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

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

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

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

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

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

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

8. Answer Louisiana Tech University Ruston, LA 71272

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: Louisiana Tech University Ruston, LA 71272

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

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

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

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

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

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

16. 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

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

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

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

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 Louisiana Tech University Ruston, LA 71272

21. 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

22. 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

23. 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

24. 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

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

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

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). Louisiana Tech University Ruston, LA 71272

28. Louisiana Tech University Ruston, LA 71272

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