living subjects.

Figure 2

Sampling of synthetic PWI data to mimic experimental measurements. The colormaps represent the simulated wall displacements, where the center image is the full 170mm simulation with bounds representing the simulated ultrasound measurements indicated, and the right panel shows the sampled incremental displacement data.

Figure 3

Dimensions and size of silicone vessel phantoms, and a photograph of the finished phantom. Two vessel phantoms were in each box, with four vessel phantoms used in total. Phantom 1 was half stiff and half soft, whereas phantoms 2, 3 and 4 had discrete stiff inclusions of size 15, 10 and 7mm, respectively.

Figure 4

Panel A: Forward model result for a Ricker wavelet inlet BC, where the colormap represents the wall displacement (m). Reflections are generated by the 3x increased stiffness inclusion between 25 and 35mm. Progress of a wave at the theoretical Moens-Korteweg (MK) speed is compared to the 50% upstroke markers commonly used to determine the PWV. Panel B: Stiffness estimated through equation (1) with c_mk computed from piecewise fits to the 50% upstroke markers, compared to the true stiffness distribution used in the model.

Figure 5

Inversion of synthetic PWI data with no added noise. Panel A shows that the PWIP recovered the compliance distribution specified in the simulation, and panel B shows that the correct inlet boundary condition was also recovered. Panel C shows the reduction in the error metric (Φ), as well as the recovered values of K_R and the outlet BC, R. Panels D and E compare the simulated measured wall displacement data, u_m with the final model predictions from the PWIP, u(θ) (colormap in units of μm), and panel F shows the relative error between the spatiotemporal maps of panels D and E.

Figure 6

Synthetic PWI data with a very high level of temporally correlated Gaussian noise (200% of mean, correlated at a scale of 20 temporal measurements). Panel A shows that the PWIP compliance distribution is reasonably close to what was specified in the simulation, and panel B shows that the correct inlet boundary condition was also recovered. Panel C shows the reduction in the error metric (Φ), as well as the recovered values of K_R and the outlet BC, R. Panels D and E compare the simulated measured wall displacement data, u_m with the final model predictions from the PWIP, u(θ) (colormap in units ofμm), and panel F shows the relative error between the spatiotemporal maps of panels D and E.

Figure 7

Example of static pressure testing for a phantom (15mm stiff inclusion centered at x=20mm). Panel A: B-mode image with typical result from automated segmentation algorithm (the green line is the identified top wall, and blue is the bottom wall). The inclusion boundaries are depicted with red lines. Panel B: Pressure radius relationship is approximately linear (a few representative locations are shown). Panel C: Compliance (k_p) along the imaged portion, where the error bars are the 95% confidence interval on the linear fit to the pressure-radius curves.

Figure 8

PWIP results for a homogenous section of soft material from phantom 1. Panel A: The PWIP (blue) correctly identified constant compliance over this section, with an overestimation of around 40% compared to static testing. The error bars are the variation over 4 repeated measurements, and the dotted red lines indicate the uncertianty in the static k_p estimate. Panel B: Estimate of inlet boundary conditions (solid) and the initial estimate (dotted) for the first dataset. Each of the three pulses is depicted by a different color. Panels C and D: Comparison of experimentally measured incremental wall dispalcement (C) and the final model wall dispalcements from the PWIP (D) for the first dataset (colormap in units of μm). The green lines separate each of the three pulses.

Figure 9

PWIP results for a homogenous section of stiff material from phantom 1. Panel A: The PWIP (blue) correctly identified constant compliance over this section, which was lower than the soft portion in Figure 8. See the caption of figure 8 for an explanation of panels B–D.

Figure 10

PWIP for phantom 1 with the pulse wave travelling from the soft to stiff side. Panel A: The PWIP (blue) correctly identified the location and contrast of the interface. See the caption of figure 8 for an explanation of panels B–D.

Figure 11

PWIP for phantom 1 with the pulse wave travelling from the stiff to soft side. Panel A: The PWIP (blue) correctly identified the location and contrast of the interface, with an overestimation of compliance of around 40%. See the caption of figure 8 for an explanation of panels B–D.

Figure 12

PWIP across the 15mm inclusion in Phantom 2. Panel A: The PWIP (blue) correctly identified the location and contrast of the stiff inclusion. See the caption of figure 8 for an explanation of panels B–D.

Figure 13

PWIP across the 10mm inclusion in Phantom 3. Panel A: The PWIP (blue) correctly identified the location and contrast of this smaller stiff inclusion. See the caption of figure 8 for an explanation of panels B–D.

Figure 14

PWIP across the 7mm inclusion in Phantom 4. Panel A: Although the PWIP (blue) correctly identified the location, contrast and size of the smallest inclusion, the error bars in this case are much wider due to one of the 4 inversions producing a much higher compliance estimate than expected. Compliance is overrestimated by 30–80% compared to static testing. See the caption of figure 8 for an explanation of panels B–D.

Figure 15

PWIP in the carotid artery of a healthy volunteer, using wall dispalcements from two primary pulses (PP1 and PP2) and two dicrotic notches (DN1 and DN2). Panel A: The PWIP (blue) indicates a gradual increase in compliance approaching the carotid bifurcation. Panel B: Inlet pressure BC estimate for each pulse. Panels C and D: Wall displacements for each primary pulse (PP1 and PP2), and dicrotic notch (DN1 and DN2). The final PWIP model dispalcements (D) were a good match to the measured displacements (C), despite the highly simplified reflection condition which was assumed at the carotid bifurcation.