Analyzing Model Misspecification in Quantitative MRI: Application to Perfusion ASL

TL;DR

This work tackles the problem of model misspecification in quantitative MRI ($qMRI$) by adopting the misspecified Cramér–Rao bound ($\mathrm{MCRB}$) as a quantitative metric for estimator variance and bias under mismatch. It introduces two statistically principled tests: (i) monitoring the asymptotic convergence of the empirical $\mathrm{MCRB}$ to the conventional $\mathrm{CRB}$ as the number of repeated measurements grows, and (ii) assessing parameter-estimate consistency across equal data subsets against the $\mathrm{CRB}$. The Buxton arterial spin labeling ($ASL$) model is used as a case study, revealing that the model is well specified in the brain but moderately misspecified in the kidney, with fixed-parameter misspecification (voxelwise $T_1$) contributing to residual uncertainty. The framework provides a general, theoretically grounded approach for validating $qMRI$ models and guiding model refinement and acquisition strategies, such as voxelwise parameter mapping or additional $T_1$ mapping to reduce uncertainty.

Abstract

Quantitative MRI (qMRI) involves parameter estimation governed by an explicit signal model. However, these models are often confounded and difficult to validate in vivo. A model is misspecified when the assumed signal model differs from the true data-generating process. Under misspecification, the variance of any unbiased estimator is lower-bounded by the misspecified Cramer-Rao bound (MCRB), and maximum-likelihood estimates (MLE) may exhibit bias and inconsistency. Based on these principles, we assess misspecification in qMRI using two tests: (i) examining whether empirical MCRB asymptotically approaches the CRB as repeated measurements increase; (ii) comparing MLE estimates from two equal-sized subsets and evaluating whether their empirical variance aligns with theoretical CRB predictions. We demonstrate the framework using arterial spin labeling (ASL) as an illustrative example. Our result shows the commonly used ASL signal model appears to be specified in the brain and moderately misspecified in the kidney. The proposed framework offers a general, theoretically grounded approach for assessing model validity in quantitative MRI.

Figures (4)

• Figure 1: Simulated and in vivo asymptotic convergence as the number of repeated measurements increases. Subject 1 represents the better-conditioned case for both organs. In simulation, as $m$ becomes sufficiently large, all metrics converge to their theoretical values. In vivo, the eigenvalues $\lambda_{\max}$, $\lambda_{\min}$ do not converge to 1 as $m$ increases, especially in the kidney.
• Figure 2: Parameter estimates from two PLD subsets and their average relative error over all voxels (white text). Errors in the kidney are larger and more spatially heterogeneous.
• Figure 3: Empirical MLE variance compared with the theoretical CRB (dotted lines). In the brain, the variance of $f$ remains tightly bounded by the CRB, and $\mathrm{ATT}$ shows a similar but slightly looser bound. In contrast, in the kidney the empirical variance exceeds the CRB by a larger margin, indicating that the CRB underestimates the estimation uncertainty.
• Figure 4: Eigenvalues and condition number obtained with a global $T_1$ (left), a voxelwise $T_1$ (middle), and their difference (right). Voxelwise $T_1$ lowers both eigenvalues, yet the condition number remains large, suggesting that fixed $T_1$ and model mismatch both contribute to misspecification.