Download The cardiovascular system, centerpiece of our study, is completely

Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
The cardiovascular system, centerpiece of our study, is completely analogous
to the electrical circuits in which we are now interested. In fact, for every closed fluid
system, there is an electrical circuit whose behavior is identical (up to conversion
factors).
2.1 Basic Components and Laws for Cardiovascular and Electrical system.
The heart develops a pressure difference which moves the blood through the system.
Similarly, a battery (or other power supply) develops a voltage "Potential Difference"
which moves electrons around the circuit (to a position of least energy). Thus a fluid
pressure drop (energy per unit volume) corresponds to a potential difference, or
"voltage drop" (energy per unit charge). Likewise, the flow rate of a fluid is analogous
to the "current" in a circuit, which is the rate of movement of charge.
Poiseuille's Law corresponds to "Ohm's Law" for electrical circuits. If we solve
Poiseuille's Law for the pressure drop, and replace pressure drop by voltage V and flow
rate by current I, we have
V = I (8 η l / π r 4), where r is radius of vessel.
(2.1)
The quantity in parentheses has the characteristics of a "resistance": for a given voltage
drop, a larger value of l results in less flow, while a larger value of r results in greater
flow.
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
In fact, the electrical analog of this fluid quantity is the resistance
R = ρ L / A,
(2.2)
where ρ is the "resistivity", L is the length of the conductor, and A is its radius. The
fluid analog of the resistivity is essentially the ratio of viscosity to cross sectional area.
Fluid conservation principles are also applicable to electrical circuits. In the
cardiovascular case, the pressure drop from the ventricle to the atrium was equal to the
pressure at the ventricle; in the electrical case, the voltage drop around any closed loop
is equal to the voltage of the source (battery or power supply) in that loop (or zero if
there is no source in that loop). This is essentially the conservation of energy. In the
cardiovascular case, the flow into a branch equaled the sum of the flows out; in the
electrical case, the same is true for currents at a junction of conductors: conservation of
charge. In the electrical parlance, these principles are known as "Kirchhoff’s Laws".
These laws lead to the following rules for treating voltages and currents in series and
parallel configurations:
1. The voltage drop across two components in series is equal to the sum of the
voltage drops across each component. Think of the pressure drops in the arteries
and in the arterioles. This implies that for any loop containing one or more
resistors and a battery, the resistors "use up" the potential energy provided by
the battery. Note that they do not change the current: kinetic energy is not
affected.
2. The current flowing through two components in series are equal. If an arteriole
and a venule were joined by a single capillary, the fluid flow through them all
would be the same.
3. The voltage drops across two components in parallel are equal. This
corresponds to a branching of an artery into two identical arterioles, as does:
4. The current flowing through two components in parallel is equal to the sum of
the currents flowing through each component. This implies that the resistances
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
-through alternate paths determine how much current flows through each of
them: electricity tends to the path of least resistance.
These rules make possible the simplification of circuits with resistors in series or
parallel by replacing them with equivalent resistors. Since voltage and resistance are
proportional (in Ohm's Law), rule 1 tells us that we can replace two resistors in series
by one whose resistance equals the sum of the original two. Since resistance and current
are inversely proportional, rule 4 gives us the following equation for a resistor
equivalent to two in parallel: 1/ R eq. = 1 / R 1 + 1 / R 2.
(2.3)
For example, consider the following circuit:
Fig.2.1 A Circuit
The two resistors on the right are in series. They can be replaced by an equivalent
resistor:
Fig.2.2 Equivalent circuit to Fig 2.1
The two resistors on the right in this new equivalent circuit are now in parallel. They
can be replaced by another equivalent resistor:
Fig.2.3 Equivalent circuit to Fig. 2.2
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
Note that the equivalent resistor has a Smaller value than either of the originals;
this is a general feature of resistors in parallel, since the current has alternate paths.
Finally, the last two resistors are in series and can be replaced by one 5 Ω resistor.
Since the 5 Ω resistor is in parallel with the battery, the voltage drop across it is 10 V.
