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where Ce is the concentration of agent in ve, Cp is the concentration of agent in Vp, and Ktrans is the volume transfer constant between Vp and ve (ToFts et al. 1999). If the delivery of contrast medium to the tissue is ample (meaning that the rate of extraction of contrast agent via the leaky capillary wall is small compared with the rate of replenishment via perfusion), Ktrans is equal to the product of the capillary wall permeability and capillary wall surface area per unit volume, PSp. PS is the permeability surface area product per unit mass of tissue (ml g1 min1), and p is the tissue density (g ml1). However, if the delivery of the contrast medium to a tissue is insufficient, blood perfusion will be the dominant factor determining contrast agent kinetics, and Ktrans approximates Fp(1-Hct), where F is blood flow [units ml blood (g tissue)1 min1]. Thus in regions with a poor blood supply, low transfer coefficient values may be observed despite high intrinsic vessel permeability (Su et al. 1994; DeGANi et al. 1997). Consideration of a general mixed perfusion- and permeability-limited regime leads to Ktrans being equal to EFp(1-Hct) (ToFts et al. 1999), where E is the extraction fraction of the tracer (i.e. the fraction of tracer that is extracted from vp into ve in a single capillary transit) (Crone 1963):

The relationships described above form the basis of the models used to describe contrast agent kinetics by a number of researchers, and the conventions for the names and symbols used are now generally accepted (Tofts et al. 1999). In normal tissues, the vascular volume is a small fraction of the total tissue volume (approximately 5%, although it can be considerably higher in some tissues), and it is sometimes assumed (largely as a matter of convenience) that the tracer concentration in the tissue as a whole, Ct, is not influenced to a large degree by the concentration in the vessels (i.e. that Ct ~ veCe) (Tofts and Kermode 1991). Whilst this assumption is acceptable in abnormalities with no large increase in blood volume that are situated in tissues with a relatively low normal blood volume, it is invalid in many contexts, especially as blood volume can increase markedly in tumours. Models of additional sophistication are required to allow these cases to be described, and a number of investigators [see for example Parker (1997); Tofts

(1997); Tofts et al. (1999); Daldrup et al. (1998); Fritz-Hansen et al. (1998); Henderson et al.

(1998)] have attempted to incorporate the effects of a significant vascular signal. Perhaps the most straightforward approach is to extend Eq. 7 to include the concentration of contrast agent in the blood plasma, giving Ct = VpCp + veCe. Using this relationship and by rearrangement of Eq. 7 we then have m = v,C,(O+^Jq.COexp^ (9)

which may be re-expressed as c,(0 = v,c,(0+c,(0®tf(/>'

where H(t) is the impulse response (or residue) function,

and ® represents the convolution operation.

It has been noted that the special case of analysis only of the first passage of a contrast agent bolus is amenable to a simplified form of kinetic model (Tofts 1997; Li et al. 2000; Vonken et al. 2000). It is assumed that during the first passage of a bolus of contrast agent through a tissue (a period that lasts up to approximately 1 min), that the flux of contrast agent from the vascular compartment into the interstitial space, ve, will not be enough to make the concentration of contrast agent in ve significant when compared with the extremely large concentrations seen in the vascular space during this period (that is Cp(t) >> Ce(t)). Under these conditions, it can be shown that (Patlak et al. 1983; Li et al. 2000):

This model variant is therefore attractive in situations where a limited period of acquisition is either necessary or desirable. This may include the study of restless patients or studies in regions that may benefit from the use of a breath-hold (for example lung or liver tumours) (Jackson et al. 2002). Note that this convenience is achieved at the expense of determining the parameter ve, and that for situations where the ratio Ktrans/ve (Eq. 9) is large the assumptions underlying this approach break down.

Model Variants and the Influence of Acquisition Method

The study of contrast agent kinetics using MRI has seen a range of compartmental modelling approaches suggested [see for example Larsson et al. (1990); Brix et al. (1991); Tofts and KermodE 1991; Li et al. 2000)]. Most of these have been shown to be theoretically compatible, after some manipulation, with the general compartmental contrast agent kinetic equation given in Eq. 9 (Tofts 1997; Tofts et al. 1999). However, there have historically often been significant differences in the assumptions used in the application of each model. The determination of an arterial input function (AIF) to define Cp(t) is an example where differences frequently occur. The early model proposed by Tofts and Kermode (1991) used an assumed AIF of biexponential form drawn from literature regarding elimination of Gd-DTPA in the normal population (Weinmann et al. 1984). The model originally proposed by Brix et al. (1991) attempted to define an exponentially decaying AIF on a patient-by-patient basis by including this as a free fitting parameter. The approach proposed by Larsson et al. (1990) utilised an AIF measured from blood samples drawn from the brachial artery at intervals of 15 s during the DCE-MRI data acquisition. Often such differences may be traced to the fact that the development of many analysis methods has proceeded in tandem with a specific data acquisition programme, and the modelling assumptions frequently reflect limitations imposed by the data. Care must therefore be taken in applying these methods in settings other than those originally intended and in comparing apparently compatible results from different studies using different models and/or data acquisitions.

