Fractional kinetics in multi-compartmental systems.

Area of Science:

• Pharmacokinetics
• Fractional Calculus
• Mathematical Modeling

Background:

• Fractional calculus, dealing with non-integer order derivatives, has been applied to one-compartment pharmacokinetic (PK) models.
• Extending fractional calculus to multi-compartmental models presents challenges in maintaining mass balance and system consistency.

Purpose of the Study:

• To extend fractional calculus to multi-compartmental pharmacokinetic models.
• To develop a consistent framework for fractional differential equations (FDEs) in multi-compartment systems.
• To apply a numerical method for solving these complex fractional models.

Main Methods:

• Developed a rationale for fractionalizing ordinary differential equations (ODEs) to ensure consistent multi-compartmental FDE systems.
• Employed a numerical inverse Laplace transform algorithm for solving the derived FDEs.
• Investigated two-compartment PK models with fractional transfer rates and simulated single IV and multiple oral doses.

Main Results:

• Fractional transfer rates in PK models lead to non-exponential terminal phases, indicating anomalous kinetics.
• Multi-dose and constant infusion fractional systems exhibit indefinite drug accumulation without reaching steady state.
• The proposed numerical method aligns with analytical solutions where available and allows parameter estimation from PK data.

Conclusions:

• The developed approach consistently formulates multi-compartment FDE systems with mixed fractional orders.
• The numerical inverse Laplace transform method provides a viable solution for analyzing these complex fractional PK models.
• This framework enhances the understanding of drug disposition, particularly in cases of deep tissue trapping.