Abstract
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Objectives GA is an efficient method for estimating total tissue distribution volume (β1) from PET uptake data. The original GA produces a negative bias in β1. Estimates of β1 using other GA forms have less bias but less precision. Bias can also be removed from GA with an instrumental variable (IV).We propose to use the estimates from four GA methods including IV to obtain a best estimate of β1.
Methods Forms of GA used are (1) ∫Rdt'/R(t)=β1∫Cpdt'/R(t)+β2 (2) ∫Cpdt'/R(t)=(1/β1)∫Rdt'/R(t)-β2/β1 and (3) R(t)=-(β1/β2)∫Cpdt'+∫Rdt'/β2 (Ichise, 2003)(Cp and R are plasma and tissue radioactivity at time t). The IV solution of (1) in vector form is β=[∑ΦΘT]-1[∑ΦY]. ΘT=[∫Cpdt'/R(t),1], Y=∫Rdt'/R(t) and Φ is the IV. Choose Φ to be uncorrelated with error. Let Φ=[∫Cpdt'/TAC(t),1] where TAC is noise-free (ie global or reference region). Simulations used a 2 compartment model with K1, k2, k4.35.086, and.0475min-1 and k3 .075, .25, .42, .7 min-1 (β1 10.5, 25.5, 40, and 64 mL/cm3) with added Gaussian noise, N=1000 for each β1.
Results The error term at time t (V(t)) for (1), (2) and (3) are related as V(t), V(t)/β1, and V(t)/β2 so that if |β2| > β1 the bias is least for (3) and if β1 > 1 the bias for (2) is less than for (1). Bias from simulations given as avg over N of ( β1EST - β1TRUE) for β1=40 (64) are -6.8 (-18.5), -1.7 (-2.2), 1.6 (2.4) and 1.3 (1.2) for Eqs (1), (2), (3) and IV. (Eq(3) produces more outliers > 100 for β1 = 64).
Conclusions Noise contributes to bias error differently for (1), (2) and (3) so they provide somewhat different estimates for β1, with (1) generally an underestimation and (3) with a number of outliers. The IV method of estimating β1, also provides an estimate with little bias. By using the median value of these four methods we can decrease the bias compared to (1) and increase the precision compared to (3) by eliminating the low and high values. However since the distribution of values is broader compared to (1), some method of noise removal may be needed such as the principal component method proposed by Joshi et al (2008)