Hierarchical Bayesian framework for regularizing intracardiac 4D Flow

4D Flow MRI is a powerful non-invasive technique for capturing timeresolved, three-directional blood velocity within a 3D volume. However, measurement noise, artifacts, and limited resolution often produce velocity fields that violate the Navier–Stokes equations and boundary conditions, complicating data interpretation and downstream analyses.

We present a hierarchical Bayesian framework for jointly reconstructing intracardiac 4D Flow MRI velocity fields and anatomical masks, incorporating physics-based regularization and uncertainty quantification. The regularization priors enforce zero divergence, endocardial free-slip, and spatial smoothness. These priors are expressed as exponentials of quadratic penalty terms, discretized using centered finite differences, resulting in a multivariate quasi-normal posterior with a block-diagonal precision matrix governed by the priors’ precision hyperparameters. In the hierarchical formulation, these hyperparameters are drawn from noninformative hyperpriors.

Inference is performed via Markov Chain Monte Carlo (MCMC), sampling the conditional posteriors of both latent fields and hyperparameters to generate chains from which posterior expectations and variances are estimated. Given the non- normality of the posterior, we develop a hybrid Gibbs sampler that leverages a maximum a posteriori (MAP) estimate of the latent fields and a Laplace approximation of the posterior covariance, derived from the discrete log- posterior Hessian. Latent variables are sampled from the resulting Gaussian approximation, while hyperparameters are drawn from their gamma conditionals via conjugacy.

This approach effectively mitigates noise in 4D Flow MRI by integrating physical knowledge, anatomical priors, and statistical inference, while also providing uncertainty estimates. The proposed Bayesian framework accommodates complex priors, supports multi-physics data integration, and is readily extensible to incorporate complementary imaging modalities.