ENTC 4350

COMPONENTS 2

THE BLOOD FLOW ANALOGY

To begin the analogy, let’s consider the heart as a pump,• which produces a maximum pressure P

during systole.

This pressure induces a flow of blood through the aorta and the arteries that lead from the aorta to various parts of the body.

Note that the arteries involved— the brachiocephalic, the left common carotid, and the left subuclavian—are of different diameters.

This suggests that the flow of blood, in milliliters per minute (ml/min), is different in each artery, with the largest flow passing through the largest artery. • We will symbolize the rate of blood flow with the

letterQ.

Pressure and Resistance

Being scientific types, we wish to go beyond this mere observation and seek a mathematical relationship between the blood pressure, the arterial diameter, and the flow of blood.

This relation, or equation, for blood flow is analogous to what we shall call Ohm’s law, and we will write it as:

)resistanceR

pressurePminmlQ

(

)()/(

Here we have introduced the new symbols P for pressure (in, say, millimeters of mercury, or mm Hg) and R for flow resistance.

The term flow resistance denotes the combined effects of the arterial diameter, the frictional losses as the blood moves through the artery, and so on. • It should be obvious that larger vessels offer less

resistance, in other words, ~ the ~irterial diameter d goes up.

In fact, since R is inversely proportional to the cross-sectional area (A) of the artery. • It is inversely proportional to the square of the

diameter (1/d2), because A = ¼d2 (where r = 3.14).

• The point to remember here is that the blood flow increases with arterial pressure P and decreases with flow resistance R.

The point is that this pressure drop is caused by the flow resistance R of the body. • The heart uses energy to pump the blood

pressure from 4 to 20 mm Hg. • This pressure pushes the blood through the body,

but it is gradually used up in the process, so that the blood returns to the heart at its original pressure of 4 mm Hg.

Rate of Energy Loss or Heat Production

The heart, then, is a pump whose task is to raise the blood pressure so that the blood can flow through the body. • The next questions we might ask are how is

the pressure drop related to the flow resistance, and what is the rate of energy loss, or heat production, as a function of flow resistance?

The drop in pressure, which we write as P = P1 P2, is equal to the product of the rate of flow times the flow resistance R, or:

RQP

For a given flow, the pressure drop is greatest across the greatest resistance.

The loss of energy law is a little more complex, and here we must ask you to accept the idea that the rate of energy loss is proportional to the square of the flow, of Q2, times the flow resistance R. • We write this as:

RQlossenergyofrateW 2)(

We write this as:

where W is expressed in watts. • Watts are often used as units of heat; for

example, electrical heaters are rated in watts.

RQlossenergyofrateW 2)(

The fact that the flow rate of blood is related to heat production has been known since the beginning of medicine. • Hippocrates noted that inflamed, infected, or injured

areas could be detected by their temperature; it was higher than that of other areas of the body.

• We know now that the body sends a higher flow of blood plus necessary white cells and fibrinogen to such injured areas in order to fight infection and promote healing.

• This extra flow of blood is to a large extent responsible for the higher temperature.

If the blood flow Q goes up, W must go up much faster, since W goes up with the square of the flow rate.

Power Output

Power is the rate at which work is done. The heart is a pump, and as such it

obtains its energy from the blood supply that is provided by the coronary arteries.

To permit an adequate flow of blood, we must have, first an adequate blood pressure P, and second, the necessary flow Q.

If the arteries are blocked by fatty deposits, the flow will be inadequate no matter how much pressure is provided, i.e., even if hypertension exists.

An even worse situation Occurs when the heart cannot provide an adequate pressure, as in fibrillation. • Even the best of arteries are no help in this case, and

you have some three minutes to get the heart going again before it is too late.

It follows from all this that the rate at which the heart (or any other organ) can work—i.e., its rate of energy production—is determined by the product of P, the blood pressure, and Q, the rate at which blood flows to the organ.

This relationship may be written as:

where W is also expressed in watts.

QPW

Summary of Relationships

1. Given a blood pressure P and an artery with flow resistance R, the flow of blood will be Q = P/R.

2. As blood is pushed through the body, the pressure P drops at a rate given by P = QR.

The greater the blood flow or the greater the arterial resistance, the larger will he the pressure drop.

