Shu Xing^{1} and Ives R. Levesque^{1,2}

Advanced diffusion-weighted MRI allows the characterization of cancer tumours noninvasively by estimating cell radius R and volume fraction v_{in}. Existing methods map the apparent R and v_{in} , under the assumption that a given voxel contains one cell population. This work investigates the feasibility of estimating the radii and volume fractions of 2 cell populations co-existing in the same voxel. This method could be useful in studying biphasic tumours like round cell/myxoid liposarcoma, which consist of high-grade and low-grade tumour cells, where the percentage of high-grade cells is strongly related to the risk of metastasis and changes the course of treatment.

The IMPULSED method combines the conventional PGSE and OGSE sequence^{1}. The diffusion MR signal of a single cell population is modeled as $$$ S = v_{in}S_{in} + (1-v_{in})S_{ex} $$$ [1] , where $$$S_{in}$$$ and $$$S_{ex}$$$ are the normalized signal from intra- and extra-cellular space, respectively. Equation 1 can be extended to 2-cell populations: $$$ S = v_{in,1}S_{in,1} + v_{in,2}S_{in,2}+(1-v_{in,1}-v_{in,2})S_{ex} $$$ [2]. Unfortunately, brute force fitting of Eq.2 produces unstable fits and poor results.

The detection sensitivity of DW-MRI depends on the effective diffusion time Δ_{eff}^{1}. With some prior knowledge on the anticipated cells sizes, we can sensitize the measurement to different length scales, allowing the separation of the 2 populations in the following 3 scenarios:

*Scenario 1: Separating large cells from small cells*

The Δ_{eff} range can be selected to remove signal sensitivity to small cells (pink box, Fig. 1a), where the intracellular diffusion coefficient D_{in,s }≈ 0 (e.g. R=1µm). Eq. 2 can be simplified to:

$$ S = v_{in,s} + v_{in,l}\cdot S_{in,l} + (1- v_{in,l}-v_{in,s})\cdot S_{ex}$$

allowing the estimation of R_{l} and v_{in,l} of the large cells and v_{in,s} of the small cells, but without sensitivity to R_{s}. This is termed the constrained 2-P IMPULSED method.

*Scenario 2: Separating small cells from large cells*

The Δ_{eff }can also be selected in the high frequency OGSE range to desensitize the signal to large cells (blue box, Fig. 1b), where D_{in,l} ≈ constant (e.g. R = 10µm). Eq. 2 becomes:

$$ S = v_{in,l}\cdot exp(-b\cdot D_{in,l}) + v_{in,s}\cdot S_{in,s} + (1- v_{in,l}-v_{in,s})\cdot S_{ex} $$

allowing the estimation of R_{s}, v_{in,s} and v_{in,l}, but without sensitivity to R_{l}. This is termed the constrained 2-P OGSE method.

*Scenario 3: Separating cells of close radius*

For cells of similar radius, the signal cannot be selectively desensitized. We choose Δ_{eff }(green box, Fig. 1c) in the PGSE range . This decreases the number of fitted parameters by eliminating the frequency dependence of D_{in} and D_{ex}. This method is termed the 2-P PGSE method.

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