Synopsis

Cellular microstructural parameters mapping can provide quantitative information for tumor diagnosis and treatment monitoring. Particularly, voxel-wise cell size distribution may provide critical clinical biomarkers to characterize heterogeneity of tumors. Currently, MRI-cytometry may measure such a distribution, but it “blurs” the distribution peaks and prevents differentiating different cell populations with different cell sizes. In this work, we develop a new approach to improve this and hence make it more practical to distinguish different cells. Detection of T cell infiltration for assessment of early response to immunotherapy may be a potential application of this method.

Introduction

Diffusion MRI (dMRI) provides a non-invasive approach to probe the microstructural information by utilizing time dependency of diffusion sensitivity to different time length scales¹. Several methods have been developed to estimate voxel-wise mean cell sizes, including VERDICT^2,3, IMPULSED^4-6, and POMACE⁷ methods. However, this is inadequate to characterize heterogeneous cell sizes in tumors. To solve this problem, MRI-cytometry has been proposed recently to estimate the intra-voxel cell size distribution and has shown potential in breast cancer patients⁸. Nevertheless, this framework still cannot differentiate the distribution peaks of cell populations with different cell sizes. This can be clinically critical to monitor T cell infiltration in the early response to immunotherapy⁹

Essentially, the extraction of cell size distribution based on dMRI signals is to solve an inverse problem in mathematics. Usually, deconvolution methods are effective for processing inverse problems. For example, orientation distribution function estimators based on spherical deconvolution (SD) are commonly used in tractography. In this study, we proposed a Fourier expansion-based deconvolution solver for microstructural parameters mapping (FED-MPM) to improve the estimation of intra-voxel cell size distribution. More importantly, this solver can discriminate multiple cell populations with different cell sizes within an image voxel without prior knowledge of the distribution function. The performance was validated by simulations and in vitro cell experiments. The results may hold great promise for monitoring the immune cell infiltration in immunotherapy.

Theory

Base on the generalized two-compartment model^4,6, namely:
$$S_{\mathrm{DW}}=f_{i n} S_{i n}+\left(1-f_{i n}\right) S_{e x}=f_{i n} S_{i n}+f_{e x} S_{e x}\tag{1}$$
we further consider the distribution of intra-voxel cell sizes and formulate the FED-MPM solver as following.
First, a cell size probability density function (CS-PDF) is defined to describe the intra-voxel distribution of cell diameters. The graphic diagrams of the CS-PDF for different cell diameters are shown in Fig. 1. With given acquisition parameters, intracellular signal vector, S_in, has a one-to-one correspondence with CS-PDF. Then, we expand CS-PDF into Fourier series and keep the first finite expansion coefficients as a vector, f. The vectors, S_in and f, also correspond to each other. We assume that they satisfy the following linear mappings:
$$S_{i n}=K f\tag{2}$$
where K is the deconvolution matrix, which can be determined by the similar computation in the SD methods. For the extracellular part, we can express the unknown signal, S_ex, as a linear combination of a series of known terms. Using a pre-defined dictionary about D_ex: D_ex(1),..., D_ex(Q), f_exS_ex can be expressed as the following linear combination:
$$f_{e x} \boldsymbol{S}_{e x}=f_{e x}\left[\exp \left(-D_{e x} \boldsymbol{b}\right)\right] \approx \sum_{i=1}^{Q} d_{i}\left[\exp \left(-D_{e x}(i) \boldsymbol{b}\right)\right]\tag{3}$$
where b is the b-value vector and d_i (i=1,2,3,..,Q) are combination coefficients. Defining f* = f_in f, we have:
$$S_{\mathrm{DW}}=K \boldsymbol{f}^{*}+\sum_{i=1}^{Q} d_{i}\left[\exp \left(-D_{e x}(i) b\right)\right]=A \boldsymbol{x}\tag{4}$$
where A=[K, exp(-D_ex(1)b), exp(-D_ex(2)b) ,..., exp(-D_ex(Q)b) ]. Through solving the above linear equations, the unknown parameter vector, x=[ f*^T, d₁, d₂,...,d_Q]^T, can be obtained. Then, CS-PDF is computed by the Fourier series according to the expansion coefficients.
Based on Fourier deconvolution, the estimation of cell size distribution and microstructure parameters can be obtained by simple linear computations, which may be efficient in practical applications. Furthermore, we can add constrained optimization to f and obtain an improved CS-PDF. Finally, CS-PDF (denoted as y_CS-PDF ) are normalized as:
$$y_{\mathrm{CS}-\mathrm{PDF}}^{*}=\frac{y_{\mathrm{CS}-\mathrm{PDF}}(d)-\min \left(y_{\mathrm{CS}-\mathrm{PDF}}\right)}{\int y_{\mathrm{CS}-\mathrm{PDF}}(d)-\min \left(y_{\mathrm{CS}-\mathrm{pDF}}\right)}\tag{5}$$

Methods

The dMRI signals with different cell sizes within one voxel were simulated as previously reported⁸. The in-vitro cell experiments were performed on a Varian/Agilent 4.7T MRI spectrometer (Agilent Technologies, Santa Clara, CA) similar to the approach described previously^4,10 The histological ground truths about cell sizes were obtained by light microscopy. The acquisition parameters of both experiments are shown in Table 1. We compared our method with MRI-cytometry.

Results and Discussion

Fig. 2 shows the simulation results of MRI-cytometry and FED-MPM for various continuous distributions of cell diameters without noise. For the normal distribution and platform-normal cases, the peaks of the computed distribution functions using both methods are consistent with that of the ground truths. For the binormal and tri-normal distributions, MRI-cytometry method 'blurs' the distribution peaks. In contrast, FED-MPM can still resolve multiple peaks for these complex distributions. Additionally, the position and width of peaks are consistent with the ground truths.

Fig. 3 shows the effect of different levels of noise (SNR) on the performance of solvers for (a) single-cell normal distribution and (b) two-cell binormal distribution. Although MRI-cytometry and FD-MPM show similar results in the single-cell case, the former failed to discriminate the two peaks when two cell populations coexist in one image voxel.

Fig. 4 demonstrates the results from the in-vitro cell experiments. The obtained b0 image is also shown in the lower right. FED-MPM shows highly consistent distribution curves with the histological ground truths. Further validations on in-vitro mixed-cells and cancer patients in treatments are expected to better evaluate the clinical value of this method.

Conclusion

In this study, a new mathematical solver for microstructural parameters mapping, FED-MPM, was proposed to estimate intra-voxel cell size distributions and further identify cell populations with different cell sizes. Numerical simulations and cell experiments in vitro were performed to validate the efficacy of the proposed solver. The results indicate that FED-MPM may provide a promising tool to improve both tumor diagnosis and treatment monitoring.

Acknowledgements

No acknowledgement found.

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