Proceedings of the meeting - Department of Physics - University of ...

A New Functional Form for Arterial Input Function ModellingMatthew ORTON 1 , James D’Arcy 1 , Simon Walker-Samuel 1 , Martin Leach 1 , David Hawkes 2 , David Collins 11 Institute of Cancer Research, Sutton, Surrey, UK. SM2 5NG.2 Centre for Medical Image Computing, University College London, WC1E 6BT, UK.O23IntroductionThe use of dynamic MR methods for first-pass kinetic modelling of diffusibletracers has received much recent attention. The holy-grail is the accuratequantification of various pharmacokinetic parameters which have importantdiagnostic and prognostic capabilities. A crucial component of such analyses isthe Arterial Input Function (AIF), which is the concentration of tracer in theblood plasma as a function of time.Since the AIF is not fixed, either between patients, or for the same patient ondifferent examinations, it is of primary importance to use an AIF that isappropriate to the data being considered. Failure to do this will have an obviousimpact on the estimation of the pharmacokinetic parameters, and this has beenwidely reported by many researchers. Current solutions include directmeasurement in a blood vessel (1), reference region methods (2), or using afunctional form of some kind (3,4) – each of these methods have advantages anddisadvantages.The approach proposed here is the use of a particular functional form for theAIF, which is given in the next section. The ultimate aim of this work will be toestimate the parameters of the proposed AIF directly from the data within theROI, removing the need for data from either blood vessels or reference regions.This abstract presents the form for the AIF along with a theoretical justification,discusses its properties, and presents some preliminary results where the AIFparameters are estimated from the tissue uptake curves taken from a primarybreast carcinoma.MethodsFunctional form for AIF: The proposed AIF is given by2 −m1t−m2t−m1tcp( t)= a1m1t e + a2( e − e ).This function has some similarities with that described by Weinmann et al. (3), inthat it contains exponentially decaying terms with two rate constants, and twoamplitude parameters. However, figure 3 shows that it has a very different initialshape from the bi-exponential form of Weinmann. An important difference isthat c p (0) = 0, so that this function correctly models the fact that the tracerconcentration gradually reaches its maximum level, rather than instantaneously.An advantage of this form is that when combined with the Tofts model for tracerdiffusion from the plasma (5), the resulting convolution has an analyticalsolution. The inclusion of m 2 1 in the first term means that a 1 is the area under thatpart of the curve.Theoretical justification for AIF: The AIF can be split into several phases – theinitial bolus, possible recirculation peak(s), equilibration with the whole bodyextravascular space, and finally renal excretion. Renal excretion occurs overseveral hours, so for first-pass modelling this effect can be neglected. Separaterecirculation peaks are also neglected since even though they can sometimes beseen in the MR signals from blood vessels, they have a diminishing effect at thetissue level. By the time the bolus reaches the aorta, its initial rectangular shapewill have been dispersed. A gamma function of the form c b (t) = α 1 t n e -m t 1is knownto be a reasonable model for this process (6), and that is the form used here.Increasing n forces the function to be near-zero for a longer initial period, whileincreasing m 1 makes both the rise and the fall faster. In principle n could takeany positive value, but for simplicity it is fixed at n = 1, which still retains themost important features of the function. The results of Weinmann et al. indicatethat the equilibration between the blood plasma and the whole body extravascularspace takes of order 10-20 minutes, during which time the initial bolus willbecome well mixed with the blood plasma. A simple dual-compartment model istherefore adequate to model the equilibration process, and this is characterised bya residue function of the form r(t) = α 2 e -m t 2. The complete AIF is the sum of theinitial bolus and the equilibrium phase, and is given bycp( t)= cb(t)+ r(t)⊗ cb(t),where ⊗ represents convolution. Some simple calculus and algebra recovers theexplicit form given above, where the amplitude terms have been condensed intothe parameters a 1 and a 2 .ResultsInitial results were obtained by using the Bayesian Fitting algorithm reportedbriefly in (7), that uses a probabilistic model to jointly estimate the AIFparameters and the pharmacokinetic parameters for a given ROI. The outputfrom the algorithm is a large collection of random samples from the inferredposterior distribution that can be used to provide estimates for the parameters inmany different forms. In figure 1, images of posterior mean estimates of thepharmacokinetic parameters k ep and K trans are shown for a primary breastcarcinoma, overlaid on the anatomical image. Of more interest in this context arethe results for the AIF parameters, and this is presented in two forms. Firstly infigure 2 estimates of the posterior distributions for each parameter are shown,which were obtained by smoothing a histogram of the samples from the fittingalgorithm. The peak of this distribution can be used as a point estimate for the0.0 1.0 2.0kep (min -1 )Figure 1: Images of pharmacokinetic parameters overlaid on the anatomicalimage of a breast carcinoma.p(a 1| data)p(m 1| data)c p(t) (mM)6543210.8 0.9 1a 1(mM min)13 14 15m 1(min −1 )1 1.1 1.2a 2(mM)Figure 2: Smoothed Posterior histograms for the four AIF parameters.Figure 3: Estimated AIF drawn as a scatter plot to represent uncertainty.0.3 0.6K trans (min -1 )00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2time (min)p(a 2| data)p(m 2| data)0.00.095 0.1 0.105 0.11m 2(min −1 )parameter, while the width gives a measure of the posterior uncertainty in theparameter. A more direct description is the AIF itself, and each sample from thefitting algorithm can be used to give an AIF curve that is a possible fit to the data.This is shown in figure 3 where each sample is used to generate a few pointsalong the AIF curve. The density of points in any region gives a measure of howlikely the curve is to pass through that region, given the uncertainty in theparameter estimates. This indicates that there is more uncertainty in the bolusthan the equilibration phase, which is not surprising since there is less data fromwhich to estimate the bolus parameters.ConclusionThe preliminary results presented here demonstrate that this functional form forthe AIF is reasonably realistic, and has some theoretical justification. When it isused in a fitting algorithm that estimates the parameters from tissue uptake data,the resulting curve is seen to be plausible, and commensurate with directlymeasured AIFs that have been previously reported (1,4). In particular, theestimate of m 2 is very similar to published values for the rate of equilibrationbetween the blood plasma and the whole body extravascular space (3).References(1) Fritz-Hansen T, et al. Mag. Res. Med. 3, 225-231 (1996)(2) Yang C, et al. Mag. Res. Med. 52, 1110-7 (2004)(3) Weinmann HJ, et al. Physiol. Chem. Phys. Med. NMR. 16, 167-172 (1984)(4) Parker GJ, et al. Proc. Intl. Soc. Mag. Res. Med. 13, 2100 (2005)(5) Tofts PS, et al. J. Magn. Reson. Imag. 10, 223-32 (1999)(6) Starmer CF, et al. J. Appl. Physiol. 28, 219-20 (1970)(7) Orton M, et al. Proc. Intl. Soc. Mag. Res. Med. 14, 3490 (2006)This project was funded by EPSRC grants GR/T20434/01 and GR/T20427/01(P),and CRUK grant C1060/A808.