Balancing Bias and Variance in Deep Learning-Based Tumor Microstructural Parameter Mapping

Time-dependent diffusion MRI enables quantification of tumor microstructural parameters useful for diagnosis and prognosis. Nevertheless, current model fitting approaches exhibit suboptimal bias-variance trade-offs; specifically, nonlinear least squares fitting (NLLS) demonstrated low bias but high variance, whereas supervised deep learning methods trained with mean squared error loss (MSE-Net) yielded low variance but elevated bias. This study investigates these bias-variance characteristics and proposes a method to control fitting bias and variance.

Random walk with barrier model was used as a representative biophysical model. NLLS and MSE-Net were reformulated within the Bayesian framework to elucidate their bias-variance behaviors. We introduced B2V-Net, a supervised learning approach using a loss function with adjustable bias-variance weighting, to control bias-variance trade-off. B2V-Net was evaluated and compared against NLLS and MSE-Net numerically across a wide range of parameters and noise levels, as well as in vivo in patients with head and neck cancer.

Flat posterior distributions that were not centered at ground truth parameters explained the bias-variance behaviors of NLLS and MSE-Net. B2V-Net controlled the bias-variance trade-off, achieving a 56% reduction in standard deviation relative to NLLS and an 18% reduction in bias compared to MSE-Net. In vivo parameter maps from B2V-Net demonstrated a balance between smoothness and accuracy.

We demonstrated and explained the low bias-high variance of NLLS and the low variance-high bias of MSE-Net. The proposed B2V-Net can balance bias and variance. Our work provided insights and methods to guide the design of customized loss functions tailored to specific clinical imaging needs.