Parametric maps of spatial two-tissue compartment model for prostate dynamic contrast enhanced MRI - comparison with the standard tofts model in the diagnosis of prostate cancer

The spatial two-tissue compartment model (2TCM) was used to analyze prostate dynamic contrast enhanced (DCE) MRI data and compared with the standard Tofts model. A total of 29 patients with biopsy-confirmed prostate cancer were included in this IRB-approved study. MRI data were acquired on a Philips Achieva 3T-TX scanner. After T2-weighted and diffusion-weighted imaging, DCE data using 3D T1-FFE mDIXON sequence were acquired pre- and post-contrast media injection (0.1 mmol/kg Multihance) for 60 dynamic scans with temporal resolution of 8.3 s/image. The 2TCM has one fast (K1trans\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{K}}_{\text{1}}^{\text{trans}}$$\end{document} and kep1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{k}}_{\text{ep}}^{\text{1}}$$\end{document}) and one slow (K2trans\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{K}}_{\text{2}}^{\text{trans}}$$\end{document} and kep2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{k}}_{\text{ep}}^{\text{2}}$$\end{document}) exchanging compartment, compared with the standard Tofts model parameters (Ktrans and kep). On average, prostate cancer had significantly higher values (p < 0.01) than normal prostate tissue for all calculated parameters. There was a strong correlation (r = 0.94, p < 0.001) between Ktrans and K1trans\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{K}}_{\text{1}}^{\text{trans}}$$\end{document} for cancer, but weak correlation (r = 0.28, p < 0.05) between kep and kep1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{k}}_{\text{ep}}^{\text{1}}$$\end{document}. Average root-mean-square error (RMSE) in fits from the 2TCM was significantly smaller (p < 0.001) than the RMSE in fits from the Tofts model. Receiver operating characteristic (ROC) analysis showed that fast K1trans\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{K}}_{\text{1}}^{\text{trans}}$$\end{document} had the highest area under the curve (AUC) than any other individual parameter. The combined four parameters from the 2TCM had a considerably higher AUC value than the combined two parameters from the Tofts model. The 2TCM is useful for quantitative analysis of prostate DCE-MRI data and provides new information in the diagnosis of prostate cancer.


Introduction
Multi-parametric MRI (mpMRI) plays an important role in the detection and grading of prostate cancer (PCa) [1][2][3]. Although T2-weighted (T2W) imaging and diffusion-weighted imaging (DWI) are the two main components of prostate mpMRI, dynamic contrast enhanced (DCE) MRI remains an integral part of the Prostate Imaging-Reporting and Data System (PI-RADS) guidelines [4][5][6]. Over the last several years, many studies have compared prostate mpMRI and biparametric MRI (bpMRI) (consisting of T2W and DWI only) and demonstrated that bpMRI is as good as mpMRI in the diagnosis of PCa [7][8][9][10][11]. The bpMRI is a powerful clinical screening tool for PCa with short scan time and without the use of contrast agent [12,13]. DCE-MRI assists T2W imaging and DWI in the detection of high-risk PCa, and in surveillance status post prostatectomy, radiotherapy or focal ablation [14][15][16][17][18][19]. In addition, DCE-MRI has shown the closet extent of the cancer volume compared to ground truth prostatectomy results, which is highly underestimated on T2W and DWI [20]. However, Please note that part of the work has been published in the Proceedings of 2020 ISMRM Virtual Conference (p0138). compared to T2W imaging and DWI, qualitative analysis of DCE-MRI is the least standardized [5,21]. There is often some overlap in enhancement patterns between benign and malignant tissues in prostate peripheral and transition zones. Some benign lesions, such as benign prostatic hyperplasia (BPH) nodules, show strong early enhancement, while some malignant lesions do not show early enhancement or washout [22]. Therefore, there are often false positives in the diagnosis of PCa using DCE-MRI [23].
Prostate DCE-MRI data is also often analyzed quantitatively using pharmacokinetic models, such as the standard Tofts model, to extract the volume transfer rate constant (K trans ) (exchange between blood plasma and the extravascular extracellular space (EES)) and fractional volume of EES (v e ) [24,25]. However, tumors often show a high degree of heterogeneity at the microvasculature level [26], and tumor heterogeneity is linked to malignancy and aggressiveness, as well as therapeutic resistance [27,28]. The standard Tofts model may not be compatible with the heterogeneous characteristics of the tumor micro-environment that results in an initial rapid uptake of contrast agent followed by a less rapid, but prolonged, uptake of the contrast agent [29,30]. As a result, tumor heterogeneity at the microscopic level could cause poor fits to DCE-MRI data for the standard Tofts model and errors in extracting K trans and v e . This would limit the diagnostic accuracy when using the standard Tofts model to analyze DCE-MRI data. Therefore, more complex models should be used to analyze DCE-MRI data, such as the multicompetent models [31], "shutter-speed" model [32], two-compartment exchange model 2CXM [33,34], and spatial two-tissue compartment model (2TCM) [35]. In contrast to the standard Tofts model with only one tissue compartment, the 2TCM has one slow and one fast exchanging tissue compartment. A previous study demonstrated that MRI contrast agent concentration curves as function of time in a heterogeneous tumor could be more accurately fitted with the 2TCM than the standard Tofts model, especially at tumor margins [35].
