Hierarchical Bayesian myocardial perfusion quantification

Purpose: Tracer-kinetic models can be used for the quantitative assessment of contrast-enhanced MRI data. However, the model-fitting can produce unreliable results due to the limited data acquired and the high noise levels. Such problems are especially prevalent in myocardial perfusion MRI leading to the compromise of constrained numerical deconvolutions and segmental signal averaging being commonly used as alternatives to the more complex tracer-kinetic models. Methods: In this work, the use of hierarchical Bayesian inference for the parameter estimation is explored. It is shown that with Bayesian inference it is possible to reliably fit the two-compartment exchange model to perfusion data. The use of prior knowledge on the ranges of kinetic parameters and the fact that neighbouring voxels are likely to have similar kinetic properties combined with a Markov chain Monte Carlo based fitting procedure significantly improves the reliability of the perfusion estimates with compared to the traditional least-squares approach. The method is assessed using both simulated and patient data. Results: The average (standard deviation) normalised mean square error for the distinct noise realisations of a simulation phantom falls from 0.32 (0.55) with the least-squares fitting to 0.13 (0.2) using Bayesian inference. The assessment of the presence of coronary artery disease based purely on the quantitative MBF maps obtained using Bayesian inference matches the visual assessment in all 24 slices. When using the maps obtained by the least-squares fitting, a corresponding assessment is only achieved in 16/24 slices. Conclusion: Bayesian inference allows a reliable, fully automated and user-independent assessment of myocardial perfusion on a voxel-wise level using the two-compartment exchange model.

💡 Research Summary

This paper addresses the longstanding challenge of reliably estimating the parameters of the two‑compartment exchange model (2CXM) for myocardial perfusion quantification from dynamic contrast‑enhanced MRI (DCE‑MRI). Conventional non‑linear least‑squares (LS) fitting often fails in this context because the data are sparse, noisy, and the model exhibits multiple local minima, strong parameter correlations, and sensitivity to initial conditions. To overcome these limitations, the authors propose a hierarchical Bayesian framework that treats the kinetic parameters (myocardial blood flow (Fb), plasma volume fraction (vp), interstitial volume fraction (ve), permeability‑surface area product (PS), and time‑delay (τ0)) as random variables with informative priors.

Key innovations include: (1) Physiological priors that constrain each parameter to biologically plausible ranges; (2) Spatial regularisation via a generalized Gaussian Markov random field (GMRF) prior on the differences between neighbouring voxels, encouraging smoothness while preserving edges (p = 1 yields a Laplace prior with edge‑preserving properties); (3) Hierarchical modelling where the hyperparameters governing the priors themselves have hyper‑priors, enabling partial pooling across voxels and allowing the data to inform the degree of regularisation.

Because the posterior distribution p(Θ|y) is analytically intractable, the authors employ Markov chain Monte Carlo (MCMC) sampling (Metropolis‑Hastings) to draw representative samples for each voxel. The posterior mean provides a point estimate, while the posterior variance quantifies uncertainty.

The methodology is first validated on a synthetic 6 × 6 voxel phantom representing three physiological states: rest, stress, and stress‑induced ischemia. Ground‑truth parameters are forward‑simulated, and Rician noise is added to achieve a realistic signal‑to‑noise ratio of 15. For each phantom, 20 independent noise realizations are generated, and the hierarchical Bayesian approach is compared against a gradient‑based LS optimiser (L‑BFGS with box constraints). The normalized mean‑square error (NMSE) across all voxels drops from 0.32 ± 0.55 (LS) to 0.13 ± 0.20 (Bayesian), a statistically significant improvement (Mann‑Whitney U, p < 0.01). Moreover, the Bayesian method yields far fewer failed fits (≈0.3 per phantom versus 4.2 for LS) and outperforms a non‑hierarchical Bayesian variant, demonstrating the benefit of learning hyper‑parameters from the data.

Clinical feasibility is examined in eight patients (four with ischemia, four without) undergoing adenosine stress perfusion MRI at 3 T. After motion correction and dual‑bolus AIF calibration, the same Bayesian pipeline is applied voxel‑wise. The resulting MBF maps perfectly match the expert visual assessment in all 24 short‑axis slices, whereas LS‑derived maps agree in only 16/24 slices. The Bayesian approach also dramatically reduces the number of outlier parameter estimates and failed fits, confirming its robustness in real‑world conditions where ground truth is unavailable.

The authors acknowledge the computational burden of MCMC, which limits immediate bedside deployment. They suggest future work on GPU‑accelerated sampling, variational inference approximations, and broader multi‑center validation to refine prior specifications across diverse populations. Nonetheless, the study convincingly demonstrates that hierarchical Bayesian inference, combined with spatial priors, can deliver reliable, fully automated, and uncertainty‑aware voxel‑wise myocardial perfusion quantification using the physiologically rich 2CXM, outperforming traditional LS methods both in simulation and in clinical practice.