Modeling Vascular Networks
with Applications
Instructor: Van Savage
Winter 2012 Quarter
Meeting time: Monday and
Wednesday, 10:00-11:50 am
Lumped Models
Ignore details of spatial structure and
use “lumped” or circuit model
Rules for constructing differential
equation for elements in series versus
elements in parallel
Basic circuit elements: resistor, capacitor, inductor
All are considered to be independent even though they really
are not for blood flow.
In SERIES: Sum of pressure drops across each element.
Volume flow rate is same across each element.
In PARALLEL: Sum across volume flow rates for each element
Pressure drop is same across each element.
Each element has its own relation for pressure drop and flow
rate, and since these are independent, we just sum across
them in different ways for different circuit configurations.
Relationship for each element in
isolation, as derived last class
Resistor:
Capacitor:
!
Inductor:
!
"p = Q˙ R (t)Z
1
"p =
C
˙
dt
Q
# C (t)
dQ˙ R
"p = L
dt
Fluid analogy
But in reality, can’t really draw separate elements like this for fluid.
Examples of “lumped” models
1.  Windkessel model: Resistor and capacitor in parallel
2. Viscoelasticity: Resistor in parallel with a resistor and
capacitor in series
3. Inertia: Resistor-Inductor pair in series and resistor-capacitor
in series with these two pairs in parallel with one another.
4. Back pressure: Resistor-back pressure pair in series and
resistor-capacitor in series with these two pairs in parallel
with one another.
Windkessel model
Windkessel model
1.  First developed by Otto Frank in 1899 and is German for
elastic reservoir
2.  Heart working as pump has systolic (contraction and high
pressure) and diastolic (relaxation and low pressure)
3.  Represent systole by capacitor (balloon) that absorbs the
high pressure and flow rate by expanding elastic vessels
4.  Represent diastole by forward (axial flow down vessel with
flow from discharging capacitor deflating balloon) that is
resisted/dissipated so represented by resistor
5.  At any time flow can go to expansion or forward movement,
so these two elements are considered to be in parallel
Windkessel diagram
!
Equivalent impedance for whole
circuit
Represent pressure from heart as simple plane wave
"p = "p0e
i#t
Impedance from capacitor is thus found by using Ohm’s law
"p
1
! d"p
˙
QC = C
= Ci#"p $ ZC =
=
dt
Q˙ C i#C
Equivalent impedance for whole
circuit
Since resistor and capacitor are in parallel we add by inverses
1
1
1
1
=
+
= + i"C
Z eq Z R ZC R
and thus
!
R
R
Z eq =
=
1+ i"CR 1+ i"tC
Recall definitions for complex
impedance
Z = Z diss + iZ ref
From our equation when C=0
Z diss = R
!
From our equation when R=0
!
Z ref
1
=
i"C
Captures some aspects of wave reflections and this term is also
called reactance
Recall definitions for complex
impedance
Z = Z diss + iZ ref
For tC<<1 we make the approximation
!
Z eq = R e
"i#tC
so
Z diss = R cos("tC )
!
and
Z ref = "R sin(#tC )
Dissipation and reflection depend on both circuit elements
Dynamics of flow
tC=0.01s
Dynamics of flow
tC=0.2 s
Capacitive flow depends on slope
of pressure
tC=0.01 s
Capacitive flow depends on slope
of pressure
tC=0.2 s
Uses for Windkessel model
1.  Modeling systemic blood flow
2.  Modeling coronary blood flow
3.  Modeling atherosclerosis by changing values of capacitance
and resistance of vessels
4.  Modeling effects of medication that change capacitance and
resistance of vessels
5.  Modeling pathologies that affect viscosity of blood
(resistance) or compliance/elasticity of vessels (capacitance)
6.  Modeling grafts and stents and their resistance and
capacitance (which is lower than normal vessels)
Problems with Windkessel model
In the 1950’s it became clear there are problems with this
method
1.  Essentially ignores wave reflections at junctions
2.  There is a large capacitive flow reserve beyond what is filled/
expanded during systole
3.  Multi-faceted regulatory mechanisms beside just capacitance
help control flow through vascular network
4.  Most “waves” occur during diastole
No model yet deals with all of these issues successfully.
