## Quantitative Approaches Using Pharmacokinetic Models

The optimal analysis of DCE-MRI data would be designed to identify specific quantitative physiological parameters which describe the tissue microvascu-lature being observed. We have discussed previously that the distribution of contrast material will be governed by regional blood flow, blood volume, vessel shape and size, endothelial permeability, endothelial surface area, and the size of the EES. In theory an optimal analysis should allow independent assessment of each of these descriptive variables. In practice this is extremely difficult to achieve but many groups have described approaches which try to derive some of these parameters in a principled manner from the changes in signal intensity that occur during contrast passage (Larsson et al. 1990; Brix et al. 1991; Tofts 1996, 1997; Tofts et al. 1995; St LawrencE and Lee 1998). This type of analysis requires the application of a quantitative mathematical model which describes the pharmacokinetics of the contrast agent. The model must include a description of each of the parameters listed above and describe their potential relationships in terms of the effect that they will have on the flux of contrast between compartments and the relative contrast concentrations seen within them.

In order to apply pharmacokinetic models of contrast distribution to imaging-based data the first essential step is to use the signal changes observed in the dynamic acquisition to calculate quantitative parametric images of contrast concentration at each time point. Since the relationship between signal intensity and contrast concentration may be non-linear this adds an additional complication and often requires the measurement of the pre-contrast T1 values for each of the voxels to be studied. There are several ways to achieve this and these are discussed in detail in Chap. 6. All these methods produce a parametric image of T1 which corresponds to the dynamic time course which is to be obtained (Fig. 1.5). Using these quantitative T1 images together with the observed signal change it is possible to transform each of the images within the dynamic time course series to a parametric image of contrast concentration. It is these contrast concentration time course curves for individual voxels that are then used as the substrate for pharmacokinetic models.

A wide range of pharmacokinetic models have been described and applied to the analysis of DCE-MRI data. It should be appreciated that this is not because the pharmacokinetics of the contrast agent are in doubt but rather because the true pharmaco-kinetics are complex and applications of an idealised pharmacokinetic model lead to logistical problems which make the analysis unstable or unreliable. Almost all of the pharmacokinetic analysis techniques use curve fitting methods to estimate the parameters of the pharmacokinetic models being studied. For the non mathematician this can be visualised as an algorithm that changes the parameters you are trying to measure until it finds the best combination of parameters to describe the relationship that has been observed between the vascular input function curve and the tumour voxel curve. The inherent problem of these curve fitting methods is that the more complex the description of the curve and the more unknown parameters that have to be estimated the more likely it is that a range of different solutions can be found. The less specific the solution the less accurate and reliable the estimate of the underlying parameters will be. In basic terms it can therefore be seen that the use of a complex multiparametric model describing all of the physiological features that we have listed above will lead to instabilities in the analysis and increasing number of errors in the estimated parameters. This has led to the development of a range of simplified models which combine the effects of several parameters into one in order to reduce the number of variables used in the curve fit analysis.

Most workers have concentrated on the calculation of the contrast transfer coefficient Ktrans (Larsson et al. 1990; Brix et al. 1991; Tofts 1996, 1997; Tofts et al. 1995; St Lawrence and Lee 1998; Li et al. 2003). Papers describing variations or changes in Ktrans are now commonplace but it is often not appreciated that the meaning of Ktrans will differ depending on the analytical model that has been applied. A simple model such as that described by Tofts and Kermode (1991) will estimate only two parameters, the first of these is the size of the EES (ve) and the second is Ktrans. Where this model is used therefore Ktrans will be affected by flow, endothelial permeability, endothelial surface area product and by the proportional blood volume of the voxel (Fig. 1.9). High values of Ktrans will therefore be seen where there is high flow, high permeability, high capillary surface area or a large proportion of intravascular contrast within the voxel. Although this measurement is highly non-specific it is relatively reproducible, provides a quantitative measurement of microvascular structure and function and has been widely used in many studies.

A further level of complexity in the analysis is introduced in other models which attempt to separate out the effect of the vascular fraction which is one of the main problems with the simple Tofts and Kermode type approach. These models will calculate vascular fraction (Vp), ve and Kians which will now be affected by flow, capillary surface area and endothelial permeability (Fig. 1.9). In practice it is also extremely desirable to separate out the effects of flow from those of capillary endothelial permeability and surface area (St Lawrence and Lee 1998; Li et al. 2003). There are two main reasons for this. Firstly, many pharmaceutical agents currently in development directly affect the angiogenic process. These imaging techniques are widely used in the study of such compounds and there is a specific interest in the changes in vascular endothelial permeability which occur in response to antiang-

iogenic drugs. The use of measurements of Ktrans, which are unable to differentiate these parameters from flow-based effects, is therefore potentially confusing. Secondly, it must be appreciated that measurements of Ktrans with this type of model are essentially flow limited. This is an important concept which can be illustrated as follows. If a tumour has very high capillary endothelial surface area and capillary endothelial permeability then the leakage of contrast from the plasma into the EES will be rapid and large. However, if the blood flow to this tumour is extremely low then this fast leakage out of the vascular system will deplete the contrast in the vascular tree which will not be replenished sufficiently quickly to keep up with the rate of leakage. Under these circumstances the measured Ktrans will not reflect the capillary endothelial permeability or surface area but will reflect only the flow. A measurement of Ktrans which is low may therefore represent an area of relatively high flow and low permeability or could represent low flow and high permeability. The value Ktrans is commonly considered to be synonymous with endothelial permeability and it can be seen that this is clearly not true unless and appropriate model is used. Such a model has been described and is available for analysis but as described above the presence of multiple parameters makes it relatively unstable and highly sensitive to poor signal-to-noise ratio in the underlying data (St Lawrence and Lee 1998).

Fig. 1.9. Maps of Ktrans calculated using a standard two compartment model (left) and a modified first past model designed to exclude erroneous high values at the sight of blood vessels. Note that the high value seen in the left image (arrows) corresponding to a large draining vein adjacent to the tumour is not seen on the right k k fp

Fig. 1.9. Maps of Ktrans calculated using a standard two compartment model (left) and a modified first past model designed to exclude erroneous high values at the sight of blood vessels. Note that the high value seen in the left image (arrows) corresponding to a large draining vein adjacent to the tumour is not seen on the right

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