Quantitative cerebral H₂ ¹⁵O perfusion PET without arterial blood sampling, a method based on washout rate

Abstract

The quantitative determination of regional cerebral blood flow (rCBF) is important in certain clinical and research applications. The disadvantage of most quantitative methods using H₂ ¹⁵O positron emission tomography (PET) is the need for arterial blood sampling. In this study a new non-invasive method for rCBF quantification was evaluated. The method is based on the washout rate of H₂ ¹⁵O following intravenous injection. All results were obtained with Alpert's method, which yields maps of the washin parameter K ₁ (rCBF_K1) and the washout parameter k ₂ (rCBF_k2). Maps of rCBF_K1 were computed with measured arterial input curves. Maps of rCBF_k2* were calculated with a standard input curve which was the mean of eight individual input curves. The mean of grey matter rCBF_k2* (CBF_k2*) was then compared with the mean of rCBF_K1 (CBF_K1) in ten healthy volunteer smokers who underwent two PET sessions on day 1 and day 3. Each session consisted of three serial H₂ ¹⁵O scans. Reproducibility was analysed using the rCBF difference scan 3−scan 2 in each session. The perfusion reserve (PR = rCBF_{acetazolamide}−rCBF_baseline) following acetazolamide challenge was calculated with rCBF_k2* (PR_k2*) and rCBF_K1 (PR_K1) in ten patients with cerebrovascular disease. The difference CBF_k2*−CBF_K1 was 5.90±8.12 ml/min/100 ml (mean±SD, n=55). The SD of the scan 3−scan 1 difference was 6.1% for rCBF_k2* and rCBF_K1, demonstrating a high reproducibility. Perfusion reserve values determined with rCBF_K1 and rCBF_k2* were in high agreement (difference PR_k2*−PR_K1=−6.5±10.4%, PR expressed in percentage increase from baseline). In conclusion, a new non-invasive method for the quantitative determination of rCBF is presented. The method is in good agreement with Alpert's original method and the reproducibility is high. It does not require arterial blood sampling, yields quantitative voxel-by-voxel maps of rCBF, and is computationally efficient and easy to implement.

Appendix

Quantitative parametric maps representing regional cerebral blood flow (rCBF) were calculated using the time-weighted integral method described by Alpert et al. [5] and a method not requiring arterial blood data described by Watabe et al. [8]. Both are based on the one-tissue compartment model for H₂ ¹⁵O represented by the differential equation:

$$ {{dC(t)} \over {dt}} = K_1 C_a (t) - k_2 C(t) $$

(1)

where C(t) denotes tissue activity concentration, C _a (t) the measured arterial input function (AIF), K ₁=rCBF and k ₂=rCBF/p (flow/partition coefficient). Assuming no tissue activity before tracer application at time zero, the solution of Eq. 1 is given by:

$$ C(t) = K_1 C_a (t) \otimes e^{ - k_2 t} = K_1 \int\limits_0^t {C_a (u)e^{k_2 (u - t)} du} $$

(2)

Alpert's method represents a computationally efficient implementation of the flow calculations based on the time course of the activity concentration in each voxel and the AIF.

Essentially, a lookup table r is calculated from the blood data as follows:

$$ r = {{\int\limits_0^T {\int\limits_0^t {C_a (u)e^{k_2 (u - t)} dudt} } } \over {\int\limits_0^T {t\int\limits_0^t {C_a (u)e^{k_2 (u - t)} dudt} } }} $$

(3)

where T denotes the acquisition duration, and k ₂ is varied in 400 increments between 0 and 200 (ml/min/100 g tissue) to cover the range of k ₂ occurring in physiological conditions. A similar operation is performed with the PET data in each voxel:

$$ \hat r = {{\int\limits_0^T {C(u)dudt} } \over {\int\limits_0^T {tC(u)dudt} }} $$

(4)

From $\hat r$, the actual k₂ is obtained by a lookup in the r table. The flow is finally calculated by entering k ₂ into the equation:

$$ rCBF = K1 = {{\int\limits_0^T {C(u)dudt} } \over {\int\limits_0^T {t\int\limits_0^t {C_a (u)e^{k_2 (u - t)} dudt} } }} $$

(5)

Compared with the true input curve in the brain, the measured time course of arterial activity in the radial artery is time-shifted and distorted due to dispersion. Both time shift and dispersion were corrected before rCBF calculation by the method described by Meyer [20]. To this end, a flow model including the delay and an exponential dispersion as parameters was fitted to the averaged time-activity curve of all brain voxels with integrated activity above 40% of the maximum. The measured AIF was then shifted by the estimated delay, and deconvolved by the exponentially decaying dispersion using a Fourier transform approach.