Viscoelasticity model
Viscoelastic materials
1.  Viscoelastic materials that exhibit both elastic and viscous
properties when pressure is applied. Their response to
pressure changes in time.
2.  Elastic materials respond to changes in pressure at a rate
that is proportional to the volume flow rate. Due to bonds
stretching in an ordered solid but without the order changing.
Like a rubber band or spring.
3.  Viscous materials respond to pressure at a rate proportional
to the time derivative of volume flow rate (i.e., volume flow
acceleration). Diffusion through amorphous solid. Like
molasses or syrup.
4.  All materials are viscoelastic to some degree
http://www.vilastic.com/FAQ_Blood.htm
http://www.vilastic.com/tech10.html
Viscoelastic model
1.  Difficult to add this to structured or unlumped model without
doing some serious fluid mechanics
2.  A capacitor can be used to represent the elastic properties of
vessel
3.  A resistor can be used to represent the viscous or dissipative
properties of a vessel.
4.  Replace elastic vessel walls (capacitor) with viscoelastic
vessel walls (capacitor and resistor in series).
5.  This is the Maxwell model of viscoelasticity. Other versions,
such as the Kelvin-Voigt model (resistor and capacitor in
parallel) with other variations also exist
Viscoelastic Windkessel diagram
Equivalent impedance for whole
circuit
Represent pressure from heart as simple plane wave
"p = "p0e
i#t
If resistance and capacitance are constant (approximation), then
!
d"p Q˙ C
dQ˙ C
=
+ Rv
dt
C
dt
Equivalent impedance for whole
circuit
Using Fourier transforms we derive that the impedance through
the resistor-capacitor pair in series is
Z eq,C
1
= Rv +
i"C
and thus combining with the other resistor in parallel
1+
i
"
t
1+
i
"
R
C
C ,v
v
Z eq = R!diss
= Rdiss
1+ i"C(Rv + Rdiss )
1+ i" (tC ,v + tC )
Recall definitions for complex
impedance
When tv,C and tC are small we can express this as
Z eq ~ Rdiss
e
e
i"t v,C
i" (t v,C +tC )
= Rdisse#i"tC
So, in this limit this is just like the Windkessel model because
there is almost no flow through capacitive element in series to
start so the dissipation through that part of circuit can be
ignored.
!
Outside of this regime the dissipative (real part/resistance) and
reflective (imaginary part/reactance) is more complicated but
can be determined.
Dynamics of flow
tC=0.2s and Rv/Rdiss=2
Dynamics of flow
tC=0.2s and Rv/Rdiss=0.5
Dynamics of flow
tC=0.2s and Rv/Rdiss=0.1
Uses for Viscoelastic Windkessel model
Several diseases, such as cardiovascular disease, peripheral
vascular disease, sickle cell anemia, diabetes, stroke, etc.
can change the values of Rv and C and allow us to model the
effects of these various pathologies
Viscoelastic Windkessel model
with inertia/inductance
Viscoelastic-Inertial models
1.  Inductors are typically ignored in lumped models, probably
because people are interested in steady-state conditions
2.  Inductor and dissipative resistor are both effects that occur in
rigid tubes and since acceleration must occur before steady
state, these effects occur (partially) sequentially, so the
inductor is placed in series with the dissipative resistor
Viscoelastic-Inertial Windkessel diagram
Equivalent impedance for whole
circuit
Represent pressure from heart as simple plane wave
"p = "p0e
i#t
For resistor and inductor in series
!
Z eq,L = Z diss + Z L = Rdiss + i"L
For resistor and capacitor in series
!