Using Ohm's Law, the current through that resistor is 2 A. Now, working backwards,
we know that the current across both of the above resistors is 2 A, since they are in
series. This means that the voltage drop across the 2 Ω resistor is 4 V, and that through
the 3 Ω resistor is 6 V. Note that the sum of the voltage drops is equal to the battery
voltage. Returning now to the previous diagram, the voltage drop across the 2 resistors
on the right are both 4 V (since they are in parallel). This means that the currents running
through them are 1 1/3 A (for the 3 Ω) and 2/3 A (for the 6 Ω). Note that the sum of
these currents is equal to the original current, as it should be. Returning now to the
original diagram, the current through both right-hand resistors is 2/3 A, since they are
in series, and so their voltage drops are both 2 V. Once again, the sum of the voltage
drops is equal to the original voltage drop. The final voltages and currents are:
Fig.2.4 Final voltage and Current Circuit
We can verify that Kirchoff's Laws are obeyed by unique answers.. It is possible (and
in general necessary) to use the techniques of linear algebra (solving multiple
simultaneous linear equations expressing Kirchoff's Laws
The steps to solving one of these circuit problems are then:
1. Simplify all of the obvious series and parallel resistors. Remember that unlike
components in parallel can be interchanged.
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
2. Repeat 1 untill all you have is one or more resistors all in parallel with the
battery. You will now have a series of more and more simplified equivalent
circuits.
3. Use Ohm's Law to find the current through the resistor(s) in the most simplified
circuit.
4. Use that current with Ohm's Law to find the voltage drops across all series
resistors you simplified in the previous circuit.
5. Use the voltage drops you found in 4 to find the currents in all of the parallel
resistors you simplified in the circuit before.
6. Repeat 4 and 5 on the successively more complex circuits until you have found
all of the currents and voltage drops.
Capacitors and RC Circuits:
The role of the capacitor in the electrical / fluid analogy is a little more complicated
than that of the resistor. A capacitor is a device which allows us to store charge, and in
this sense is analogous to the arterial walls, which store energy during pulsatile flow.
However, the electrical analog to pulsatile flow is "alternating current" ("A.C."), and
we would need to use calculus to adequately understand it. Fortunately, we can learn a
great deal about electrical systems without getting into alternating current.
Capacitors store separated electric charges, and therefore energy. The "capacitance" of
a capacitor is stated in terms of the amount of charge (Q) stored at a given voltage drop
(across the capacitor):
C = Q / V.
(2.4)
The unit of capacitance is the "Farad", which is equal to a C / V. It is a very large unit;
typical capacitances are measured in microfarads or picofarads. The canonical
capacitor consists of a dielectric substance sandwiched between two parallel
conducting plates. It has capacitance
C = ε A / d,
(2.5)
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
where A is the plate surface area and d is the separation distance between the plates. In
general, the capacitance of any capacitor depends only on the dielectric constant and
the geometry of the capacitor. This is equivalent to the dependence of the energy of a
charge configuration on the permittivity and the separation distances. The stored energy
in a given capacitor at a given voltage is
C V2 / 2
(2.6)
Since the capacitor stores potential energy, there is a force trying to pull the plates
together.
For a membrane, the capacitance C is measured per unit length. The membrane must
be strong enough to resist the force resulting from the separation of charges on either
side. Biological membranes typically have a capacitance of from 0.5 to 1.3 x 10 - 10 F /
m. This means that they must store a charge on the order of 4 x 10 7 ions / m to create a
rest potential. This corresponds to a stored energy of about 2 x 10 - 13 J / m. For a
membrane thickness of 10 - 8 m, the force is on the order of 2 x 10 - 5 N / m.
Kirchoff's Laws also lead to rules for simplifying circuits with capacitors in series or
parallel. Since capacitance is inversely proportional to voltage, capacitors in series are
treated like resistors in parallel:
1/ C eq. = 1 / C 1 + 1 / C 2.
(2.7)
Since capacitance is proportional to charge, which is proportional to current, the
capacitance of a pair of parallel capacitors is simply the sum of their capacitances.
A circuit consisting of a battery, a switch, a resistor and a capacitor in a series loop is
called an "RC" circuit. Kirchhoff’s voltage law for this circuit is
V=IR+Q/C
(2.8)
When expressed purely in terms of the charge, this becomes
V = (δQ/δt) R + Q / C
(2.9)
This is a "differential" equation, whose solution is an exponential function.
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
2.2 Circuit Topology
A "circuit" is any arrangement of "components" and conductors (i.e. wires) which don't
have "loose" ends. That is, they are composed of closed loops which are connected
together in various ways. For examples:
Fig.2.5 Circuit Topology
In circuits, two (or more) components are said to be in "series" if any electrons which
go through the second one must have come from the first. They are said to be in parallel
if they provide alternate paths for the electrons to go when starting and coming together
again at common points.
