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Ultrasonic Medical Tissue Imaging Using Probabilistic Inversion: Leveraging Variational Inference for Speed Reconstruction and Uncertainty Quantification

Qiang Li, Heyu Ma, Chengcheng Liu, Dean Ta

TL;DR

This work introduces a probabilistic inversion framework for ultrasonic imaging that integrates Stein Variational Gradient Descent ($SVGD$) with Full Waveform Inversion ($FWI$) to achieve fast, uncertainty-quantified reconstructions of tissue properties. By deriving the posterior gradient and employing an $SVGD$ particle-based update, the method yields accurate squared slowness distributions $m=1/c^2$ with spatially coherent uncertainty maps, and demonstrates faster convergence and improved ROI fidelity relative to conventional $FWI$ and stochastic variational inference ($SVI$), while remaining comparable to or approaching $MH$-MCMC benchmarks. Across synthetic linear- and ring-array phantoms and a breast-tissue model, $SVGD$-FWI consistently improves imaging quality and provides reliable uncertainty quantification, even with modest particle counts. The results suggest that probabilistic inversion via $SVGD$ is a promising tool for clinical ultrasound, enabling better decision-making through quantified uncertainty and enabling scalable, parallelizable computations.

Abstract

Full Waveform Inversion (FWI) is a promising technique for achieving high-resolution imaging in medical ultrasound. However, conventional FWI methods suffer from issues related to computational efficiency, dependence on initial models, and the inability to quantify uncertainty. This study aims to enhance inversion performance and provide a reliable method for uncertainty quantification in medical FWI imaging. This study integrates the Stein Variational Gradient Descent (SVGD) algorithm into the FWI framework by deriving the posterior gradient for probabilistic inversion. To evaluate the proposed method, numerical experiments are conducted on synthetic datasets, including a breast tissue model with realistic anatomical structure. Imaging accuracy and uncertainty quantification are assessed to compare the performance of SVGD-based FWI with conventional FWI and Stochastic Variational Inference (SVI) methods. Markov Chain Monte Carlo (MCMC) is implemented as a benchmark to evaluate the quality of uncertainty estimates. For synthetic data, the SVGD-based FWI framework yields more precise estimates in the region of interest (ROI) and demonstrates faster convergence compared to the conventional FWI. For the anatomically realistic breast tissue simulation, SVGD produces a maximum relative error of 1.10\% and a mean relative error of 0.09\% in the ROI. The estimated uncertainty is spatially consistent, with most values below 0.01 and a mean of approximately 0.003. Compared to SVI, SVGD provides improved structural resolution and stronger agreement with the MCMC benchmark, indicating more reliable uncertainty quantification. The SVGD-based FWI method improves inversion quality, enhances uncertainty quantification. These findings indicate that probabilistic inversion is a promising tool for addressing the limitations of traditional FWI methods in ultrasonic imaging of medical tissues.

Ultrasonic Medical Tissue Imaging Using Probabilistic Inversion: Leveraging Variational Inference for Speed Reconstruction and Uncertainty Quantification

TL;DR

This work introduces a probabilistic inversion framework for ultrasonic imaging that integrates Stein Variational Gradient Descent () with Full Waveform Inversion () to achieve fast, uncertainty-quantified reconstructions of tissue properties. By deriving the posterior gradient and employing an particle-based update, the method yields accurate squared slowness distributions with spatially coherent uncertainty maps, and demonstrates faster convergence and improved ROI fidelity relative to conventional and stochastic variational inference (), while remaining comparable to or approaching -MCMC benchmarks. Across synthetic linear- and ring-array phantoms and a breast-tissue model, -FWI consistently improves imaging quality and provides reliable uncertainty quantification, even with modest particle counts. The results suggest that probabilistic inversion via is a promising tool for clinical ultrasound, enabling better decision-making through quantified uncertainty and enabling scalable, parallelizable computations.

