Correcting pulse duration effects in the diffusional kurtosis of the multi‐compartment Kärger model

Abstract

Purpose

To demonstrate a method of reducing pulse duration effects for the diffusional kurtosis of the multi‐compartment Kärger model (KM) as estimated with a Stejskal‐Tanner DWI sequence.

Theory and Methods

An effective diffusion time is introduced that corrects errors in the apparent diffusional kurtosis arising from a nonzero pulse duration δ for the multi‐compartment KM. The correction is exact to first order in the diffusion time Δ, and numerical calculations are used to assess how well it reduces pulse duration effects. Specifically, for the two‐compartment KM, the deviations of the apparent kurtosis obtained with the Stejskal‐Tanner sequence from the exact kurtosis are calculated for the full range of δ and Δ, and similar calculations are performed for the deviation in the derivative of the kurtosis with respect to Δ. For the general multi‐compartment KM, upper bounds on the maximum magnitude of the deviations are determined. Application of the correction to estimation of intercompartmental exchange rates is illustrated with several examples.

Results

For the two‐compartment KM, the correction reduces the deviation of the apparent kurtosis and its time derivative for most values of δ and Δ. For the general multi‐compartment KM, the maximum deviation magnitude, relative to the initial kurtosis, is 2.26% for the uncorrected kurtosis and 0.57% after correction. The correction reduces the maximum deviation magnitude of the derivative from 46% to less than 1%.

Conclusion

Pulse duration effects for the kurtosis of the multi‐compartment KM can be strongly suppressed by applying the effective diffusion time correction.

1. INTRODUCTION

The time dependence of the diffusional kurtosis as estimated with diffusion MRI (dMRI) has been used to quantify intercompartmental water exchange rates in both brain and tumor tissue. ¹ , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ Typically, some version of the Kärger model (KM) is fit to kurtosis measurements for several different diffusion times, ⁹ , ¹⁰ , ¹¹ , ¹² , ¹³ , ¹⁴ with a characteristic exchange rate being determined from the model parameters. However, in most cases, the effect of the gradient pulse duration $\delta$ for the diffusion weighted sequence has not been considered. This is well‐justified when $\delta$ is small in comparison to the diffusion time $\mathrm{\Delta }$, but significant pulse duration effects may occur when $\delta$ is comparable to $\mathrm{\Delta }$, as is often the case on human scanners.

In this paper, we consider the effect of pulse duration on the time dependence of the apparent kurtosis, $K_{\mathrm{app}}$, as predicted by the general multi‐compartment KM for a Stejskal‐Tanner dMRI sequence. ¹⁵ , ¹⁶ An effective diffusion time is derived that corrects for pulse durations effects exactly to first order in $\mathrm{\Delta }$ for all values of $\delta$. This correction is then tested numerically for the full range of $\mathrm{\Delta }$ and $\delta$ values. The correction is straightforward to implement and applies to all KMs. It may be useful for improving the accuracy of intercompartmental water exchange rates estimated from dMRI measurements of the kurtosis in brain and other biological tissues.

2. THEORY

2.1. Effective diffusion time for two‐compartment KM

For a two‐compartment KM, the exact kurtosis is given by ¹⁷ , ¹⁸

$$K(\mathrm{\Delta })=K_0\mathrm{{\rm Y}}\left(\frac{\mathrm{\Delta }}{\tau }\right),$$ (1)

where $\tau$ is a water exchange time, $K_0$ is the initial kurtosis, and

$$\mathrm{{\rm Y}}(X)\equiv \frac{2}{X}\left(1-\frac{1-e^{-X}}{X}\right).$$ (2)

Note that $\mathrm{{\rm Y}}(0)=1$. This formula has been previously used to model the time dependence of the kurtosis in brain tissue and tumors. ¹ , ² , ³ , ⁴ , ⁵ , ⁶ In contrast, the diffusivity $D$ for the KM is independent of both $\mathrm{\Delta }$ and $\tau$, making it insensitive to water exchange effects. ⁷ , ¹⁸ , ¹⁹ It should be emphasized that Eq. (1), as well as those that follow, apply to any diffusion direction even if the kurtosis is anisotropic. Therefore, the application of our results to the axial, radial, and mean kurtoses is straightforward.

