Parameter estimation in dynamic contrast-enhanced MRI (DCE MRI) is usually performed by non-linear least square fitting of a tracer-kinetic model to a measured concentration-time curve. The two-compartment exchange model (2CXM) describes the compartments „plasma“ and „interstitial volume“ with fractional volume v_p and v_e and their exchange in terms of plasma flow F_p and permeability-surface area product PS. The model function is defined by a system of two coupled differential equations^{1, 2}. A closed-form analytic solution as convolution of a biexponential residue function with the arterial concentration (AIF derives from indicator dilution theory. Several studies on model feasibility and comparison between different models have been performed^{3, 4, 5}, however, little attention was given to technical aspects of the fitting routine⁶ . Parameters are usually estimated by fitting the model function to measured data using standard non-linear least squares methods. The residuals can be calculated by either numeric convolution of the closed-form solution^₃ or by directly solving the differential equations² (e.g. using the Runge Kutta algorithm). We studied both strategies in terms of accuracy, robustness and computation speed by fitting simulated concentration time curves in image space, and quantitative and qualitative evaluation of the resulting parameter maps.

Time-resolved concentration images with gaussian noise and contrast-to-noise ratio CNR=300 (expressed as ratio between the maximum of the AIF and the standard deviation of the noise) were simulated, using a measured arterial input function resulting from a double-bolus injection (temporal resolution 2.11s, 169 timepoints, Ref. 7). All curves were fitted with an in-house written software module developed within the Medical Imaging and Interaction Toolkit MITK⁸, which uses the levenberg-marquard optimizer implementation of the visual numeric library (VNL). The differential equations were solved using the boost implementation of the Runge-Kutta method. For the convolution approach, the arterial input function was interpolated linearly. Parameter constraints were applied to assure model consistency : 0<v_p,v_e< 1, v_p+v_e<1, 0<F_p<100 ml/min/100ml, 0<PS< 100 ml/min/100ml. The optimizer configuration and start parameters were kept constant for all fits.

Accuracy and robustness were evaluated on concentration-curves using 5 different parameter combinations (Figure 1a). For each of these combinations, a homogeneous 30x30 pixel concentration image of 169 time points with random noise was simulated. The resulting fitted parameters were compared to the input value by calculating the bias, e.g. the difference between input value and the mean of the fitting result, and the variance of the mean.

For illustration a 50x50 pixel concentration image of 169 timepoints with different lettering for F_p and v_p and constant PS, v_e was fitted with both approaches.

The convolution and the differential equation approach yielded similar results for a given parameter combination (Figure 2). The mean estimates of F_p compared to the true value showed no considerable deviation except for low F_p. (Figure 3). For the reference situation as well as low PS and low v_e the parameter estimates of PS and v_p were close to the original values. The fitting result for v_e yielded a poor estimate with many outliers. The mean and median parameter estimates differed strongly. For low values of F_p, both fitting strategies yielded poor fit stability, large uncertainties and many outliers. The convolution approach excelled in terms of computational time (Figure 4).

In the visual approach (Figure 5) the text „ISMRM“ and „Singapore“ from the input parameter maps could be reproduced by both fitting routines. However, the value of v_p showed interference with parameter estimate of PS and v_e as fragments of the textline „Singapore“ could be observed in the corresponding maps. PS showed good fit stability in the rest of the image, whereas v_e again presented poor fitting results.

The convolution and differential equation definition of the model function of the 2CXM yield very similar results in terms of accuracy and precision. Fitting with the closed form solution is superior in computational time. Both approaches are limited in accuracy for situations with low blood flow. The model parameter v_e shows great instability and little reliability in all cases. This is expected to improve considerably by usage of longer total acquisition time. More parameter combinations need to be evaluated for a detailed error estimation. Further studies should include variations in the starting parameters as well as different configurations for the optimizer.