^{1}

^{2}

^{3}

^{4}

^{*}

^{1}

^{2}

^{3}

^{4}

Edited by: Pedro Antonio Valdes-Sosa, Centro de Neurociencias de Cuba, Cuba

Reviewed by: Jim Voyvodic, Duke University, USA; Philippe CIUCIU, Commissariat à l'Energie Atomique et aux Energies Alternatives, France

*Correspondence: Kevin J. Black

This article was submitted to Brain Imaging Methods, a section of the journal Frontiers in Neuroscience

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We recently described rapid quantitative pharmacodynamic imaging, a novel method for estimating sensitivity of a biological system to a drug. We tested its accuracy in simulated biological signals with varying receptor sensitivity and varying levels of random noise, and presented initial proof-of-concept data from functional MRI (fMRI) studies in primate brain. However, the initial simulation testing used a simple iterative approach to estimate pharmacokinetic-pharmacodynamic (PKPD) parameters, an approach that was computationally efficient but returned parameters only from a small, discrete set of values chosen

Measuring the sensitivity of an organ to a drug _{50}, the plasma concentration of drug that produces half the maximum possible effect _{max}. The iterative approach was computationally efficient but could only select _{50} from a short list of parameter values chosen

Here we revisit the simulation testing using a Bayesian method to provide continuous estimates of the PKPD parameters. The Bayesian approach also identifies data too noisy to produce meaningful parameter estimates (using a model selection package described below). Bayesian methods have been used successfully in other PKPD analyses (Lavielle,

We used a standard sigmoid PKPD model (Holford and Sheiner, _{max} = 0) and five with varying sensitivities to drug: _{max} = 10 and

As in the previous work, the concentration of drug in plasma over time is modeled as
_{k}, are given at times _{k}, _{s} (for “time shift”) is a fixed delay between drug concentration and effect, and _{1∕2} is the elimination half-life of drug from plasma (Black et al.,

The full model is then

The test curves were generated using _{1} = _{2} = _{3} = _{4} = the dose of drug that produces a peak plasma concentration of 1 (arbitrary concentration units), _{s} = 0.5 min, _{1∕2} = 41 min, _{0} = 1000, _{1} = 2∕(40 min), and _{2}=0. The 6 resulting curves are shown in Figure

_{50}, i.e., the test data before adding noise

Finally we added Gaussian noise to each time point. This was done 1000 times for each of the 6 curves above and for each of 8 noise levels from _{max} to 2_{max}, resulting in 48,000 noisy time–signal curves plus the original 6 “clean” curves (see Supplemental Data).

In the simulated data described above, each of the 48,006 time courses were analyzed using the “Image Model Selection” package from the Bayesian Data-Analysis Toolbox (Bretthorst,

indicated the full model, _{50}, _{s}, _{max}, _{0}, _{1}, and _{2}. The software returns both the mean parameter values and the values from the simulation with maximum likelihood; the present report uses the latter. This analysis was repeated for each of the 48,006 time courses.

To provide more even sampling of parameter space across the conventional logarithmic abscissa for concentration-effect curves, _{50} was coded as 10^{q}, where _{10} _{50}, and a uniform prior probability was assumed for _{50} values from 0.001 to 20.0. A uniform prior with range [0, 1] min was used for the time shift parameter _{s}. The Hill coefficient _{1∕2} = 41 min. _{max} and the coefficients of the signal drift function

Since tissues with high values of _{50} respond less to a given dose of drug, i.e., _{max}, the ratio _{max} ≪ _{50} and noise. We defined “signal” as the maximum value of _{50} = 7.5 to about 9 for _{50} = 0.25. We define SNR as the ratio of this signal to the standard deviation of the added noise.

We tested the model described above using the same phMRI (pharmacological fMRI) data we analyzed previously with the iterative method, namely, regional BOLD-sensitive fMRI time-signal curves from midbrain and striatum in each of two animals (Black et al., _{1} agonist SKF82958 was given intravenously, divided into 4 equal doses on one day and into 8 equal doses on the other day (see Table 4 and Figure 10 in Black et al.,

The iterative analysis had allowed only values of 5 or 30 min for the half-life of drug disappearance from the blood during the scan session; here we used a uniform prior probability over [2, 60] min for _{1∕2}. Prior probabilities for all other parameters were the same as described above for the simulated data.

Figure

_{max} = 10.0, _{s} = 0.50, added to 1000 + 0.05^{2} and Gaussian noise with _{max} = 10.6, _{50} = 1.43, _{s} = 0.451, _{1} = 0.0553, _{2} = −0.000149. For this time course, prob(model) was estimated as 0.540, and the SD of the residuals was 2.04.

The full PKPD model explained the data better than a simpler model, i.e., prob(model) >0.5, except when signal was low (higher _{50}) or noise was substantial (Figures

_{50} and SD used to generate the time courses

_{50} and noise as a function of that combination's SNR as defined in Section Methods

For the data sets containing no intentional signal, i.e., noise added to the _{max} = 0 line, the Toolbox never returned

Accuracy of the _{50} estimate was considered for time courses with prob(model) >0.5. Figure _{50} as a function of the input _{50}; as expected, accuracy is best with higher SNR. Figure _{50} to input _{50} in terms of SNR. Perfect accuracy would produce a ratio of 1.0, and values >1.0 indicate overestimation of _{50}, i.e., underestimation of the sensitivity to drug.

