Anders Garpebring^{1}, Max Hellström^{1}, Mikael Bylund^{1}, and Tommy Löfstedt^{1}

The purpose of this work was to develop a method that simultaneously reduces and estimates the uncertainty in the T1 maps obtained with the VFA method while also avoiding the need for any manual tuning of regularization parameters. A Markov Chain Monte Carlo-based algorithm was implemented and evaluated on real and synthetic data. The results show that the method can be used to reduce both noise and noise-induced bias and simultaneously give information about the uncertainty in the estimates.

In dynamic contrast-enhanced MRI the acquisition of a T1 map prior to the injection of contrast agent is important for quantification of pharmacokinetic parameters^{1}. Commonly, the T1 map is obtained using the variable flip angle (VFA) method^{2}, which may result in noisy and also biased T1 maps. A solution to this issue is to add a spatial regularizer in the data fitting to reduce the noise^{3}, and the bias introduced from nonlinear noise propagation. Unfortunately, there is usually no information about the uncertainty in the results produced by these methods and one needs to set regularization parameters manually. The purpose of this work was to develop a method that simultaneously reduces and estimates the uncertainty in the T1 maps obtained with the VFA method while also avoiding the need for any manual tuning of regularization parameters.

**Statistical model**

A Bayesian hierarchical model was used in this work to model the data and the parameter maps. In the model, the data likelihood is given by

$$p(\boldsymbol{y}|\boldsymbol{\theta}, \sigma^2)=(2\pi\sigma^2)^{-MV/2}e^{-\frac{1}{2\sigma^2}\sum_{m=1}^{M}\sum_{v = 1}^{V}\left( s_{m}(\boldsymbol{\theta}_v) - y_{m,v}\right)^2}$$

where $$$\boldsymbol{\theta}$$$ is the parameter maps, $$$v$$$ is an index for the pixels, $$$V$$$ is the number of pixels in the analyzed region, $$$m$$$ is an index for the acquired flip angles, $$$M$$$ is the number of flip angles and $$$\sigma^2$$$ is the noise variance in each pixel. The elements of $$$\boldsymbol{y}$$$, denoted $$$y_{m,v}$$$, corresponds to the measured signals in each voxel for each flip-angle and $$$s_m(\boldsymbol{\theta}_v)$$$ is the signal intensity given by the standard spoiled gradient echo (SPGR) signal equation^{4}. A total variation based prior

$$p(\boldsymbol{\theta} | \boldsymbol{\lambda}) \propto \prod_{i=1}^2 \lambda_i^V e^{-\lambda_i TV(\boldsymbol{\theta}^i)}$$

was used for the parameter maps in which $$$TV(\cdot)$$$ is the 2D total variation operator. For the regularization parameters $$$\boldsymbol{\lambda}$$$ and the variance

$$p(\boldsymbol{\lambda},\sigma^2)\propto \prod_{i=1}^2\lambda_i^{-\kappa V}\sigma^{-1}$$

was used as prior. Together these constitute spatial model of how the parameter maps are likely to behave. The index $$$i$$$ in the above equation corresponds to the T1 and proton density maps. The parameter $$$\kappa$$$ was used to find an optimal balance between trusting the data and the prior.

**Estimation**

To estimate parameter maps from the model, the Markov Chain Monte Carlo (MCMC) method by Goodman and Weare^{5} was implemented in Matlab. To find the optimal trade-off between the data and prior, the $$$\kappa \in [0,1]$$$ that gave the smallest average difference between the model and the data was selected.

**Evaluation**

The method was evaluated on synthetic images generated using data from the BrainWeb project^{6}, the SPGRE signal equation with settings $$$TR = 5.0$$$ ms, $$$FA = \{2^\circ, 10^\circ,19^\circ \}$$$ and Rician noise conditions corresponding to that of a 3 Tesla MRI scanner using a head-coil and 2 mm thick slices. The method was also tested on the RIDER dataset^{7,8} from a 1.5 T MRI scanner with imaging settings $$$TR = 4.43$$$ ms and $$$FA = \{5^\circ, 10^\circ,15^\circ, 20^\circ \}$$$.

