Mario Nagase: No financial relationships to disclose
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Objectives: Linear compartment models, often applied to PK analysis, may entail identifiability issues, represented by “Flip-Flop kinetics”, in which multiple combinations of parameter values can give the same likelihood. The issue stems from a permutation of the rank order of the eigenvalues (EVs) associated with the coefficient matrix, K, say. One solution would be to set constraints directly on the parameter space, e.g., the absorption constant “ka” is set larger than the elimination constant “k”. However, once a model becomes more complex, e.g., with three or more compartments, setting constraints on the parameter space corresponding to the order of EVs becomes extremely challenging. Here, a novel method is proposed where parameters become globally identified by simply specifying the order of EVs corresponding to each compartment.
Methods: The solution for each compartment is expressed as a linear combination of exponential functions, with the EVs (multiplied by time) of K as exponents. Provided that there are no repeated EVs, their quantity equals the number of compartments. However, mathematically, the ordering of these EVs is immaterial with respect to the resulting solution, in other words, there exists no explicit bijective mapping between EVs and compartments in math, while researchers can intuitively associate -ka with the absorption compartment and -k with the main compartment, in the previous example. In the proposed method, after establishing a bijective mapping between each eigenvalue and each compartment, the procedure commences by imposing an ordering on the EVs based on the anticipated magnitude of the elimination capacity associated with each compartment. For example, the drug elimination capacity of the absorption compartment can be set as being higher than that of the main compartment. In the estimation, along with the parameters, EVs and eigenvectors are also included as auxiliary parameters in the model and estimated simultaneously. Furthermore, constraints ensuring that the auxiliary parameters are indeed those of K are imposed using the Bayesian constraint relaxation. The method was applied to synthetic data and its performance was compared with existing methods with respect to the following criteria: convergence behavior, estimation bias, and computational time. NUTS algorithm in numpyro 0.18.0(Python 3.12.4, jax 0.5.3) was used for the analysis.
Results: In contrast to the existing method, which failed to converge without appropriate initial values or converged to another local minima if the initial values were not relevant, the proposed method achieved convergence solely by specifying the eigenvalue ordering. While the existing method with appropriate initial values completed in 10 seconds, the proposed method required approximately 2 minutes. Upon comparison of estimated values, the maximum discrepancy with those obtained from the existing method was less than 0.63%, the difference deemed negligible.
Conclusions: The method is useful in situations where reasonable relevant initial values cannot be specified, or reasonable constraints cannot be imposed. A ready application of this method is to mitigate misspecification risk for analysis flip-flop type kinetics.
Citations: [1] Kuan, I.H., Wright, D.F. and Duffull, S.B., 2023. The influence of flip‐flop in population pharmacokinetic analyses. CPT: Pharmacometrics & Systems Pharmacology, 12(3), p.285. [2] Duan, L.L., Young, A.L., Nishimura, A. and Dunson, D.B., 2020. Bayesian constraint relaxation. Biometrika, 107(1), pp.191-204. [3] Margossian, C.C., Zhang, Y. and Gillespie, W.R., 2022. Flexible and efficient Bayesian pharmacometrics modeling using Stan and Torsten, part I. CPT: Pharmacometrics & Systems Pharmacology, 11 (9), 1151–1169
