Introduction

Cancerous tumours are characterized by their cellular composition and cell size, both linked to tumour type and grade. Advanced diffusion MRI methods have been proposed to map the average cancer cell radius $$$R$$$ and the relative cell volume fraction $$$v_{in}$$$ covering the entire tumour volume(1,2). Existing methods assume that tumours feature a single cell population, which is not valid in many applications. We have previously proposed a two-cell population microstructure model (2P-MM) that can map $$$Rs$$$ and $$$v_{in}s$$$, when 2 cell populations of different radii and volume co-exist in the same voxel(3). Here, we further study the robustness of the 2P-MM under the influence of noise and investigate methods to improve the stability of parameter estimation. The ability to reliably estimate $$$Rs$$$ and $$$v_{in}s$$$ for tissue with 2 cell populations opens exciting avenue of potential applications in tumor diagnosis and treatment monitoring. For instance, round cell/myxoid liposarcoma is composed of high-grade cells (R~10μm) and low-grade cells (R~4μm). The amount of the high-grade component is strongly related to tumor grade and changes the course of treatment (4). The 2P-MM could also potentially quantify the infiltrating T-cells (R~4μm) to cancer cells (R~10μm) during immunotherapy (5).

Theory

The normalized diffusion signal of a single cell population is modeled with the one cell population microstructure model (1P-MM):
$$ S/S_0=\boldsymbol{v_{in}}S_{in}(\boldsymbol{R,D_{in}})+(1-v_{in})S_{exo}(\boldsymbol{D_{exo}}) \hspace{0.5cm}[1]$$
where S_in and S_exo describe the intracellular restricted diffusion and extracellular hindered diffusion signal, respectively (6). Equation 1 can be extended to a 2P-MM by including an additional restricted diffusion compartment:$$S/S_0=\boldsymbol{v_{in,l}}S_{in,l}(\boldsymbol{R_l,D_{in}})+\boldsymbol{v_{in,s}}S_{in,s}(\boldsymbol{R_s},D_{in})+(1-v_{in,l}-v_{in,s})S_{exo}(\boldsymbol{D_{exo}}) \hspace{0.5cm}[2]$$
where the subscript $$$l$$$ and $$$s$$$ present large and small cell populations. Instead of four fitting parameters (bolded) in Eq. 1, the 2P-MM has six fitting parameters. Stabilization of the fitting quickly becomes the primary challenge. Here, we propose two techniques to reduce the number of fitting parameters, thus stabilizing the fit.

Stabilization by signal suppression from one cell population: special case scenario
The diffusion signals over a range of effective diffusion times Δ_eff show strong dependence on cell radius (Figure 1). Signal from cells of R=1,2 μm shows little change with Δ_eff. The Δ_eff range can be strategically selected to remove signal sensitivity to small cells, where $$$S_{in,s}\approx 1$$$. Eq. 2 is then simplified to:
$$S/S_0=\boldsymbol{v_{in,s}}\cdot1 +\boldsymbol{v_{in,l}}S_{in,l}(\boldsymbol{R_l,D_{in}})+(1-v_{in,l}-v_{in,s})S_{exo}(\boldsymbol{D_{exo}}) \hspace{0.5cm}[3]$$
Stabilization by fixing fitting parameters: general case scenario

• 2-step fitting

For cells of similar radii, the signal can no longer be selectively desensitized. Our simulation results suggest that by fitting 1P-MM to diffusion signal from tissue with two cell populations, the estimated $$$v_{in}$$$ tends to yield the total volume $$$v_{in,tot}$$$ (Figure 2 and 3, blue dots). We can exploit this observation by first fitting the 1P-MM to the signal, then fix $$$v_{in,tot}$$$ to reduce the number of free parameters in the 2P-MM: $$ S/S_0=(v_{in,tot}-v_{in,s})\cdot S_{in,l}(\boldsymbol{R_l},D_{in}) +\boldsymbol{v_{in,s}}S_{in,s}(\boldsymbol{R_s,D_{in}})+(1-v_{in,tot})S_{exo}(\boldsymbol{D_{exo}}) \hspace{0.5cm}[4]$$

• Fixed Din

Previous literature has suggested that fixed D_in improves the robustness of fitting (7,8). The influence of fixed D_in in the 2P-MM was also evaluated.

Method

The matrix method (9) was used to simulate PGSE diffusion signal for a two-cell population input with different $$$Rs$$$ and $$$v_{in}s$$$, using D_in = 1μm²/ms and D_exo = 2μm²/ms. Both 1P-MM and 2P-MM were fitted to the simulated signal with 100 starting points. Estimated $$$Rs$$$ and $$$v_{in}s$$$ were compared to ground truth values specified in the simulation. This process was repeated for a range of simulation parameters. Rician noise was added to yield SNR of 20, 50, 80 in the b = 0 ms/μm² signal and the same fitting process was repeated for 1500 noisy signals. Fits that were not within 1% of the fit constraints were considered acceptable fits.

Results and Discussion

The 2P-MM accurately separated the large and small cells for the noiseless fits over a range of radii and volume fractions in both the special and general case (Figure 2). With added noise, the precision of the fitted parameters improved as SNR increased for both scenarios (Figure 3). In this work, interquartile ranges of 2 μm for the fitted Rs and 25% for fitted $$$v_{in}s$$$ were considered acceptable. Thus, a SNR of 50 is required for reliable parameter estimation. Fixing fitting parameters, with the 2-step fitting or fixed D_in, generally improved robustness and accuracy (Figure 4). Both techniques produced higher number of acceptable fits and lower median % error on the fitted parameters, with the exception of $$$R_s$$$. The choice of fixed D_in values had a noticeable effect on the accuracy of fitted parameters (Figure 5). The fixed D_in had to be close to or greater than the true value for the fitted $$$Rs$$$ and $$$v_{in}s$$$ to remain within $$$\pm 1\mu m$$$ and $$$\pm 10\%$$$ of the ground truth. These observations are consistent with prior studies that a relatively large fixed can provide robust fitting with an acceptable sacrifice in accuracy (7,8).

Conclusion

In this work, we studied the robustness of the 2P-MM under the influence of noise and proposed methods to improve the stability of the fitting, where microstructure parameters $$$R_l,R_s,v_{in,l}$$$ and $$$v_{in,s}$$$ can be reliably estimated when two cell populations co-exist in the same voxel. This method could be useful in studying biphasic tumours like round cell/myxoid liposarcoma and for treatment monitoring.

Acknowledgements

The authors acknowledge Dr. Junzhong Xu, Dr. Xiaoyu Jiang, Zaki Ahmed, and Véronique Fortier for valuable discussions. This work is supported by Medical Physics Research Training Network Grant of the Natural sciences and Engineering Research Council (Grant number: 432290), and the Fonds de Recherche en Santé du Quebec.