If estimated from DWI data obtained with a Stejskal‐Tanner sequence, the apparent kurtosis is ¹⁹

$$K_{\mathrm{app}}(\mathrm{\Delta },\delta )=K_0{\mathrm{{\rm Y}}}_{\mathrm{app}}\left(\frac{\mathrm{\Delta }}{\tau },\frac{\delta }{\tau }\right),$$ (3)

where

$$\begin{array}{ll}{\mathrm{{\rm Y}}}_{\mathrm{app}}(X,Y)& \equiv \frac{2}{15{(X-Y/3)}^2{Y}^4}\\ & \left[15X{Y}^4-9Y^5-40Y^3+60Y^2-120\right.\\ & +120(Y+1)e^{-Y}+120(Y-1)e^{-X}.\\ & \left.+60{(Y-1)}^2{e}^{-X+Y}+60e^{-X-Y}\right].\end{array}$$ (4)

One may verify that ${\mathrm{{\rm Y}}}_{\mathrm{app}}(X,0)=\mathrm{{\rm Y}}(X)$, which implies $K_{\mathrm{app}}(\mathrm{\Delta },0)=K(\mathrm{\Delta })$. Here it is assumed that $K_{\mathrm{app}}$ is obtained from the usual cumulant expansion formula ¹⁷ , ¹⁸

$$\ln [S(b)]=\ln [S(0)]-b{D}_{\mathrm{app}}+\frac{1}{6}{K}_{\mathrm{app}}{\left(b{D}_{\mathrm{app}}\right)}^2+O\left(b^3\right),$$ (5)

where $S(b)$ is the signal intensity as a function of b‐value and $D_{\mathrm{app}}$ is the apparent diffusivity. The b‐value is taken to be sufficiently small so that the $O\left(b^3\right)$ term in Eq. (5) can be neglected. For any pulse duration, $D_{\mathrm{app}}=D$, which is just a constant. ¹⁹ Inherent to the design of the Stejskal‐Tanner sequence is the constraint $0\le \delta \le \mathrm{\Delta }$. ¹⁵ , ¹⁶

For small diffusion times, we have the power series approximations

$$K(\mathrm{\Delta })=K_0\left[1-\frac{\mathrm{\Delta }}{3\tau }+O\left({\mathrm{\Delta }}^2\right)\right]$$ (6)

and

$$K_{\mathrm{app}}(\mathrm{\Delta },\delta )=K_0\left[1-\frac{\mathrm{\Delta }}{3\tau }\eta \left(\frac{\delta }{\mathrm{\Delta }}\right)+O\left({\mathrm{\Delta }}^2\right)\right],$$ (7)

where the ratio $\delta /\mathrm{\Delta }$ is regarded as fixed in deriving the expansion of Eq. (7). The function $\eta (x)$ is defined by

$$\eta (x)\equiv \frac{3}{7}·\frac{21-21x+14x^2-4x^3}{{(3-x)}^2}.$$ (8)

Clearly, $\eta (0)=1$ so that Eq. (7) reduces to Eq. (6) for $\delta /\mathrm{\Delta }=0$. The minimum value of $\eta (x)$ is $\approx 0.9240$ and occurs at $x\approx 0.4373$ while the maximum is $15/14$ for $x=1$. A plot of $\eta (x)$ is given in Figure 1. The approximation of Eq. (7) was previously derived by Ning and coworkers ²⁰ and applied to the assessment of ischemic stroke by Lampinen and coworkers. ²¹

FIGURE 1.

The function $\eta (x)$ used to define the effective diffusion time for the KM diffusional kurtosis as estimated with a Stejskal‐Tanner sequence.

Eqs. (6) and (7) motivate defining an effective diffusion time

$${\mathrm{\Delta }}_{\mathrm{eff}}(\mathrm{\Delta },\delta )\equiv \eta \left(\frac{\delta }{\mathrm{\Delta }}\right)\mathrm{\Delta }.$$ (9)

Then the approximation

$$K_{\mathrm{app}}(\mathrm{\Delta },\delta )\approx K\left[{\mathrm{\Delta }}_{\mathrm{eff}}(\mathrm{\Delta },\delta )\right]$$ (10)

is exact to first order in $\mathrm{\Delta }$. In this paper, we compare the accuracy of Eq. (10) with the conventionally used approximation of $K_{\mathrm{app}}(\mathrm{\Delta },\delta )\approx K(\mathrm{\Delta })$.

