Extended graphical model for analysis of dynamic contrast-enhanced MRI

trans within acquisitions of short duration. By exploring the mathematical relationship between the modified Kety/Tofts model and the Patlak model, we found that the latter is a linear approximation based on an infinite expansion of the former. A higher order approximation based on this expansion is found to correct for reflux, while retaining the central advantages of the Patlak model - it is linear in the parameters to be estimated, leads to a simple graphical interpretation, and has a stable fitting procedure. Simulated and in vivo experimental data were used to investigate whether the new "extended graphical model" can outperform the Patlak and modified Kety/Tofts models and under what circumstances. Methods and Materials Theory With the assumption of bi-directional exchange of contrast agent between plasma and extravascular extracellular space (EES), the modified Kety/Tofts model is given by (1): ∫ − − + = t ep p trans p p t d t k + = t p trans p p t d C K t C v t C 0 ) ( ) ( ) ( τ τ . (2) If we expand Eq. (1) using integration by parts, we observe that the Patlak model (2) accounts for the first two terms in the expansion: ∑ ∫∫ − + ∫ + = ∞ = = + + n i t i i p i ep trans t p trans p p t d d C k K d C K t C v t C i 1 0 1 0 1 1 0 ) ( ) ( ) ( ) ( ) ( τ τ τ τ τ τ L L . (3) Thus, the infinite sum term in Eq. (3) corrects for contrast agent reflux. As with other expansions, a more accurate approximation can be expected if more additional terms are included from Eq. (3). In particular, if only the first order correction term is chosen, the resulting model is our linear extended graphical model, given by: ∫∫ − ∫ + = t p trans ep t p trans p p t d d C K τ τ τ τ τ . (4) Simulation To test this model, we simulated an arterial input function using an existing model (4): Cp(t)=(A1+A2)(t/t0) when t<7s Cp(t)=A1exp(-K1(t-t0))+A2exp(-K2(t-t0)) when t≥7s (5) where A1=3.4mM, A2=1.81mM, K1=2.7min -1 , K2=0.09min -1 . Tissue uptake curves were generated with Eq. (1) using kep=0.3 min -1 , ve =0.5, vp =0.05 and sampling interval =3s. The fitting procedure of each model was then repeated 5000 times with 5% random noise added to each sample (standard deviation is 5% of the maximum Cp value). The mean values and root mean square error (RMSE) of the K trans estimates were computed for duration times (TDur) ranging from 60s to 300s with a step of 6s. In vivo DCE MRI of carotid artery To evaluate the performance of the model in vivo, the dependence of parameter estimation on TDur was also tested with DCE-MRI of carotid plaque. We obtained 8 MRI data sets from subjects with advanced carotid artery disease consisting of axial 2D spoiled gradient recalled echo images (8 slices, slice thickness 2 mm, FOV 140*140 mm, Matrix 256*256, TR= 95ms, TE=4.6ms). Coincident with the second image in the sequence, 0.1 mmol/kg of a gadolinium-based contrast agent was injected at a rate of 2 ml/s by a power injector. A spatial saturation band was applied to induce a T1-dependent blood signal on images prior to contrast bolus arrival. After bolus arrival, images were acquired at 20 time points separated by a repetition interval of 13s. These data were analyzed using the method described by Kerwin et al. (5) to obtain average measurements of K trans