The "topology" of a circuit defines how its various parts are connected (in series
and parallel). The reason it is tricky is that it is based on connectivity, and not on visual
appearance. For instance, the components in these circuit fragments are all in "series":
Fig.2.6 Component in Series
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
while these are all in "parallel":
Fig.2.7 Components in Parallel
2.3 Fluid Flow and Circuit:
There are several obvious analogies between circulatory systems and electrical circuits.
Current is analogous to the flow of blood, resistors dissipate energy just as viscosity
does and batteries provide the energy for the movement of charge carriers in a rough
analogy to the beating heart. Of course, batteries are a source of direct current, whose
direction is constant corresponding to a constant potential difference. In contrast, the
beating heart causes the flow of blood to be pulsatile: the blood pressure is a periodic
function of time. This pulsatile flow persists throughout the arterial system, where the
periodic voltage of alternating current provides a much better analogy. In the venous
system, however, the flow of blood is nearly laminar, and we do not need alternating
current to make our analogy with electrical circuits more precise.
We start by observing that the units of pressure are the same as the units of energy
density: energy per unit volume. If we equate a volume of blood to a quantity of charge,
we see that pressure is strictly analogous to voltage. The same correspondence means
that flow is strictly analogous to current: volume per unit time corresponds to charge
per unit time. The fluid analogy to resistance can then be seen if we solve Poiseuille's
Equation for the ratio of pressure drop to flow, which is analogous to the ratio of voltage
to current from Ohm's Law:
ΔP / F = 8 η l / (π r4)
(2.10)
The resistance of a length of wire is greater for a longer piece of wire, and less for a
thicker piece of wire, just as the flow of water from a hose is less for longer hoses and
greater for wider hoses. Thicker wires allow more charge carriers to flow
simultaneously, while it takes more work to move the charge carriers a longer distance.
This is all consistent with an equation you constructed earlier:
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
R=ρl/A
(2.11)
where ρ is the resistivity, l is the length of the wire and A is its cross-sectional area.
Comparing this equation to the one above, we see that if
ρ = 8 η / r2
(2.12)
the analogy is complete. If you have a network of veins whose pressure drops are equal
in each hierarchical layer in the network (a not unreasonable assumption, left):
Fig.2.8 Vessel – Circuit Analogy
we can compute pressure drops and flows by replacing it with the electrical circuit on
the right and equating currents to flows and voltage drops to pressure drops.
We can also see that Kirchhoff's Laws apply equally well to circulatory systems:
if a vessel branches into two, or if two vessels join, the total flow through the two must
equal the flow through the single vessel. What was conservation of charge is now
conservation of matter (blood). Similarly, in a closed fluid system, the sum of the
pressure drops must equal zero: the increased blood pressure at the entrance of the aorta
from the heart at systole (the peak of the blood pressure increase during a heartbeat) is
completely dissipated by the time the blood returns from the vena cava into the heart:
the blood there is barely moving, and in fact requires help from the movement of
surrounding muscles and a system of one-way valves. Of course, circulatory systems
are not strictly closed: fluid can pass through the membranes of the blood vessels and,
of course, the vessels themselves can be broken. But we can ignore these small and
exceptional events in the analyses we are about to undertake.
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
In real circulatory systems, one vessel may split into two or two vessels may
join into one. Hence while the entire system is a vastly complicated network of billions
of vessels, flow problems ultimately come down to three vessels at a time.
Most of the electrical systems can be modelled by three basic elements:
Resistor, Inductor, and Capacitor. Circuits consisting of these three elements are
analyzed by using Kirchhoff's Voltage law and Current law.
2.4 Mathematical Modeling for RLC Circuit:
Circuit model of resister is shown in following fig.
Fig.2.9 Resister
The mathematical model is given by the Ohm's law relationship:
V(t) = i(t) R, i.e. i(t) = V(t) | R
(2.13)
Circuit model of inductor is shown in following fig.
Fig. 2.10 Inductor
The input output relations are given by Faraday's law,
𝑉(𝑡) = 𝐿
𝑑𝑖(𝑡)
𝑑𝑡
(2.14)
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
1
𝑖(𝑡) = 𝐿 ∫ 𝑉(𝑡)𝑑𝑡
(2.15)
Circuit model of capacitor is shown in following fig.
Fig. 2.11 Circuit Model of Capacitor
1
𝑉(𝑡) = 𝐶 ∫ 𝑖(𝑡)𝑑𝑡
Or 𝑖(𝑡) = 𝐶
𝑑𝑉(𝑡)
𝑑𝑡
(2.16)
(2.17)
∫ 𝑖(𝑡)𝑑𝑡 is known as the charge on the capacitor and is denoted by 'q’.