2.2. Extension to multi‐compartment KM

Recently, the kurtosis of a KM with an arbitrary number of compartments has been shown to be

$$K(\mathrm{\Delta })=\sum _{n=2}^N{\kappa }_n\mathrm{{\rm Y}}\left(\frac{\mathrm{\Delta }}{{\tau }_n}\right),$$ (11)

where $N$ is the number of compartments, ${\tau }_n$ is an exchange time associated with the nth eigenvector of the exchange rate matrix, and ${\kappa }_n$ is a partial initial kurtosis associated with the nth eigenvector. ⁷ Explicit formulae for calculating ${\tau }_n$ and ${\kappa }_n$ for any KM are provided in Ref. 7. Formally, the exchange time ${\tau }_1$, associated with the first eigenvector, is infinite, but this eigenvector does not contribute to the sum in Eq. (11) due to an automatic cancellation. Thus, the physical exchange times are numbered beginning with $n=2$. In all cases, these constants satisfy ${\tau }_n\ge 0$ and ${\kappa }_n\ge 0$. By evaluating Eq. (11) at $\mathrm{\Delta }=0$ and applying $\mathrm{{\rm Y}}(0)=1$, one sees that the initial kurtosis is given by

$$K_0=\sum _{n=2}^N{\kappa }_n.$$ (12)

The same methods used to determine the apparent kurtosis for the two‐compartment KM are straightforward to extend to the multi‐compartment case, which yields

$$K_{\mathrm{app}}(\mathrm{\Delta },\mathrm{\delta })=\sum _{n=2}^N{\kappa }_n{\mathrm{{\rm Y}}}_{\mathrm{app}}\left(\frac{\mathrm{\Delta }}{{\tau }_n},\frac{\mathrm{\delta }}{{\tau }_n}\right).$$ (13)

In addition, we have $D_{\mathrm{app}}=D$ so that $D_{\mathrm{app}}$ is again a constant.

For the multi‐compartment KM, Eqs. (6) and (7) generalize to

$$K(\mathrm{\Delta })=K_0\left[1-\frac{\mathrm{\Delta }}{3}{R}_{\mathrm{KM}}+O\left({\mathrm{\Delta }}^2\right)\right]$$ (14)

and

$$K_{\mathrm{app}}(\mathrm{\Delta },\delta )=K_0\left[1-\frac{1}{3}{\mathrm{\Delta }}_{\mathrm{eff}}(\mathrm{\Delta },\delta )R_{\mathrm{KM}}+O\left({\mathrm{\Delta }}^2\right)\right],$$ (15)

where

$$R_{\mathrm{KM}}=\sum _{n=2}^N\frac{{\kappa }_n}{K_0{\tau }_n}$$ (16)

is the mean KM exchange rate. As a consequence, the approximation of Eq. (10) continues to hold exactly to first order in $\mathrm{\Delta }$ and is just as applicable to the multi‐compartment KM as it is to two compartments. This is important for applications to tissues with complex microstructure, such as brain, which may not be well described by an elementary two‐compartment KM.

3. METHODS

3.1. Pulse duration effects for two‐compartment KM

In order to test how well the effective diffusion time correction indicated by Eq. (10) reduces pulse duration effects in the kurtosis, we calculate, for the two‐compartment KM, the difference between the apparent and exact kurtosis both with and without this correction. Pulse duration effects are then quantified in terms of the percent deviation relative to the initial kurtosis $K_0$. Without the correction, the percent deviation is defined by

$$\xi \left(\frac{\mathrm{\Delta }}{\tau },\frac{\delta }{\tau }\right)\equiv \frac{100}{K_0}\left[K_{\mathrm{app}}(\mathrm{\Delta },\delta )-K(\mathrm{\Delta })\right],$$ (17)

and with the correction, by

$${\xi }_{\text{corr}}\left(\frac{\mathrm{\Delta }}{\tau },\frac{\delta }{\tau }\right)\equiv \frac{100}{K_0}\left\{K_{\mathrm{app}}(\mathrm{\Delta },\delta )-K\left[{\mathrm{\Delta }}_{\mathrm{eff}}(\mathrm{\Delta },\delta )\right]\right\}.$$ (18)

Eqs. (17) and (18) differ only in that the exact kurtosis is evaluated at the diffusion time $\mathrm{\Delta }$ without the correction and at the effective diffusion time ${\mathrm{\Delta }}_{\mathrm{eff}}$ with the correction. A smaller percent deviation indicates a greater accuracy. That the percent deviations $\xi$ and ${\xi }_{\text{corr}}$ depend only on the ratios $\mathrm{\Delta }/\tau$ and $\mathrm{\delta }/\tau$ follows directly from Eqs. (1), (3), and (9). For plotting purposes, we employ the quantities $\delta /\mathrm{\Delta }$ and $\mathrm{\Delta }/(\tau +\mathrm{\Delta })$ to cover this 2D parameter space since both of these range conveniently between 0 and 1.