_{50} for time courses with prob(model) >0.5 is shown as a function of the input _{50}

_{50} for time courses with prob(model) >0.5 is shown as a function of SNR as defined in Section Methods_{50} divided by the input _{50}. The full-width horizontal lines indicate perfect accuracy (ratio = 1.0) and 3/2 and 2/3 of perfect accuracy. The accuracy of the estimated _{50} is superb when SNR > about 6.5, and tends to be accurate for SNR as low as 0.9.

The full PKPD model was selected for 6 of the 8 regional time-signal curves (see Table

_{max} |
_{1∕2} |
_{s} |
||||||
---|---|---|---|---|---|---|---|---|

A | 4 | 1 | Midbrain | 1.00 | 12.59 | 3.44 | 58.33 | 0.98 |

B | 4 | 1 | Striatum | 1.00 | −13.58 | 4.15 | 59.48 | 0.81 |

C | 4 | 2 | Midbrain | 1.00 | 29.27 | 6.32 | 3.93 | 0.23 |

D | 4 | 2 | Striatum | 1.00 | −2.48 | 0.001 | 40.58 | 0.01 |

E | 8 | 1 | Midbrain | 0.00 | – | – | – | – |

F | 8 | 1 | Striatum | 0.02 | – | – | – | – |

G | 8 | 2 | Midbrain | 0.76 | 7.38 | 0.418 | 13.16 | 0.18 |

H | 8 | 2 | Striatum | 1.00 | −13.9 | 1.63 | 2.00 | 0.72 |

_{50} to the peak concentration C_{max} after a single 25 μg/kg dose of drug. E_{max} is in BOLD signal units, and t_{1∕2} and t_{s} are in min.

Bayesian parameter estimation for the QuanDyn™ quantitative pharmacodynamic imaging method produced excellent results in simulated data: first, the Model Select method very accurately identified time courses with a meaningful drug-related signal, until noise overwhelmed signal, i.e., when SNR < about 3.5. The Bayesian Data-Analysis Toolbox successfully avoided false positives, correctly refraining from identifying a signal in every noise-only time course, even where sensitivity was 100%. In time courses with a signal, mean accuracy was reasonable even in the face of low SNR, as shown in Figures _{50} usually erring on the high side (Figure

This simulation used a simple noise model that may be best suited to a temporally stable, quantitative outcome measure, such as positron emission tomography, arterial spin labeling, or quantitative BOLD. However, because the PKPD model

Similar comments hold for the signal as well as for noise: the QuanDyn™ quantitative pharmacodynamic imaging method will perform less well if the PKPD model does not realistically model the data. However, prior to initiating an expensive imaging study, one would determine the appropriate family of PKPD models for the drug to be tested, based on traditional dose-response experiments. We discuss this point further in Black et al. (

Even with the relatively simple signal and noise models adopted for this initial testing, the tested method appeared to handle reasonably the

The QuanDyn™ method described here has several potential advantages compared to the traditional approach to quantifying a drug effect, which is to estimate the population _{50} by sampling a wide range of doses, one dose per subject and several subjects per dose. That approach is an excellent choice when the population under study is homogeneous (e.g., an inbred rodent strain), but does not apply well to single human subjects. One might adapt the traditional approach by repeatedly scanning a single subject, one dose per scan session, but that option brings its own complications, including scientific concerns such as sensitization or development of tolerance with repeated doses in addition to the practical and ethical consequences of repeated scanning sessions in each subject. That option, like the population method, would also require that subjects receive doses substantially higher than the _{50}, which may often be inappropriate in early human studies. Specifically, to estimate _{50}, traditional population PKPD studies require drug doses that produce effects of at least ~95% _{max} (Dutta et al.,

The following information was supplied regarding the deposition of related data: The simulated data sets (1000 time courses for each set of parameter values and noise level) are available at the journal web site as Supplementary Data.

JK performed the experiments, analyzed the data, contributed analysis tools, reviewed and critiqued the manuscript. MV performed the experiments, analyzed the data, reviewed and critiqued the manuscript. GB contributed analysis tools, reviewed and critiqued the manuscript. KB conceived and designed the experiments, performed the experiments, analyzed the data, wrote the paper.

Supported by the U.S. National Institutes of Health (NIH), grants R01 NS044598, 1 R21 MH081080-01A1, 3 R21 MH081080-01A1S1, K24 MH087913 and R21 MH098670, and by the McDonnell Center for Systems Neuroscience at Washington University in St. Louis. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Authors KB and JK have intellectual property rights in the QuanDyn™ method (U.S. Patent #8,463,552 and patent pending 13/890,198, “Novel methods for medicinal dosage determination and diagnosis.”). KB is an Associate Editor for the Brain Imaging Methods section of Frontiers in Neuroscience. The other authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Some of these results were presented previously (JK, GB, KB: A novel analysis method for pharmacodynamic imaging. Program #504.1, annual meeting, Society for Neuroscience, Chicago, 20 Oct 2009), and a preprint was posted on bioRxiv (DOI: 10.1101/017921).

The Supplementary Material for this article can be found online at:

_{max}model with truncated data typical of clinical studies?