1. Molecules in Space Continuum and Compartmental Approaches

2. Review: There are just two things molecules can do:  React:  Move: discrete motion continuous motion  Here we consider motion

3. Review: there are two kinds of motion  Convection: molecules move because they are carrried by a moving medium.  Diffusion: molecules move independently of the motion of the medium  Convection and diffusion (typically parallel)  Convective diffusion (typically orthogonal)

4. Molecular motion is driven by potential – not concentration

5. Motion to, from, and between compartments

6. Compartments are entered by flow streams (mostly convection) or through permeable areas (mostly diffusion – ordinary or forced)  Convection, general case.  Convection (liquid, fixed volume)

7. Diffusion and Permeation  Permeability  Saturable transport (permeases)

8. Most compartments have fixed volume  Some don’t:

9. Steady State  Balance among three processes:  Reaction  Permeation  Convection Usually between two of the three –

10. Reaction-Permeation

11. Convection-Reaction Notice that the outflow concentration must equal the compartment concentration

12. Permeation-Convection What are the units of each term – with and without the units of c, which is common to each term?

13. The clearance (Cl) model ( always steady state) Extraction of a solute by an organ (reactive, diffusive) is modeled as producing two outflows that sum to the inflow: one at the inlet concentration, one at zero concentration. Cl is the flowrate of the (virtual) stream at zero concentration. Q > Cl > 0. Cl [=] flow (l 3 / t)

14. Multi-compartment Systems  Simple Artificial Kidney models  The body  Single compartment  Multi-compartment – ‘rebound’  The artificial kidney  The quasi-static assumption  A very simple compartmental model  (The continuum model comes later)  When quasistatic behavior won’t suffice.

15. The body (solutes) [single compartment] Simple exponential fall in concentration with time

16. The body (solutes) [two compartments] Bi-exponential decay. Post-treatment “rebound” For Simulink, try V1 = 15 L, V2 = 35 L, Cl = 0.2 L/min, PA between compartments 0.15 L/min. Treatment time 3 hr. Observation time 5 hr.

17. Quasi-static Assumption  Kidney example:  The dialyzer responds far faster than the body  The dialyzer is always in steady state.  Assumption is general and widely used.

18. A simple kidney  Two compartments separated by a membrane. Notice that the direction of flow is immaterial  Compartment volume is immaterial in quasi-static steady state.  Equations:

19. Which, with a little algebra, gives the neat result (If any of q A, q B, or PA becomes too small, it limits the clearance.)

20. Cascades: the ‘controlling’ resistance  The bathtub metaphor  Applies to similar as well as different processes in the cascade.

21. Dialysate recirculation:  The effect of recirculation pattern on dynamics.

22. Compartmental Modeling  The tracer concept  The traced substance (tracee)  The tracer  A superposition of the steady (or quasi- steady) and the unsteady state.

23. Compartmental Modeling  Functional Compartments

24. Compartmental Modeling  Spatial Compartments

25. Compartmental Modeling  Overlaying spatial and functional compartments

26. Compartmental Modeling  Recirculation phenomena  Regional perfusion

27. Continuum Problems  One-dimensional steady state problems  Flow along a line contacting a uniform medium.  Flow along a line that contacts flow along another line.  Flow with reaction along a line  Axial dispersion along the flow axis  Molecular diffusion is negligible  Taylor dispersion is not negligible

28. Flow along a line contacting a uniform medium

29. Flow along a line that contacts flow along another line

30. Flow with reaction along a line

31. Axial dispersion  The general effect and its asymptotes  Taylor dispersion

32. Diffusion in Tissue  Cellular aggregates

33. The Krogh Tissue Cylinder