Recently, there has been a growing interest in using the time dependence of the diffusional kurtosis to estimate intercompartmental water exchange rates. ¹ , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ For such applications, the derivative of the kurtosis with respect to the diffusion time is relevant. Therefore, we also consider the percent deviations in the kurtosis derivatives. Without correction, we define this by

$${\xi }^{\prime }\left(\frac{\mathrm{\Delta }}{\tau },\frac{\delta }{\tau }\right)\equiv \frac{300\tau }{K_0}·\frac{\partial }{\mathrm{\partial \Delta }}\left[K_{\mathrm{app}}(\mathrm{\Delta },\delta )-K(\mathrm{\Delta })\right]$$ (19)

and with correction by

$${\xi }_{\text{corr}}^{\prime }\left(\frac{\mathrm{\Delta }}{\tau },\frac{\delta }{\tau }\right)\equiv \frac{300\tau }{K_0}·\frac{\partial }{\mathrm{\partial \Delta }}\left\{K_{\mathrm{app}}(\mathrm{\Delta },\delta )-K\left[{\mathrm{\Delta }}_{\mathrm{eff}}(\mathrm{\Delta },\delta )\right]\right\}.$$ (20)

These are both normalized relative to the magnitude of the initial time derivative for $K(\mathrm{\Delta })$ of $K_0/(3\tau )$, as follows from Eq. (6).

3.2. Upper bounds on pulse duration effects for multi‐compartment KM

For the multi‐compartment KM, the uncorrected percent deviation is

$$\xi =\frac{100}{K_0}\sum _{n=2}^N{\kappa }_n\left[{\mathrm{{\rm Y}}}_{\mathrm{app}}\left(\frac{\mathrm{\Delta }}{{\tau }_n},\frac{\mathrm{\delta }}{{\tau }_n}\right)-\mathrm{{\rm Y}}\left(\frac{\mathrm{\Delta }}{{\tau }_n}\right)\right],$$ (21)

as follows from Eqs. (11), (12), and (17). Since this depends on the $N-1$ different exchange times ${\tau }_n$, it is impractical to examine $\xi$ for all possible cases when $N>2$. Therefore, we instead compute upper bounds on the magnitude of $\xi$ in order to characterize the maximum possible deviations. To do this, we note that the triangle inequality ²² implies

$$\left|\xi \right|\le \frac{100}{K_0}\sum _{n=2}^N{\kappa }_n\left|{\mathrm{{\rm Y}}}_{\mathrm{app}}\left(\frac{\mathrm{\Delta }}{{\tau }_n},\frac{\mathrm{\delta }}{{\tau }_n}\right)-\mathrm{{\rm Y}}\left(\frac{\mathrm{\Delta }}{{\tau }_n}\right)\right|.$$ (22)

Now imagine maximizing each term of the sum in Eq. (22) over all possible values of ${\tau }_n$. This leads to

$$\left|\xi \right|\le \mu \left(\frac{\delta }{\mathrm{\Delta }}\right),$$ (23)

where

$$\mu \left(\frac{\delta }{\mathrm{\Delta }}\right)\equiv 100·\max \left|{\mathrm{{\rm Y}}}_{\mathrm{app}}\left(\frac{\mathrm{\Delta }}{\tau },\frac{\mathrm{\delta }}{\tau }\right)-\mathrm{{\rm Y}}\left(\frac{\mathrm{\Delta }}{\tau }\right)\right|$$ (24)

and we have used Eq. (12) to carry out the sum over compartments. Thus, for any given value of $\delta /\mathrm{\Delta }$, $\mu$ provides an upper bound on the magnitude of the uncorrected percent deviation for any KM. Similarly, an upper bound for the magnitude of the corrected percent deviation is given by

$$\left|{\xi }_{\text{corr}}\right|\le {\mu }_{\text{corr}}\left(\frac{\delta }{\mathrm{\Delta }}\right),$$ (25)

where

$${\mu }_{\text{corr}}\left(\frac{\delta }{\mathrm{\Delta }}\right)\equiv 100·\max \left|{\mathrm{{\rm Y}}}_{\mathrm{app}}\left(\frac{\mathrm{\Delta }}{\tau },\frac{\mathrm{\delta }}{\tau }\right)-\mathrm{{\rm Y}}\left[{\mathrm{\Delta }}_{\mathrm{eff}}\left(\frac{\mathrm{\Delta }}{\tau },\frac{\mathrm{\delta }}{\tau }\right)\right]\right|.$$ (26)

