Improving the accuracy and precision of DCE-MRI tracer kinetic modelling by imposing inter-variable constraints

2 , explore how often those constraints are violated in an example clinical data set, propose a simple way in which those constraints can be imposed and, using simulations, compare the proposed solution to a conventional approach. We show that the proposed approach provides constrained parameter estimates that are more accurate and precise. Eqn. 1 shows the extended Tofts model, which is parameterised in terms of the bulk transfer coefficient K trrans (min -1 ), the relative extravascular extracellular volume ve, and the relative plasma volume vp; Cp(t) is the concentration of contrast agent in the arterial blood plasma at time t (in min), and Ct(t) is the measured contrast agent concentration at time t. The model is subject to the following constraints: K trans ≥ 0, and 0 ≤ ve + vp ≤1 (the total relative volume of a voxel is unity, and the model does not parameterise the intracellular space, vi). Fig. 1 shows the latter constraint: parameter values outside the shaded area are non-physiological. Most fitting software, e.g., Matlab's lsqcurvefit, allows constraints like 0 ≤ ve ≤ 1 and 0 ≤ vp ≤ 1 to be enforced, but not inter-variable constraints. th time point (of N), Ct(ti), and the sum of the model's prediction of those contrast agent concentrations, Ct(ti; Θ'), and the penalty term. The penalty term is a piecewise function that is zero for physiological parameters, and a sigmoid when they are not. The sigmoid has three parameters, c1, c2, & c3 that control its shape; we set these to 500, 7,600, & 6.4, respectively, but have not systematically identified optimal values. The sigmoid was chosen so that the loss function changes smoothly at the boundary of the physiological and non-physiological spaces. (The conventional SSD loss function is identical to Eqn. 2 except it lacks the piecewise term.) We evaluated the modified loss function's ability to enforce the constraint at the boundary by: choosing known parameters (K trans =0.9 min -1 , ve=½, vp=½); generating 1000 corresponding contrast agent concentration time series (using a population averaged 3 Cp(t)); adding noise to the resulting time series; and then estimating the model parameters using the conventional SSD loss function and the method described above. The accuracy and precision of the two methods were compared by computing 95% confidence intervals on the mean and variance of the absolute difference between the estimated and known values of ve and vp. Finally, the mean time required to estimate parameters for a single time series was calculated for each method. All work was performed using Mathematica v7 (Wolfram Research Inc., Champaign, IL). Fig. 2 The topology of the conventional (a) and penalised (b) SSD loss function: see text. Estimated values of ve and vp for the conventional (c) and penalised (d) SSD loss functions. Histograms of ve+vp for the conventional (e) and penalised (f) loss functions. RESULTS Of the ~38,000 vowels studied, ~550 had non-physiological parameter estimates. Figs. 2(a) & 2(b) show the topologies of the conventional SSD and penalised loss functions, respectively, in the absence of noise: optimal values of ve and vp correspond to the functions' minima; Fig. 2(b) illustrates the smooth penalisation of non-physiological estimates. Figs. 2(c) & 2(d) show the results of the evaluation. Fig. 2(c) shows unconstrained fitting results: the variation around the true parameter values is the result of the noise added to the simulated contrast agent concentration time series. Fig. 2(d) demonstrates that the penalty term eliminates non-physiological estimates. Figs. 2(e) & 2(f) show histograms of the quantity ve + vp for the two methods, demonstrating the elimination of non-physiological estimates and the improved accuracy and precision of the constrained fitting method. The table below presents the 95% confidence intervals on the accuracy and precision for the two methods for the parameter ve: the constrained fitting method gave parameter estimates that were both more accurate and precise; results for vp were similar. The mean time required to estimate parameters for one time series was 4.6 s for the unconstrained method and 8.5 s for the constrained method. Accuracy (95% Confidence Intervals) Precision (95% Confidence Intervals) Unconstrained