Diffusional kurtosis time dependence and the water exchange rate for the multi-compartment Kärger model

PURPOSE

To demonstrate an analytic formula giving the time dependence of the diffusional kurtosis for the Kärger model (KM) with an arbitrary number of exchanging compartments and its application in estimating the mean KM water exchange rate.

THEORY AND METHODS

The general formula for the kurtosis is derived from a power series solution for the multi-compartment KM. A lower bound on the exchange rate is established from the observation that the kurtosis is always a logarithmically convex function of time. Both the kurtosis time dependence and the lower bound are illustrated with numerical calculations. The lower bound is also applied to previously published data for the time dependence of the kurtosis in both brain and tumors.

RESULTS

The kurtosis for the multi-compartment KM is given by a sum in which each term is associated with an eigenvector of the exchange rate matrix. The lower bound is determined from the most negative value for the logarithmic derivative of the kurtosis with respect to time. In the cerebral cortex, the lower bound is found to vary from 15 to 76 s-1 , depending on the experimental details, while for the tumors considered, it varies from 2 to 4 s-1 .

CONCLUSION

The time dependence of the kurtosis for the multi-compartment KM has a simple analytic solution that allows a lower bound for the mean KM water exchange rate to be determined directly from experiment. This may be useful in tissues with complex microstructure that is difficult to model explicitly.

Time-dependent diffusivity and kurtosis in phantoms and patients with head and neck cancer

PURPOSE

To assess the reliability of measuring diffusivity, diffusional kurtosis, and cellular-interstitial water exchange time with long diffusion times (100-800 ms) using stimulated-echo DWI.

METHODS

Time-dependent diffusion MRI was tested on two well-established diffusion phantoms and in 5 patients with head and neck cancer. Measurements were conducted using an in-house diffusion-weighted STEAM-EPI pulse sequence with multiple diffusion times at a fixed TE on three scanners. We used the weighted linear least-squares fit method to estimate time-dependent diffusivity,D ( t ) $$ D(t) $$ , and diffusional kurtosis,K ( t ) $$ K(t) $$ . Additionally, the Kärger model was used to estimate cellular-interstitial water exchange time (τ ex $$ {\tau}_{ex} $$ ) fromK ( t ) $$ K(t) $$ .

RESULTS

Diffusivity measured by time-dependent STEAM-EPI measurements and commercial SE-EPI showed comparable results with R2 above 0.98 and overall 5.4 ± 3.0% deviation across diffusion times. Diffusional kurtosis phantom data showed expected patterns: constantD $$ D $$ andK $$ K $$  = 0 for negative controls and slow varyingD $$ D $$ andK $$ K $$ for samples made of nanoscopic vesicles. Time-dependent diffusion MRI in patients with head and neck cancer found that the Kärger model could be considered valid in 72% ± 23% of the voxels in the metastatic lymph nodes. The median cellular-interstitial water exchange time estimated for lesions was between 58.5 ms and 70.6 ms.

CONCLUSIONS

Based on two well-established diffusion phantoms, we found that time-dependent diffusion MRI measurements can provide stable diffusion and kurtosis values over a wide range of diffusion times and across multiple MRI systems. Moreover, estimation of cellular-interstitial water exchange time can be achieved using the Kärger model for the metastatic lymph nodes in patients with head and neck cancer.

The Kärger vs bi-exponential model: Theoretical insights and experimental validations

We revise three common models accounting for water exchange in pulsed-gradient spin-echo measurements: a bi-exponential model with time-dependent water fractions, the Kärger model, and a modified Kärger model designed for restricted diffusion, e.g. inside cells. The three models are compared and applied to experimental data from yeast cell suspensions. The Kärger model and the modified Kärger model yield very close results and accurately fit the data. The bi-exponential model, although less rigorous, has a natural physical interpretation and suggests a new experimental modality to estimate the water exchange time.