Compartmental Models

1. Compartmental Models Juan M. Lopez BIEN 501 Friday, May 09, 2008 Louisiana Tech University Ruston, LA 71272

2. Compartment Models Well Mixed Conservation of Mass D is the mass injected, and and Dd(t) is rate of injection. Initial Conditions: Louisiana Tech University Ruston, LA 71272

3. Compare to Distributed Model Distributed Concentration varies spatially within the compartment, according to Fick’s Law. Compartmental (lumped) Concentration is the same at all locations in the compartment. Louisiana Tech University Ruston, LA 71272

4. Alternative Mathematical Description Delta function: Injection is not instantaneous, but with respect to the larger time scale it can be treated that way. Conservation of Mass Solution: Time for dosage to reduce to half it’s initial value. Louisiana Tech University Ruston, LA 71272

5. Review Assumptions • Rate of clearance is proportional to concentration • Well-mixed system Note the relationship to a lumped-parameter analysis. Louisiana Tech University Ruston, LA 71272

6. Other Physiological Definitions Body Clearance: Rate of drug elimination relative to the drug’s plasma concentration. “Area Under the Curve” For a constant dose: Louisiana Tech University Ruston, LA 71272

7. Two Compartment Model Peripheral Compartment Central Compartment C2 C1 Clearance Conservation of Mass Louisiana Tech University Ruston, LA 71272

8. Two Compartment Model In terms of the volume ratio Conservation of Mass Initial Conditions Solve the two ODEs for C1 Louisiana Tech University Ruston, LA 71272

9. ICs in terms of C1 Louisiana Tech University Ruston, LA 71272

10. Solution The solution to: With Is Where: Louisiana Tech University Ruston, LA 71272

11. Two Compartment Model Slow Release Rapid Release One Compartment Louisiana Tech University Ruston, LA 71272

12. Two Compartment Model The two-compartment model obeys the same differential equations as the simple RLC circuit. It is useful to compare the individual components to the RLC circuit: Damping Transfer from L to C Louisiana Tech University Ruston, LA 71272

13. Two Compartment Model One might expect that overshoot (ringing) could happen. However, ringing will only happen for imaginary values of l. In our case: As you increase k2 or ke, you must also increase (k1+k2+k3). And for the RLC Circuit: Can make the square root imaginary with small R or large C. Louisiana Tech University Ruston, LA 71272

14. Two Compartment Model To see if the square root can become imaginary, minimize it’s argument w.r.t. ke and see if it can be less than 0. Louisiana Tech University Ruston, LA 71272

15. Two Compartment Model What value does the argument of the square root take on at the minimum? Since k2 and k1 cannot be negative, the argument of the square root can never be negative. I.e. no ringing. Louisiana Tech University Ruston, LA 71272

16. Pharmacokinetic Models Vascular Interstitial Q: Plasma Flow L: Lymph Flow Js, q: Exchange rates Cellular PBPK: Physiologically-Based Pharmocokinetic Model Louisiana Tech University Ruston, LA 71272

17. Pharmacokinetic Models Z: Equilibrium concentration ratio between interstitium and lymph. Louisiana Tech University Ruston, LA 71272

18. More Complicated Models Plasma Liver G.I. Track Kidney Muscle Louisiana Tech University Ruston, LA 71272

19. Note on Complexity • While the equations become more complicated as more components are added, the basic concepts remain the same, and the systems can be analyzed with the same tools you would use to analyze a linear system in electrical engineering (e.g. transfer functions, Laplace transforms, Mason’s rule). Louisiana Tech University Ruston, LA 71272

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