3. The heat produced in the tissue by the flow of blood is given by W = Q2R.

If the blood flow doubles, the rate of heat generation goes up by a factor of four.

4. The power that the heart can deliver is given by the product PQ, which we called W (power output) or W = PQ.

ELECTRICAL RELATIONSHIPS AND OHM’S LAW

To transfer the knowledge expressed in the previous relationships to the world of electricity, we need only note that with electricity, • the unit of pressure will be V (volts) instead of P (mm

Hg),

• the unit of flow will be I (amps) instead of Q (mI/min), and

• best of all, the symbol for resistance. R, will remain the same and its unit will acquire a name. “ohms.”

We can rewrite all our fluid equations so they can be used with electricity just by substituting symbols: • P = V,

• Q = I, and

• R = R.

The following are the fundamental relationships that you will be using throughout this book.• I suggest that you memorize these few rules.

• The first and foremost electrical relationship that we will focus upon is Ohm’s law:

• where

• I = amps (current),

• V = volts (voltage), and

• R = ohms (resistance)

R

VI

Ohm’s Law is the most fundamental law of electricity. • It may also be written as:

where V means the difference in voltage between two points, or V2 V1. • You might just as well commit Ohm’s law to memory;

• we will be using it over and over again.

RIV

The next relationships are our electrical power equations. • The first is the power loss equation:

• where W, expressed in watts, is the power loss (through heat production) and I and R mean the same as in Ohm’s law.

RIW 2

The second is the power output equation:

• where W means the rate of work output, which may also be expressed in watts, and I and V are as before.

IVW

• A watt, as a unit of power, is used for any kind of power, • whether it is the power output of an organ or a

machine,

• the rate at which energy is lost through heat, or even electrical power.

• A watt is a watt, regardless of its source.

It thus provides us with a concept for comparing different kinds of energy: • for example, an electrical motor may require

so many watts of electrical power (1000 watts or 1 kilowatt),

• it may put out so many watts of mechanical power (say, 750 watts, or about I horsepower), and,

• in running, it may dissipate so many watts of heat (250 watts).

For electricity, it is convenient to remember watts = amps x volts (W = IV).

It should be kept in mind that engineers make a distinction between power (watts) and energy. • Power is the rate of energy loss or production:

• It may be expressed as energy per unit time.

If we multiply power times a unit of time (say, seconds), we cancel out the unit of time, leaving units of energy (A/B x B = A). • Thus, a common measure of energy is the

watt-second (watts seconds), • You will see this unit again when we discuss the

defibrillator.

Energy is always conserved: • What goes in, must come out, and vice versa.

• Energy, being conserved, is never really lost or produced, but when we speak of energy loss or energy production, we mean its loss or production for useful purposes.

The purpose of the heart as a pump is to move blood, and we spoke of the heat produced by the flowing blood as energy lost. • It is energy lost from the body, and it is lost in

that it can no longer serve to move more blood, but this energy is in no way destroyed.

Summary of Basic Units

A voIt is a measure of electrical force or pressure, and it is defined as the difference in electrical potential between two points a and b.

An ampere, or amp, is a measure of the flow of electricity, or current, and is defined as the number of electrical charges flowing through a conductor per unit time.• Physicists think of current as a flow of electrons, which

are small, electrically charged particles.

An ohm is a measure of the resistance to such a flow of electricity, and this resistance is property of the material that the current flows through.• Certain materials, like metal, are good

conductors, whereas others, like plastics, are poor conductors, or nonconductors.• Nonconductors are called insulators.

Now that we have told you what these terms mean, • You can forget all these definitions.

• All you need to remember is Ohm ‘s Law, I = V/R, and the formulas relating to it.

We are concerned with using these concepts, not their meaning.

In discussing the definition of volt, we use the terms positive and negative in connection with the terminals on a battery.• You will often see these marked as + and - .

• The designations positive and ‘negative, which were originally introduced by Benjamin Franklin (believe it or not) are arbitrary conventions;

• It does not make any difference which terminal is called positive and which one negative, as long as you keep them straight.

