Patlak Model¶

The Patlak model describes irreversible uptake: contrast agent leaks from plasma into the extravascular extracellular space and, over the duration of the acquisition, does not return. Removing the backflux term reduces the model to two parameters and makes it linear in those parameters, which gives it stability that the reversible models cannot match.

Equations¶

The impulse response is a constant leakage term plus an instantaneous plasma term:

\[ h(t) = K^{trans} + v_p \, \delta(t) \]

giving

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

This is the limiting case of the extended Tofts model as \(k_{ep} \to 0\), that is, as \(v_e\) becomes large enough that efflux is negligible over the observation window. Because \(v_e\) no longer appears, it cannot be estimated.

Linear form¶

Dividing through by \(C_p(t)\) makes the model linear:

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

Plotting \(C_t/C_p\) against the ratio of the plasma integral to \(C_p\), known as the Patlak plot, yields a straight line whose slope is \(K^{trans}\) and whose intercept is \(v_p\). A linear least squares solution is available and is used by default, as it requires no initial values, cannot fail to converge, and returns the global optimum directly.

Departure from linearity

Curvature in the Patlak plot indicates that backflux is not in fact negligible. Where the plot bends at later timepoints, either restrict the analysis to the early linear portion or move to a reversible model. The fitting window can be restricted with the restrict_fit_start_min and restrict_fit_end_min options.

Parameters¶

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

When to use it¶

The Patlak model is the standard choice for low permeability tissue, where \(K^{trans}\) is small enough that backflux does not become measurable within a typical acquisition. Its principal application is the quantification of blood–brain barrier integrity, where the transfer constants of interest are two to three orders of magnitude below those seen in tumors and where the stability of a linear estimator is decisive.

It is also useful as a robust fallback for any tissue in which the nonlinear models converge unreliably, at the cost of not estimating \(v_e\).

Configuration¶

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

Patlak, C.S., et al. Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Journal of Cerebral Blood Flow and Metabolism, 3(1), 1-7 (1983).

Patlak, C.S. and Blasberg, R.G. Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Generalizations. Journal of Cerebral Blood Flow and Metabolism, 5(4), 584-590 (1985).