Simplification of a pharmacokinetic model for red blood cell methotrexate disposition

Methotrexate (MTX, MTXGlu1) is the gold standard treatment of rheumatoid arthritis. The time course of clinical outcomes is difficult to predict after commencing oral or subcutaneous MTX treatment. A 9-compartment pharmacokinetic (PK) model is available for describing the time course of the concentrations of MTXGlu1 and its polyglutamated metabolites (MTXGlu2-5) in red blood cells (RBCs) [1]. Previous work showed that MTXGlu3-5 have the greatest activity and may predict clinical response [2]. However, the current design of MTX PK experiments requires the measurements of all 5 RBC MTX polyglutamates. A more parsimonious MTX PK design with only the sum of the measurements of MTXGlu3-5 is desirable.

The overall aim of this work is to simplify the existing population MTX RBC PK model for potential use as a driver of MTX response. The specific objectives for the current work are:
1) to simplify the MTX RBC PK model for the sum of RBC MTXGlu3-5 concentrations.
2) to assess the identifiability of the simplified model.

Materials & Methods:
For objective 1), a formal lumping technique was applied using MATLAB® [3].
Firstly, RBC MTXGlu3-5 stayed unlumped. Other states were combined one at a time while maintaining the relative difference between the sum of the predicted RBC MTXGlu3-5 concentrations from the lumped and the original model to be within 20% over time. Secondly, RBC MTXGlu3-5 were considered as a single state. The parameters were adjusted heuristically to provide an acceptable output. The clinical data used for parameter adjustment was the same as for the original MTX RBC PK model development.
For objective 2), the structural identifiability of the simplified model was assessed using the software DAISY [4]. Parameters were fixed gradually until the simplified model became identifiable.

When RBC MTXGlu3-5 stayed unlumped, the original MTX RBC PK model was simplified to a 5-compartment model and the relative difference was 0.3% over time. When RBC MTXGlu3-5 were lumped, the 5-compartment model was further simplified to a 3-compartment model and the relative difference was 6.3% over time.
The 3-compartment model was not structurally identifiable. The identifiability issue was solved by fixing 2 disposition parameters in the model.

The existing 9-compartment MTX RBC PK model was simplified to a 3-compartment model. In future work, the identifiable simplified model will be used as the basis for the development of a population model. Subsequently, a design for future MTX RBC PK experiments will be optimised for estimation.

[1] Korell J et al. Clin Pharmacokinet. 2013;52(6): 475-485.
[2] Pan S et al. Arthritis Rheum. 2013;(manuscript in preparation).
[3] Gulati A et al. CPT Pharmacometrics Syst Pharmacol. 2013;(manuscript submitted).
[4] Bellu G et al. Comput Methods Programs Biomed. 2007;88(1): 52-61.