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

A B C D (power-law kinetics; enzyme kinetics)

G + H M (equilibrium)

Any change that alters relevant activity is a reaction

n m p

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

95% Oxygen

Blood

Silicone Membrane

POTENTIAL(PARTIALPRESSURE)

CONCENTRATION(MOLS/LITER)

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)

overall: component:

( );in out in in out out

dV d Vcq q q c q c

dt dt

0;in out in out

dcq q q c c V

dt

Diffusion and Permeation

Permeability

Saturable transport (permeases)

mass, molecules/ ;

moles/(area time) i j i j i jJ PA c c K Px

D

0

( )1

2j i i j j j i i i j

i j i j

k pJ

K K K K K K

c c c c

Most compartments have fixed volume

Some don’t:

in in out out

dcq c q c V

d

dVcdtt

Steady State

Balance among three processes:

ReactionPermeationConvection

Usually between two of the three –

Reaction-Permeation

Transport

Reaction

env

dcV Vkc P c cdt

Convection-Reaction

in

dcV Vkc q c cdt

ReactionTransport in Transport out

Notice that the outflow concentration must equal the compartment concentration

Permeation-Convection

Convection out

Convection in

Permeation

( ) ( )in env

dcV q c c PA c cdt

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)

Permeation(or Reaction)

ci

ci

0

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 (l3/ t)

1 o

i

cCl q

c

Multi-compartment Systems

Simple Artificial Kidney modelsThe body

Single compartmentMulti-compartment – ‘rebound’

The artificial kidneyThe quasi-static assumptionA very simple compartmental model(The continuum model comes later)When quasistatic behavior won’t suffice.

The body (solutes)[single compartment]

body

q

c(t)BW

dcV Cl c

dt

Simple exponential fall in concentration with time

The body (solutes)[two compartments]

Bi-exponential decay. Post-treatment “rebound”

body compart-

ment 1c1 (t)

body compart-

ment 2c2 (t)

q, c1

For Simulink, try V1 = 15 L, V2 = 35 L, Cl = 0.2 L/min, PA between compartments0.15 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:

( ) 0

( ) 0

d di do do bo

d di do b bi bo

q c c PA c c

q c c q c c

qb, cbi

qb, cbo

qd, cdi

qd, cdo

PA

Which, with a little algebra, gives the neat result

11 1 1

A B

Cl

q q PA

(If any of qA, qB, 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 conceptThe 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 phenomenaRegional perfusion

Continuum Problems

One-dimensional steady state problemsFlow along a line contacting a uniform

medium.Flow along a line that contacts flow along

another line.Flow with reaction along a lineAxial dispersion along the flow axis

Molecular diffusion is negligibleTaylor 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 asymptotesTaylor dispersion

Diffusion in Tissue

Cellular aggregates

The Krogh Tissue Cylinder