Z eq,C
1
= ZV + ZC = RV +
i"C
Equivalent impedance for whole
circuit
Combining these two lumped elements in parallel into one final
lump,
1
1
1
1
1
=
+
=
+
Z eq Z eq,C Z eq,L Rdiss + i"L R + 1
v
i"C
after some algebra
2
(1" #$ tC t L ) + i$ ( #tC + t L )
Z eq = Rdiss
;
2
(1" $ tC t L ) + i$tC (1+ # )
Rv
#=
Rdiss
Resonance, normal modes, and
magic norms
We can check various limits, corresponding to eliminating
elements of the circuit, and everything checks out.
Note that in the special case where
1.  λ=1 so Rv=Rdiss
2.  tC=tL so L=Rdiss2C
We obtain
Z eq = Rdiss
So, total flow is same as if it was just flow through single,
dissipative resistor. Thus, pressure drop and volume flow
rate fall right on top of one another and are in phase. More
!
generally, resonance
depends on value of frequency, ω
Dynamics of flow
tC=0.2s, λ=Rv/Rdiss=2, tL=1.0s
Dynamics of flow
tC=0.2s, λ=Rv/Rdiss=2, tL=0.1s
Dynamics of flow: Normal modes
tC=0.2s, λ=Rv/Rdiss=1, tL=0.2s
Back-pressure model
Back-pressure model
1.  Coronary blood flow is highest during diastolic/low pressure
phase. This is not captured by our models thus far, and
seems counterintuitive.
2.  When heart contracts (systole/high pressure) muscles
around heart in coronary region contract and increase
pressure around vessels, preventing expansion and flow.
3.  Can model this by adding back pressure that works against
axial flow down the vessels. (We ignore inductance.)
4.  Note that this violates the notion of truly being in parallel, and
is analogous to case of asymmetric branching. Could use
this kind of circuit to model asymmetric branching junction.
5.  Forms of impedances are unchanged.
Back-pressure Windkessel diagram
Composite of forward and back
pressure
Dynamics of total flow with no back
pressure-flow is low during diastole
tC=0.1s, λ=Rv/Rdiss=1
Dynamics of total flow—flow is high
during diastole
tC=0.1s, λ=Rv/Rdiss=1
Dynamics of resistive and
capacitive flow with back pressure
tC=0.1s, λ=Rv/Rdiss=1
Waterfall analogy
Topics we did not cover
1.  Numerical simulations: Charles Taylor, Peskin, Olufsen
2.  Fourier components to represent composite pressure and
flow waves
3.  Plant vascular models
4.  Lungs and air transport
5.  Neural architecture and sleep
6.  Intracellular transport
What you have learned
1.  Fractal geometry
2.  Symmetric and asymmetric branching networks
3.  Fluid mechanics—rigid tube, Newtonian fluids and laminar
flow, pulsatile flow, elastic vessel walls, pressure, volume
flow rates, complex impedance
4.  Energy loss to dissipation and wave reflections at junctions
5.  Using energy minimization, space-filling, and constant size of
terminal units to determine fractal-like structure of vascular
network
6.  Structure and flow of network used to explain biological
scaling
What you have learned
7.  Applications to angiogenesis and tumor growth
8.  How to use circuit elements—resistance, capacitance,
inductance—to describe properties—dissipation, elastic
expansion of vessel walls, inertia/mass—of fluid flow through
network
9.  Combining circuit elements in series and in parallel
10. Common versions of lumped models—Windkessel,
viscoelastic, inertial, and back pressure
11. How to analyze data for vascular networks and identify
various types of patterns
What mathematical methods did we
learn?
1. Working with several coupled partial differential equations
2. Separation of variables
3.Fourier transform
4. Bessel functions
5. Dirac delta function
6. Asymptotic expansions and Taylor series
7. Complex analysis
8. Wave-like equations and wave functions
9. Diffusion-type equations
10. Navier-Stokes equations