Whilst quantification of contrast agent kinetics in absolute physical terms apparently promises inter-study comparability, in practice this is rarely the case, although intra-study and, to a lesser degree, same-model comparisons are more likely to be valid. For example, a transfer coefficient K derived by a particular model will rarely be directly comparable to K derived using another. There is also a wide range of data-imposed limitations that can lead to differences, including temporal resolution (Henderson et al. 1998), T1 contrast dynamic range (Roberts 1997) and spatial resolution (this can lead to differences in partial volume averaging, which may be a significant issue in very heterogeneous tumours). Therefore, results derived with the same model but under different image acquisition conditions may not be comparable. Perhaps the most significant data-related influences on model output are the ability to quantify T1 (Tofts et al. 1995; Roberts 1997; Evelhoch 1999) and the ability to measure the arterial input function non-inva-sively for each patient (Tofts et al. 1995; ParkeR 1997; Parker et al. 1996). Although these measurements are today more practical than previously, due to sequence and hardware improvements, they are still technically challenging to perform accurately and reproducibly. For this reason many workers make an assumption of a predictable linear relationship between contrast agent concentration and signal intensity change and may assume an input function for all individuals. Often such assumptions are a pragmatic answer to the need for kinetic analysis in a clinical environment when the technical resources to allow a more sophisticated acquisition are lacking. Modelling under these conditions is likely to produce less accurate kinetic parameters than may be obtained under ideal measurement conditions, often with a bias in the results obtained. Such assumptions may be particularly dangerous in a study that is wishing to assess treatment-related changes in kinetic parameters, as a treatment or disease progression that could alter tumour T1 (for instance by inducing oedema) or affect the AIF (for instance by altering kidney function or heart output rate) could cause apparent changes in kinetic parameters that are in fact due to changes in tissue composition or unrelated physiological state changes. Whilst observations of apparent changes in kinetic parameters in these conditions may well indicate an effect due to intervention they will not be specific to the microvascular parameters that are the aim of the study, leading to severe difficulties in data interpretation.

The Importance of the Arterial Input Function

Some early attempts to model contrast agent kinetics did not use an explicitly measured AIF. Rather a population-averaged AIF was used, which lacked the detail of a true AIF, but provided a convenient solution to a potentially difficult measurement problem. However, if it is possible to measure an AIF there may be benefits for the accuracy of derived kinetic parameters. AIF measurement is feasible given a suitable image acquisition protocol [see for example Fritz-Hansen et al. (1998); Li et al. (2000)], and can be shown to be reproducible - Fig. 6.3 shows AIF measurements repeated in six glioma patients (Parker et al. 2003b). Whilst it is clear that reproducible measurement of the AIF is possible for a given patient, it is also apparent that the exact shape may vary between patients, as it is a function of injection timing and dose, heart output rate, distribution of the contrast agent about the body and kidney function.

Figure 6.4 shows model fits using Eq. 9 to the contrast agent concentration time course observed in a lung tumour (Parker et al. 2003a). It is clear that when a standard AIF, as used by Tofts and KermodE (1991), and originally described in Weinmann et al. (1984), is employed the fitted function fails to describe the time course adequately. In particular the concentration changes immediately after contrast agent arrival in the tissue are poorly modelled when an obvious 'first pass' peak is observed. The net result of these poor fits is that the various parameters in Eq. 9 are inaccurately estimated (Parker et al. 1996; Parker 1997). However, the fit of Eq. 9 to the data is improved when an explicitly measured

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Fig. 6.3. Arterial input functions measured in the middle cerebral artery of six glioma patients. Patients were scanned on average 1.5 days apart (series 1 and series 2). Manual injection used for all cases, with an automated arterial input function definition algorithm (Parker et al. 2003b)

time (min)

Fig. 6.4. Top, arterial input function measured in the aorta. Right, tissue response curve measured in a region of interest placed in a lung tumour (crosses) with fits using Eq. 9. Dashed line, fit with assumed AIF. Solid line, fit with AIF as measured in the left graph

AIF is utilised, and the derived parameter estimates are more likely to be more accurate.

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