態對比增強磁共振影像技術在評估腫瘤特性中扮演著重要的角色，利用一些動力學參數，例如體積轉移常數和速率常數，可以達到定量的分析。為了獲得這些動力學參數，研究者通常利用非線性最小平方法擬合時間濃度曲線到動力學模型上。然而此方法是十分耗時的。先前的研究證明，對於修改後的Tofts and Kermode 模型，線性最小平方法可能是更準確且快速的方法。本研究我們利用電腦模擬評估線性最小平方法和非線性最小平方法在解Tofts and Kermode時動力學參數的正確性。此外我們也使用這兩種方法分析八個有腦腫瘤的病人數據。當訊雜比較低時，利用線性最小平方方法得到的參數將會比非線性最小平方方法準確。比起非線性最小平方法，利用線性最小平方法得到的參數的準確性和精密度較不易受體積轉移和速率常數變動影響。此外線性最小平方法的計算速度比非線性最小平方法快約十七倍。在實際臨床應用時，對於四個多形惡性神經膠質瘤的病人，線性最小平方法和非線性最小平方法在評估動力學參數上有顯著的差異。

Dynamic contrast enhanced (DCE) MRI plays an important role for quantitative assessment of tumor characterization using the kinetic parameters, such as volume transfer constant (Ktrans) and rate constant (kep). Nonlinear least square (NLSQ) method is often used to fit concentration time curves to kinetics models for obtaining kinetic parameters. However, one disadvantage of this method is time-consuming. The previous study demonstrated that liner least-squares (LLSQ) method may be a more accurate and fast method than NLSQ method in solving modified Tofts and Kermode model. In this study, we performed computer simulations to assess the accuracy of estimated the kinetic parameters for LLSQ and NLSQ methods in solving Tofts and Kermode model. In addition, the two methods were used to analyze the data from eight patients with brain tumors. At lower signal-to-noise (SNR), the accuracy of parameters estimated with LLSQ method was better than NLSQ method. Compare with NLSQ method, effect of varying Ktrans and kep on accuracy and precision of the kinetic parameters with LLSQ method was smaller. Besides, the calculation velocity of LLSQ method was seventeen times faster than NLSQ method. In clinical application, there were significant differences between LLSQ and NLSQ estimated kinetic parameters on the all patients with glioblastoma multiforme (GBM).

1.Zhu, X.P., et al., Quantification of endothelial permeability, leakage space, and blood volume in brain tumors using combined T1 and T2* contrast-enhanced dynamic MR imaging. J Magn Reson Imaging, 2000. 11(6): p. 575-85.
2.Eliat, P.A., et al., Magnetic resonance imaging contrast-enhanced relaxometry of breast tumors: an MRI multicenter investigation concerning 100 patients. Magn Reson Imaging, 2004. 22(4): p. 475-81.
3.Furman-Haran, E., et al., Magnetic resonance imaging reveals functional diversity of the vasculature in benign and malignant breast lesions. Cancer, 2005. 104(4): p. 708-18.
4.Alonzi, R., A.R. Padhani, and C. Allen, Dynamic contrast enhanced MRI in prostate cancer. Eur J Radiol, 2007. 63(3): p. 335-50.
5.Harry, V.N., et al., Use of new imaging techniques to predict tumour response to therapy. Lancet Oncol, 2010. 11: p. 92–102.
6.Armitage, P.A., et al., Quantitative assessment of intracranial tumor response to dexamethasone using diffusion, perfusion and permeability magnetic resonance imaging. Magn Reson Imaging, 2007. 25(3): p. 303-10.
7.Durukan, A., et al., Post-ischemic blood-brain barrier leakage in rats: one-week follow-up by MRI. Brain Res, 2009. 1280: p. 158-65.
8.Vlachos, F., Y.S. Tung, and E.E. Konofagou, Permeability assessment of the focused ultrasound-induced blood-brain barrier opening using dynamic contrast-enhanced MRI. Phys Med Biol, 2010. 55(18): p. 5451-66.
9.Padhani, A.R., Dynamic contrast-enhanced MRI in clinical oncology: current status and future directions. J Magn Reson Imaging, 2002. 16(4): p. 407-22.
10.Tofts, P.S. and A.G. Kermode, Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging. 1. Fundamental concepts. Magn Reson Med, 1991. 17(2): p. 357-67.
11.Tofts, P.S., Modeling tracer kinetics in dynamic Gd-DTPA MR imaging. J Magn Reson Imaging, 1997. 7(1): p. 91-101.
12.Murase, K., Efficient method for calculating kinetic parameters using T1-weighted dynamic contrast-enhanced magnetic resonance imaging. Magn Reson Med, 2004. 51(4): p. 858-62.
13.Coleman, T.F. and Y.Y. Li, An interior trust region approach for nonlinear minimization subject to bounds. Siam Journal on Optimization, 1996. 6(2): p. 418-445.
14.Sun, W. and Y.X. Yuan, Optimization theory and methods. springer, 2006. Chapter 6: p. 303
15.Lawson, C.L. and R.J. Hanson, Solving Least Squares Problems. Prentice-Hall, 1974. Chapter 23: p. 161.
16.Singh, A., et al., Quantification of physiological and hemodynamic indices using T(1) dynamic contrast-enhanced MRI in intracranial mass lesions. J Magn Reson Imaging, 2007. 26(4): p. 871-80.
17.Medved, M., et al., Semiquantitative analysis of dynamic contrast enhanced MRI in cancer patients: Variability and changes in tumor tissue over time. J Magn Reson Imaging, 2004. 20(1): p. 122-8.
18.Larsson, H.B., et al., Measurement of brain perfusion, blood volume, and blood-brain barrier permeability, using dynamic contrast-enhanced T(1)-weighted MRI at 3 tesla. Magn Reson Med, 2009. 62(5): p. 1270-81.
19.Ethofer, T., et al., Comparison of longitudinal metabolite relaxation times in different regions of the human brain at 1.5 and 3 Tesla. Magn Reson Med, 2003. 50(6): p. 1296-301.
20.Li, K.L., et al., Improved 3D quantitative mapping of blood volume and endothelial permeability in brain tumors. J Magn Reson Imaging, 2000. 12(2): p. 347-57.
21.Buckley, D.L., Uncertainty in the analysis of tracer kinetics using dynamic contrast-enhanced T1-weighted MRI. Magn Reson Med, 2002. 47(3): p. 601-6.
22.Calamante, F., D.G. Gadian, and A. Connelly, Delay and dispersion effects in dynamic susceptibility contrast MRI: simulations using singular value decomposition. Magn Reson Med, 2000. 44(3): p. 466-73.
23.Yankeelov, T.E., et al., Quantitative pharmacokinetic analysis of DCE-MRI data without an arterial input function: a reference region model. Magn Reson Imaging, 2005. 23(4): p. 519-29.
24.St Lawrence, K.S. and T.Y. Lee, An adiabatic approximation to the tissue homogeneity model for water exchange in the brain: I. Theoretical derivation. J Cereb Blood Flow Metab, 1998. 18(12): p. 1365-77.