**tab below). The first of these assumes that the plasma component (**

*Advanced Discussion**vp*) is small and makes little contribution to the total tissue MR signal. The second (revised) formulation of the Tofts model includes this plasma contribution, and (although adding complexity) is probably more appropriate for representing highly vascular lesions such as malignant tumors. The meanings and practical use of the Tofts DCE parameters will be explored in the next several Q&A's.

### Advanced Discussion (show/hide)»

**Pharmacokinetic analysis using the Tofts Model**

The original Tofts and Kermode (1991) model assumes that the equilibrium concentration of gadolinium in whole tissue, *C _{t}*, is driven by simple passive diffusion based on its concentration difference between plasma (

*C*) and the tissue extravascular extracellular space (

_{p}*C*). Since gadolinium contrast does not enter cells,

_{e}*C*is higher than

_{e}*C*by the factor (

_{t}*1/v*), where

_{e}*v*is the dimensionless volume fraction (0<

_{e}*v*<1) of the tissue extravascular extracellular space (EES). The defining Tofts-Kermode equation is therefore

_{e}where *K ^{trans}* 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 *k _{ep}* defined as

*K*, representing the (reverse) transfer rate for contrast from the EES back into plasma. Note that

^{trans}/v_{e}*K*,

^{trans}*v*, and

_{e}*k*are interrelated and thus not all independent parameters. If two are known, the third can be computed.

_{ep}In a DCE study we are able to determine the time-dependent gadolinium concentration in plasma, *C _{p}*, 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,

*C*. We do not know either

_{t}*K*or

^{trans}*v*, however. These are potentially important physiological parameters of interest that can be estimated by fitting the measured data to the Tofts model.

_{e}*v*is very small and disregarded in the analysis. The simple TK model thus has only a single compartment with two free parameters, whose differential equation can solved by integration to yield

_{p}Using nonlinear least squares estimation on a voxel-by-voxel basis, *K ^{trans}* and

*v*can then be computed.

_{e}The above formulation assumes that the intravascular contribution to total tissue gadolinium concentration is negligible and that *C _{t}* 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 *v _{p}*, the volume fraction of total tissue containing plasma. Like its counterpart

*v*,

_{e}*v*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 (

_{p}*v*)

_{p}C_{p}**Other Tissue Models for DCE Imaging**

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 *K _{12}* and

*K*, respectively, corresponding roughly to

_{21}*K*and

^{trans}*k*of the Tofts model. However, the Brix model explicitly separated flow from permeability effects (which are confounded in the

_{ep}*K*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.

^{trans}*is assumed to exist along the length of each capillary. This requires incorporation of transit time into the model but allows the potential of extracting blood flow information directly from the analysis.*

__gradient__The Patlak model has recently become popular for the analysis of subtle areas of increased blood-brain-barrier permeability (which may be a clue to the pathophysiology of Alzheimer disease). Like the others, the Patlak model describes a highly perfused two-compartment model. The major difference is that contrast diffusion from the blood plasma to the extravascular extracellular space (EES) is considered to be unidirectional. The tracer concentration in tissue *C _{t}(t)* is given by

*C*

_{t}(t) = v_{p}C_{p}(t) + K^{Trans}∫C_{p}(τ) dτA linear graphical analysis of the ratio *C _{t}(t)/C_{p}(t) regressed against ∫C_{p}(τ)dτ/C_{p}(t) produces a straight line with slope K^{Trans} and y-intercept v_{p}.*

**References**

Brix G, Semmler W, Port R, et al. Pharmacokinetic parameters in CNS Gd-DTPA enhanced MR imaging. J Comput Assist Tomogr 1991; 15:621-628.

Heye AK, Culling RD, del C Valdés Hernández M, et al. Assessment of blood-brain barrier disruption using dynamic contrast-enhanced MRI. A systematic review. NeuroImage: Clinical 2014; 6:262-274. [DOI LINK]

Manning C, Stringer M, Dickie B, et al. Sources of systematic error in DCE-MRI estimation of low-level blood-brain barrier leakage. Magn Reson Med 2021; (in press) [DOI LINK] (Casts doubt on accuracy of Patlak model for measuring BBB disruption)

Tofts PS. Modeling tracer kinetics in dynamic Gd-DTPA MR imaging. J Magn Reson Imaging 1997; 7:91-101.

Tofts PS, Brix, G, Buckley DL, et al. Estimating kinetic parameters from dynamic contrast-enhanced T1-weighted MRI of a diffusable tracer: standardized quantities and symbols. J Magn Reson Imaging 1999; 10:223-232.

Tofts PS, Kermode AG. Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging. 1. Fundamental concepts. Magn Reson Med 1991; 17:357-367

Zaharchuk G. Theoretical basis of hemodynamic MR imaging techniques to measure cerebral blood volume, cerebral blood flow, and permeability. AJNR Am J Neuroradiol 2007; 28:1850-8.

**Related Questions**

*Is Ktrans the same as permeability?***How do calculated DCE parameters relate to patterns of enhancement we see on clinical images?**