Gregory Lemberskiy^{1,2}, Els Fieremans^{1}, Jelle Veraart^{1}, Andrew B. Rosenkrantz^{1}, and Dmitry S Novikov^{1}

We identify the contributions of prostate cellular and glandular compartments to the overall diffusion time-dependent diffusion tensor, by varying echo time and relying on their different T2 values. We test the functional form of compartment tensor eigenvalues with respect to the diffusion time against a variety of tissue models, and find the glandular tissue being best described by the short-time surface-to-volume limit, whereas the random permeable barrier model is most applicable in cellular tissue, and for tumors of various grade. Our framework allows quantification of glandular sizes, cellular fiber diameters, membrane permeability, compartment T2-values, and volume fractions.

The apparent diffusion coefficient (ADC) is a highly sensitive biomarker for identifying prostate cancer^{1}. Though much of the literature reports ADC^{2} as a measurement of cellularity, a side-by-side comparison^{3} of cellularity and volumetric changes in epithelium, stroma, and gland compartments revealed that the volume fractions alone explain most of the ADC contrast. Moreover, changes in compartment volume fraction correlate significantly with prostate cancer progression^{3}. Though ADC is sensitive, it is not specific to the relative compartment contributions of the microstructural changes. In this study, we utilize the large differences^{4} in T2 to isolate tissue compartments in order to study their diffusion properties separately.
We model multi-compartment signal:$$S(b,t,T_\text{E})=S_0\sum_c\,f_c\,e^{-T_\text{E}/T_2^c-bD_c(t)+\mathcal{O}(b^2)}\tag{1}$$in 3 dimensions of parameter space: diffusion time $$$t$$$, diffusion-weighting $$$b$$$, and echo-time $$$T_\text{E}$$$, that allows us to study compartmental diffusion and derive physical properties of each tissue compartment’s microstructure.

**MRI: **A multi-parametric diffusion STEAM acquisition, with $$$b=500\,\text{s/mm}^2$$$ along$$$\,17\,$$$directions and$$$\,2\text{ nominal}\,b=0$$$ images, which actually ranged from $$$[b=3-102\,\text{s/mm}^2],\,8\,$$$diffusion times:$$$\,t=[25.2-740\text{ms}]\,$$$using 8 mixing times$$$\,T_{\text{M}}\,=\,[6.38-719.32]\,\text{ms}$$$, and $$$3\,T_\text{E}=[52,115,180]\,\text{ms}$$$, was used to image$$$\,3\,$$$volunteers, on a Siemens 3T PRISMA system.