Although there are more complicated models available, complex model parameters may be overfitting the data and introducing redundant parameters that do not correlate with actual tissue or biophysical process and may not be stable enough to provide parameter estimates [36,37]. It is a challenge to select an appropriate model to extract physiological parameters in analysis of DCE-MRI data [38]. In this study, the 2TCM of DCE-MRI was selected to test the hypothesis that there would be two tissue compartments in the prostate with different kinetic properties of contrast agent exchange rates with plasma. This was supported by the study using x-ray fluorescence microscopy (XFM) imaging to demonstrate that IV injection of gadodiamide leaks from the cellular tissue into the lumen of murine prostate glands, which validated the need to use a two-tissue compartment model with different kinetic properties and exchange rates [39]. On one hand, the 2TCM is only two parameters more than the standard Tofts model. But on the other hand, the 2TCM is very close to the standard Tofts model mathematically. When the exchange rates are the same or when one of the tissue volumes vanishes, the 2TCM becomes the standard Tofts model [35]. Therefore, the 2TCM was used to analyze prostate DCE-MRI data and prove the hypothesis that diagnostic accuracy could be improved for DCE-MRI. The results obtained from 2TCM were compared with those from the standard Tofts model as a DCE model evaluation.

Patients
This retrospective study was approved by the Ethics Committee of the University of Chicago, Institutional Review Boards (IRB), with the approval number IRB13-0756. Informed patient consent was compliant with the Health Insurance Portability and Accountability Act (HIPAA). The patients imaged and followed by subsequent radical prostatectomy, were recruited for this study between March 2014 and May 2015. Twenty-nine patients (mean age 57 years, range 40-70 years; and mean PSA 7.4 ng/ml, range 1.8-26.1 ng/ml) with biopsy-confirmed prostate cancer were included in this study. Patients who received prior radiation or hormonal therapy were excluded.

Data analysis
DCE-MRI data were analyzed using Matlab (Mathworks, Natick, MA, USA) with an in-house software package. Whole mount sectioning of the prostate was performed in approximately the same plane as the MR images. H&Estained whole mount radical prostatectomy sections were matched with corresponding prostate MR images by an expert radiologist (AO − 20 years' experience with prostate MRI) and expert pathologist (TA − 17 years' experience) to guide the selection of regions-of-interests (ROIs) of cancer and normal prostate tissues in the peripheral zone (PZ), transition zone (TZ), central zone (CZ), and anterior fibromuscular stroma (AFMS). The cancer ROIs were marked by the radiologist on areas of visible cancer on MRI confirmed by pathology. A medical physicist (XF − 20 years' experience with DCE-MRI) manually traced ROIs of blood vessels, used for arterial input function (AIF), on the iliac artery on a slice with cancer on DCE images. On slices with ROIs, pixel contrast agent concentration (C(t)) as function of time (t) was calculated from the non-linear model using the gradient echo signal equation [40] with measured T1 value from VFA sequence.
For each C(t), physiological parameters were extracted from the standard Tofts model, refer to Eq. (24) in [25]: as well as from the 2TCM, refer to Eq. (2) in [35]: where C p (t) = C b (t)/(1 − Hct) is the AIF, C b (t) is contrast media concentration in blood, Hct is the hematocrit (= 0.42), k ep =K trans /v e and k i ep = K trans i ∕v i e (i = 1, 2) is the efflux rate constant from the EES to the plasma. A schematic diagram of the standard Tofts model and the 2TCM is shown in Fig. 1. The root-mean-square error (RMSE) was calculated to evaluate data fitting for the standard Tofts model and the 2TCM. In addition, the Akaike information criterion (AIC) and the associated Akaike weights (AW) was calculated to rank pharmacokinetic models on the basis of goodness-of-fit and number of parameters. The AIC has been applied in DCE-MRI models by several studies [33,41,42].
