Synopsis

The purpose of this work is to develop a generalized formulation to study the parameter estimation of complex MR signals and its multi-species components.

Introduction

Generalized Formulation

The MR signal evolution of $$$M$$$ chemical species can be written as$$s(t)=\sum_{m=1}^M\rho_m e^{i\phi_m}e^{i(\omega_m+ir_m)t_n}$$ where $$$\rho_m,\phi_m\in\mathrm{R}$$$ are the amplitude and the phase and $$$\omega_m,r_m\in \mathbb{R}$$$ the chemical frequency shift and the transverse relaxation rate of the $$$m$$$-th species, respectively. For a multi-echo sampling with $$$N$$$ points we can write$$\vec{s}=AP\vec{\rho},$$where we introduced$$\vec{s}_{N\times1}=[s(t_1),s(t_2),...,s(t_N)]^T$$

$$A_{N \times M}=\begin{pmatrix}e^{i(\omega_1+ir_1)t_1}&...&e^{i(\omega_M+ir_M)t_1}\\\vdots& &\vdots \\e^{i(\omega_1+ir_1)t_N} &...&e^{i(\omega_M+ir_M)t_N}\\\end{pmatrix}$$

$$\rho_{M\times 1}=\begin{pmatrix}\rho_1 \\\vdots \\\rho_M \\\end{pmatrix}.$$

These equations can be thought of as the signal model of $M$ species where all model parameters $$\vec{\beta}=[\vec{\rho}^T, \vec{\phi}^T, \vec{\omega}^T, \vec{r}^T]^T \in \mathbb{R}^{4M}$$ are unknowns and no constrains linking different species are introduced. Most often such a model is practically unfeasible in MRI and parameters are fixed by a priori assumptions and constrains, which reduce the number of unknowns and allows to fit the model to real experimental data where $N$ is typically not very large. Such constrains can be conveniently introduced on the level of derivatives describing how the model behaves for varying parameters. These form the full model Jacobian

$$J=\left[ d\vec{s}/d\vec{\rho},d\vec{s}/\vec{\phi},d\vec{s}/d\vec{\omega},d\vec{s}/\vec{r}\right]$$

$$d\vec{s}/d\vec{\rho}=APC_{\rho}$$

$$d\vec{s}/d\vec{\phi}=iAP(\vec{\rho}\vec{u}^T\circ C_{\phi})$$

$$d\vec{s}/d\vec{\omega}=iTAP(\vec{\rho}\vec{u}^T\circ C_{\omega})$$

$$d\vec{s}/d\vec{r}=-TAP(\vec{\rho}\vec{u}^T\circ C_r)$$

with $$$T=\diag(t_1,...,t_N)$$$, $$$\vec{u}^T=[1,1,...,1]_{M\times 1}$$$, and $$$\circ$$$ being element-wise multiplication.

Cramer-Rao Analysis

In the case of complex white Gaussian noise with variance $$$\sigma^{2}$$$, it can be shown that the Fisher information matrix of the model⁶ with given constrains $$$\{C_{\rho},C_{\phi},C_{\omega},C_r\}$$$ is given by $$$F=\frac{1}{\sigma^{2}}J^{\dagger}J$$$. Together with Equation above there is thus a general expression of the Cramer-Rao lower bound for each chosen signal model, as$$\text{CRLB}\leq F^{-1}.$$

Algorithm

Note that Equation above also allows to develop a parameter estimation algorithm that is independent of an a priori choice of the signal model and iteratively solves for the unknown parameters $$$\vec{\beta}'$$$ via the variable projection method (VARPRO)⁵ using a Gauss–Newton update

$$\vec{\beta}^{'(k)}_{\text{nonlinear}}= (J^{\dagger}J)^{-1}J^{\dagger} (\vec{s}_{\text{measured}}-\vec{s}(\vec{\beta}^{'(k-1)}_{\text{nonlinear}})),$$

for the nonlinear parameters $$$\vec{\beta}_{\text{nonlinear}}$$$. A fast MATLAB mex implementation of this algorithm, together with methods to estimate the noise behavior of different signal models via Equation above will be made available.

Results

In the following we demonstrate the flexibility of the model-open formulation in deriving a known result from literature in chemical-shift encoded water–fat imaging and show one new application in a different, less well-studied water–fat model.

Single-$$$T_2^{*}$$$ Multi-peak Water--Fat Separation

A widely used class of water--fat models^2,3 use an a priori fixed $M$-peak fat spectrum with frequencies shifts $$$\{\Delta f_m\}$$$ and relative peak amplitudes $$$\{\alpha_m\}$$$, with $$$\sum_m^M \alpha_m=1$$$, which is often written as

$$s_n=(W+c_nF)e^{i2\pi f_Bt_n}e^{-R_2^*t_n},\qquad c_n=\sum_m^M\alpha_me^{2\pi i(\Delta f_m)t_n},$$

where each multiple species obeys a common $$$R_2^{*}$$$-decay and the frequency of the water peak $$$f_B$$$ is named "field map", which governs also the phase behavior of all fat peaks. Such a model can be derived by setting the $$$M\times M$$$ constraining matrices as given in Figure 1. The NSA of each model parameter based on the theoretical AA is also depicted in Figure 1, where 6 echos with an echo spacing of 1ms were assumed. Figure 2 shows an in vivo result of estimating the model parameters in an invivo thigh scan with the same echo times that was reconstructed with an algorithm using the generalized formulation.

Single-$$$T_2^{*}$$$ Multi-peak Water--Fat Separation in the Presence of an Additional Fat-Spectrum Shift

Previously a water–fat model with an additional parameter was reported⁴ that models a variable shift $$$x$$$ of the whole fat spectrum with respect to the water peak like

$$s_n=(W+c_nF)e^{i2\pi f_Bt_n}e^{-R_2^*t_n}, \qquad c_n=\sum_m^M\alpha_me^{2\pi i(\Delta f_m+x)t_n}.$$

In the generalized formulation this can comfortably implemented with switching only two subcolumns in the matrix $$$C_{\omega}$$$. Figure 3 shows in the same CRLB analysis that noise performance is only reduced in the phases of the water and fat signal and not in the other parameters, while effectively increasing the model match as the residual of the parameter estimation fit is reduced compared to the model above.

Discussion & Conculsion

Acknowledgements

References

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