Tissue Uptake Model¶

The tissue uptake model describes a plasma compartment of finite transit time feeding an extravascular space from which contrast agent does not return. It sits between the Patlak model, which treats plasma transit as instantaneous, and the two-compartment exchange model, which allows backflux. By retaining plasma transit while discarding backflux it separates flow from permeability using one fewer parameter than the full exchange model.

Equations¶

The impulse response is a plasma exponential plus a constant retention term. Writing the extraction fraction as \(E = K^{trans}/F_p\),

\[ h(t) = F_p \left[ E + (1 - E)\, e^{-t/T_p} \right] \]

Expanding gives the form ROCKETSHIP evaluates:

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

The first term is the irreversibly retained contrast agent, identical to the leakage term of the Patlak model. The second describes contrast agent passing through the plasma compartment with mean transit time \(T_p\).

Parameters¶

Parameter	Symbol	Units	Default initial value	Default bounds
Volume transfer constant	\(K^{trans}\)	min\(^{-1}\)	2 × 10\(^{-4}\)	10\(^{-7}\) to 2
Plasma flow	\(F_p\)	min\(^{-1}\)	0.35	10\(^{-4}\) to 20
Plasma mean transit time	\(T_p\)	min	0.12	0 to 1.5

Two further quantities follow from the fitted parameters:

\[ v_p = F_p \, T_p, \qquad PS = \frac{K^{trans} F_p}{F_p - K^{trans}} \]

The permeability surface area product follows from inverting \(K^{trans} = E F_p\) with \(E = PS/(F_p + PS)\). It is poorly determined when \(K^{trans}\) approaches \(F_p\), since the denominator then approaches zero.

Model-specific defaults

The initial values and bounds above differ from those used by the Tofts family. The tissue uptake model has its own settings, named with a _tissue_uptake suffix, for example voxel_initial_value_fp_tissue_uptake. Notice in particular that \(T_p\) is bounded above at 1.5 minutes; plasma transit times far above that are not physiological and generally indicate the fit has drifted to a degenerate solution.

When to use it¶

The tissue uptake model is appropriate where flow and permeability need to be distinguished but the acquisition is too short, or the tissue too impermeable, for backflux to be observable. It requires temporal resolution sufficient to resolve the first pass of the bolus, since \(T_p\) is estimated from the shape of that passage.

Where backflux is measurable, the model will absorb it into the other parameters and bias them. Comparing against the two-compartment exchange model on the same data is the direct test of whether the irreversibility assumption is defensible.

Configuration¶

Enable the model with the tissue_uptake entry in model_flags. Reported outputs are \(K^{trans}\), \(F_p\), \(T_p\), the sum of squared errors, and the ninety-five percent confidence interval for each parameter. Fitting is performed internally in minutes with rate constants per minute, and returned parameters are converted to match the units of the supplied time vector. An accelerated implementation is available on GPU and multi-core CPU backends.

References¶

Sourbron, S.P. and Buckley, D.L. Tracer kinetic modelling in MRI: estimating perfusion and capillary permeability. Physics in Medicine and Biology, 57(2), R1-R33 (2012).

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).