molecules in space continuum and compartmental approaches
TRANSCRIPT
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