The original Tofts and Kermode (1991) model assumed that the equilibrium concentration of gadolinium in whole tissue, Ct, is driven by simple passive diffusion based on its concentration difference between that in plasma and in the tissue extravascular extracellular space (EES). Since gadolinium contrast does not enter cells, its actual concentration in the EES, Ce, is higher than that Ct by the factor (1/ve), where ve is the dimensionless volume fraction (0<ve<1) of the EES for tissue. The defining Tofts-Kermode equation is therefore
where Ktrans is the mass transfer influx constant measured in units of min−1. Note that the right side of the equation can be re-written as
with kep defined as Ktrans/ve, representing the transfer rate for contrast from the EES back into plasma. Note that Ktrans, ve, and kep are interrelated and thus not all independent parameters. If two are known, the third can be computed.
In a DCE study we are able to determine the time-dependent gadolinium concentration in plasma, Cp, using either an arterial input function (AIF) or estimating it assuming bi-exponential decay from an initial value resulting from renal clearance and whole body extracellular space distribution. Likewise, by measuring the signal intensities of tissue we can determine the tissue gadolinium concentration, Ct. We do not know either Ktrans or ve, however. These are potentially important physiological parameters of interest that can be estimated by fitting the measured data to the Tofts model.
Using nonlinear least squares estimation on a voxel-by-voxel basis, Ktrans and ve can then be computed.
The above formulation assumes that the intravascular contribution to total tissue gadolinium concentration is negligible and that Ct arises purely from gadolinium in the EES. Although this may be true for lesions such as multiple sclerosis plaques and low-grade tumors, it is certainly not the case for many malignant tumors and other highly vascular lesions.
A common modification of the original Tofts model takes this vascular contribution into account, introducing a new parameter vp, the volume fraction of total tissue containing plasma. Like its counterpart ve, vp is dimensionless as it represents blood plasma volume per unit volume of tissue. This expanded Tofts model now has 2 compartments and 3 parameters, admitting to the following solution including a new term (vpCp)
A pharmacokinetic model was developed by Brix and colleagues in Germany at the same time Tofts and Kermode were working in England. The original Brix formulation, now expanded, was an open-exchange model with central (vascular/plasma) and peripheral (tissue extracellular space) compartments. Forward and reverse gadolinium exchange rates between the compartments were denoted K12 and K21, respectively, corresponding roughly to Ktrans and kep of the Tofts model. However, the Brix model explicitly separated flow from permeability effects (which are confounded in the Ktrans concept). The original Brix model also used a rather inaccurate method of estimating gadolinium concentration from percent signal changes, but had the advantage that it did not require precontrast tissue T1 mapping nor measurement of an arterial input function.
Advanced Discussion (show/hide)»
What causes magnetism?