Area Under the Curve¶

The area under the curve (AUC) is a model-free summary of enhancement. It makes no assumption about compartments, permeability or flow, and requires no fitting, so it is defined for every voxel regardless of whether a kinetic model would converge there. It is correspondingly less specific: it aggregates flow, permeability and volume effects into a single number that cannot be decomposed.

Equations¶

Four quantities are reported. Two are integrals of the tissue curves, in concentration and in signal intensity units respectively:

\[ \mathrm{AUC}_c = \int_{t_0}^{t_{end}} C_t(\tau)\,\mathrm{d}\tau, \qquad \mathrm{AUC}_s = \int_{t_0}^{t_{end}} S_t(\tau)\,\mathrm{d}\tau \]

The other two normalize these by the corresponding integrals of the arterial curves:

\[ \mathrm{NAUC}_c = \frac{\mathrm{AUC}_c}{\displaystyle\int_{t_0}^{t_{end}} C_p(\tau)\,\mathrm{d}\tau}, \qquad \mathrm{NAUC}_s = \frac{\mathrm{AUC}_s}{\displaystyle\int_{t_0}^{t_{end}} S_p(\tau)\,\mathrm{d}\tau} \]

Integration is performed by the trapezoidal rule over the acquired timepoints. The lower limit \(t_0\) is the resolved start of the injection, so the pre-contrast baseline is excluded. The signal intensity curves have their own baseline subtracted before integration, so that both the concentration and the signal forms measure enhancement above baseline rather than absolute level.

Reported values¶

Output	Symbol	Basis
AUC conc	\(\mathrm{AUC}_c\)	Tissue concentration curve
AUC sig	\(\mathrm{AUC}_s\)	Tissue signal intensity curve
NAUC conc	\(\mathrm{NAUC}_c\)	Normalized by the arterial concentration integral
NAUC sig	\(\mathrm{NAUC}_s\)	Normalized by the arterial signal integral

No parameters are fitted, so no confidence intervals or sum of squared errors are reported for this model.

Interpretation¶

The concentration form has physical units of mM·min and depends on the accuracy of the relaxivity, the \(T_1\) map and the conversion described in signal to concentration. The signal form is in arbitrary units and is not comparable between scans, subjects or sessions.

The normalized forms are the more useful of the four for comparison across subjects. Dividing by the arterial integral removes the dependence on injected dose, cardiac output and injection rate, which are the largest sources of between-subject variability in the unnormalized measures.

Dependence on the integration window

AUC has no fixed endpoint. Its value depends directly on how long the acquisition ran and where the integration window was placed, so values are comparable only between acquisitions with the same timing. Where AUC is used as an endpoint, fix the window explicitly with the restrict_fit_start_min and restrict_fit_end_min options rather than relying on the acquisitions happening to match.

When to use it¶

AUC is appropriate as a robust descriptive measure where model fitting is unreliable: low signal to noise ratio, coarse temporal resolution, an unreliable arterial input function, or tissue whose enhancement pattern no available model describes well. It is also useful as a sanity check alongside a fitted model, since a region with substantial enhancement but a degenerate \(K^{trans}\) estimate usually indicates a fitting problem rather than a physiological one.

It is not a substitute for a kinetic parameter where one can be estimated reliably, because it confounds the processes a kinetic model separates.

Configuration¶

Enable the measure with the auc entry in model_flags. It requires the tissue and arterial signal intensity curves from the earlier pipeline stages in addition to the concentration curves.

References¶

Evelhoch, J.L. Key factors in the acquisition of contrast kinetic data for oncology. Journal of Magnetic Resonance Imaging, 10(3), 254-259 (1999).

Walker-Samuel, S., et al. Evaluation of response to treatment using DCE-MRI: the relationship between initial area under the gadolinium curve (IAUGC) and quantitative pharmacokinetic analysis. Physics in Medicine and Biology, 51(14), 3593-3602 (2006).