The uncorrected percent deviation for the kurtosis derivative of the multi‐compartment KM can be defined as

$${\xi }^{\prime }=\frac{300}{K_0{R}_{\mathrm{KM}}}\sum _{n=1}^N{\kappa }_n\frac{\partial }{\partial \Delta }\left[{\mathrm{{\rm Y}}}_{\mathrm{app}}\left(\frac{\mathrm{\Delta }}{{\tau }_n},\frac{\mathrm{\delta }}{{\tau }_n}\right)-\mathrm{{\rm Y}}\left(\frac{\mathrm{\Delta }}{{\tau }_n}\right)\right],$$ (27)

where this has been normalized relative to the magnitude of the initial time derivative for $K(\mathrm{\Delta })$ of $K_0{R}_{\mathrm{KM}}/3$, as follows from Eq. (14). Applying the triangle inequality to Eq. (27) gives

$$\left|{\xi }^{\prime }\right|\le \frac{300}{K_0{R}_{\mathrm{KM}}}\sum _{n=1}^N{\kappa }_n\left|\frac{\partial }{\mathrm{\partial \Delta }}\left[{\mathrm{{\rm Y}}}_{\mathrm{app}}\left(\frac{\mathrm{\Delta }}{{\tau }_n},\frac{\mathrm{\delta }}{{\tau }_n}\right)-\mathrm{{\rm Y}}\left(\frac{\mathrm{\Delta }}{{\tau }_n}\right)\right]\right|.$$ (28)

With the help of Eq. (16), we then obtain

$$\left|{\xi }^{\prime }\right|\le {\mu }^{\prime }\left(\frac{\delta }{\mathrm{\Delta }}\right),$$ (29)

where

$${\mu }^{\prime }\left(\frac{\delta }{\mathrm{\Delta }}\right)\equiv 100·\max \left|\tau \frac{\partial }{\mathrm{\partial \Delta }}\left[{\mathrm{{\rm Y}}}_{\mathrm{app}}\left(\frac{\mathrm{\Delta }}{\tau },\frac{\mathrm{\delta }}{\tau }\right)-\mathrm{{\rm Y}}\left(\frac{\mathrm{\Delta }}{\tau }\right)\right]\right|$$ (30)

with the maximum again being taken over all possible values for $\tau$. Extending this argument to the corrected percent deviation of the kurtosis derivative yields

$$\begin{array}{l}{\xi }_{\text{corr}}^{\prime }=\frac{300}{K_0{R}_{\mathrm{KM}}}\sum _{n=1}^N{\kappa }_n\frac{\partial }{\mathrm{\partial \Delta }}\left\{{\mathrm{{\rm Y}}}_{\mathrm{app}}\left(\frac{\mathrm{\Delta }}{{\tau }_n},\frac{\mathrm{\delta }}{{\tau }_n}\right)-\mathrm{{\rm Y}}\left[{\mathrm{\Delta }}_{\mathrm{eff}}\left(\frac{\mathrm{\Delta }}{{\tau }_n},\frac{\mathrm{\delta }}{{\tau }_n}\right)\right]\right\}\end{array}$$ (31)

and the corresponding upper bound

$$\left|{\xi }_{\text{corr}}^{\prime }\right|\le {\mu }_{\text{corr}}^{\prime }\left(\frac{\delta }{\mathrm{\Delta }}\right),$$ (32)

with

$$\begin{array}{l}{\mu }_{\text{corr}}^{\prime }\left(\frac{\delta }{\mathrm{\Delta }}\right)\equiv 100·\max \left|\tau \frac{\partial }{\mathrm{\partial \Delta }}\left\{{\mathrm{{\rm Y}}}_{\mathrm{app}}\left(\frac{\mathrm{\Delta }}{\tau },\frac{\mathrm{\delta }}{\tau }\right)-\mathrm{{\rm Y}}\left[{\mathrm{\Delta }}_{\mathrm{eff}}\left(\frac{\mathrm{\Delta }}{\tau },\frac{\mathrm{\delta }}{\tau }\right)\right]\right\}\right|.\end{array}$$ (33)

As with the bounds of Eqs. (23) and (25), Eqs. (29) and (32) apply to any KM no matter how complicated.

This set of four upper bounds on $\left|\xi \right|$, $\left|{\xi }_{\text{corr}}\right|$, $\left|{\xi }^{\prime }\right|$, and $\left|{\xi }_{\text{corr}}^{\prime }\right|$ provides a means of assessing how well the effective time correction of Eq. (10) compensates pulse duration effects for the multi‐compartment KM at any given value of $\delta /\mathrm{\Delta }$. Broadly speaking, a smaller upper bound can be regarded as indicating a better approximation.

