Combining machine learning and mathematical modeling in inverse problems: Application to estimation of T1 relaxation time from cardiac magnetic resonance imaging data

<h2>Abstract</h2> A method for estimating tissue parameters using cardiac magnetic resonance imaging (MRI) by combining a Bloch equation-based model, neural network (NN) and numerical optimization (NO) is presented. The method is applied to the <mml:math><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:math> relaxation time estimation from Modified Look-Locker Inversion Recovery (MOLLI) image series. The estimation is performed in two stages: the first-stage estimate <mml:math><mml:mi>T</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:math> was obtained by the NN trained on synthetic data generated by the Bloch simulator; the final estimate <mml:math><mml:mi>T</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mi>O</mml:mi></mml:mrow></mml:math> was obtained by numerical minimization of the similarity criterion between the simulated and measured MOLLI series. The method was tested on a set of 10,000 synthetic MOLLI series generated using a wide range of parameters. The most efficient optimization strategy was then used on real MRI data. MOLLI data were acquired for: eight phantoms on 1.5 and 3 T MRI systems; twelve patients on 1.5 T and three patients on 3 T systems (native and after administering <mml:math><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:math>-reducing contrast agent). In the phantom study, additionally, inversion recovery turbo spin echo (IR-TSE) data provided a pseudo-ground truth of the <mml:math><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:math> relaxation time. The MOLLI images were post-processed by the scanner (<mml:math><mml:mi>T</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> estimate) and by the proposed two-stage method. In the phantom study on 3 T, the results were closer to the pseudo-ground truth in 7 out of 8 phantoms. On 1.5 T, the difference between the proposed method and the scanner was not statistically significant. The NO stage improved the accuracy and decreased the variation of <mml:math><mml:mi>T</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:math>. For the <i>in vivo</i> data, the pre-contrast <mml:math><mml:mi>T</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mi>O</mml:mi></mml:mrow></mml:math> in blood and myocardium was higher than <mml:math><mml:mi>T</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> in most subjects measured on 1.5 T and 3 T systems. The post-contrast <mml:math><mml:mi>T</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mi>O</mml:mi></mml:mrow></mml:math> was higher than <mml:math><mml:mi>T</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> in blood for subjects measured on both systems. In the case of post-contrast myocardium, <mml:math><mml:mi>T</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> was higher than <mml:math><mml:mi>T</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mi>O</mml:mi></mml:mrow></mml:math> in most subjects measured. The results are in line with the phantom experiment and are expected due to the known underestimation of <mml:math><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:math> from MOLLI. NO initialized by NN, which can be trained using simulated data, was shown to have the potential to increase the efficiency and robustness of tissue parameter estimation from image data by using biophysical models.