Modeling Vascular Networks with

Applications

Instructor: Van Savage Spring 2015 Quarter

Unlumped Models: Structure Matters

Bulk properties in unlumped model that describe flow

a.  Pressure difference driving flow

b.  Volume flow rate through network

c.  Impedance of network

All encapsulated in basic relationship (Ohm’s Law)

€

Δp = ˙ Q Z

Lumped Models

1. Combine (i.e., “lump”) vessel impedances into a single or several

equivalent black boxes a.  For vessels in series, sum impedances together to obtain impedance of equivalent black box b. For symmetric branching vessels in parallel, sum inverse of impedances together to obtain impedance of equivalent black box

c. For asymmetric vessels in parallel, things are more complicated, but there is still some equivalent impedance for a black box that can replace those vessels.

Slightly more complicated model with vessels in series and parallel

Ignore details of spatial structure and use “lumped” or circuit model

2. Calculate pressure drop across and volume flow rate through these

equivalent black boxes Using our standard relationships, we can now talk about the pressure and flow through one or several of these black boxes

€

Δp = ˙ Q ZWe will see that other dynamical properties can be calculated as well by introducing circuit elements that describe the elasticity of the vessel walls and the inertial/time delay effects due to the mass of the fluid.

Resistors represent impedances of different parts of the cardiovascular system

Lumped model, windkessel model, circuit model

3. Differences between lumped and unlumped models

a.  Replace cardiovascular geometry and structure with one or several black boxes

b. Talk about bulk properties rather than properties at level of vessel

c. Lose a lot of information about wave reflections at vessel junctions because knowledge of vessel junctions is lost. However, we are still able to capture analog of wave reflections between black boxes with umatched impedances and reactance.

3. Differences between lumped and unlumped models

d. Knowledge of specific vessel sizes and relative scale is lost, so most information about allometry is lost unless “put in by hand” into flow rates, pressures, and impedances. Not predictive e. For models of heart disease or within a specific organism or organ, these lumped models can be a very useful way for probing how changes in resistance or capacitance in different vascular regions (represented by different black boxes) can affect flow or pressure and create pathological situations. f. Empirical measures at the regional (or blackbox) level are much easier than at the individual vessel level for parameterizing a specific problem of interest.

Derivation of general lumped-model differential equation in

series and introduction to capacitance (elasticity) and inductance (inertial effects)

Time derivative of our basic flow equation

€

dΔpdt

=d( ˙ Q Z)

dt=

d ˙ Q dt

Z + ˙ Q dZdt

Recall that the volume flow rate depends on the linear velocity and vessel cross-sectional area

€

˙ Q = u A

Time derivative of our basic flow equation

Changes in flow rate due to changing cross-sectional area due to changing vessel walls

Changes in flow rate due to changes in linear velocity (i.e., acceleration) because of mass of fluid and inertial effects

Changes in impedance due to changing geometry €

dΔpdt

= A du dt

Z + u dAdt

Z + ˙ Q dZdt

We now discuss each of these in turn and what circuit elements they represent

Inductance—inertial terms

We effectively have studied systems at steady states in rigid tubes or steady state oscillation in elastic tube. We have not studied acceleration of fluid to establish steady state. These are due to inertial effects.

Inertial terms

rigid tube

€

dΔpL

dt= A du

dtZ =

d(AR u )dt

Z =d ˙ Q Rdt

Z

By Newton’s laws

€

F = ma⇒ΔpL A =mV

Va = ρl d(AR u )dt

= ρl d ˙ Q Rdt

Definition of Inductance

€

L =ρlA

By analogy with circuits, we identify the inductance €

ΔpL =ρlA

$

% &

'

( )

d ˙ Q Rdt

≡ L d ˙ Q Rdt

Just depends on density (mass) of fluid and geometry of vessel

Characteristic time constant

€

tL =LZ

Using our basic flow equation and assuming impedance is not changing with time for rigid tube

€

ΔpL = L d ˙ Q Rdt

=LZ

dΔpdt

≡ tLdΔpdt

This allows us to identify the time constant

Total pressure drop, not just that due to inductance

Ignoring changes in area (capacitance)

Integrating this equation

the subscript R is for rigid tube

€

dΔpdt

=ddt

L d ˙ Q Rdt

#

$ %

&

' ( + ˙ Q R

dZdt

€

Δp = L d ˙ Q Rdt

+ ˙ Q R Z

Solving the differential equation

€

˙ Q R (t) =e− t / tL

LdtΔp∫ e− t / tL

Solve the homogeneous equation, and then use that solution to solve the inhomogeneous part

Example 1: Constant pressure drop

€

Δp = Δp0

€

˙ Q R (t) =Δp0

Z1− e− t / tL( )

As t>>1, the exponential piece goes to zero, and we get the steady-state result of p/Z. Thus, tL really is the time constant that determines the time delay of the response. The higher the density (the more massive the fluid), the greater L and tL and the slower the approach. When length (l) is larger, inertial effects are distributed, and when the area (A) is larger, more of the fluid is right in front of you.

Decay rates for different values of impedance time constant

Example 2: oscillatory pressure drop

€

Δp = Δp0 sin ωt( )

€

˙ Q R (t) =Δp0

Z 2 +ω 2L2sin ωt −θ( ) − ωL

Z 2 +ω 2L2e−t / tL

&

' (

)

* +

As t>>tL, exponential piece goes to zero, and we get a steady oscillatory flow as is expected. Also, as density goes to zero, we get a simpler oscillatory flow that is what we expect. Note that now there is a “phase lag” that depends on tL.

€

θ = tan−1 ωtL( )

Phase lags for different values of impedance time constant

Capacitance

Capacitance (elasticity) terms

Capacitance term due to elasticity of walls and changing cross-sectional area

€

u dAdt

Z =d ˙ Q Cdt

Z

€

d ˙ Q Cdt

=˙ Q R + ˙ Q C − ˙ Q R

tC

=˙ Q CtC

Change from relaxed “rigid” vessel to expanded vessel can be written as

Definition of Capacitance

Relationship to elasticity

€

C =ΔVΔ(Δp)

≡˙ Q CΔtΔ(Δp)

Recall that higher values of E correspond to more rigidity. Thus, small values of E correspond to higher ability to expand to hold fluid and thus higher capacitance.

€

E =Δ(Δp)ΔV /V( )

=VC

Characteristic time constant

From previous two slides

Therefore, we identify the time constant as

€

tc

ZdΔpc

dt= ˙ Q C = C dΔpc

dt

€

tc = CZ

Ignoring inertial terms (inductance)

Integrating this equation

the subscript R is for rigid tube

€

dΔpdt

=˙ Q CC

+ ˙ Q RdZdt

€

Δp =1C

dt∫ ˙ Q C (t) + ˙ Q R Z

Conservation of fluid

€

˙ Q TOT = ˙ Q C + ˙ Q R

expanded tube

rigid tube

Example 1: Constant pressure drop

€

Δp = Δp0

€

C = 0⇒ ˙ Q C = 0⇒ ˙ Q R =Δp0

Z

If the pressure drop is not changing and we are ignoring inertial terms, then no expansion is occurring, so the steady state can be considered the fixed or “rigid” tube.

Example 2: linearly increasing pressure drop over some time T

€

Δp = Δp0tT

€

˙ Q C (t) = C Δp0

T=

CZT

˙ Q (t) ≡ tC

T˙ Q (t)

Allows us to see that the time constant does exactly determine the rate of the dynamics. Higher values of tC lead to higher values of capacitance and flow rate into expanding part of vessel.

Effects on flow rate for oscillatory pressure gradient