Two-Compartment Exchange Model¶

The two-compartment exchange model (2CXM) describes tissue as two well mixed compartments, plasma and extravascular extracellular space, with bidirectional exchange between them and finite plasma flow. It is the most complete of the models available in ROCKETSHIP and the only one that estimates flow, permeability and both volume fractions independently.

Equations¶

The impulse response is a biexponential:

\[ h(t) = F_p \left[(1 - A)\, e^{-K_{+} t} + A \, e^{-K_{-} t}\right] \]

so that

\[ C_t(t) = F_p \int_{0}^{t} C_p(\tau) \left[(1 - A)\, e^{-K_{+}(t - \tau)} + A\, e^{-K_{-}(t - \tau)}\right] \mathrm{d}\tau \]

The rate constants and the mixing coefficient derive from three characteristic times. Writing the extraction fraction as \(E = K^{trans}/F_p\):

\[ T_p = \frac{v_p (1 - E)}{F_p}, \qquad T_e = \frac{v_e (1 - E)}{E \, F_p}, \qquad T_b = \frac{v_p}{F_p} \]

where \(T_p\) is the plasma mean transit time, \(T_e\) the extravascular mean transit time, and \(T_b\) the plasma volume divided by flow. The exponential rate constants are the roots

\[ K_{\pm} = \frac{1}{2}\left[ \left(\frac{1}{T_p} + \frac{1}{T_e}\right) \pm \sqrt{\left(\frac{1}{T_p} + \frac{1}{T_e}\right)^{2} - \frac{4}{T_e T_b}} \;\right] \]

and the mixing coefficient is

\[ A = \frac{K_{+} - 1/T_b}{K_{+} - K_{-}} \]

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	\(v_e\)	—	0.15	0.05 to 1
Plasma volume fraction	\(v_p\)	—	0.02	0.001 to 1
Plasma flow	\(F_p\)	min\(^{-1}\)	0.35	10\(^{-4}\) to 20

The permeability surface area product follows from the fitted parameters as

\[ PS = \frac{K^{trans} F_p}{F_p - K^{trans}} \]

Fitting is carried out in an extraction fraction parametrization, with \(E = K^{trans}/F_p\) in place of \(K^{trans}\), and the supplied \(K^{trans}\) bounds are mapped into \(E\) space accordingly. This parametrization is better conditioned, because it keeps the fitted variable bounded in \((0, 1)\) and decouples it from the flow estimate.

Model-specific defaults

The 2CXM uses its own initial values and bounds, named with a _2cxm suffix, for example voxel_lower_limit_ve_2cxm. It also uses higher iteration limits than the simpler models, reflecting the larger parameter space.

Temporal resolution¶

The 2CXM is the model most sensitive to temporal resolution. Plasma mean transit times are routinely one to two seconds, which is shorter than a typical DCE frame interval, so evaluating the model directly on the acquired timebase substantially under-resolves the plasma exponential.

ROCKETSHIP therefore evaluates the model on a dense internal grid. Following the OSIPI convention, the arterial input function is interpolated to a 0.1 second grid, the forward model is evaluated there, and the resulting curve is sampled back at the acquired timepoints for comparison with the data. This removes the discretization error introduced by evaluating on a coarse grid.

Interpolation is not information

Dense evaluation removes discretization error; it cannot recover detail the acquisition never recorded. Where the frame interval is long relative to the plasma transit time, the residual error from an unresolved first pass dominates every other error source in this model, and \(F_p\) and \(v_p\) should be treated as poorly determined. Confidence intervals are computed with degrees of freedom based on the number of acquired samples, not the number of points on the interpolation grid.

When to use it¶

Use the 2CXM where the scientific question requires flow and permeability separately, and where the acquisition was designed to support it: high temporal resolution through the first pass, good signal to noise ratio, and a duration long enough for backflux to become measurable. Given data that meet those conditions it is the most informative model available.

Given data that do not, its extra parameters will be poorly constrained, and one of the simpler models will produce more reproducible results.

Configuration¶

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

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).

Brix, G., et al. Microcirculation and microvasculature in breast tumors: pharmacokinetic analysis of dynamic MR image series. Magnetic Resonance in Medicine, 52(2), 420-429 (2004).