Louisiana Tech University Ruston, LA Slide 1 Review Steven A. Jones BIEN 501 Friday, May 14, 2007
Louisiana Tech University Ruston, LA Slide 2 Simple Flow Field What is the pathline?
Louisiana Tech University Ruston, LA Slide 3 Simple Flow Field
Louisiana Tech University Ruston, LA Slide 4 Simple Flow Field Pathline follows the particle
Louisiana Tech University Ruston, LA Slide 5 Simple Flow Field What is the streakline?
Louisiana Tech University Ruston, LA Slide 6 What is the Differential Equation that Describes a Streamline? Assume we know that: Answer: Since So
Louisiana Tech University Ruston, LA Slide 7 Continuity For a two-dimensional flow: Use the equation of continuity to determine v.
Louisiana Tech University Ruston, LA Slide 8 Answer
Louisiana Tech University Ruston, LA 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:
Louisiana Tech University Ruston, LA Slide 10 Example Assume that: Answer: Is continuity satisfied?
Louisiana Tech University Ruston, LA Slide 11 What is the equation for a pathline? Answer: Assume that: What is the equation for the pathline through (1,2)?
Louisiana Tech University Ruston, LA Slide 12 What is the equation for a pathline? Write:
Louisiana Tech University Ruston, LA Slide 13 What is the equation for a pathline? so
Louisiana Tech University Ruston, LA Slide 14 Answer (Continued)
Louisiana Tech University Ruston, LA Slide 15 Two Compartment Model Conservation of Mass C1C1 C2C2 Clearance Central Compartment Peripheral Compartment
Louisiana Tech University Ruston, LA Slide 16 Two Compartment Model Conservation of Mass In terms of the volume ratio Initial Conditions Solve the two ODEs for C 1
Louisiana Tech University Ruston, LA Slide 17 ICs in terms of C 1
Louisiana Tech University Ruston, LA Slide 18 Solution The solution to: With Is Where:
Louisiana Tech University Ruston, LA Slide 19 Two Compartment Model Rapid Release Slow Release One Compartment
Louisiana Tech University Ruston, LA 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
Louisiana Tech University Ruston, LA 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 ).
Louisiana Tech University Ruston, LA 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.
Louisiana Tech University Ruston, LA 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.
Louisiana Tech University Ruston, LA Slide 24 Pharmacokinetic Models Vascular Interstitial Cellular PBPK: Physiologically-Based Pharmocokinetic Model Q : Plasma Flow L : Lymph Flow J s, q: Exchange rates
Louisiana Tech University Ruston, LA Slide 25 Pharmacokinetic Models Z : Equilibrium concentration ratio between interstitium and lymph.
Louisiana Tech University Ruston, LA Slide 26 More Complicated Models Plasma Liver Kidney Muscle G.I. Track
Louisiana Tech University Ruston, LA 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).
Louisiana Tech University Ruston, LA Slide 28
Louisiana Tech University Ruston, LA Slide 29 What is the Differential Equation that Describes a Streamline?