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