Chu-Yu Lee^{1}, In-Young Choi^{1,2,3}, and Phil Lee^{1,3}

Cell compartments can present non-uniform T2 relaxations, depending on tissue properties. The non-uniform T2 relaxation has been shown to affect measurements of apparent diffusion coefficients (ADC). In additional to ADC, diffusion heterogeneity measured by modeling non-monoexponential decay has been used to characterize water diffusion in tissue microstructure. The purpose of this study is to study the effects of non-uniform T2 relaxations on the diffusion heterogeneity using a Monte Carlo simulation. The results demonstrate that the diffusion heterogeneity may provide more information about water diffusion within microstructure with non-uniform T2 relaxations.

Non-monoexponential decay of
diffusion-weighted imaging (DWI) signals can be phenomenologically modeled by assuming
multiple diffusion components. The bi-exponential model fits the data well, but
it is difficult to accurately estimate the fitted parameters [1]. More generally, the signal decay
is assumed to arise from a continuous distribution of diffusion coefficients [2],
for example, using stretched exponential model [3] or gamma distribution model
[2]. The shape of the distribution, such as mean and standard deviation
(STD), may provide information about microstructure, but their relationship is
unclear. Previously,
we have developed a 3-dimensional (3-D) Monte Carlo simulation to study the
relationship between tumor malignancy-related tissue changes and a
non-monoexponential model assuming uniform T2 relaxations [4]. Non-uniform T2
relaxations between cell compartments have been shown to affect measurements of
apparent diffusion coefficient (ADC) [5-7]. In this
study, we investigate how the non-uniform T2 relaxations affect the non-monoexponential decay
described by gamma distribution model with b-values up to 6000 s/mm^{2}. We
evaluate changes in the fitted parameters with simulated microstructural changes and
the fitting accuracy.

Microstructure:

A 3-D
microenviroment was simulated using randomly packed spheres [4]. The baseline
microstructural parameters were derived from previous measurements on biological
tissues and low-grade tumors. Four
independent microstructural changes related to tumor malignancy were simulated
as described previously [4] (Fig. 3 and 4), including increased cell density,
nuclear volume, extracellular volume fraction (VF_{ex}), and
extracellular tortuosity (λ_{ex}). Water diffusivities for nuclear (Dnu), cytoplasmic (Dcyto), and extracellular (Dex) compartments were: 1.2, 0.3, and 1.8 × 10^{-3} mm^{2}/s, respectively. T2nu, T2cyto, and T2ex were 78, 29.2, and 150 ms [5-7]. Furthermore, two additional values
of membrane permeability (P_{mem}): 2.4 × 10^{-3} and 2.4 × 10^{-1} mm/s were
applied to study the effects of P_{mem}.

DWI experiment:

DWI signals
were simulated using a Monte Carlo simulation of random walkers and pulsed
gradient spin-echo sequence as described previously [4]. The b-values ranged
from 0 to 6000 s/mm^{2} in increment of 500 s/mm^{2}. Each simulation was repeated 5
times. The simulated signals were fitted with gamma distribution model: (S(b) =
βα /(β+b)α) using the Levenberg-Marquardt algorithm in
Matlab (Mathworks, Inc.). The mean and STD of the gamma distribution (mean_{gamma} and STD_{gamma}) were computed. Apparent diffusion coefficient (ADC) was also
computed using the simulated signals with b-values of 0 and 1000 s/mm^{2}.

Statistics:

The goodness
of fit was assessed using the reduced chi-square statistic (χ_{ν}^{2})
and the Akaike Information Criterion (AIC) with a correction for finite sample
sizes [8]. The changes of the fitted parameters (ADC, mean_{gamma}, and STD_{gamma}) with simulated
microstructural changes were evaluated using Wilcoxon rank-sum test with
*p*-value < 0.05.

Fitting assessment:

The gamma distribution model fit the simulated DWI signals of all experiments using the χ_{ν}^{2} test; χ_{ν}^{2} = 0.03-1.33, ν = 9 (Fig. 1). The AIC values were higher for the signal decay with high P_{mem} and non-uniform T2 relaxations (Fig. 2). The STD of the fitted parameters across five repeated experiments was 1% for ADC and mean_{gamma}, and was 2% for STD_{gamma}.

Effects of uniform T2 relaxations:

Compared with the fitted parameters obtain with uniform T2 relaxations (Fig. 4), the non-uniform T2 relaxations resulted in higher ADC, mean_{gamma}, and STD_{gamma} (Fig. 3). Changes in the fitted parameters obtained with uniform and non-uniform T2 relaxations were similar in responding to increased cell density, VF_{ex} and λ_{ex}. However, ADC and mean_{gamma} increased with increased nuclear volume with uniform T2 relaxations (Fig. 4), whereas they showed a decrease or no changes with non-uniform T2 relaxations (Fig. 3). STD_{gamma} increased with nuclear volume with uniform T2 relaxations (Fig. 4) but showed a decrease with non-uniform T2 relaxations (Fig. 3). Furthermore, with increased P_{mem}, ADC and mean_{gamma} showed an increase with uniform T2 relaxations but a decrease with non-uniform T2 relaxations. STD_{gamma} increased with decreased P_{mem} with both uniform and non-uniform T2 relaxations.

Consistent with previous studies on ADC measurements, non-uniform T2 relaxations led to T2 filtering effect and an increase in diffusion coefficients [5,6]. It also caused reduced diffusion coefficients when P_{mem} was increased [5]. Additionally, we demonstrate that non-uniform T2 relaxations caused increased diffusion heterogeneity measured by STD_{gamma}. Interestingly, when P_{mem} was increased, the diffusion heterogeneity decreased in both cases of uniform and non-uniform T2 relaxations. These results suggest that, compared with ADC, the diffusion heterogeneity may provide more information about changes in heterogeneous microstructure with non-uniform T2 relaxations.

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[2]. Yablonskiy DA, et al, NMR Biomed 23, 661-681, 2010.

[3]. Bennett KM, et al, MRM 50, 727-34, 2003.

[4]. Lee CY, et al, ISMRM, 3080, 2016.

[5]. Harkins KD, et al, MRM 62,1414-1422, 2009.

[6]. Xu J, et al, MRI 29, 380–390, 2011.

[7]. White NS, et al, MRM 72, 1435-43, 2014.

[8]. Wittsack HJ, et al, MRM 64, 616-622, 2010.

**Figure 1:** Simulated DWI signal decay (linear scale (**a**) and logarithmic scale (**b**)) assuming uniform and non-uniform T2 relaxations between cell compartments for the baseline microstructure with cell
diameter of 10 ± 8 µm (gamma distributed), nuclear-to-cytoplasmic volume ratio
(NC ratio) of 6.4 %, intracellular volume fraction of 67 %, and membrane
permeability (P_{mem}) of 2.4 × 10^{-2} mm/s. The χ_{ν}^{2 }of gamma distribution fit was 0.21 ± 0.04 (uniform T2) and 0.23 ± 0.11 (non-uniform T2).