Molecules in Space Continuum and Compartmental Approaches

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

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)

Molecular motion is driven by potential – not concentration

Motion to, from, and between compartments

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

Diffusion and Permeation  Permeability  Saturable transport (permeases)

Most compartments have fixed volume  Some don’t:

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

Reaction-Permeation

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

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

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)

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.

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

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.

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.

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:

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.)

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

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

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

Compartmental Modeling  Functional Compartments

Compartmental Modeling  Spatial Compartments

Compartmental Modeling  Overlaying spatial and functional compartments

Compartmental Modeling  Recirculation phenomena  Regional perfusion

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

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  The general effect and its asymptotes  Taylor dispersion

Diffusion in Tissue  Cellular aggregates

The Krogh Tissue Cylinder