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

PURPOSE: 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. METHODS: 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. RESULTS: 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. CONCLUSION: 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.