Robust and efficient pharmacokinetic parameter non-linear least squares estimation for dynamic contrast enhanced MRI of the prostate

Abstract

Purpose: To describe an efficient numerical optimization technique using non-linear least squares to estimate perfusion parameters for the Tofts and extended Tofts models from dynamic contrast enhanced (DCE) MRI data and apply the technique to prostate cancer.

Methods: Parameters were estimated by fitting the two Tofts-based perfusion models to the acquired data via non-linear least squares. We apply Variable Projection (VP) to convert the fitting problem from a multi-dimensional to a one-dimensional line search to improve computational efficiency and robustness. Using simulation and DCE-MRI studies in twenty patients with suspected prostate cancer, the VP-based solver was compared against the traditional Levenberg-Marquardt (LM) strategy for accuracy, noise amplification, robustness to converge, and computation time.

Results: The simulation demonstrated that VP and LM were both accurate in that the medians closely matched assumed values across typical signal to noise ratio (SNR) levels for both Tofts models. VP and LM showed similar noise sensitivity. Studies using the patient data showed that the VP method reliably converged and matched results from LM with approximate 3× and 2× reductions in computation time for the standard (two-parameter) and extended (three-parameter) Tofts models. While LM failed to converge in 14% of the patient data, VP converged in the ideal 100%.

Conclusion: The VP-based method for non-linear least squares estimation of perfusion parameters for prostate MRI is equivalent in accuracy and robustness to noise, while being more reliably (100%) convergent and computationally about 3× (TM) and 2× (ETM) faster than the LM-based method.

Keywords: Dynamic-contrast-enhanced magnetic resonance imaging; Multi-parametric magnetic resonance imaging; Perfusion; Pharmacokinetic modeling; Prostate cancer.

Figure 1

(a) Axial slice of the pelvis (Patient #1) through the prostate showing the bilateral femoral arteries, prostate (within yellow oval) and endo-rectal coil (yellow circle, not active for DCE-MRI sequence). The arteries are enhanced at time 38.2 s post-injection. (b) Graph of the average (bold curve) arterial input function selected from the left deep femoral artery as shown. The average is from nine (3×3) individual voxels (non-bold curves). (c–e) Source DCE-MRI images demonstrate an area of hyperenhancement of cancerous tissue in the right peripheral zone at the apex compared to the non-cancerous tissue in the contralateral side at times 44.6 s (c), 51.0 s (d), and 127.5 s (e). Red and blue points show the representative voxels in the cancer and non-cancerous regions respectively. These are used in Figure 2.

Figure 2

Cost function (J) vs. k_ep shown for one voxel in (a) the cancerous region (red line) and one voxel in (b) the non-cancerous region (blue line) in the same patient shown in Figure 1. Black dots correspond to points selected each iteration by Golden Section Search (GSS) method. (c) Estimated perfusion (dashed line) vs. acquired perfusion (solid line) for the two representative voxels shown in Figure 1c–e. The optimum k_ep values (cost function minimum) found via GSS method and the subsequently computed K^trans are indicated.

Figure 3

The results of the numeric simulation using Monte Carlo simulation. K^trans (top) and k_ep (bottom) box-and-whisker plots are depicted for VP and LM for six SNR levels (10, 15, 20, 25, 30, 35). The boxes depict the median, first and third quartiles and the whiskers mark the 10^th and 90^th percentiles across 500 Monte Carlo samples for the specific combination of K^trans =0.2 min⁻¹, k_ep =1.0 min⁻¹, v_p =0.01. Figure 3a, b show the results for Tofts and extended Tofts model respectively.

Figure 4

(a) Absolute computation time for convergence for each point (total of ten points for each patient) shown for VP and LM using the Tofts model. The VP-based method converged for all 100 points. At the top of each plot, the number of points (out of ten) that failed for LM method is noted. The failed points are not shown in the plots. (b) the equivalent plots for the extended Tofts model. (c) Absolute computation time for convergence for all the analyzed points in (a, b). 15 out of 100 points failed to converge for the LM method using the Tofts model and 14 failed using the extended Tofts model. Box-and-whisker plots depict median (dashed line), mean (solid line), first and third quartiles, and 10^th and 90^th percentiles.

Figure 4

(a) Absolute computation time for convergence for each point (total of ten points for each patient) shown for VP and LM using the Tofts model. The VP-based method converged for all 100 points. At the top of each plot, the number of points (out of ten) that failed for LM method is noted. The failed points are not shown in the plots. (b) the equivalent plots for the extended Tofts model. (c) Absolute computation time for convergence for all the analyzed points in (a, b). 15 out of 100 points failed to converge for the LM method using the Tofts model and 14 failed using the extended Tofts model. Box-and-whisker plots depict median (dashed line), mean (solid line), first and third quartiles, and 10^th and 90^th percentiles.

Figure 5

(a) The K^trans range estimated for normal and cancer region in twenty patients for Tofts and extended Tofts model via the VARPRO technique. (b) The ROC curve based on K^trans, for Tofts and extended Tofts model. The equivalent information for k_ep is shown in (c, d). The cut-off values (solid dot on the ROC curve), sensitivity, specificity, and the AUC are noted on the figures (b, d) for TM and ETM.

Figure 6

66-year-old male (Patient #1) with PSA of 8.7 ng/mL. Prostate biopsy revealed adenocarcinoma, Gleason 4 + 3 = 7 on the right and benign tissue on the left.: (a) T₂-weighted spin echo (T2SE) image, and (b) apparent diffusion coefficient (ADC) map demonstrate an ill-defined heterogeneous tumor of hypointensity in the right posterolateral peripheral zone at the mid gland (b, arrow). (c–f) Corresponding perfusion abnormality is comparably shown on K^trans and k_ep maps from VARPRO (VP) using Tofts model (TM) and extended Tofts model (ETM). The 10 representative points used for computation speed comparisons, are shown in the major tumor and the contralateral side in (c) and (d).

Figure 7

64-year-old male (Patient #2) with PSA of 11 ng/mL.: (a) T₂ signal abnormalities and (b) diffusion restriction (ADC map) in the right posterior peripheral zone (b, arrow). (c, e) K^trans and (d, f) k_ep maps from VARPRO (VP) using Tofts model (TM) and extended Tofts model (ETM), reveal hyperenhancing lesion in the right posterolateral peripheral zone. The 10 representative points used for computation speed comparisons are shown in the major tumor and the contralateral side in (c) and (d). Prostate biopsy yielded adenocarcinoma, Gleason score 4+3=7 disease on the right.