• If you get them mixed up, you may get what is called a short circuit, which, among other adverse side effects, is demonstrated by a lot of sparks.

SERIES AND PARALLEL

Resistance The figure shows a typical freeway traffic situation

where four lanes are squeezed into two by construction.

The distribution of automobiles is shown by the black dots. • The density of the dots is a result of the local resistances, R

that the different road segments offer to traffic. • We have marked the resistance of the four-lane road segment

R1 and the construction area as R2.

Anyone who has driven on the freeway knows that the resistance R2 is greater than R1, and that the flow of vehicles throughout this entire section of highway will be controlled primarily by R2.

To write this in mathematical form, we define I as vehicles passing R2 per minute,V as vehicles per mile of highway, and R as the resistance to vehicle flow. • We can now write:

21 RR

VI

This simply means that the flow of vehicles per minute is equal to the number of vehicles per mile divided by the sum (total) of the different resistances (in some units or other).

• The point is that if R2 equals some large number, say 10 k,

• while R1 is a much smaller number, say, 1,

• then we can write I = VIR2. because 1000 + 1 1000. • You should have no trouble recognizing the equation I = V/R by

now.)

This confirms our intuitive hunch that the traffic flow will be controlled primarily by the greatest resistance, R2.

The pressure drop across R2, in terms of the values of V (vehicles per mile) on the upstream and downstream side of R2. is large.

The energy loss at the bottleneck—which may be described by W = PR—appears as heat, wasted gasoline, lost tempers, crying children, and banged fenders.

Obviously. W will go up as both I2 and R go up.

The Voltage Divider—Resistors in Series

Two resistors and a battery are shown in a series circuit below. • The word series simply means that the

current flows through the resistors one after the other.

The battery can be thought of as a pump with an output pressure of V volts. • The pressure of the returning electricity will be taken as

zero.

• We know that:

• Because the individual drops in pressure (V1 and V2) must equal the total pressure drop, which is the difference between the output of the battery (V) and the returning pressure to the battery (zero).

21 VVV

Also

• because the electrical situation is analogous to the previous traffic flow example.

The flow of current is impeded by the resistors just as the flow of traffic was impeded by the construction bottleneck.• The small resistor (R1) corresponds to the normal

resistance of the four-lane highway, and the larger resistance (R2) corresponds to the two-lane detour.

21 RR

VI

If R2 = 10 R1, intuition tells us that the voltage pressure drop across R2 will be greater than that across R1.• Since the same current flows through each

resistor,

11 RIV

22 RIV

Since the current is the same through both resistors, I, is the same in both equations.

The current is same throughout the circuit and is given by the equation:

21 RR

VI

Substituting in the equations for voltages yields:

• This the voltage divider equation.

• The voltage across any resistor in a series circuit is the supply voltage (V) times the resistor divided all the resistors in the series circuit.

21

11

21111 RR

VRR

RR

VVRIV

21

22

21222 RR

VRR

RR

VVRIV

Note that the largest drop in pressure occurs across the largest resistance in the circuit.

THE CURRENT DIVIDER—RESISTORS IN PARALLEL

Let’s look at the external iliac and femoral arteries and show both the deep femoral and femoral arteries as they really exist.

The flow in the femoral artery (Q1) parallels that in the deep femoral artery (Q2).• The term parallel flow implies that the blood

flows through both arteries at the same time.

Note that the same pressure P is applied to both arteries at the same time.• So,

and

11 R

PQ

22 R

PQ

If the femoral artery is partially blocked by a clot, R1 will go up.• This means that Q1 will go down because:

To keep total flow Q = Q1 + Q2 constant, Q2 must go up.• This is called collateral circulation.

11 R

PQ

Suppose we want to calculate Q1 and Q2 when we know Q, R1, and R2.

We noted that Q = Q1 + Q2, so

22

11 R

PQ

R

PQ

R

PQ

21 R

P

R

P

R

P

Note that,

• This is the equation for parallel resistances.• Note the differences between resistances in series.

2121

111

RRRR

P

R

P

R

P

21 RRR

21

111

RRR

21

1

21

21

RR

R

RR

R

R

21

121

RR

RR

R

12

21

21

12

1

/1

1

RR

RR

RRRRR