1. Louisiana Tech University Ruston, LA 71272 Slide 1 Review Steven A. Jones BIEN 501 Friday, May 14, 2007

2. Louisiana Tech University Ruston, LA 71272 Slide 2 Simple Flow Field What is the pathline?

3. Louisiana Tech University Ruston, LA 71272 Slide 3 Simple Flow Field

4. Louisiana Tech University Ruston, LA 71272 Slide 4 Simple Flow Field Pathline follows the particle

5. Louisiana Tech University Ruston, LA 71272 Slide 5 Simple Flow Field What is the streakline?

6. Louisiana Tech University Ruston, LA 71272 Slide 6 What is the Differential Equation that Describes a Streamline? Assume we know that: Answer: Since So

7. Louisiana Tech University Ruston, LA 71272 Slide 7 Continuity For a two-dimensional flow: Use the equation of continuity to determine v.

8. Louisiana Tech University Ruston, LA 71272 Slide 8 Answer

9. Louisiana Tech University Ruston, LA 71272 Slide 9 What is the equation for a pathline? A pathline follows a fluid particle. Assume that you know the entire velocity field: and that the particle passes through the point at time 0. Answer:

10. Louisiana Tech University Ruston, LA 71272 Slide 10 Example Assume that: Answer: Is continuity satisfied?

11. Louisiana Tech University Ruston, LA 71272 Slide 11 What is the equation for a pathline? Answer: Assume that: What is the equation for the pathline through (1,2)?

12. Louisiana Tech University Ruston, LA 71272 Slide 12 What is the equation for a pathline? Write:

13. Louisiana Tech University Ruston, LA 71272 Slide 13 What is the equation for a pathline? so

14. Louisiana Tech University Ruston, LA 71272 Slide 14 Answer (Continued)

15. Louisiana Tech University Ruston, LA 71272 Slide 15 Two Compartment Model Conservation of Mass C1C1 C2C2 Clearance Central Compartment Peripheral Compartment

16. Louisiana Tech University Ruston, LA 71272 Slide 16 Two Compartment Model Conservation of Mass In terms of the volume ratio Initial Conditions Solve the two ODEs for C 1

17. Louisiana Tech University Ruston, LA 71272 Slide 17 ICs in terms of C 1

18. Louisiana Tech University Ruston, LA 71272 Slide 18 Solution The solution to: With Is Where:

19. Louisiana Tech University Ruston, LA 71272 Slide 19 Two Compartment Model Rapid Release Slow Release One Compartment

20. Louisiana Tech University Ruston, LA 71272 Slide 20 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

21. Louisiana Tech University Ruston, LA 71272 Slide 21 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 ).

22. Louisiana Tech University Ruston, LA 71272 Slide 22 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.

23. Louisiana Tech University Ruston, LA 71272 Slide 23 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.

24. Louisiana Tech University Ruston, LA 71272 Slide 24 Pharmacokinetic Models Vascular Interstitial Cellular PBPK: Physiologically-Based Pharmocokinetic Model Q : Plasma Flow L : Lymph Flow J s, q: Exchange rates

25. Louisiana Tech University Ruston, LA 71272 Slide 25 Pharmacokinetic Models Z : Equilibrium concentration ratio between interstitium and lymph.

26. Louisiana Tech University Ruston, LA 71272 Slide 26 More Complicated Models Plasma Liver Kidney Muscle G.I. Track

27. Louisiana Tech University Ruston, LA 71272 Slide 27 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).

28. Louisiana Tech University Ruston, LA 71272 Slide 28

29. Louisiana Tech University Ruston, LA 71272 Slide 29 What is the Differential Equation that Describes a Streamline?