Center for Biomedical Imaging, Department of Radiology, New York University School of Medicine, New York City, New York, USA.
Center for Advanced Imaging Innovation and Research (CAI2R), Department of Radiology, New York University School of Medicine, New York City, New York, USA.
Magn Reson Med. 2022 Jul;88(1):436-448. doi: 10.1002/mrm.29206. Epub 2022 Mar 28.
To improve the performance of neural networks for parameter estimation in quantitative MRI, in particular when the noise propagation varies throughout the space of biophysical parameters.
A theoretically well-founded loss function is proposed that normalizes the squared error of each estimate with respective Cramér-Rao bound (CRB)-a theoretical lower bound for the variance of an unbiased estimator. This avoids a dominance of hard-to-estimate parameters and areas in parameter space, which are often of little interest. The normalization with corresponding CRB balances the large errors of fundamentally more noisy estimates and the small errors of fundamentally less noisy estimates, allowing the network to better learn to estimate the latter. Further, proposed loss function provides an absolute evaluation metric for performance: A network has an average loss of 1 if it is a maximally efficient unbiased estimator, which can be considered the ideal performance. The performance gain with proposed loss function is demonstrated at the example of an eight-parameter magnetization transfer model that is fitted to phantom and in vivo data.
Networks trained with proposed loss function perform close to optimal, that is, their loss converges to approximately 1, and their performance is superior to networks trained with the standard mean-squared error (MSE). The proposed loss function reduces the bias of the estimates compared to the MSE loss, and improves the match of the noise variance to the CRB. This performance gain translates to in vivo maps that align better with the literature.
Normalizing the squared error with the CRB during the training of neural networks improves their performance in estimating biophysical parameters.
提高神经网络在定量磁共振成像中参数估计的性能，特别是在噪声传播在生物物理参数空间中变化时。
提出了一种理论上合理的损失函数，该函数将每个估计的平方误差与相应的克拉美-罗界（CRB）——无偏估计方差的理论下限——进行归一化。这避免了参数空间中难以估计的参数和区域的主导地位，这些区域通常没有什么兴趣。与相应的 CRB 进行归一化可以平衡基本噪声较大的估计的大误差和基本噪声较小的估计的小误差，从而使网络更好地学习估计后者。此外，所提出的损失函数提供了性能的绝对评估指标：如果网络是最大效率的无偏估计器，则其平均损失为 1，这可以被认为是理想的性能。所提出的损失函数在拟合幻影和体内数据的八参数磁化传递模型的示例中展示了性能增益。
使用所提出的损失函数训练的网络接近最优，即其损失收敛到大约 1，并且其性能优于使用标准均方误差（MSE）训练的网络。与 MSE 损失相比，所提出的损失函数减少了估计的偏差，并提高了噪声方差与 CRB 的匹配程度。这种性能增益转化为与文献更一致的体内图谱。
在神经网络的训练过程中，用 CRB 对平方误差进行归一化可以提高其生物物理参数估计的性能。
