Maximum Likelihood Estimation of the Fat Fraction Using Magnetic Resonance Imaging at High Signal-to-Noise Ratio

Nonalcoholic fatty liver disease (NAFLD) is becoming one of the most common liver conditions in western society since the rates of obesity and diabetes are rising.  The standard method for diagnosis of NAFLD is by a liver biopsy.  Yet, a liver biopsy is invasive and not representative since it samples 1/50,000^th of the liver.  Our project involves the use of magnetic resonance imaging (MRI) as a noninvasive method to diagnose NAFLD.  NAFLD is diagnosed by quantifying the fat fraction, which is the amount of fat over the amount of fat and water.  We defined the true fat fraction as Ρ_true = μ_fat/(μ_fat + μ_water) where μ_fat and μ_water is the mean of the fat and water signal.  The magnitude and the maximum likelihood estimation (MLE) methods estimate the fat fraction at high signal-to-noise ratio (SNR).  SNR is the ratio of the mean over the standard deviation of the signal and is high when this ratio is at least 3.  With the magnitude method, the fat-fraction estimate is Ρ_magnitude = |F|/(|F| + |W|). The magnitude of the fat and water signals, denoted as |F| and |W|, follow the Rician distribution.  However, at high SNR, they approximately follow a normal distribution.  With the MLE method, the fat-fraction estimate is Ρ_MLE = F_MLE/(F_MLE + W_MLE) where F_MLE and W_MLE is the estimated fat and water signal. Finally, we will compare both methods by computing the mean squared error (MSE) to show that the MLE method estimates the fat fraction more accurately since it has a smaller MSE.