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

    Some dont:

  • Steady StateBalance among three processes: ReactionPermeationConvection

    Usually between two of the three

  • Reaction-Permeation

  • 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.

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