In order to obtain unique results for fitting C(t) using MATLAB, Eq. 2 was written in an asymmetric form as follows: where K trans 2 = ⋅ K trans 1 and k 2 ep = k 1 ep + . Please note that for different pixels either K trans 1 or K trans 2 could be large or small depending on whether ε is smaller or larger than 1.0. In order to consistently compare calculated parameters, we selected K trans 1 as the larger one of them (=max K trans 1 , K trans 2 ) with its corresponding k 1 ep or k 2 ep as k 1 ep . Therefore, for each pixel K trans 1 is always larger than K trans 2 in our study. Parametric maps were only generated for the slices with ROIs. For each ROI, the average value was calculated for all physiological parameters. Generally, PCa shows earlier and faster enhancement and earlier contrast agent washout compared to normal prostate tissue [43]. Based on K trans map and the matching histopathology slice, normal prostate tissue regions with high K trans value (similar to nearby cancer) were also manually traced with a slightly larger ROI on the map. Then, a cutoff value was used to determine final 'selected false positive' ROI of K trans . The cutoff value was determined from the average K trans value of a nearby cancer ROI minus its standard deviations. With this method, the selected false positive ROI would have an average K trans value close to the average cancer K trans value.
A paired t-test was performed to determine whether there was a significant difference for the RMSE and Akaike weights between two models. Pearson's correlation coefficient was calculated to test whether there are linear correlations between physiological parameters obtained from the standard Tofts model and the 2TCM. One-way ANOVA with post-hoc Tukey HSD (Honestly Significant Difference) Test was performed to determine whether there was significant difference for all calculated physiological parameters between cancer, normal tissue and false positive region. Receiver operating characteristic (ROC) analysis was used to evaluate performance in differentiating between cancer and normal tissue. Binary logistic regression was used to assess combinations of parameters from the 2TCM. All statistical analysis was calculated using SPSS (IBM Corporation, Armonk, NY).  Figure 2 shows a prostate DCE image, plot of the AIF (red line) and plots of measured C(t) (black dots), as well as corresponding fits with the standard Tofts model (orange line) and the 2TCM (blue line) for three tumor pixels (a, b and c) and one normal tissue pixel (d). It can be seen that the 2TCM fits are better than those of the standard Tofts model. On average (± standard deviation), the RMSE (= 0.013 ± 0.009) obtained from the 2TCM was significantly smaller (p < 0.001) than the RMSE (= 0.022 ± 0.024) obtained from the Tofts model, which shows better fitting for the data using the 2TCM. This observation is confirmed by analysis of the Akaike weights, which was 0.79 ± 0.34 and 0.21 ± 0.34 for the 2TCM and the standard Tofts model, respectively. is similar to K trans but with higher value only in cancer region. By comparing with histopathology slice, the K trans and K trans 1 maps show extent of cancer better than the T2W image and ADC map [20]. However, similar to the results of other studies [16,44,45], the v e and v i e (i = 1, 2) could not discriminate between cancer and normal tissue. Figure 5 shows scatter plots of averaged physiological parameters over the ROIs obtained from the Tofts model and the 2TCM for (a, d) cancer and (b, c) normal tissue with K trans and K trans i (i = 1, 2) in the top row, and k ep and k i ep (i = 1, 2) in the bottom row. There are strong correlations (r = 0.82 to 0.94, p < 0.001) between K trans and K trans i (i = 1, 2) for cancer ( Fig. 5 (a)), and moderate to strong correlations (r = 0.69 to 0.93, p < 0.001) for normal tissue (Fig. 5 (b)). There was weak correlation (r = 0.28, p < 0.05) between k ep and k 1 ep , but strong correlation (r = 0.83, p < 0.001) between k ep and k 2 ep for cancer (Fig. 5c). There were moderate correlations (r = 0.67 to 0.77, p < 0.001) between k ep and k i ep (i = 1, 2) for normal tissue (Fig. 5 (d)). The differences between K trans 1 and K trans 2 , and between k 1 ep and k 2 ep indicate that there were slow and fast exchanging tissue compartments in PCa. Figure 6 shows box-plots of all physiological parameters compared between cancer (red), normal tissue (green) and selected false positive (black) ROIs for (a) K trans , (b) K trans 1 , (c) K trans 2 , (d) k ep , (e) k 1 ep , (f) k 2 ep . ANOVA with post-hoc Tukey HSD test showed significant difference (p < 0.01) for all the physiological parameters between cancer and normal tissue. There is a difference (p < 0.03) between cancer and false positive ROIs for K trans 1 , but not for other parameters. When comparing normal tissue with the false positive ROIs, there is significant difference (< 0.01) between K trans and K trans 2 , but not for K trans 1 . There is no statistical difference (p ≥ 0.06) between normal tissue and false positive ROIs for k ep and k i ep (i = 1, 2). By performing receiver operating characteristics (ROC) analysis between cancer and false positive ROIs for the parameters K trans and K trans 1 , the area under the curve (AUC) of ROC was 0.43 and 0.63 for K trans and K trans 1 , respectively. This indicates that there was no difference between cancer and false positive for K trans , but there was a difference for K trans 1 . Finally, Table 1 shows ROC analysis results for all individual parameters and combined parameters obtained from the Tofts model and the 2TCM when differentiating between cancer and normal tissue. It shows that the parameter K trans 1 has the highest AUC of 0.787, which is at least ~ 7% higher than any other individual parameter. The AUC value calculated from the combined four parameters of the 2TCM was 0.800, compared with AUC value of 0.742 calculated from the combined two parameters of the Tofts model.