3.3. Numerical examples

To demonstrate the application of the effective time correction of Eq. (10), we consider simple numerical examples of the two‐ and three‐compartment KM that are qualitatively similar to recent experiments investigating the time dependence of the kurtosis. ¹ , ²³ For the two‐compartment KM, we assume the exact kurtosis is given by Eq. (1) with $K_0=1$ and $\tau =$ 20, 40, or 80 ms. Correspondingly, the simulated experimental data (i.e., the apparent kurtosis) are calculated from Eq. (3) with the same values of $\tau$ and for diffusion times of $\mathrm{\Delta }=$ 20, 25, 30, 35, and 40 ms. The pulse duration is set to $\delta =$ 15 ms in all cases. For the three‐compartment KM, we take Eq. (11) with ${\kappa }_2=0.8$, ${\tau }_2=10$ ms, ${\kappa }_3=0.2$, and ${\tau }_3=80$ ms as the exact kurtosis while the apparent kurtosis is obtained from Eq. (13) for $\delta =$ 15 ms and $\mathrm{\Delta }=$ 20, 25, 30, 35, 40, 100, 200, and 300 ms. A broader of range of diffusion times is used here in order to adequately constrain this model's larger number of adjustable parameters.

A conventional approach for estimating the intercompartmental exchange rate is to fit experimental data to the exact KM expression of Eqs. (1) or (11). ¹ , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ However, the estimated exchange time will then generally be biased by pulse duration effects. Here we compare this standard method with fits to Eq. (10), which approximately corrects for pulse duration effects. To apply Eq. (10), we simply associate each apparent value of the kurtosis with the effective diffusion time of Eq. (9) and then fit to Eqs. (1) or (11) as if the effective diffusion time were the true diffusion time. In other words, Eq. (9) is just used to adjust the diffusion times with the fitting procedure otherwise being unchanged. The exchange times obtained with this modified approach will then be compared with those obtained when pulse duration effects are ignored. Note that this effective diffusion time correction does not impact estimates of the diffusivity for the KM since the KM diffusivity is independent of both $\mathrm{\Delta }$ and $\delta$, as discussed above in Sections 2.1 and 2.2.

3.4. Cortical gray matter

As an example of the impact of the effective time correction on real experimental data obtained on a clinical scanner, we apply it to estimate the mean KM exchange rate $R_{\mathrm{KM}}$ in cortical gray matter using data from Ref. 1 and following the analysis described in Ref. 7. Here a lower bound, $R_{\mathrm{KM}}^{*}$, is found from the logarithmic derivative of the apparent kurtosis with respect to the diffusion time, and we compare $R_{\mathrm{KM}}^{*}$ obtained with and without the correction. For these data, the diffusion times vary from $\mathrm{\Delta }=$ 21.2 to 100 ms with the pulse duration set to $\delta =$ 15 ms. The estimate $R_{\mathrm{KM}}^{*}$ is calculated from the first four diffusion times in order to have the tightest lower bound.

4. RESULTS

4.1. Pulse duration effects for two‐compartment KM

Contour plots of the uncorrected and corrected kurtosis percent deviations, as defined by Eqs. (17) and (18), are shown in Figure 2A,B, respectively. These plots encompass the full parameter space for the two‐compartment KM. Although the percent deviations are small in all cases, the corrected percent deviations are substantially smaller than the uncorrected percent deviations for most parameter choices. The analogous contour plots for the uncorrected and corrected percent deviations in the kurtosis derivatives, as defined by Eqs. (19) and (20), are shown in Figure 2C,D. In this case, the uncorrected deviation can be rather large particularly for $\delta /\mathrm{\Delta }>0.5$, but the magnitude of the corrected deviation is always less than 1%.

FIGURE 2.

Contour plots showing percent deviations in the apparent kurtosis and its derivative as functions of $\mathrm{\Delta }$ and $\delta$ for a two‐compartment KM model with an exchange time $\tau$. (A) Plots for the uncorrected deviation $\xi$ for the apparent kurtosis defined by Eq. (17). (B) Plots for the corrected deviation ${\xi }_{\text{corr}}$ for the apparent kurtosis defined by Eq. (18). (C) Plots for the uncorrected deviation ${\xi }^{\prime }$ for the derivative of the apparent kurtosis defined by Eq. (19). (D) Plots for the corrected deviation ${\xi }_{\text{corr}}^{\prime }$ for the derivative of the apparent kurtosis defined by Eq. (20). The magnitudes of the corrected deviations are substantially smaller than those of the uncorrected deviations for the vast majority of $\mathrm{\Delta }$ and $\delta$ values showing the improved accuracy obtained when the effective diffusion time of Eq. (9) is utilized.