Thus 𝑉(𝑡) =
𝑞(𝑡)
(2.18)
𝐶
Example 2.1 Consider the network in following Fig. 2.12. Obtain the relation between
the applied voltage and the current in the form of Differential equation.
Fig. 2.12 An R, L, C series Circuit excited by a voltage Source
Solution: By Kirchhoff's voltage law equation for the loop, we have
𝑅 𝑖(𝑡) + 𝐿
𝑑𝑖(𝑡)
𝑑𝑡
1
𝑡
+ ∫−∞ 𝑖(𝑡)𝑑𝑡 = 𝑉
𝐶
(2.19)
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
𝑡
Denoting ∫−∞ 𝑖(𝑡)𝑑𝑡 = 𝑞(𝑡) , above equation can also written as
𝐿
𝑑2 𝑞(𝑡)
𝑑𝑡 2
+𝑅
𝑑𝑞(𝑡)
𝑑𝑡
+
𝑞(𝑡)
𝐶
=𝑉
(2.20)
Which is second order linear differential equation with constant coefficient.
Example 2.2 Consider the parallel RLC network excited by a current source (Fig. 2.13).
Find the Differential equation representation of the system..
Fig.2.13 Parallel RLC circuit excited by a current source
Solution: Applying Kirchhoff's current law at the node,
𝑉(𝑡)
𝑅
+𝐶
𝑑𝑉(𝑡)
𝑑𝑡
1
+ 𝐿 ∫ 𝑉(𝑡)𝑑𝑡 = 𝑖(𝑡)
(2.21)
Replacing∫ 𝑉(𝑡)𝑑𝑡 = 𝜓(𝑡), flux Linkages, we have,
𝐶
𝑑2 𝜓(𝑡)
𝑑𝑡 2
1 𝑑𝜓(𝑡)
+𝑅
𝑑𝑡
+
𝜓(𝑡)
𝐿
= 𝑖(𝑡)
(2.22)
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
Analogy of Cardiovascular System with Electrical circuit:
Table: 2.1 Analogy of Cardiovascular System with Electrical circuit.
Cardiovascular
System
Electronic
Circuit
& Symbols
Equation
Relations
For Electrical
Circuit
Equation
Relations
For
Cardiovascula
r System
Vessel
Resistance (R c )
Electrical
Resistance(R e )
v = I Re
V = Voltage
I = Current Flow
R e = Circuit’s
Resistance
P = FR c
P = Pressure
F = blood Flow
R c = vessel’s
Resistance
Vessel
Compliance
(Cc )
Capacitance (Ce )
dv
=I
dt
Ce = Circuit
Capacitor
dp
=F
dt
Cc = Vessel’s
compliance
Blood inertia (Lc )
Inductance (Le )
Valve
Diode
Ce
Le
dI
=v
dt
0 if v < 0
I = { v if v ≥ 0
R
e
Cc
Lc
dF
=P
dt
0 if P < 0
F ={ P if P ≥ 0
R
c
To illustrate the use of the circuit analogy, we will model a single uniform
Section of an artery. The artery will be modeled as flexible, and it will be connected to
a simulated end load. A steady state solution for pressure and flow rate will be
calculated and interpreted.
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
Fig. 2.14 Equivalent electrical circuit for Blood vessel
2.5 Lumped Parameter Model:
A mathematical model of a physical system which is simplified from spatially
distributed variables in a single scale variable is called a lumped parameter model.
Lumped Parameter Model is OD model. Lumped parameter models are in common use
for studying the factors that affect pressure and flow waveforms.
Rideout et al [20], [21],[22],[23] used a lumped parameter model concept in
studying various parts of circulatory systems in their research. While developing a
model of the human systemic arterial tree, Snyder et al. (1968) [25] used an equal
volume modeling feature in the simulation. According to the equal volume feature, the
arterial system has been divided into sections in which length and cross sectional area
was inversely proportional [25]. The model was based on lumped parameter
approximation and has been studied with the use of analog computers and electrical
network analogs.
A lumped parameter model is one in which the continuous variation of the
system’s state variables in space is represented by a finite number of variables, defined
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Chapter 2 Analogy of Cardiovascular System with Electrical Circuit – Lumped Parameter Model
at special points called nodes. The model would be less computationally expensive,
with a correspondingly lower spatial resolution, while still providing useful information
at important points within the model.
Lumped parameter models are good for helping to study the relationship of
cardiac output to peripheral loads, for example, but because of the finite number of
lumped elements, they cannot model the higher spatial-resolution aspects of the system
without adding many, many more elements.
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