The use of a physiologically based pharmacokinetic model to evaluate deconvolution measurements of systemic absorption

Abstract

Background: An unknown input function can be determined by deconvolution using the systemic bolus input function (r) determined using an experimental input of duration ranging from a few seconds to many minutes. The quantitative relation between the duration of the input and the accuracy of r is unknown. Although a large number of deconvolution procedures have been described, these routines are not available in a convenient software package.

Methods: Four deconvolution methods are implemented in a new, user-friendly software program (PKQuest, http://www.pkquest.com). Three of these methods are characterized by input parameters that are adjusted by the user to provide the "best" fit. A new approach is used to determine these parameters, based on the assumption that the input can be approximated by a gamma distribution. Deconvolution methodologies are evaluated using data generated from a physiologically based pharmacokinetic model (PBPK).

Results and conclusions: The 11-compartment PBPK model is accurately described by either a 2 or 3-exponential function, depending on whether or not there is significant tissue binding. For an accurate estimate of r the first venous sample should be at or before the end of the constant infusion and a long (10 minute) constant infusion is preferable to a bolus injection. For noisy data, a gamma distribution deconvolution provides the best result if the input has the form of a gamma distribution. For other input functions, good results are obtained using deconvolution methods based on modeling the input with either a B-spline or uniform dense set of time points.

Figure 1

Gamma distribution fit to intestinal absorption rate. The absorption rates were determined by applying PKQuest to experimental measurements of venous concentrations after an oral dose of ethanol (fasting or with a meal), standard propranolol or long acting propranolol. The open squares are the experimental data points and the solid line is a 3-parameter gamma distribution fit to these points.

Figure 2

"Simple" PBPK model with exact data sampled at high resolution. Accuracy of 2-exponential bolus input function determined using model venous concentration data generated using the "simple" PBPK organ model. The data was sampled (black squares) at the "high resolutions" time points (2.5, 5, 10, 15, 20, 25, 30, 60, 90, 120, 180, 300, 480 and 720 minutes). The left hand column shows the results for the case where a 30-second constant infusion IV infusion was used. The right hand column results are for a 10-minute constant IV infusion. The first 3 columns compare the model venous concentration (black) with the concentration determined using the bolus input function (red). The first row is for short times, the second for long times and the third is a semi-log plot for all times. The last row compares the deconvolution intestinal input rate (black) with the model input rate (green).

Figure 3

"Simple" PBPK model with exact data sampled at low resolution. Same as figure 2 except the data was sampled at the "low" resolution time points (10, 20, 30, 60, 90, 120, 180, 300, 480 and 720 minutes).

Figure 4

"Binding" PBPK model with exact data sampled at high resolution. Same as figure 2 except that the "binding" PBPK organ model was used to generate the model data.

Figure 5

"Binding" PBPK model with exact data sampled at low resolution. Same as figure 4 except the data was sampled at the "low" resolution time points.

Figure 6

"Binding" PBPK model with exact data sampled at high resolution using 3-exponential response function. Same as figure 4 except that a 3-exponential bolus response function was used.

Figure 7

"Binding" PBPK model with exact data sampled at low resolution using 3-exponential response function. Same as figure 6 except the data was sampled at the "low" resolution time points.

Figure 8

"Binding" PBPK data for 30 minute constant infusion with first data point at 30 minutes.

Figure 9

Evaluation of the "gamma distribution" deconvolution method for "noisy" data. The green line (left column) shows the gamma distribution intestinal input to the "binding" PBPK model. The PBPK model venous concentration produced by this input was made "noisy" by adding a normally distributed random error (black squares, right column) and these data points were then used to determined an intestinal input rate (black line) and the corresponding venous concentration (red line) using the gamma distribution deconvolution method. Each row shows the results for a different set of random data.

Figure 10

Evaluation of the "analytical" deconvolution method for exact data. Same as figure 9 except that the "analytical" deconvolution method was used and applied only to the exact venous concentration data (squares). Each row corresponds to a different analytical deconvolution smoothing parameter.

Figure 11

Evaluation of the "analytical" deconvolution method for "noisy" data. Comparison of intestinal input rate determined by "analytical" deconvolution (black line), "gamma distribution" deconvolution (red line) and the PBPK model input (green line). Each column is for a different noisy data set, and each row is for a different value of the analytical deconvolution smoothing parameter.

Figure 12

Importance of choice of "breakpoints" on "spline function" deconvolution. Comparison of intestinal absorption rate determined by the "spline function" deconvolution method (black line) with the PBPK model input (red line) for one set of "noisy" data. The two columns correspond to different sets of B-spline "breakpoints" and the rows are for different value of the spline deconvolution smoothing parameter.

Figure 13

Evaluation of the "spline function" deconvolution method for "noisy" data. Same as figure 9 except that the "spline function" deconvolution method was used. The "default" breakpoints and smoothing parameters were used for each noisy data set (rows). The red line (left column) is the gamma distribution deconvolution rate and is identical to the black line in fig. 9.

Figure 14

Evaluation of the "uniform" deconvolution method for "noisy" data. Same as figure 11 except that the "uniform" deconvolution method was used.

Figure 15

Evaluation of deconvolution methods for two component intestinal input with exact data. The green line is the composite intestinal input to the PBPK model consisting of the same gamma input used in figs. 9 to 14, plus a second, smaller, delayed gamma input. The black line is the intestinal input determined by application of the four deconvolution methods to the exact venous concentration data. The "spline" and "uniform" deconvolution methods use the "default" input parameters and the "analytical" method uses a smoothing parameter of 0.

Figure 16

Evaluation of deconvolution methods for two component intestinal input with "noisy" data. Same as figure 15 except that a random error was added to the venous concentration data. Each column corresponds to a different deconvolution method and each row to a different set of "noisy" data. The default parameters were used for all 3 deconvolution methods.