An effective approach for obtaining optimal sampling windows for population pharmacokinetic experiments
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1 An effective approach for obtaining optimal sampling windows for population pharmacokinetic experiments Kayode Ogungbenro and Leon Aarons Centre for Applied Pharmacokinetic Research School of Pharmacy and Pharmaceutical Sciences The University of Manchester, Manchester United Kingdom
2 Introduction Population pharmacokinetics involves collection of blood samples Sample collection at specific times may not be feasible less informative Size of the trial (Phase III elays in seeing the medical personnel Poor patient compliance with respect to dosing times More immediate medical procedure Sampling windows controlling sampling times Sampling within some time intervals Gives flexibility, informative data and satisfactory parameter estimation
3 Population PK modelling Level Individual level yi f ( θi, ti + ε i,..., ni, i,..., N Level Population level θ i g( θ, bi ~ N(0, Ω Parameters Ψ ( 0, σ σ f ( θ, t ε ~ N + i i i b i ω Ω ω. [ θ,..., θ, ω,..., ω, ω,..., ω,..., ω, σ σ ] p p p pp, ω ω... ω pp Sampling Times ξ i [ t,..., t ] i in i 3
4 4 Population Fisher information Matrix Log-likelihood of observations Expressions for PFIM based on mixed effects modelling by maximum likelihood method Ψ Ψ Ψ Ψ T i i y l E F ; (, ( ξ Approximations based on linearization of the model Ψ Ψ + Ψ, ( V V V V tr J V J F z a T z a az ξ i Population design Ξ Q Q N N,...,,..., ξ ξ Ψ Ψ Ψ Ξ Q q q q Q q N i i F N F F q q q, (, (, ( ξ ξ
5 5 Optimal esign -Efficiency, ( max arg Ξ Ψ Ξ F χ / dim(, (, (,, ( Ψ Ψ Ξ Ψ Ξ Ξ Ξ Ψ F F eff Optimality criteria - -optimality Population -optimal design
6 Sampling windows determination - approaches uffull et al. (Pharmaceutical Research 00 Sampling windows design obtained as marginal windows associated with each time points varying one sampling time at a time until determinant is reduced by 5% Graham and Aarons (Statistics in Medicine 006 Two stage approach and a quadratic loss function Sampling windows design result in specified loss of efficiency compared to fixed -optimal time points Patan and Bogacka (Advances in Model-Oriented esign and analysis 007 Based on equivalence theorem for -optimal continuous designs using the variance function 6
7 Sampling windows determination - approaches Summary of attributes Attribute uffull G&A P&B Parameter Sensitivities Yes No Yes Assessment of Sampling Windows Efficiency No Yes Yes Exact Yes Yes No esign Continuous Yes Yes Yes 7
8 Sampling windows determination new approach Main features Very efficient and effective Can be applied to different types of design Exact Continuous Reflects parameter sensitivities Less flexibility (narrow window for high parameter sensitivities (important to sample close to optimal time point More flexibility (wide window for low parameter sensitivities (less important to sample close to optimal time point The efficiency of the sampling windows design can be assessed ointly 8
9 9 Sampling windows determination Sampling windows population design Fixed time population -optimal design Ξ L n L U n U W t t t t,...,,..., L U t t t t δ δ + [ ] δ,...,δ n Δ [ ] n t t N,..., Ξ Ξ ξ
10 Sampling windows determination Efficiency functions conditional and oint Uniform or loguniform distribution eff W / dim( Ψ E F( Ψ, Ξ ( δ W ( Ψ, Ξ ( δ / dim( Ψ F( Ψ, Ξ eff E F( Ψ, Ξ / dim( Ψ W ( Ψ, Ξ ( Δ / dim( Ψ W F( Ψ, Ξ ( Δ 0
11 Sampling windows determination 3 stage approach Stage : Optimisation of fixed -optimal time points Stage : Assuming a distribution uniform or loguniform efine a target mean efficiency level, eff 0 Optimise one window length at a time using a quadratic function δ arg min δ δ Δ [( ] W eff ( Ψ, Ξ ( eff 0
12 Sampling windows determination Stage 3: Evaluate the efficiency of the oint sampling windows eff ( Ψ, Ξ W ( Δ Check if eff is greater than eff 0 if not (ideally reduce the window lengths by equal percentage, % and obtain a new vector for the lengths of the sampling windows δ δ ( new ( old ( old ( δ * 0.0 Repeat Stage 3 until the required efficiency level is obtained Take the last vector of sampling windows lengths as the optimal sampling windows lengths
