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

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

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

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

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

6. Louisiana Tech University Ruston, LA 71272 Slide 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:

7. Louisiana Tech University Ruston, LA 71272 Slide 7 Two Compartment Model Conservation of Mass C1C1 C2C2 Clearance Central Compartment Peripheral Compartment

8. Louisiana Tech University Ruston, LA 71272 Slide 8 Two Compartment Model Conservation of Mass In terms of the volume ratio Initial Conditions Solve the two ODEs for C 1

9. Louisiana Tech University Ruston, LA 71272 Slide 9 ICs in terms of C 1

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

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

12. Louisiana Tech University Ruston, LA 71272 Slide 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

13. Louisiana Tech University Ruston, LA 71272 Slide 13 Two Compartment Model One might expect that overshoot (ringing) could happen. However, ringing will only happen for imaginary values of. In our case: And for the RLC Circuit: Can make the square root imaginary with small R or large C. As you increase k 2 or k e, you must also increase (k 1 +k 2 +k 3 ).

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

15. Louisiana Tech University Ruston, LA 71272 Slide 15 Two Compartment Model What value does the argument of the square root take on at the minimum? Since k 2 and k 1 cannot be negative, the argument of the square root can never be negative. I.e. no ringing.

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

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

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

19. Louisiana Tech University Ruston, LA 71272 Slide 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).

20. Louisiana Tech University Ruston, LA 71272 Slide 20