Watabe's method uses a reference tissue approach. Equation 1 is integrated twice for the tissue of interest C(t), yielding:

$$ \int\limits_0^T {C(t)dt = } K_1 \int\limits_0^T {\int\limits_0^t {C_a (u)dudt - {{K_1 } \over p}} } \int\limits_0^T {\int\limits_0^t {C(u)dudt} } $$

(6)

The same operation is performed for a reference tissue C ^Ref(t). From the two resulting expressions C _a(t) can be algebraically eliminated, yielding equation:

$$ \int\limits_0^T {C^{{\mathop{\rm Re}\nolimits} {\rm f}} (t)dt = } {{K_1 ^{{\mathop{\rm Re}\nolimits} {\rm f}} } \over {K_1 }}\int\limits_0^T {C(u)dt} + {{K_1 ^{{\mathop{\rm Re}\nolimits} {\rm f}} } \over p}\int\limits_0^T {\int\limits_0^t {C(u)dudt} } - {{K_1 ^{{\mathop{\rm Re}\nolimits} {\rm f}} } \over {p^{{\mathop{\rm Re}\nolimits} {\rm f}} }}\int\limits_0^T {\int\limits_0^t {C^{{\mathop{\rm Re}\nolimits} {\rm f}} (u)dudt} } $$

(7)

which contains the tracer concentration in the two tissue regions as measured quantities and the flows (K ₁, K ₁ ^Ref), and partition coefficients ( p, p ^Ref) as the unknowns. This equation can be solved for K ₁:

$$ {\rm rCBF} = K_1 = {{\int\limits_0^T {C(t)dt} } \over {{1 \over {K_1 ^{{\mathop{\rm Re}\nolimits} {\rm f}} }}\int\limits_0^T {C^{{\mathop{\rm Re}\nolimits} {\rm f}} (t)dt} + {1 \over {p^{{\mathop{\rm Re}\nolimits} {\rm f}} }}\int\limits_0^T {\int\limits_0^t {C^{{\mathop{\rm Re}\nolimits} {\rm f}} (u)dudt} } - {1 \over p}\int\limits_0^T {\int\limits_0^t {C(u)dudt} } }} $$

(8)

In an initial pre-processing step, time-activity curves were calculated for two clusters of brain voxels: the "low-flow" voxels defined as the 5,000 voxels with the lowest integrated activity [C(t)], and the "high-flow" voxels, serving as the reference and defined as the 2,000 voxels with the highest integrated activity [C ^Ref(t)]. Equation 7 evaluated at all acquisition times (T=T _i , i=1...18) then served as the operational equation from which the unknowns (K ₁, K ₁ ^Ref, p, p ^Ref) were estimated in an iterative optimisation using a Marquardt-Levenberg algorithm. Subsequently, Eq. 8 was applied to calculate flow for all voxel-wise time-activity curves C(t) in a single step. Hereby, a fixed partition coefficient p=0.86 ml/ml [16] was assumed, and K ₁ ^Ref and p ^Ref resulting from pre-processing were inserted.

The pre-processing step is crucial for the results of the flow calculations. As four parameters are estimated in a fit to a smooth curve, a dependency of the results on the starting parameters was noted, indicating that the method is prone to local minima. Therefore the flow and the partition coefficient were determined for the time-activity curves of the low-flow and the high-flow voxel clusters in one volunteer study by means of a full one-tissue compartment model analysis using the AIF. The result parameters were then used as a standard set of starting parameters for K ₁, K ₁ ^Ref, p and p ^Ref (13, 63, 0.3 and 1.0 respectively).

All computations were performed with the dedicated software PMOD (www.pmod.com) [21]. This JAVA-based software allows the easy implementation of the necessary models to calculate rCBF, including the corrections of the input curve.