Nima Gilani^{1}

T2-weighted MR signal of the cartilage knee has shown to be better explained by the biexponential relaxation model. The short and long T2 signal components presumably describe tightly bound and loosely bound macromolecular water components of the knee, respectively. More precise estimation of these two parameters might help in the better diagnosis of Osteoarthritis in reasonable scanning times. Here, Cramér-Rao Lower Bound method was used to find optimum echo times that improve estimation of these relaxation components. It was shown that using maximum echo times of twice as much as what is routinely used might substantially improve the biexponential estimates. Echo time optimization might play a role as important as increasing acceleration factors in reducing acquisition times.

It has been shown that T_{2}-weighted MR signal of the cartilage knee could be better explained by the biexponential relaxation model^{1}:

$$$ S(T_E)=S_{s}e^{-\frac{T_E}{T_{2s}}}+S_{l}e^{-\frac{T_E}{T_{2l}}}=S_{0}(f_{s}e^{-\frac{T_E}{T_{2s}}}+f_{l}e^{-\frac{T_E}{T_{2l}}})$$$

where T_{E} is the echo time, S_{0} is signal at T_{E} = 0, *T*_{2s} and *T _{2l}* are the short and long

Most of the works on echo time optimization of biexponential acquisitions (e.g. ^{2,3}) have not considered the effect of acceleration factor or multi-coil acquisition on parameter estimation errors ^{4}. This is similar and confirmatory to the argument in a previous study by Bouhrara and Spencer ^{5}, where, it has been shown that echo-time optimization using Cramér-Rao Lower Bound requires accounting for the non-central or non-Gaussian behavior of noise in multi-coil diffusion acquisitions.

First, this argument is mostly applicable to diffusion-weighted imaging where SNR values are typically less than 10-20 and Rician ^{6} nature of noise becomes more dominant for each coil.

Second, uncorrected Cramér-Rao Lower optimization becomes substantially less accurate if the acceleration factor is greater than 4-8 ^{5}. With an acceleration factor of 3 and SNR of around 60 for *in vivo* imaging of the knee cartilage ^{1}, none of the two conditions above apply. Hence echo time optimization could be performed using Cramér-Rao Lower optimization without correction for non-centrality of noise, which has been previously validated with Monte Carlo methods ^{2,7} for 3 different models fitted on the MR signal (i.e. monoexponential, biexponential, and kurtosis).

The short (7-9 ms) and long (40-50) *T*_{2} components of the signal from the knee cartilage presumably describe tightly and loosely bound macromolecular water components of the knee cartilage. More precise estimation of these two parameters might help in the better diagnosis of Osteoarthritis at early stages. This requires using optimum echo-times that improve estimation of these two relaxation components in addition to their fraction. Here, Cramér-Rao Lower Bound method was used to find these optimum echo times similar to a study on T2-weighted imaging of the prostate ^{2}.

Optimum echo times were found by searching over uniform n-dimensional grids of values between 0.5 to 200 ms with 0.5 ms spacing, where, n was the number of echoes and varied between were 5 to 8. The search was performed to find the echoes that minimized the coefficient of variation in estimating parameter i ($$$CoV_i=\frac{\sqrt{Q_i}}{i}$$$), where, $$$Q_i$$$ was the diagonal element of covariance matrix corresponding to parameter *i *(either of *T _{2s}*,

Table 1 gives a summary of errors in estimating the biexponential parameters for optimized and non-optimized sets of echo times. It was observed that increasing the echo times nearly to double the value of *T _{2l}* increased the efficiency in estimating all of the four biexponential parameters.

Figures 1-3 show how sensitive the optimization was with regards to variations in *T _{2s}*,

Using eight optimized echoes gives smaller errors compared to the acquisition of ^{1} with fifteen acquisitions. Using an optimized acquisition with five echoes gives smaller errors compared with the case of 15 equally distanced echoes from 0.5-50 ms. Similar to ^{2}, optimization is highly dependent on *T _{2s}*, or

In conclusion, this study suggests that using optimized echo times prior to increasing acceleration factors might substantially improve biexponential estimations.

1. Sharafi, A., Chang, G. & Regatte, R. R. Biexponential T2 relaxation estimation of human knee cartilage in vivo at 3T. Journal of Magnetic Resonance Imaging 47, 809-819 (2018).

2. Gilani, N., Rosenkrantz, A. B., Malcolm, P. & Johnson, G. Minimization of errors in biexponential T2 measurements of the prostate. Journal of magnetic resonance imaging : JMRI 42, 1072-1077, doi:10.1002/jmri.24870 (2015).

3. Shrager, R., Weiss, G. & Spencer, R. Optimal time spacings for T2 measurements: monoexponential and biexponential systems. NMR in Biomedicine: An International Journal Devoted to the Development and Application of Magnetic Resonance In Vivo 11, 297-305 (1998).

4. Zibetti, M. V. W., Sharafi, A., Otazo, R. & Regatte, R. R. Compressed sensing acceleration of biexponential 3D-T1rho relaxation mapping of knee cartilage. Magn Reson Med 0, doi:10.1002/mrm.27416 (2018).

5. Bouhrara, M. & Spencer, R. G. Fisher information and Cramér-Rao lower bound for experimental design in parallel imaging. Magnetic Resonance in Medicine 79, 3249-3255, doi:doi:10.1002/mrm.26984 (2018).

6. Gudbjartsson, H. & Patz, S. The Rician distribution of noisy MRI data. Magn Reson Med 34, 910-914 (1995).

7. Gilani, N., Malcolm, P. N. & Johnson, G. Parameter Estimation Error Dependency on the Acquisition Protocol in Diffusion Kurtosis Imaging. Appl Magn Reson 47, 1229-1238, doi:10.1007/s00723-016-0829-x (2016).

Table 1 Coefficients of variation (CoV) for estimating short and long *T*_{2} values (*T*_{2s} and *T*_{2l}, respectively) and signal fractions. The optimization was performed for target values of *T*_{2s }= 8ms, *T*_{2l }= 48 ms, and a fraction of 0.5 for each component. CoV values correspond to SNR = 100 and are linearly correlated to 1/SNR.

Fig. 1 Changes in CoV values of the four biexponential parameters with regards to variations in *T*_{2s} if the optimized eight echoes are used. Note, *T*_{2l} and signal fractions are fixed.

Fig. 2 Changes in CoV values of the four biexponential parameters with regards to variations in *T*_{2l} if the optimized eight echoes are used. Note, *T*_{2s} and signal fractions are fixed.

Fig. 3 Changes in CoV values of the four biexponential parameters with regards to variations in the signal fractions if the optimized eight echoes are used. Note, *T*_{2s} and *T*_{2l} are fixed.