**Modeling: **The diffusion tensor from each acquisition is calculated to determine un-weighted$$$\,S|_{b=0}(T_\text{M},T_{\text{E}})$$$, which is modeled (Fig.2) using a$$$\,5\,$$$parameter model consisting of proton density$$$\,S_0$$$, cellular compartment fraction,$$$\,f$$$, cellular (fast) and glandular (slow)$$$\,T_2$$$, and single compartment$$$\,T_1$$$:
$$S|_{b=0}=S_{0}e^{-T_\text{M}/T_1}\bigg(\underbrace{fe^{-T_\text{E}/T_2^{(C)}}}_{\textbf{C}}+\underbrace{(1-f)e^{-T_\text{E}/T_2^{(G)}}}_{\textbf{G}}\bigg)\tag{2}$$
These parameters were then recombined to estimate $$$T_\text{E}$$$-specific weights,$$W^{(C)}(T_\text{E})=\frac{C}{C+G},\quad\,W^{(G)}=1-W^{(C)}\tag{3}$$
which were subsequently used to extract diffusion tensors$$$\,D^{(C)}(t)\,$$$and$$$\,D^{(G)}(t)\,$$$corresponding to the cellular and glandular compartment from the overall diffusion tensor$$D(t,T_{\text{E}})=W^{(C)}(T_{\text{E}})\cdot\,D^{(C)}(t)+W^{(G)}(T_\text{E})\cdot\,D^{(G)}(t)\tag{4}$$$$\binom{D^{(C)}(t)}{D^{(G)}(t)}=\text{pinv}\left(\begin{bmatrix}W^{(C)}(T_{\text{E}_1})\,&\,W^{(G)}(T_{\text{E}_1})\\W^{(C)}(T_{\text{E}_2})\,&\,W^{(G)}(T_{\text{E}_2})\\\vdots&\vdots\\\,W^{(C)}(T_{\text{E}_\text{N}})\,&\,W^{(G)}(T_{\text{E}_\text{N}})\,\\\end{bmatrix}\right)\begin{bmatrix}\,D(t,T_{\text{E}_1})\\\,D(t,T_{\text{E}_2})\\\vdots\\\,D(t,T_{\text{E}_\text{N}})\,\\\end{bmatrix}\tag{5}$$
Where$$$\,t\sim\,T_\text{E}/2+T_\text{M}\,$$$and N=3 for our experiment. Eqn.5 separates the dependence of $$$T_\text{E}$$$ and$$$\,t\,$$$from$$$\,D\,$$$allowing for the evaluation of microstructural properties arising from those parameters. Probabilistic fiber tracking^{5}, via MRTRIX3 (mrtrix.org), was performed on a dyadic tensor^{6} across 8 diffusion times on the cellular compartment.Compartmental tensors were then diagonalized for each $$$t$$$ to generate a set of eigenvectors and eigenvalues. The time-dependence of the eigenvalues from cellular, glandular, and overall-diffusion tensors at individual $$$T_\text{E}$$$ is evaluated [Table.1] using the short-time limit^{7}, which depends on the free diffusivity,$$$\,D_0$$$, and surface-to-volume ratio, $$$S/V$$$, as well as with the long time limit, which depends on the finite bulk diffusion constant, $$$D_\infty$$$, the slope of the power law tail, $$$A$$$, and the power law scaling associated with the disorder geometry^{8}, $$$\vartheta$$$:$$D(t)\simeq\,D_\infty+At^{-\vartheta}\tag{6}$$
For all assessments of the long-time limit,$$$\,\vartheta$$$, is fixed to study well established disorder geometries: extended disorder [Table.1,Eqn.[8]], hyper-uniform disorder [Table.1,Eqn.[9]], and short range disorder in 2-dimensions [Table.1,Eqn.[10]]. Additionally, the patient data (at low $$$T_\text{E}$$$) from ref.(7), which featured a diffusion STEAM acquisition at multiple $$$t$$$, is used to assess the power-law scaling in prostate tumors.

The functional form of $$$D(t)$$$ at various$$$\,T_\text{E}\,$$$changes [Figure.1(A)] from being well described by the long-time limit (stroma) at$$$\,T_\text{E}=52\,\text{ms}$$$ [$$$\rho=0.91$$$], to being poorly described by those same models at$$$\,T_\text{E}=180\,\text{ms}$$$ [$$$\rho=0.63$$$]. Conversely, correlation with the short time limit improves with longer $$$T_\text{E}=[52,115,180]$$$ resulting in $$$\rho=[0.91,0.93,0.96]$$$. Moreover, the value of the diffusion coefficient increases by as much as 59% between $$$\,T_\text{E}=52$$$ and $$$\,T_\text{E}=180$$$ ms. There is a factor of $$$\sim\,4$$$ difference in $$$T_2$$$ from cellular and glandular compartments [Fig.2] that facilitated the separation of diffusion compartments.

The cellular compartment and the patient tumor data is well described by extended disorder [Fig.3], prompting the use of the random permeable
barrier model (RPBM)^{10,11} to derive cellular diameters
and cellular permeability [Fig.1(C), Fig.4)]. On the other hand, the poor
fit of the gland compartment with long-time limits ($$$\rho=0.56$$$) and much better agreement
with short time limit: $$$\rho=0.95$$$, prompted the use of the latter
to derive glandular diameters [Fig.1(B), Fig.4].

The extracted stromal $$$(17.7 ± 8.8)$$$ and glandular diameters $$$(128.6
± 88.0)$$$ were in good agreement with those reported in
histopathology^{12}. Moreover, the cell membrane
permeability $$$(0.048 ± 0.02)$$$ is similar to permeability measured from the red blood cell
membrane^{13}.

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