Introduction

Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) can be used for the non-invasive assessment of myocardial perfusion. Currently, the clinical evaluation of such image series is performed visually but such an assessment is difficult and time-consuming. Hence, the quantification of perfusion through tracer-kinetic modelling would be desirable and would enable the clinical translation of the technique. Quantification is performed by fitting tracer-kinetic models to the observed imaging data in order to estimate the model parameters. However, this model fitting is difficult as a result of the high noise level and limited number of acquisition points and temporal resolution, potentially leading to unreliable model parameter estimates.^1,2 To improve the robustness of the estimates, much of the work in this field performs segment-wise averaging to improve SNR or uses simpler but unphysiological models such as the Fermi function which leads to the estimation of fewer physiological parameters. However, the averaging of signal across segments compromises the spatial information present in the data and leads to the loss of important diagnostic information. The use of physiologically motivated models such as the two-compartment exchange model would also be desirable. The 2CXM model has four parameters, all of which are physiologically interpretable (compared to one in the Fermi function). In this work we propose the use of Bayesian inference to estimate the model parameters and demonstrate the reliable fitting of the 2CXM using both simulated and patient data.

Methods

Traditionally, the parameter estimation is done using a least-squares fitting which finds the set of parameters that minimises the least-squares between the model and the observed data. However, such parameter estimation problems are susceptible to convergence to local optima (Figure 1). This leads to unreliable parameter estimates that are highly sensitive to the initial conditions of the optimisation and the specific noise realisation present in the data. It has been proposed that Bayesian inference can improve the reliability of the tracer-kinetic parameter estimation.³ Bayesian inference provides a natural framework for the inclusion of prior knowledge of the kinetic parameters and by using a Markov random field prior it can take advantage of the fact that neighbouring voxels are likely to have similar kinetic properties. A hierarchical Bayesian model is employed in this study. Hierarchical Bayesian modelling does not use prior distributions with fixed hyperparameters but rather with hyperparameters governed by another distribution, a hyperprior. That is, rather than having $$$F_b \sim \mathcal{N}(\mu,\sigma^2)$$$ with a fixed value of $$$\mu$$$ instead $$$\mu$$$ is governed by a hyperprior. This choice is natural for myocardial perfusion quantification as it is difficult to specify hyperparameters without first knowing if a patient is healthy or diseased. The posterior distribution for the parameters is given through the application of Bayes’ theorem. The posterior distribution is not analytically tractable and must be approximated using Markov chain Monte Carlo (MCMC) techniques. This Markov chain is constructed using the Metropolis-Hastings algorithm. The proposed method was first tested using simulated image series as this is the only situation where estimates can be compared to ground-truth values. The error of proposed parameter estimation method is compared in a Monte-Carlo study for distinct noise realisations to the least-squares fitting. The technique was subsequently tested in eight patients suspected of having coronary artery disease. We test whether the kinetic parameter values that are estimated can identify perfusion defects that match the visual assessment found in the clinical reports.

Results

The average (standard deviation) mean square error with the ground-truth values for the 25 distinct noise realisations is 4.05 (1.13) This error falls to 1.32 (0.37) when the fitting is repeated with 5 different random initial conditions. The described Bayesian inference method reduces the error significantly to 0.078 (0.029) (Figure 2). A comparison of the best fit with 100 random initialisations of least-squares fitting and the Bayesian inference for an example patient is shown in Figure 3. The assessment of the patients’ health based purely on the quantitative flow 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 (Figure 4).

Conclusion

Hierarchical Bayesian inference allows the reliable voxel-wise quantification of the kinetic parameters of the 2CXM from myocardial perfusion MRI data. The high resolution perfusion maps can aid the diagnosis of myocardial ischaemia. The method however needs to be validated more thoroughly in comparison with established techniques such as PET or microspheres. Additionally, the Bayesian framework also allows the quantification of uncertainty which may prove beneficial in the clinic.