Discussion
The 2TCM was compared with the standard Tofts model using prostate DCE-MRI data. For all calculated parameters, PCa had significantly higher values than normal tissue. Our results also demonstrated that prostate cancer is heterogeneous, involving both the fast ( K trans 1 ) and the slow ( K trans 2 ) exchange compartments. Mathematically, when k 1 ep and k 2 ep are the same, two additive exponential terms can be combined as one exponential term, which is used in the standard Tofts model. However, Fig. 5c and d showed that obtained from the 2TCM showed better separation between cancer and normal tissue than K trans obtained from the standard Tofts model. Combined 2TCM parameters had a higher AUC value than combined Tofts model parameters, suggesting potential advantages for diagnosis of PCa.
PI-RADS version 2.1 favors qualitative analysis of DCE-MRI data, which is the least standardized compared to T2W imaging and DWI [5]. On the other hand, the two-compartment (i.e. blood plasma and EES) pharmacokinetic Tofts model is the most commonly used quantitative analysis technique in clinic for diagnosis of PCa. However, the standard Tofts model often could not fit cancer contrast agent concentration curve accurately and there were false positives when using calculated parameters to detect cancer. Based on AIC, the 2TCM was found to be a more likely model representation rather than the commonly used the standard Tofts model. This suggests that the 2TCM is better model to fit and describe the DCE data than the standard Tofts model, and does not over fit the data despite having Our results showed that PCa has higher K trans and k ep than normal prostate tissue, which is consistent with previous studies [24,46]. Similarly, all the parameters calculated from the 2TCM were also higher in cancer than normal tissue. The fast K trans 1 is similar to K trans in the diagnosis of cancer, but K trans 1 has lower values at selected false positive regions where K trans is higher.
As we demonstrated in this study, using the 2TCM is just as simple as using the standard Tofts model, but the results obtained from the 2TCM were richer than the standard Tofts model. There were several limitations to this study. First, our sample size was relatively small. The 2TCM should be tested in a much larger and more diverse group of patients. Second, the dataset used in this study was old and could have low signal-to-noise ratio (SNR) and contrast-to-noise ratio (CNR), which would influence parameter estimation more adversely for the 2TCM, which has more parameters [38]. More studies should be conducted comparing the 2TCM with the standard Tofts model using new datasets acquired with high SNR/CNR and improved temporal/spatial resolutions. Third, there was no automated/perfect alignment of DCE-MRI itself, as well as with histopathology slices. Although it is desirable to follow contrast agent for a longer time, 5 to 10 min [16,47], there could be motion effects to cause low data quality and consequently to have effects on calculated physiological parameters, especially on the 2TCM as well as correlations results shown in Fig. 5 [48]. In future, motion correction of DCE image series and better rad-path correlation should be performed in further validation studies [49]. In addition, special techniques should be used in preparing histopathology slices [50,51]. Fourth, there was no reliable analysis for the drawn ROIs. Since both the standard Tofts model and the 2TCM shared the same ROIs, the result of comparisons between two models should be valid. Fifth, the 2TCM was only compared with the standard Tofts model. In the future, more models should be compared with the 2TCM. Finally, we did not follow the contrast agent for a longer period of time (≤ 8 min). If DCE-MRI data were acquired for a longer period, we believe the advantage of the 2TCM would be more obvious, as the errors in fitting curves using the standard Tofts model would be higher. Nevertheless, more studies are needed to further explore the 2TCM in analysis of prostate DCE-MRI data.

Conclusion
Our study demonstrated that the 2TCM of DCE-MRI is useful for quantitative analysis of prostate DCE-MRI. We compared the 2TCM with the standard Tofts model to demonstrate an advantage when using the 2TCM. As a two-tissue compartment has different kinetic properties and exchanges contrast agent with plasma at distinct rates, the 2TCM provides new diagnostic information for PCa.