Parisa Movahedi^{1,2}, Harri Merisaari^{1,3}, Ileana Montoya Perez^{1,2}, Jussi Toivonen^{1,2}, Pekka Taimen^{4}, Peter J. Boström^{5}, Janne Verho^{1}, Hannu J. Aronen^{1,6}, Tapio Pahikkala^{2}, and Ivan Jambor^{1,6}

The aim of this study was to evaluate various mathematical methods for enhanced parameter estimation of bi-exponential DWI (12 b values 0-2000 s/mm2) of prostate cancer. Least Squares (LSQ), Bayesian Shrinkage (BS) and Maximum Penalized Likelihood Estimation (MPLE) fitting methods were evaluated in the terms of Coefficients of Variation (CV), Contrast to Noise Ratio (CNR) and the Area under the curve (AUC) between tumor and non-tumor prostate tissue. BS and MPLE methods improved AUC and CNR values of bi-exponential model parameters and also decreased CV values in comparison with the commonly used LSQ fitting method.

The study was approved by institutional review
board and each patient with historically confirmed prostate cancer (PCa) included
in the study provided written informed consent. The DWI was performed using a 3T MR
scanner with a single shot spin-echo echo planar imaging, monopolar diffusion
gradient scheme, and the following parameters: TR/TE 3141/51 $$$ms$$$, FOV 250 × 250 $$$mm^{2}$$$, slice thickness 5 $$$mm$$$, diffusion gradient timing (Δ) 24.5 $$$ms$$$,
diffusion gradient duration (δ) 12.6 $$$ms$$$, diffusion time (Δ − δ/3) 20.3 $$$ms$$$, 12 b
values of [number of signal averages]: 0 [2], 100 [2], 300 [2], 500 [2], 700
[2], 900 [2], 1100 [2], 1300 [2], 1500 [2], 1700 [3], 1900 [4], 2000 [4] $$$s/mm^{2}$$$.
Prostate cancer lesions were delineated
using whole mount prostatectomy sections as the “gold standard”.
Voxel-wise fitting
of bi-exponential model ($$$f, D_f, D_s$$$),
eq.1,^{6 }was performed by Least Squares fitting
(LSQ), Bayesian Shrinkage (BS)^{4} and Maximum Penalized Likelihood
estimation (MPLE)^{5} methods on whole prostate area including the
lesions.

$${\bf{Y}}=c(fe^{-{\bf{b}}D_f}+(1-f)e^{-{\bf{b}}D_s})~~~~(1)$$

where $$$N$$$ is the number of b-values, signal vector $$${\bf{Y}}=(y_n)^{N}_{n=1}$$$ is the obtained signal model values estimated from the bi-exponential model Eq.(1), $$${\bf{b}}=(b_n)^{N}_{n=1}$$$ is the b value set, $$$c$$$ is the signal without diffusion weighting. $$$D_s$$$ is the slow diffusion component, $$$D_f$$$ is the fast decay component and $$$f$$$ is the fraction between fast and slow diffusion components. While LSQ method only estimates each voxel parameters individually, both BS and MPLE utilize spatial prior characteristics of surrounding voxels as a prior knowledge when fitting the model parameters for individual voxels. MPLE hyper parameter set (kernel size, constraints and the weighting penalty) were optimized while BS and LSQ did not required any prior parameter settings. The goodness of the methods’ fit were evaluated using parametric maps and Fitting errors. Furthermore Coefficients of Variation (CV), Contrast to Noise Ratio (CNR) and the Area under the curve (AUC) between each tumor and non-tumor prostate region was calculated and compared in-between the methods.

In total, 50 patients were included in final analyses. The tumor visibility was improved in both BS and MPLE for parametric maps (Figure1). The mean values of all estimated parameter values were lowest for BS and highest for LSQ (Figure2). The CV was significantly lower in all parameters’ estimation with BS method compared to other two methods ($$$p<0.05$$$) (see Figure 3.A). The BS and MPLE provided better CNR with the model parameter values $$$D_f$$$ and $$$D_s$$$ , compared with the LSQ and BS has significantly lower CNR for $$$f$$$ parameter (Figure 3.B).

For classification performance between cancer and non-cancer tissue types when voxels of all 50 cases were pooled, the BS and MPLE method provided better AUC values than LSQ. The best classification performance was with MPLE method in $$$D_f$$$ and $$$D_s$$$ while classification performance for $$$f$$$ parameter was best with the Bayesian Shrinkage (Table 1). The AUC values calculated for each subject separately are shown in Table 2. Significant differences between methods were found in $$$f$$$ and $$$D_f$$$ parameters. Classification performance of $$$f$$$ was best with BS. For $$$D_f$$$ and $$$D_s$$$ the classification performance was better with BS and MPLE than with LSQ, where in $$$D_s$$$ the difference was statistically significant.

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