Characterization of bispecific antibodies and the ternary complex including an optimal dosing strategy
Johannes Schropp (1), Gilbert Koch (2)
(1) Department of Mathematics and Statistics, University of Konstanz, Germany (2) Paediatric Pharmacology and Pharmacometrics, University of Basel, Children’s Hospital (UKBB), Basel, Switzerland
Objectives: Bispecific monoclonal antibodies (BsMabs) are promising candidates in cancer immunotherapy. For example, BsMabs may simultaneously bind a T cell and a tumor cell. In general, a BsMab binds to two targets (e.g. receptors) forming two binary complexes. Both binary complexes further cross-bind with the same targets creating the ternary complex. Based on this binding kinetics, modeled by the law of mass action, several mathematical models [1]-[4] were developed to guide development of BsMabs. In this study, the level of occupancy of the ternary complex is considered as the produced effect of the BsMab drug, and the behaviour of the ternary complex in relation to the BsMabs concentration is investigated. We apply a BsMab model [5] that incorporates linear elimination, internalization of the complexes, and synthesis and degradation of the targets. We will demonstrate by explicit formulas and simulations that due to the underlying cross-binding the ternary complex has some special uncommon features such as (i) for escalating BsMab doses the level of occupancy of ternary complex decreases and finally vanishes, and (ii) the ternary complex is still fully available although the BsMab concentration is already below limit of quantification. Based on this behaviour a method to develop an optimal dosing strategy for a BsMab drug is presented.
Methods: To develop an optimal dosing strategy, a drug concentration-ternary complex load relationship is needed. First, Li et al. [3] revealed that the core of the dynamics in a similar full BsMab model is governed by the pure binding relations. This so-called equilibrium binding (EB) model describes the instantaneous answer of the binary and ternary complexes of the full BsMab model and its QE approximation for a constant offer of total drug concentration Ctot, and total targets RtotA and RtotB. We additionally present an explicit representation formula describing the level of occupancy for the ternary complex RCAB(C) with respect to the free drug concentration C. This representation proves that in contrast to classical concentration-effect terms, the level of occupancy for escalating doses decreases and finally vanishes. Second, following Li et al. [3] we visualize the level of occupancy of the ternary complex in a (Ctot(C), RCAB(C)) diagram. Diagrams of that type show an optimal level of occupancy for the ternary complex in the EB model, if the total amount of drug is within the range between the minimum and the maximum of RtotA and RtotB which defines the optimal working area of a BsMab. Due to our explicit representation formula, we reformulated this relation for the optimal working area in free drug concentration. Using parameter values based on literature, we show that the ternary complex is working at an optimal level of occupancy even if the level of the free drug is below the level of quantification. Third, the rapid binding assumption used in the QE approximation and singular perturbation theory [6] ensure that (i) the dynamics of the full BsMab model and its QE approximation are nearly identical, and (ii) the solutions of these models in the binary and ternary complexes move along the predictions of the EB model. This allows us to translate the EB results to the full model or QE approximation.
Results: We show that an optimal dosing strategy for the maximal level of the ternary complex occupancy has to generate a level of Ctot(t) which is between the maximum and the minimum of the total amount of targets A and B for as many time points as possible. This defines the optimal working area for the full model and its QE approximation. Within this framework optimal redosing points are points when the total amount of drug leaves the optimal working area via its lower bound and the next optimal dose has to lift the total amount of drug from the lower to the upper limit of the optimal working area. Using this principle an optimal dosing schedule for the BsMaBs model can be established provided all model parameter values are known.
Conclusions: The relationship between BsMab concentration and level of ternary complex occupancy is described by cross-binding behaviour. Simulations from a full BsMab model showed that the ternary complex has some unique behaviour which has to be confirmed by experimental data. Finally, we present an optimal dosing strategy based on a free BsMab concentration – ternary complex relationship.
References:
[1] Rhoden JJ, Dyas GL, Wroblewski VJ A (2016) Modeling and Experimental Investigation of the Effects of Antigen Density, Binding Affinity, and Antigen Expression Ratio on Bispecific Antibody Binding to Cell Surface Targets. J Biol Chem 291(21):11337-47
[2] Doldan-Martelli V, Guantes R, Miquez DG (2013) A mathematical model for the rational design of chimeric ligands in selective drug therapies. CPT Pharmacometrics Syst Pharmacol 2:e26
[3] Li L, Gardner I, Gill K (2014) Modeling the Binding Kinetics of Bispecific Antibodies under the Framework of a Minimal Human PBPK Model. AAPS Poster T2056
[4] Chudasama VL, Zutshi A, Singh P, Abraham AK, Mager DE, Harrold JM (2015) Simulations of site-specific target-mediated pharmacokinetic models for guiding the development of bispecific antibodies. J Pharmacokinet Pharmacodyn 42(1):1-18
[5] Schropp J, Koch G (2017) Target-mediated drug disposition model for a bispecific antibody: development of full model and quasi-equilibrium like approximation. PAGE Poster II-04
[6] Vasileva AB (1963) Asymptotic behaviour of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives. Russ Math Surv 18:13-83
