## Modelling Blood Flow

Equation 9 provides a straightforward model to allow the estimation of the transfer constant, Ktrans, the vascular plasma volume, Vp, and the interstitial space, ve, from a series of concentration values obtained over time. However, as was noted earlier, these approaches rely on the assumption that contrast agent is instantaneously and well-mixed in each of the compartments it occupies (this is, in fact, inherent to the definition of a compartmental model). Additionally, there is no term allowing independent determination of blood plasma flow rates (as opposed to blood plasma volume).

Two possible approaches exist to determine tumour blood flow. Methods based on the indicator dilution theory have been applied in tumours using first pass T2*-weighted data acquisitions [see Chap. 4 for a detailed description of these dynamic susceptibility contrast (DSC-MRI) approaches]. The same modelling approaches may be applied to Tl-weighted time series, if the data acquisition is rapid enough, although to date this possibility has only been exploited in normal tissues (Hatabu et al. 1999; Ohno et al. 2004). The indicator dilution theory in its standard form assumes that contrast agent does not leave the blood pool during its first passage through tissue. This assumption is likely to be an oversimplification in many tumours when using small molecular weight contrast agents (see Chaps. 2 and 3), leading to possible errors in blood flow estimates, and the missed opportunity to estimate contrast agent leakage characteristics. However, in situations where blood volume is likely to be low, and where extravasation is low, DSC-MRI methods are likely to provide more sensitive indications of blood volume than T1-weighted methods.

Recent work (St. Lawrence and Lee 1998; Henderson et al. 2000; Koh et al. 2001; Buckley 2002), using well-established concepts (Kety 1951; Renkin 1959; Crone 1963; Zierler 1963; JohnsoN and Wilson 1966), has attempted to extend the modelling processes described in Sect. 6.4 to account additionally for blood flow, thus providing a comprehensive assessment of bulk microvascular characteristics. These workers have shown that the tissue homogeneity model introduced in (Johnson and Wilson 1966) may be adapted to the dynamic contrast-enhanced MRI experiment via an adia-batic approximation (St. Lawrence and Lee 1998; Henderson et al. 2000). The tissue homogeneity model divides tissue into a vascular plasma volume, Vp, and the EES, Ve, in the same way as the previously discussed models, but differs in that it defines the tracer concentration within Vp as a function of both time and distance along the length of the capillary, while Ve is assumed still to be a well-mixed compartment. The adiabatic approximation to the tissue homogeneity model (AATH) invokes the additional assumption that the rate of change in Ce is low relative to that of Cp, an approximation that makes application of the model practically possible.

The AATH model requires the following residue functions, separating the time course of contrast agent arrival into a 'vascular' phase (t < Tc, where Tc is the transit time through a capillary), and a 'parenchymal tissue' phase (t > Tc) (St. Lawrence and LeE 1998; Koh et al. 2001):

As may be seen immediately, H(t) at t > Tc in this model is equivalent to that defined previously for the compartmental model of diffusive transport (Eq. 11) under the general mixed perfusion- and permeability-limited regime (when Ktrans = EFp(l-Hct)). The presence of the separate residue function for 0 < t < Tc is what allows the estimation of blood plasma flow as an independent variable when using this model. Note that for consistency with Eqs. 1-12, the form of Eq. 13 is slightly different to that in (St. LawrencE and Lee 1998). The tissue homogeneity approach models the tissue contrast agent concentration as:

where t is the mean capillary transit time within a voxel (typically a few seconds). t may be expressed as Vp/(Fp(1-Hct)), thus reducing Eq. 12 to Eq. 9 if t is small and by substituting Ktrans for EFp(l-Hct). It is therefore clear that the AATH model is compatible with the compartmental approaches to examining the kinetics of contrast agent accumulation in tissues, whilst providing the possibility for extracting

time (min)

Fig. 6.5. Top, axial T2-weighted turbo spin echo image showing the prostate with tumour in the posterior area of the peripheral zone. Bottom, graph showing enhancement over time in tumour (open squares) and normal prostate (filled circles) regions. The lines represent the fit of the AATH model (St. Lawrence and Lee 1998) to the data. Parameter estimates were obtained for tumour: F, 59.3 ml/100 ml/min; Vb, 1.0 ml/100 ml; PS, 19.0 ml/100 ml/min; Ve, 40.0 ml/100 ml; normal: F, 23.39 ml/100 ml/min; Vb, 0.4 ml/100 ml/min; PS, 10.3 ml/100 ml/min; Ve, 18.4 ml/100 ml (a tissue density p of 1 g/ml is assumed)

time (min)

Fig. 6.5. Top, axial T2-weighted turbo spin echo image showing the prostate with tumour in the posterior area of the peripheral zone. Bottom, graph showing enhancement over time in tumour (open squares) and normal prostate (filled circles) regions. The lines represent the fit of the AATH model (St. Lawrence and Lee 1998) to the data. Parameter estimates were obtained for tumour: F, 59.3 ml/100 ml/min; Vb, 1.0 ml/100 ml; PS, 19.0 ml/100 ml/min; Ve, 40.0 ml/100 ml; normal: F, 23.39 ml/100 ml/min; Vb, 0.4 ml/100 ml/min; PS, 10.3 ml/100 ml/min; Ve, 18.4 ml/100 ml (a tissue density p of 1 g/ml is assumed)

additional information. Figure 6.5 gives examples of fits of this model to time courses extracted from ROIs placed in normal prostate and prostate tumour. Recent observations in a group of 22 men show that it is possible to differentiate tumour from normal peripheral zone prostate based on the derived values of blood flow and EES, whilst blood volume and capillary permeability surface area product (PS) do not appear to be different (Buckley et al. 2003; Buckley et al. in press). It is clear that the use of a sophisticated modelling approach such as this is viable in a practical setting and that it can provide specific functional information, allowing a rich characterisation of tumour microvasculature.

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