4.2. Upper bounds on pulse duration effects for multi‐compartment KM

The upper bounds on the magnitudes of the uncorrected and corrected kurtosis percent deviations for the multi‐compartment KM, given by Eqs. (24) and (26), are plotted in Figure 3A as functions of $\delta /\mathrm{\Delta }$. The overall maximum possible uncorrected percent deviation is 2.26% and occurs at $\delta /\mathrm{\Delta }\approx$ 0.47. The overall maximum possible corrected percent deviation is 0.57% and occurs at $\delta /\mathrm{\Delta }=1$. For the most values of $\delta /\mathrm{\Delta }$ the upper bound is reduced by using the effective time correction of Eq. (10), with the exception being the interval $0.85\le \delta /\mathrm{\Delta }\le 0.92$.

FIGURE 3.

Maximum percent deviations as a function of the ratio $\delta /\mathrm{\Delta }$ for the multi‐compartment KM. (A) Maximum deviation for the apparent kurtosis is lower with the effective diffusion time correction (${\mu }_{\text{corr}}$) than without ($\mu$) for all positive values of $\delta /\mathrm{\Delta }$ except in the small range $0.85\le \delta /\mathrm{\Delta }\le 0.92$. (B) Maximum deviation for the apparent kurtosis derivative is lower with the effective diffusion time correction (${\mu }_{\text{corr}}^{\prime }$) than without (${\mu }^{\prime }$) for all positive values of $\delta /\mathrm{\Delta }$. Without the correction, the derivative deviation can be as high as 46% but is always less than 1% with the correction.

Figure 3B gives the upper bounds on the magnitudes of the uncorrected and corrected percent deviations for the kurtosis derivatives, as given by Eqs. (30) and (33). For all values of $\delta /\mathrm{\Delta }$, the effective diffusion time correction reduces the upper bound. With correction, the upper bound is always less than 1%, but can be as high as 46% for the uncorrected percent deviation. Therefore, the effective diffusion time correction could potentially improve the accuracy of estimates for quantities related to the time dependence of the kurtosis for any KM.

4.3. Numerical examples

Figure 4A plots the apparent kurtosis as a function of diffusion time for the two‐compartment KM with $\delta =15$ ms and $\tau =$ 20, 40, and 80 ms. The data points are calculated from Eq. (3) while the curves show least square fits to the data for the ideal kurtosis of Eq. (1). These fits lead to exchange time estimates of 22.95, 45.55, and 90.73 ms, which correspond to errors of 14.8%, 13.9%, and 13.4%, respectively. In Figure 4B, the same data are plotted as a function of the effective diffusion time defined by Eq. (9) while the curves are once again fits for Eq. (1). With this correction, the estimated exchange times become 19.84, 39.90, and 79.92 ms, corresponding to errors of $-$0.80%, $-$0.25%, and $-$0.10%. A similar improvement in accuracy is seen for the three‐compartment KM, as shown in Figure 4C,D.

FIGURE 4.

Comparison of uncorrected and corrected fits for the KM with $\delta =15$ ms. (A,B) The two‐compartment KM with $\mathrm{\Delta }=20,25,30,35,40$ ms and $\tau =20,40,80$ ms. Without correction, estimated exchange times ${\tau }_{\mathrm{fit}}$ differ from the exact exchange times by 13% to 15% while the errors are all less than 1% after correction. (C,D) The three‐compartment KM with $\mathrm{\Delta }=20,25,30,35,40,100,200,300$ ms, ${\tau }_2=10$ ms, and ${\tau }_3=80$ ms. The errors for the estimated exchange times, ${\tau }_{2,\mathrm{fit}}$ and ${\tau }_{3,\mathrm{fit}}$, are much smaller with the corrected fit.

4.4. Cortical gray matter

Data taken from Ref. 1 are shown in Figure 5A along with an estimate of $R_{\mathrm{KM}}^{*}=15{\mathrm{s}}^{-1}$ based on the four data points having the shortest diffusion times, which reproduces the analysis of Ref. 7. When the analysis is repeated with the effective diffusion time correction, as shown in Figure 5B, one obtains $R_{\mathrm{KM}}^{*}=18{\mathrm{s}}^{-1}$. This 20% increase demonstrates how the correction can have a substantial impact under experimental conditions appropriate for clinical scanners.

FIGURE 5.

