Mathematical Model

Overview

In the past, spectroscopic fitting has been possible with intensity estimates, or area integration. However, in-vivo where spectral resolution and signal to noise is limited, mathematical modeling is essential.

The model used depends on the characteristics of the data, as well as the post-processing steps that transform the data.

FitMAN incorporates two basic model functions. One describes spectroscopic data in the measurement (time) domain, while the other describes spectroscopic data in the frequency domain (after Fast Fourier Transform). Real and imaginary data must both be present for fitting. The fitting routine uses a Levenberg-Marquart minimization routine.

The time domain model function used by the fitMAN minimization routine is given by the following equation:

where

• = Sampled points along the estimated time domain signal at points n where n = 1, 2, ... , N
• k = Resonance index, where k = 1, 2, ... , K
• K = Maximum number of resonances
• N = Maximum number of discrete samples
• = Chemical shift (frequency) of resonance k
• = Amplitude of resonance k
• = Lorentzian damping of resonance k
• = Gaussian damping of resonance k
• = Phase of resonance k
• = Delay time (time between center of stimulated echo and first readout point)
• = Dwell time

The frequency domain model function used by the fitMAN minimization routine is given by the following equation:

where

• = sampled points along spectrum at frequency s
• = delay time
• , , , = real valued amplitude, Lorentzian damping factor, angular frequency, and phase of sinusoid number k

Both time and frequency domain model functions incorporate phase and delay time as parameters to be fit. The Frequency domain model function does not include a term for Gaussian damping.