Mustapha Bouhrara^{1} and Richard G. Spencer^{1}

The Cramér-Rao lower bound (CRLB) is widely used in the design of magnetic resonance (MR) experiments for parameter estimation. Previous work has considered only Gaussian or Rician noise distributions in this calculation. However, the noise distribution for multiple-coil acquisitions, such as in parallel imaging, obeys the noncentral χ-distribution under many circumstances. Here, we present the general mathematical expression for the CRLB calculation for parameter estimation from multiple-coil acquisitions. Our results indicate that the CRLB calculation must account for the noncentral χ-distribution of noise in multi-coil acquisitions, especially in the low-to-moderate signal-to-noise ratio (SNR) regime.

**PURPOSE**

**THEORY**

*Noncentral
χ-distribution
*

The probability density function (PDF) for signal intensity within a given voxel of a magnitude image reconstructed from multiple-coil acquisitions is given by (7):

$$P_χ (M,A,σ,m)=\frac{A^{1-m}}{σ^2}\ M^m\ exp(-\frac{M^2+A^2}{2σ^2})\ I_{m-1}(\frac{MA}{σ^2}),\ \ \ \ [1]$$

where *A* is the magnitude of the underlying noise-free
signal, *M* is the magnitude of the observed signal, *m* is the number of coils, *σ ^{2}* is the noise variance, and

**CRLB for the
noncentral χ-distribution
**

Calculation of the CRLB requires inversion of the Fisher matrix
(1-6). For *N* measured data values fit to a signal model
defined by a parameter vector **β**, the Fisher matrix elements for
the noncentral χ-distribution are:

$$F_{ij}=-E[\frac{∂}{∂β_i}\frac{∂}{∂β_j}\ log(L_χ (M,A,σ,m))]=-E[\frac{∂}{∂β_i}\frac{∂}{∂β_j}\log(\prod_{n=1}^NP_χ (M_n,A_n,σ,m))],\ \ \ [2]$$

where *E* stands for expectation value, and *L _{χ}* is the likelihood function. The CRLB for the
standard deviation (SD) of an unbiased parameter

$$SD(β_i )=\sqrt{F_{ii}^{-1}}.\ \ \ \ [3]$$

After some algebra, Eq.2 reduces to

$$F_{ij}=\frac{1}{σ^2} \sum_{n=1}^N\frac{∂A_n}{∂β_i}\frac{∂A_n}{∂β_j}\ R_n,\ \ \ [4]$$

where

$$R_n=-\frac{A_n^2}{σ^2}+\frac{A_n^{1-m}}{σ^4} \int_{0}^{∞}M_n^{m+2}\ \frac{I_m^2(z_n)}{I_{m-1}(z_n)}\ exp(-\frac{M_n^2+A_n^2}{2σ^2})\ dM\ \ \ [5]$$

and must be calculated numerically.

**METHODS**

**PDF and R _{n} as a function of SNR and number of coils**

The PDF of voxel intensity, given by Eq.1, was calculated for
different values of *m* as a function of *M*/*σ* and for *A*=1, 40 and 80. Results were also
obtained using a Gaussian PDF, appropriate for single coil acquisition at high
SNR. Note that *m*=1 represents the conventional
Rician noise model. In addition, the factor given by Eq.5 was calculated as a
function of SNR for different values of *m*.

**CRLB of diffusion kurtosis signal model**

We consider a model, $$$A(β,b_n )=A_0\ exp(-b_n D_{app}+(b_n^2 D_{app}^2 K_{app})⁄6)$$$, describing
diffusion kurtosis as a function of *b*-value
and parameter set **β**=(*A _{0}*,

*Monte Carlo (MC) simulation*

MC simulations were used to assess the SDs in the estimation of *D _{app}* and

**RESULTS & DISCUSSION**

Fig.1a shows that while at high SNR the noncentral
χ-distribution approaches a Gaussian, at low-to-moderate SNR it deviates
substantially from the Gaussian distribution. This departure is more pronounced
for larger values of *m*. Fig.1b shows that for *A*>>*σ* (high SNR), the value of *R _{n}* approaches 1, in which case the Fisher matrix for the
noncentral χ-distribution becomes identical to that of the Gaussian
distribution for any

Fig.2 shows
that for all *m*, the SDs in the
estimation of *K _{app}* and

Fig.3 shows that the minimal theoretical SDs as defined by the CRLB can be achieved at moderate-to-high SNR. Most importantly, the CRLB assuming a noncentral χ-distribution for noise provided a closer match to the MC simulations as compared to the Gaussian CRLB results, as expected.

**CONCLUSIONS**

1. Rao CR. Information and the Accuracy Attainable in the Estimation of Statistical Parameters. In: Kotz S, Johnson NL, editors. Breakthroughs in Statistics: Foundations and Basic Theory. New York, NY: Springer New York; 1992. p. 235-47.

2. Cramér H. Mathematical methods of statistics. Princeton University Princeton, NJ: Princeton University Press; 1946.

3. Bouhrara M, Reiter DA, Celik H, Bonny JM, Lukas V, Fishbein KW, Spencer RG. Incorporation of rician noise in the analysis of biexponential transverse relaxation in cartilage using a multiple gradient echo sequence at 3 and 7 tesla. Magnetic resonance in medicine. 2015;73(1):352-66.

4. Karlsen OT, Verhagen R, Bovee WM. Parameter estimation from Rician-distributed data sets using a maximum likelihood estimator: application to T1 and perfusion measurements. Magnetic resonance in medicine. 1999;41(3):614-23.

5. Celik H, Bouhrara M, Reiter DA, Fishbein KW, Spencer RG. Stabilization of the inverse Laplace transform of multiexponential decay through introduction of a second dimension. Journal of magnetic resonance (San Diego, Calif : 1997). 2013;236:134-9.

6. Alexander DC. A general framework for experiment design in diffusion MRI and its application in measuring direct tissue-microstructure features. Magnetic resonance in medicine. 2008;60(2):439-48.

7. Aja-Fernández S, Vegas-Sánchez-Ferrero G. Statistical Analysis of Noise in MRI: Springer International Publishing; 2016.

8. Wang C, He T, Liu X, Zhong S, Chen W, Feng Y. Rapid look-up table method for noise-corrected curve fitting in the R2* mapping of iron loaded liver. Magnetic resonance in medicine. 2015;73(2):865-71.