Predicting brain concentrations of drug using positron emission tomography and venous input: modeling of arterial-venous concentration differences

Abstract

Objective

In a positron emission tomography (PET) study, the concentrations of the labeled drug (radiotracer) are often different in arterial and venous plasma, especially immediately following administration. In a PET study, the transfer of the drug from plasma to brain is usually described using arterial plasma concentrations, whereas venous sampling is standard in clinical pharmacokinetic studies of new drug candidates. The purpose of the study was to demonstrate the modeling of brain drug kinetics based on PET data in combination with venous blood sampling and an arterio-venous transform (T_av).

Methods

Brain kinetics (C_br) was described as the convolution of arterial plasma kinetics (C_ar) with an arterial-to-brain impulse response function (T_br). The arterial plasma kinetics was obtained as venous plasma kinetics (C_ve) convolved with the inverse of the arterio-venous transform (T_av ⁻¹). The brain kinetics was then given by C_br=C_ve*T_av ⁻¹*T_br. This concept was applied on data from a clinical PET study in which both arterial and venous plasma sampling was done in parallel to PET measurement of brain drug kinetics. The predictions of the brain kinetics based on an arterial input were compared with predictions using a venous input with and without an arterio-venous transform.

Results

The venous based models for brain distribution, including a biexponential arterio-venous transform, performed comparably to models based on arterial data and better than venous based models without the transform. It was also shown that three different brain regions with different shaped concentration curves could be modeled with a common arterio-venous transform together with an individual brain distribution model.

Conclusion

We demonstrated the feasibility of modeling brain drug kinetics based on PET data in combination with venous blood sampling and an arterio-venous transform. Such a model can in turn be used for the calculation of brain kinetics resulting from an arbitrary administration mode by applying this model on venous plasma pharmacokinetics. This would be an important advantage in the development of drugs acting in the brain, and in other circumstances when the effect is likely to be closer related to the brain than the plasma concentration.

Appendices

Appendix I: The inverse of a biexponential arterio-venous transform ${\left( {T_{{av2}} = A_{1} e^{{ - a_{1} t}} + A_{2} e^{{ - a_{2} t}} } \right)}$

$$T^{{ - 1}}_{{av2}} = c_{1} \cdot \partial _{0} ^{} + c_{2} \cdot \partial _{0} + c_{3} \cdot e^{{ - c_{4} t}} $$

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$$C_{{ar}} {\left( t \right)} = c_{1} \cdot \frac{d} {{dt}}C_{{ve}} {\left( t \right)} + c_{2} \cdot C_{{ve}} {\left( t \right)} + c_{3} \cdot {\int {e^{{ - c_{4} {\left( {t - x} \right)}}} } } \cdot C_{{ve}} {\left( x \right)} \cdot dx$$

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The parameters c₁−c₄ are combinations of the arterio-venous transform parameters (A₁, A₂, a₁ and a₂):

$$c_{1} = \frac{1} {{A_{1} + A_{2} }}$$

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$$c_{2} = \frac{{A_{1} a_{1} + A_{2} a_{2} }} {{{\left( {A_{1} + A_{2} } \right)}^{2} }}$$

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$$c_{3} = \frac{{ - A_{1} A_{2} {\left( {a^{2}_{1} - 2a_{1} a_{2} + a^{2}_{2} } \right)}}} {{{\left( {A_{1} + A_{2} } \right)}^{3} }}$$

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$$c_{4} = \frac{{A_{1} a_{2} + A_{2} a_{1} }} {{{\left( {A_{1} + A_{2} } \right)}}}$$

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Appendix II: Run test

$$TS = \frac{{G - \mu _{G} + 0.5}} {{\sigma _{G} }}$$

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where G is the number of runs (a run is a sequence of residuals with the same sign), μ_G is the expected number of runs and σ_G is the standard deviation. μ_G and σ_G are calculated as following:

$$\mu _{G} = \frac{{2 \cdot m \cdot n}} {{m + n}} + 1$$

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$$\sigma _{G} = {\sqrt {\frac{{{\left( {2 \cdot m \cdot n} \right)}{\left( {2 \cdot m \cdot n - m - n} \right)}}} {{{\left( {m + n} \right)}^{2} {\left( {m + n - 1} \right)}}}} }$$

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where m and n are the number of residuals with positive and negative sign, respectively.

The test variable (TS) is compared with a critical value obtained from a z-table. Hence, randomness will be rejected at a 5% level of significance if TS is larger than 1.96 or smaller than –1.96.

Appendix III: Akaike information coefficient

$$A \mathord{\left/ {\vphantom {A C}} \right. \kern-\nulldelimiterspace} C = N \cdot \ln {\left( {\frac{{RSS}} {N}} \right)} + 2{\left( {P + 1} \right)}$$

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where N is the number of measurements, RSS is the residual sum of squares and P is the number of free parameters.