Abstract

Full Waveform Inversion (FWI) is a promising technique for achieving high-resolution imaging in medical ultrasound. However, conventional FWI methods suffer from issues related to computational efficiency, dependence on initial models, and the inability to quantify uncertainty. This study aims to enhance inversion performance and provide a reliable method for uncertainty quantification in medical FWI imaging. This study integrates the Stein Variational Gradient Descent (SVGD) algorithm into the FWI framework by deriving the posterior gradient for probabilistic inversion. To evaluate the proposed method, numerical experiments are conducted on synthetic datasets, including a breast tissue model with realistic anatomical structure. Imaging accuracy and uncertainty quantification are assessed to compare the performance of SVGD-based FWI with conventional FWI and Stochastic Variational Inference (SVI) methods. Markov Chain Monte Carlo (MCMC) is implemented as a benchmark to evaluate the quality of uncertainty estimates. For synthetic data, the SVGD-based FWI framework yields more precise estimates in the region of interest (ROI) and demonstrates faster convergence compared to the conventional FWI. For the anatomically realistic breast tissue simulation, SVGD produces a maximum relative error of 1.10\% and a mean relative error of 0.09\% in the ROI. The estimated uncertainty is spatially consistent, with most values below 0.01 and a mean of approximately 0.003. Compared to SVI, SVGD provides improved structural resolution and stronger agreement with the MCMC benchmark, indicating more reliable uncertainty quantification. The SVGD-based FWI method improves inversion quality, enhances uncertainty quantification. These findings indicate that probabilistic inversion is a promising tool for addressing the limitations of traditional FWI methods in ultrasonic imaging of medical tissues.
Paper Structure (15 sections, 28 equations, 13 figures, 1 table)

This paper contains 15 sections, 28 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Acoustic wave propagation and data generation in FWI
  4. Variational inference solves FWI problems
  5. SVGD-based posterior approximation and uncertainty quantification in FWI
  6. Results
  7. Synthetic data imaging with linear array transducer
  8. Synthetic data imaging with ring array transducer
  9. Breast tissue imaging using SVGD and FWI algorithms
  10. Uncertainty Quantification via SVI-Based FWI and Metropolis-Hastings MCMC
  11. Computational and memory cost analysis
  12. Discussion
  13. Conclusion
  14. Acknowledgements
  15. Declaration of Competing Interest

Figures (13)

  • Figure 1: Inversion setup and initial conditions for the synthetic data experiment using a linear array transducer. (a) The true model setup. The orange dots and green diamonds indicate the locations of the sources and receivers, respectively. (b) The Ricker wavelet used as the source waveform. (c) The initial model for the conventional FWI. (d) An example of the initial particles for SVGD, with a total of 20 particles used in the inversion.
  • Figure 2: Results of the inversion process for the linear array transducer simulation. (a) The SOS distribution obtained from the conventional FWI. (b) The mean result of the SVGD particles. (c) The standard deviation of the SOS distribution from SVGD. (d) The loss values plotted against the iteration number.
  • Figure 3: Results of the inversion process for the ring array transducer simulation. (a) The true model setup. The orange dots and green diamonds indicate the locations of the sources and receivers, respectively. (b) The SOS distribution obtained from the conventional FWI. (c) The mean SOS estimated by SVGD using 20 particles. (d) The standard deviation of the SOS distribution from SVGD. The pink dashed circle (radius = 3.27 cm) indicates the ROI used to evaluate the effectiveness of the inversion algorithms.
  • Figure 4: Loss and relative error analysis for the ring array transducer simulation. (a) The loss values plotted against the iteration number. (b) and (c) show the relative error for FWI and SVGD, respectively. The pink dashed circle (radius = 3.27 cm) indicates the ROI.
  • Figure 5: Sensitivity of the SVGD algorithm to the number of particles. (a) The mean SOS for different particle counts ($n = 3, 5, 10, 15$, from left to right). (b) The standard deviation of the SOS.
  • ...and 8 more figures