Catherine L Lian^{1,2}, Brendan Moloney^{2}, Eric M Baker^{2}, Greg Wilson^{3}, Erin W Gilbert^{4}, Thomas M Barbara^{2}, Charles S Springer^{2}, and Xin Li^{2}

^{1}Westview High School, Portland, OR, United States, ^{2}Advanced Imaging Research Center, Oregon Health & Science University, Portland, OR, United States, ^{3}Home, Mill Creek, WA, United States, ^{4}Surgery, Oregon Health & Science University, Portland, OR, United States

### Synopsis

Although
it is well known that diffusion-weighted imaging (DWI) is sensitive to in vivo trans-membrane water-exchange, quantitative
interpretation of the diffusion b-space decay remains difficult. Using random-walk simulated DWI data, this
study investigates the feasibility and reliability in studying the water
exchange effects with a multi-exponential fitting approach on DWI data.

### Purpose

Diffusion-weighted imaging (DWI) is sensitive to *in vivo* trans-membrane water-exchange. However, quantitative interpretation of the diffusion decay remains difficult. Recent Inverse Laplace Transformations^{1} of DWI data spanning up to b values of 6000 s/mm^{2} showed only one diffusigraphic peak for each brain tissue type (grey matter, white mater, etc.). Using [random-walk] RW-simulated DWI data, the purpose of this study is to investigate the feasibility and reliability in studying the water exchange effects on DWI using a multi-exponential fitting approach.

### Method

A Monte Carlo [MC] RW approach is used to simulate DWI data.^{2} Water molecule displacements within a 3D ensemble of 10,648 identical spheres having hexagonal close-packed symmetry are simulated. The 37 ^{o}C pure water diffusion coefficient, D_{0} = 3.0 μm^{2}/ms, is used for all particles, whether inside or outside cells. Noise-free *in silico *DWI decay curves for five steady-state cellular water efflux exchange rate constant (k_{io}) values of 0, 2, 4, 6, 10 (s^{-1}) are generated.

### Results

Figure **1a **and **1b** show simulated DWI b-space decay points with k_{io} = 0 [above] and 10 s^{-1} [below]. Blue and red curves show empirical single- (a) and biexponential (b) fittings of the k_{io} = 0 and 10 points, respectively. Based on the Akaike information criterion (AIC), each of the five noiseless data sets is better fitted by a bi-exponential model [only two are shown in Fig. 1a and in 1b]. When the natural logs of the simulated data are plotted against b-values (Fig. 1c), none of the five decays exhibit a straight line. The AIC model selection outcome could change quickly with increasing Gaussian noise present in the data. Using MC simulations, Figure 2 shows the fractions of fittings for which a biexponential model is favored, for three different SNR values: 50, 15, 5. The five bars from left to right for each SNR represent the five k_{io} values of 0, 2, 4, 6, 10 s^{-1} , respectively. When SNR diminishes to 5, there is no clear exponentiality preference.

### Discussion

Multi-exponential fitting has been used extensively in DWI and the two-site exchange Kärger model^{3} has been frequently adapted for *in vivo* MRI. However, quantitative interpretation remains controversial. Conflicting literature results could partially arise from different study designs. Besides the SNR effects mentioned above, inter-compartmental water exchange issues can be visualized with Figure 3. It plots system rate constants vs. b-values. Line AC measures the overall exchange rate constant [k = k_{io} + k_{oi} ], where k_{oi} is cellular water influx rate constant. Line OAB traces the maximum “diffusigraphic shutter-speed,” к_{D} , defined as |q_{max}^{2}ΔD|,^{4} where ΔD is (D_{2} - D_{1} ), the site ADC difference, and q_{max} is the maximum q value achieved during the b variation: b is changed by incrementing q with fixed diffusion time, Δ. Line BC indicates the diffusion-weighting with highest b-value achieved at q_{max}^{2}. For a reliable exchange-sensitive bi-exponential fitting, к_{D} has to be sufficiently greater than k.^{4} Practically, this means some DWI data points have to be measured with |q^{2}ΔD| values above the horizontal AC line. Thus, ABC represents a “golden triangle” to facilitate water exchange measurement using the Kärger model.^{3} To maximize water exchange-sensitivity, the more points inside the golden triangle the better. Unlike DCE-MRI, the degree of exchange-sensitivity in the DWI experiment is somewhat under the operator’s control. For the same b value, a pulse sequence with higher q^{2 }is more sensitive to water exchange than the one with smaller q^{2}. One exchange-insensitive example is shown by line OD, where no experimental points could reach the golden triangle due to a q_{max} that is too small. The common practice of achieving large b-values by increasing Δ can often lead to sub-optimal q_{max}. It also becomes obvious that for systems with negligible ΔD values, as implied for the human brain,^{1} attempts to extract water exchange and ADCs from bi-exponential fittings are likely to be unsuccessful. For the current study, a q_{max} value of ~51.9 mm^{-1} is achieved at maximum b-value of 4600 s/mm^{2}. With an estimated ΔD of ~ 1.3 μm^{2}/ms, the maximum к_{D} achieved is 3.5 s^{-1}, which is smaller than the overall exchange rate for almost all non-zero k_{io} cases. Therefore, the simulation for a typical clinical DWI study doesn’t reach the sufficiently large к_{D} required. If the Kärger model is used, a single exponential decay would generally be expected, especially with faster overall exchange rate. Other factors, like non-single-valued ADCs within each compartment, may contribute to outcomes where multi-exponentiality is AIC-preferred even for SNRs as low as 10 (Figure 2). A different DWI approach^{2} determines k when к_{D} = 0.

### Acknowledgements

NIH: R44 CA180425, OHSU Brenden-Colsen Center for Pancreatic Care.

### References

1. Avram, Sarlls, Basser, PISMRM, 26:5242 (2018). 2. Springer, Wilson, Moloney, Barbara, Li, Rooney, Maki, PISMRM 26:261 (2018). 3. Kärger, Adv.Colloid. Interfac. 23:129–148 (1985). 4. Lee, Springer, MRM, 49:450–458 (2003).