Extended Tofts Model¶

The extended Tofts model adds an explicit plasma compartment to the Tofts model. The measured concentration is the sum of contrast agent that has leaked into the extravascular extracellular space and contrast agent still resident in the plasma within the voxel. It is the most widely used model in clinical DCE-MRI.

Equations¶

The impulse response combines a decaying exponential with an instantaneous plasma term:

\[ h(t) = K^{trans} e^{-k_{ep} t} + v_p \, \delta(t), \qquad k_{ep} = \frac{K^{trans}}{v_e} \]

giving

\[ C_t(t) = K^{trans} \int_{0}^{t} C_p(\tau)\, e^{-\frac{K^{trans}}{v_e}(t - \tau)} \,\mathrm{d}\tau \;+\; v_p \, C_p(t) \]

The plasma term is not convolved. This is the model's defining approximation: plasma within the voxel is assumed to equilibrate with the arterial supply instantaneously, so its contribution tracks \(C_p(t)\) with no delay or dispersion.

Parameters¶

Parameter	Symbol	Units	Default initial value	Default bounds
Volume transfer constant	\(K^{trans}\)	min\(^{-1}\)	2 × 10\(^{-4}\)	10\(^{-7}\) to 2
Extravascular extracellular volume fraction	\(v_e\)	—	0.2	0.02 to 1
Plasma volume fraction	\(v_p\)	—	0.02	0.001 to 1

Interpretation¶

As in the Tofts model, \(K^{trans}\) is a composite of flow and permeability and cannot be decomposed into the two without a model that describes plasma transit explicitly. What the plasma term buys is not a separation of flow from permeability, but protection of \(K^{trans}\) and \(v_e\) from contamination by the vascular signal.

The instantaneous plasma assumption holds when the plasma mean transit time is short compared with the temporal resolution of the acquisition. Since plasma transit times are typically one to two seconds and DCE frame times are frequently longer, the assumption is usually reasonable. Where the acquisition does resolve plasma transit, the two-compartment exchange model describes it explicitly.

When to use it¶

This is the appropriate default for well vascularized tissue, and for tumor imaging in particular. It requires only that the acquisition capture the arrival of the bolus with enough fidelity to distinguish the vascular peak from subsequent leakage, and it degrades gracefully towards the Tofts model as \(v_p\) approaches zero.

Parameter identifiability

Estimating \(v_p\) requires that the acquisition resolve the first pass of the bolus. If the baseline is short or the temporal resolution is coarse, \(v_p\) may be poorly determined even where the fit as a whole appears sound. Inspect the reported confidence intervals before interpreting \(v_p\) quantitatively.

Configuration¶

Enable the model with the ex_tofts entry in model_flags. Reported outputs are \(K^{trans}\), \(v_e\), \(v_p\), the sum of squared errors, and the ninety-five percent confidence interval for each parameter. An accelerated implementation is available on GPU and multi-core CPU backends.

References¶

Tofts, P.S., et al. Estimating kinetic parameters from dynamic contrast-enhanced T1-weighted MRI of a diffusable tracer: standardized quantities and symbols. Journal of Magnetic Resonance Imaging, 10(3), 223-232 (1999).

Sourbron, S.P. and Buckley, D.L. On the scope and interpretation of the Tofts models for DCE-MRI. Magnetic Resonance in Medicine, 66(3), 735-745 (2011).