Estimates of the lower bound $R_{\mathrm{KM}}^{*}$ for human cortical gray matter obtained from kurtosis measurements with $\delta =15$ ms (data taken from Ref. 1). The logarithmic derivative for $K_{\mathrm{app}}$ is calculated from the slope of the best fit line for the first four diffusion times, as described in Ref. 7. The lower bound is then given by $-3$ times this slope. (A) The estimate for $R_{\mathrm{KM}}^{*}$ without the effective diffusion time correction is 15 ${\mathrm{s}}^{-1}$ (adapted from Ref. 7). (B) The corrected estimate for $R_{\mathrm{KM}}^{*}$ is 20% higher.

5. DISCUSSION

The notion of an effective diffusion time has a long history in NMR and MRI. ²⁴ , ²⁵ However, no single definition is sufficient to compensate for pulse duration effects in all physical quantities and pulse sequences. ²⁶ Here we have focused on the diffusional kurtosis as estimated with the Stejskal‐Tanner pulse sequence.

A primary assumption of this work is that the diffusion dynamics are well described by the KM, which is the simplest physical model of diffusion incorporating intercompartmental water exchange. ⁹ , ¹⁰ , ¹¹ , ¹² , ¹³ , ¹⁴ The KM is consistent with experimental observations in a variety of tissues, including cerebral gray matter and tumors, where it has been applied to estimate water exchange rates. ¹ , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ Nonetheless, it can break down for both very short ⁷ , ¹¹ , ²⁷ and very long ²⁷ diffusion times and is not suitable for diffusive media that lack well‐defined compartments. Therefore, it is important to verify the appropriateness of the KM prior to utilizing the results of this paper. One useful, but not definitive, test is to confirm the KM prediction that the diffusivity is independent of the diffusion time. ⁷ , ⁸

The effect of pulse duration on the dMRI signal for the KM has been considered in previous work for the two‐compartment KM, ¹⁹ , ²⁰ , ²⁸ , ²⁹ , ³⁰ but the extension to the general multi‐compartment KM is given here for the first time. This is important for applications to tissues with complex microstructure, such as brain, for which a simple two‐compartment picture may not always be fully satisfactory.

Our main result is that by regarding the apparent kurtosis as a function of the effective diffusion time of Eq. (9) rather than the nominal diffusion time $\mathrm{\Delta }$, errors due to pulse duration effects can be significantly reduced. For the value of the kurtosis itself at a given time point, the impact of this correction is modest since the uncorrected pulse duration effects are already small, as has been previously shown for the two‐compartment KM. ¹⁹ However, for the derivative of the kurtosis with respect to the diffusion time $\mathrm{\Delta }$, the correction can result in a substantial improvement, most notably for $\delta /\mathrm{\Delta }>1/2$, as illustrated by Figures 2 and 3. In particular, the magnitude of the deviation in the derivative is always less than 1% with the correction, but can be as large as 46% without it. We emphasize that this conclusion applies to the general multi‐compartment KM, no matter the choice of model parameters. As a consequence, the effective diffusion time correction can meaningfully improve the accuracy of water exchange rates estimated from the time dependence of the kurtosis as demonstrated by the examples highlighted in Figure 4. As illustrated in Figure 5, the correction method is expected to be especially helpful for estimates of the mean KM exchange rate determined from the logarithmic derivative of the kurtosis with respect to the diffusion time ⁷ , ⁸ since these are typically most accurate for small $\mathrm{\Delta }$ in which case the $\delta /\mathrm{\Delta }$ ratio tends to be relatively large.

A limitation of this study is that our results only strictly apply to the Stejskal‐Tanner pulse sequence, but this does provide a fairly good approximation for the monopolar sequences often used in practice. ¹⁶ For some other sequences, such as the twice‐refocused sequence sometimes employed to reduce eddy current distortion, ³¹ the effective diffusion time correction proposed here would, however, not be valid. Developing methods to correct pulse duration effects for the diffusional kurtosis as measured with sequences other than the Stejskal‐Tanner sequence would be a valuable extension of this work.

6. CONCLUSIONS

By plotting the apparent diffusional kurtosis measured with a Stejskal‐Tanner sequence as a function of the effective diffusion time of Eq. (9) rather than the nominal diffusion time $\mathrm{\Delta }$, pulse duration effects can be substantially reduced provided the diffusion dynamics are well‐described by the general multi‐compartment KM. This correction scheme may help improve the accuracy of estimates for intercompartmental water exchange rates determined from the time dependence of the kurtosis.

Jensen JH. Correcting pulse duration effects in the diffusional kurtosis of the multi‐compartment Kärger model. Magn Reson Med. 2025;94(5):2249‐2257. doi: 10.1002/mrm.30608