Figure 1 illustrates the use of the method on the RIDER data. In Fig. 1B a T1 map obtained with pixel-by-pixel fitting is displayed. The image is quite noisy, which hides some details. With the proposed spatial model the noise is much reduced and small details are clearer as can be seen in Fig 1C. Using MCMC sampling also gives the uncertainty, which is shown as histograms in Fig. 1A. The histograms show the uncertainty in the yellow circle in Fig. 1A and B and it is clear from the histograms that the proposed spatial model decreases the uncertainty and also shifts the T1 estimates towards lower values.

Figure 2 and 3 show the effect of applying the method to the synthetic data. The error is clearly reduced when using the proposed spatial model while edges are preserved. In Fig. 3A-C it is also apparent that both bias and variance is reduced with the proposed method. To evaluate the uncertainty estimation, the fraction of correct T1-values within a credible interval (CI) is plotted against the CI size. This should ideally result in a straight line with slope one and the result in Fig. 3D is very close to this with a small deviation when the spatial model is used.

The presented results shows that using a spatial total variation based model, noise (as well as bias caused by nonlinear noise-propagation) can be greatly reduced in T1 maps. By performing the estimation with an MCMC algorithm, samples from the posterior distribution and hence uncertainty can be obtained. Further, by searching over a single hyper-parameter, $$$\kappa$$$, to minimize the difference between model prediction and data, subjective manual parameter tuning was avoided.

1. Yankeelov, T. E. & Gore, J. C. Dynamic Contrast Enhanced Magnetic Resonance Imaging in Oncology: Theory, Data Acquisition, Analysis, and Examples. Curr. Med. Imaging Rev. 3, 91–107

2. Fram, E. K. et al. Rapid calculation of T1 using variable flip angle gradient refocused imaging. Magn. Reson. Imaging 5, 201–8 (1987).

3. Wang, H. & Cao, Y. Spatially regularized T(1) estimation from variable flip angles MRI. Med. Phys. 39, 4139–48 (2012).

4. Schabel, M. C. & Parker, D. L. Uncertainty and bias in contrast concentration measurements using spoiled gradient echo pulse sequences. Phys. Med. Biol. 53, 2345–2373 (2008).

5. Goodman, J. & Weare, J. Ensemble samplers with affine invariance. Commun. Appl. Math. Comput. Sci. 5, 65–80 (2010).

6. Aubert-Broche, B., Evans, A. C. & Collins, L. A new improved version of the realistic digital brain phantom. Neuroimage 32, 138–45 (2006).

7. Barboriak, D. Data From RIDER_NEURO_MRI. The Cancer Imaging Archive. (2015). doi:http://doi.org/10.7937/K9/TCIA.2015.VOSN3HN1

8. Clark, K. et al. The cancer imaging archive (TCIA): Maintaining and operating a public information repository. J. Digit. Imaging 26, 1045–1057 (2013).

Figure
1: T1 map
estimated using a pixel-by-pixel approach (B) and the proposed method (C). (A) Estimated
probability distribution for the pixel at the center of the yellow circle in (B) and (C).

Figure 2: Results
from applying the method to synthetic images. (A) The ground truth
T1 map. (B) T1 map estimated using a pixel-by-pixel approach. (C)
T1 map estimated using the proposed method. (D) and (E) shows the
error in the estimated values in (B) and (C), respectively.

Figure 3: (A)
T1 estimated using a pixel-by-pixel approach plotted against true
values. (B) T1 estimated using the proposed method plotted against the true values.
(C) Relative bias and standard deviation components of the error in the
estimated parameter as a function of the true T1 value. (D) Fraction of true
values within the estimated credible interval.