13 Sampling windows determination Apart from mean efficiency level, percentiles can also be used - mean efficiency can produce variation in the realized design depending on distribution of samples within the windows Sampling at or close to boundaries less efficient design Sampling at or close to fixed -optimal time points more efficient design Constraints during optimisation especially if any of the fixed -optimal time points is at the boundary or near the boundary Ensure sampling windows do not extend outside the design space (sampling at negative time points 3
14 Sampling windows determination Example Individual continuous design One compartment IV bolus model (proportional residual Ψ y ose V ( Cl / V t esign region 0 and 4 hrs e [ Cl, V, σ ] [ 3,30,0.04] ose 450mg Sampling windows P&B and new approach 95% and 90% efficiency levels Mean, 5th and 0th percentile efficiency criteria Uniform and loguniform sample distributions 4
15 Sampling windows determination Example Fixed -optimal time points 0 and 4 hr (0.5,0.5 Efficiency level Criteria P&B Uniform New Approach P&B Loguniform New Approach Mean % 0 th th Mean % 0 th th
16 Sampling windows determination Example Individual continuous design One compartment first order absorption model at steady state (the data and the model are log transformed log( y Ψ Fka log Vka Cl F, mg, τ hr e e / V kat e e Clτ / V kaτ [ Cl, V, ka, σ ] [.55,00,.08,0.05] Clt + ε ( σ esign region 0. and hr Sampling windows P&B and new approach 95% and 90% efficiency levels Mean, 5th and 0th percentile efficiency criteria Uniform and loguniform sample distributions 6
17 Sampling windows determination Example Fixed -optimal time points 0.9,.46, hr (0.33,0.33,0.33 Efficiency level Criteria P&B Uniform New Approach P&B Loguniform New Approach Mean % 0 th th Mean % 0 th th
18 Sampling windows determination Example (95% Efficiency, Mean criteria and Uniform distribution Fixed -optimal time points 0.9,.46, hr (0.33,0.33,0.33 sensitivity Equivalence theorem - sensitivity time (min 8
19 Sampling windows determination Example 3 Population exact design One compartment first order absorption model at steady state (the data and the model are log transformed Ψ Cliti / Vi kaiti Fka i e e log( y i log ε ( σ Cl τ / V ka τ + i i i i Vikai Cl i e e [ Cl, V, ka, ω, ω, ω, σ ] [.55,00,.08,0.09,0.09,0.09,0.05] Cl F, mg, hr V ka τ, b ] [ 0,] [a subects, group and 3 time points Sampling windows G&A, uffull and new approach 95% and 90% efficiency levels Mean and 0th percentile efficiency criteria Uniform and loguniform sample distributions 9
20 Sampling windows determination Example (Uniform distribution Optimal design (fixed time 0.30,.9 and hrs (det Efficiency Criterion G&A uffull New approach Mean (0.8*.6-.7 ( ( ( ( ( ( ( (4.0 det** [0.90] det 4.3 [0.74] det 7.9 [0.9] 90% 0% ( ( ( ( ( ( ( ( (3.9 det [0.90] det 48.6 [0.78] det 7.36 [0.90] Mean ( ( ( ( ( ( ( ( (.78 det [0.95] det [0.84] det [0.95] 95% 0% ( ( ( ( ( ( ( ( (.0 det [0.95] det 70.7 [0.90] det 8.7 [0.95] *Window half lengths in parentheses, **det is the normalised determinant 0
21 Sampling windows determination Example (Loguniform distribution Optimal design (fixed time 0.30,.9 and hrs (det Efficiency Criterion G&A uffull New approach 90% Mean 0% (0.54* ( (0.54 det** 7.43 [0.90] ( ( (0.40 det 70.4 [0.90] ( ( (0.7 det 9.9 [0.68] ( ( (0.44 det 45. [ ( ( (0.40 det 7.5 [0.9] ( ( (0.30 det [0.9] Mean ( ( ( ( ( ( ( ( (0.5 det 80.8 [0.95] det [0.83] det 80.5 [0.95] 95% 0% ( ( ( ( ( ( ( ( (0.7 det 80. [0.95] det [0.89] det [0.95] *Window half lengths (on the logarithmic scale in parentheses, **det is the normalised determinant
22 Sampling windows determination Example (uniform distribution, 90% efficiency level and mean criteria G&A uffull New ( ( ( ( ( ( ( ( (4.0
23 Sampling windows determination Example (simulations, RE 3 Fixed Time 3 G&A RE RE Cl V ka λ Cl λ V λ ka σ Cl V ka λ Cl λ V λ ka σ 3 uffull 3 New RE RE Cl V ka λ Cl λ V λ ka σ Cl V ka λ Cl λ V λ ka σ 3
24 Sampling windows determination Conclusion Sampling windows provide adequate flexibility (controlled flexibility for sample collection The new approach is efficient and reflects parameter sensitivities This approach can be applied to both exact and continuous design as well as multiresponse designs Choice of efficiency level, criteria function and parameter distribution must be balanced against other design properties Efficient population PK experiment can provide improved parameter estimates and can help to reduce cost and time 4
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