Freya Bachmann (1), Gilbert Koch (2), Robert Bauer (3), Britta Steffens (2), Gabor Szinnai (4), Marc Pfister (2), Johannes Schropp (1)
(1) Department of Mathematics and Statistics, University of Konstanz, Germany (2) Pediatric Pharmacology and Pharmacometrics, University Children's Hospital Basel (UKBB), University of Basel, Switzerland (3) ICON Clinical Research LLC, Gaithersburg MD, USA (4) Pediatric Endocrinology and Diabetology, University Children's Hospital Basel (UKBB), University of Basel, Switzerland
Objectives:
Determining “optimal” drug dosing for a target population is still a complex, laborious and qualitative rather than quantitative task. Recently, an optimal dosing algorithm (OptiDose [1]) was developed to compute the optimal drug doses for any PKPD model for a given dosing scenario. These doses are associated with a response that is as close as possible to a desired therapeutic goal for the target population.
The objectives of this work are to (i) introduce a reformulation of the optimal control problem of OptiDose that can be solved in NONMEM [2] solely utilizing standard commands and existing routines, and (ii) apply the OptiDose implementation in NONMEM to compute the optimal doses for four relevant but substantially different PKPD examples demonstrating its broad potential.
Methods:
To define the optimal control problem, three components are essential. First, a fully developed PKPD model including fixed model parameters is necessary. Second, the dosing scenario must be defined where doses can be grouped to realize scenarios with repeated dose administration, e.g., the same daily dose for an entire week. Third, the desired therapeutic goal to be reached must be provided by a user-defined reference function. Then, the optimal doses are computed by minimizing an objective functional, e.g., the squared difference between the response of the PKPD model and the user-defined reference function.
To reformulate the optimal control problem, the PKPD model is augmented with an additional state which provides the objective functional value for the inputted doses. This allows NONMEM to solve the reformulated problem and compute the optimal doses as briefly summarized in the following. The data file indicates the dosing time points by setting AMT = 1 in each dose record and the final time in an observation record with DV = 0. In the control stream, model parameters are fixed in $PK, whereas the doses THETA are estimated utilizing the scale factor F. Different doses are assigned to their specific dosing time points via IF-statements. In $DES, the user-defined reference function is coded, and the PKPD model equations are augmented with a differential equation computing the objective functional value. The additional state is assigned to be the output Y in $ERROR without an error model. The initial guess and bounds for the doses are given in $THETA. Finally, the optimization is carried out by $EST –2LL.
Results:
The proposed approach was applied to four optimal dosing examples covering a broad variety of dosing questions. First, an indirect response model was applied to return an elevated biomarker to the normal range. Specific choices of the reference function characterizing this goal and different IV bolus dosing scenarios regarding number of repeated dose administration and additional upper bound on the dose were investigated.
Second, the frequent task of reaching a target AUC of a drug concentration was considered, e.g., for a one compartment model with first-order absorption. The dosing scenario assumed that a fixed dose had already been administered twice, and the subsequent dose to be administered four times was optimized to reach a certain AUC.
Third, the optimal dose was computed to keep the ternary drug-receptor complex of an orally administered bispecific antibody [3] at its most beneficial level which was constant over time.
Moreover, a delay differential equation model for rheumatoid arthritis in collagen-induced arthritic mice [4] was applied for a weekly dosing of the same dose via IV bolus administration. The therapeutic goal was to ensure a moderate disease progression for both the total arthritic and the ankylosis score measured as sum of all paws.
In all examples, the optimal doses were computed within a few seconds by NONMEM and verified utilizing OptiDose implemented in MATLAB [5]. The associated responses closely followed the desired therapeutic goals providing small objective functional values.
Conclusion:
Computation of optimal drug dosing to achieve certain therapeutic goals can be realized in NONMEM as illustrated by four substantially different examples. With the OptiDose implementation in NONMEM, every user will be able to solve their own optimal dosing questions.
References: [1] Bachmann F, Koch G, Pfister M, Szinnai G, Schropp J (2021) OptiDose: Computing the Individualized Optimal Drug Dosing Regimen Using Optimal Control. J Optim Theory Appl 189:46–65
[2] Beal SL, Sheiner LB, Boeckmann AJ, and Bauer RJ (eds) NONMEM 7.5.1 Users Guides. (1989–2020). ICON plc, Gaithersburg, MD
[3] Schropp J, Khot A, Shah DK, Koch G (2019) Target-Mediated Drug Disposition Model for Bispecific Antibodies: Properties, Approximation, and Optimal Dosing Strategy. CPT PSP 8(3):177-187
[4] Koch G, Wagner T, Plater-Zyberk C, Lahu G, Schropp J (2012) Multi-response model for rheumatoid arthritis based on delay differential equations in collagen-induced arthritic mice with an anti-GM-CSF antibody. J Pharmacokinet Pharmacodyn 39(1):55-65
[5] MATLAB Release (2020a) The MathWorks, Inc. Math-Works, Natick
Reference: PAGE 30 (2022) Abstr 10143 [www.page-meeting.org/?abstract=10143]
Poster: Methodology - Other topics