molecules in space continuum and compartmental approaches
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Molecules in SpaceContinuum 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 motionConvection: molecules move because they are carrried by a moving medium.Diffusion: molecules move independently of the motion of the mediumConvection 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 PermeationPermeability
Saturable transport (permeases)
Most compartments have fixed volume
Steady StateBalance among three processes: ReactionPermeationConvection
Usually between two of the three
Convection-ReactionNotice that the outflow concentration must equal the compartment concentration
Permeation-ConvectionWhat 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 (l3/ t)
Multi-compartment Systems Simple Artificial Kidney modelsThe bodySingle compartmentMulti-compartment reboundThe artificial kidneyThe quasi-static assumptionA very simple compartmental model(The continuum model comes later)When quasistatic behavior wont suffice.
The body (solutes)[single compartment]Simple exponential fall in concentration with time
The body (solutes)[two compartments]Bi-exponential decay. Post-treatment reboundFor 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 AssumptionKidney 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 qA, qB, or PA becomes too small, it limits the clearance.)
Cascades: the controlling resistanceThe bathtub metaphor
Applies to similar as well as different processes in the cascade.
Dialysate recirculation:The effect of recirculation pattern on dynamics.
Compartmental ModelingThe tracer conceptThe traced substance (tracee)The tracerA superposition of the steady (or quasi-steady) and the unsteady state.
Compartmental ModelingFunctional Compartments
Compartmental ModelingSpatial Compartments
Compartmental Modeling Overlaying spatial and functional compartments
Compartmental ModelingRecirculation phenomenaRegional perfusion
Continuum ProblemsOne-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 axisMolecular 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 dispersionThe general effect and its asymptotesTaylor dispersion
Diffusion in TissueCellular aggregates
The Krogh Tissue Cylinder
In a given medium potential is often (not always, e.g. hemoglobin, CO2 with protein buffers) proportional to concentration.