Tofts Model¶

The Tofts model describes tissue as a single extravascular extracellular compartment supplied by plasma, with no explicit intravascular contribution to the measured signal. It is the longest established model in DCE-MRI and remains the reference against which others are compared.

Equations¶

The impulse response is a single decaying exponential:

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

giving the tissue concentration

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

Equivalently, in differential form, the rate of change of tissue concentration is the difference between influx from plasma and efflux back to it:

\[ \frac{\mathrm{d}C_t}{\mathrm{d}t} = K^{trans} C_p(t) - k_{ep} C_t(t) \]

The concentration is zero at the first timepoint by construction, since the baseline re-anchoring described in the signal to concentration conversion sets the pre-contrast concentration to zero.

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

The efflux rate constant \(k_{ep} = K^{trans}/v_e\) is not fitted independently; it is fully determined by the two fitted parameters.

Initial values and bounds are set with the voxel_initial_value_ktrans, voxel_lower_limit_ktrans, voxel_upper_limit_ktrans options and their \(v_e\) equivalents.

Interpretation¶

\(K^{trans}\) is a composite quantity. Its physiological meaning depends on which process limits contrast agent delivery to the tissue:

Where permeability is the limiting factor, that is \(PS \ll F_p\), then \(K^{trans} \approx PS\).
Where flow is the limiting factor, that is \(PS \gg F_p\), then \(K^{trans} \approx F_p\).
Between these regimes \(K^{trans} = E F_p\), where \(E = 1 - e^{-PS/F_p}\) is the extraction fraction.

The Tofts model cannot distinguish these cases. Separating flow from permeability requires the tissue uptake or two-compartment exchange model, and correspondingly higher temporal resolution.

When to use it¶

The Tofts model is appropriate where the intravascular contribution to the measured signal is genuinely negligible: weakly vascularized tissue, or an acquisition whose first timepoints do not resolve the vascular peak. It is well conditioned and converges reliably, which makes it a robust choice for data that cannot support a third parameter.

Bias in well vascularized tissue

Where a plasma compartment does contribute measurably, omitting it biases both fitted parameters. The plasma signal is absorbed into the extravascular compartment, typically inflating \(K^{trans}\) and depressing \(v_e\). In such tissue the extended Tofts model is the more appropriate choice.

Configuration¶

Enable the model with the tofts entry in model_flags. Reported outputs are \(K^{trans}\), \(v_e\), 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).

Tofts, P.S. and Kermode, A.G. Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging. Magnetic Resonance in Medicine, 17(2), 357-367 (1991).