Introduction

The image contrast in Magnetic Resonance Imaging (MRI) depends on several tissue parameters of the imaged patient. Quantitative MRI is a subset of MRI that include actual quantification of these tissue parameters. These methods require a series of MRI images and a post-imaging reconstruction process that isolates the relationship between the MRI signal and the tissue parameters of interest in the imaged region. There are several medical applications where the quantification of tissue parameters provide useful information, e.g. in oncology^1,2,3.

The tissue parameter estimation (commonly referred to as parameter mapping) is a post-imaging procedure. Conventionally, the value of the tissue parameters is calculated by a pixel-wise non-linear least squares (NLLS) regression of the MRI magnitude signal data, which can result in noisy parameter maps. Adding denoising into this process introduces spatial correlations which makes uncertainty estimation more complicated. Convolutional neural networks (CNN) are capable to address spatial correlations, but these methods require large training datasets that are highly specific for each problem of interest. Blind denoising of image data with deep image prior⁴ (DIP) has been shown successfully in several applications, including medical imaging applications^5,6,7. These methods are CNN based but do not require any training dataset which makes the methods easier to implement. Combining a DIP approach with uncertainty estimations, e.g., from a Bayesian approximation⁸, results in methods that address denoising and uncertainty estimation without the need for a training dataset. Here we extend this concept to parameter mapping in quantitative MRI by implementing task-specific loss functions.

Methods

In this work, we address parameter mapping with a convolutional neural network (CNN) based framework. With a parameter mapping dataset $$$\boldsymbol{y} \in \mathbb{R}^{V \times M}$$$, i.e., a total of $$$M$$$ magnitude signal measurement over $$$V$$$ voxels, we define task-specific loss functions
$$E(\mathbf{y}; \boldsymbol{\beta}) = \frac{1}{VM}\sum_{v=1}^{V}\sum_{m=1}^{M}\Big\Vert y_{v,m} - s(t_m, \boldsymbol{\beta}_v) \Big\Vert_2^2$$
where $$$s$$$ is the magnitude MR signal equation, which depends on both scanner settings $$$t$$$ and the tissue parameters $$$\boldsymbol{\beta}$$$. Examples of applications are variable flip angle T1 estimation with spoiled gradient echo and multi-echo spin-echo for T2 estimation.

Following the works of Ulyanov⁴ et al, we define $$$\mathbf{f}_{\boldsymbol{\theta}}(\mathbf{Z})$$$ to be the output of a CNN $$$\mathbf{f}$$$ with weights $$$\boldsymbol{\theta}$$$, when given a random tensor $$$\mathbf{Z}$$$ as input. $$$\mathbf{f}$$$ is then reinterpreted as a Bayesian Neural Network with Bernoulli variation inference⁹ by appending dropout after every convolution layer. DIP-enhanced parameter mapping is then implemented by optimizing over network weights
$$\arg \min_\boldsymbol{\theta} \frac{1}{VM}\sum_{v=1}^{V}\sum_{m=1}^{M}\Big\Vert y_{v,m} - s(t_m, \mathbf{f}_{\boldsymbol{\theta}}(\mathbf{Z})_v) \Big\Vert_2^2,$$
and after a number of iterations, estimate the tissue parameters with Monte Carlo integration of $$$N$$$ concurrent forward passes with dropout applied
$$$\hat{\boldsymbol{\beta}} = \frac{1}{N}\sum_{n=1}^{N} \mathbf{f}_{\boldsymbol{\theta}}(\mathbf{Z})$$$ with associated model uncertainty $$$\sigma_{\hat{\boldsymbol{\beta}}} = \frac{1}{N}\sum_{n=1}^{N} (\mathbf{f}_{\boldsymbol{\theta}}(\mathbf{Z}) - \hat{\boldsymbol{\beta}})^2$$$. This enables noise-reduced parameter mapping with associated model uncertainty estimation, without explicitly formulated spatial priors nor task-specific training data sets.

Our method was tested with three separate parameter mapping tasks: Variable flip-angle spoiled gradient echo for T1 mapping, multi-echo spin-echo for T2 mapping, and spin-echo based diffusion imaging for ADC mapping. For all tests, a skip-connection U-net (with about $$$2.2\times 10^{6}$$$ trainable parameters) was selected as network architecture. The method was tested on both in-vivo data and synthetically generated MRI signal with ricean distributed noise and a known ground-truth reference.

This study was conducted in accordance with the principles embodied in the Declaration of Helsinki and was ethically approved by the regional research ethics committee (dnr: 2019-02666).

Results

Tissue parameter estimation with synthetic data is presented in figure 1 (T1 estimation with VFA data) and figure 2 (T2 estimation with multi-echo spin-echo data). In both cases, the result clearly illustrates the denoising feature of DIP with a decrease in root mean squared error, but even more important, that the concept of DIP denoising extend into the domain of parameter mapping in quantitative MRI. The added benefit of the Bayesian approximation is illustrated in figure 3, which highlights the sampling distribution of the T1 samples at different spatial locations.

ADC mapping with in-vivo data is presented in figure 4, which gives similar results as in figure 1 and 2, but with no available ground truth reference.

Discussion and Conclusion

Our method successfully produces parameter maps with the added benefits of significant noise reduction and model uncertainty estimation. In contrast to conventional CNN approaches, our method does not require any training dataset due to the DIP implementation. Our method is easy to customize for different applications since the reconstruction is embedded in a loss function that is specific to the problem of interest, and the majority of the CNN architecture can be reused for different applications.

Future work and further investigations should address computational efficiency, optimization stopping criterion, and uncertainty calibration.

Acknowledgements

This research was in part funded by grants from The Swedish Research Council (grant number 2019-0432), Region Västerbotten (Central ALF, grant number RV-738491), Cancerforskningsfonden i Norrland (grant number AMP 18-912), and Lions Cancerforskningsfond (grant number LP 18-2182).