Synopsis

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.

Introduction

It has been shown that T₂-weighted MR signal of the cartilage knee could be better explained by the biexponential relaxation model¹:

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

where T_E is the echo time, S₀ is signal at T_E = 0, T_2s and T_2l are the short and long T₂ components, S_s and S_l are their respective signal components at T_E = 0, and f_s and f_l are the normalized fractions of the two components.

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) T₂ 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.

Methods

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, T_2l ,f_s, or f_l). Later, these coefficients of variation were compared for optimized sets of echoes, the echoes used in Sharafi et al. ¹, and three cases of 15 equally distanced echoes.

Results

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, T_2l and their signal fractions, respectively. With increasing T_2s or decreasing T_2l which is equivalent to having smaller differences between T₂ of the two components, errors in estimating biexponential parameters increased. Additionally, with decreasing signal fraction of either of the relaxation components, errors in estimating that component increased.

Discussion

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 T_2l and less dependent on fl. Regardless of how short or long the components were, it was shown that having more acquisitions close to double the value of the slow relaxing component improved fitting accuracy. This finding is similar to the one for the prostate ² with two components of 50-70 and 300-500 ms, where it has been shown that more acquisitions at echoes close to double the value of slow relaxing component improve biexponential fitting.

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

Acknowledgements

References

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