Ultrasound Physics & Instrumentation
4th Edition
Volume I
Companion Presentation
Frank R. Miele
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Volume I Outline
 Chapter 1: Mathematics
 Level 1
 Level 2
 Chapter 2: Waves
 Chapter 3: Attenuation
 Chapter 4: Pulsed Wave
 Chapter 5: Transducers
 Chapter 6: System Operation
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Mathematics: Level 1
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Why Mathematics Matter
Mathematics is the engine which drives physics.
Without understanding math:
 Physics becomes pure memorization
 Memorization is painful, boring, and not real knowledge
 Without physics knowledge, you will not understand ultrasound
 If you do not understand ultrasound well, your career is not as enjoyable
 Your patients do not get the best care they should receive
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What is Mathematics?
Mathematics is a collection of disciplines.
Most people incorrectly think of math as manipulation of numbers, or arithmetic.
Math is really a set of reasoning skills and tools which include:
 Numerical manipulation
 Equations and relationships
 Measurements
 Angular effects
 Logic and reasoning
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Fractions and Percentages
You should be able to write any fraction in decimal form and vice versa.
Similarly, you should be able to convert any fraction into a percentage
and vice versa.
 1/2 = 0.5 = 50%
 1/3 = 0.33 = 33%
F ractions 
 1/5 = 0.2 = 20%
num erator
denom inator
 1/50 = 0.02 = 2%
 14/100 = 0.14 = 14%
 28/200 = 14/100 = 0.14 = 14%
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Reciprocals
When reciprocals are multiplied the product is 1.
 The reciprocal of 7 is 1/7  7 x 1/7 = 1
 The reciprocal of 2,013 is 1/2,013
 The reciprocal of 1/7 is 7
 The reciprocal of seconds is 1/seconds
 The reciprocal of 1/seconds is seconds
 The reciprocal of 1 MHz is 1/(1 MHz)
 The reciprocal of x is 1/x
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Variables
A letter (abbreviation) which represents a physical quantity.
How much money do you spend on video games if each video game costs
$12.00?
Let M = money spent on video games
Let N = number of video games purchased
Equation: M = $12.00 • N
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Number Raised to a Power
Raising a number to a power is a shorthand notation for multiplication.
In the expression XA, X is called the base and A is called the exponent. When
the exponent is positive, the exponent tells you how many times the base is
used as a factor.
 23 = 2 x 2 x 2 = 8
 25 = 2 x 2 x 2 x 2 x 2 = 32
 52 = 5 x 5 = 25
 55 = 5 x 5 x 5 x 5 x 5 = 3,125
 (1/2)3= 1/2 x 1/2 x 1/2 = 1/8
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Numbers to a Negative Power
A negative exponent tells how many times to use the reciprocal of the
base as a factor.
 2-3 = 1/2 x 1/2 x 1/2 = 1/8
 2-5 = 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/32
 5-2 = 1/5 x 1/5 = 1/25
 5-5 = 1/5 x 1/5 x 1/5 x 1/5 x 1/5 = 1/3,125
 (1/2)-3= 2 x 2 x 2 = 8
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Exponential Notation
Using powers of 10 to simplify large and small numbers
 4,600,000,000 = 4.6 x 109
 0.0000063 = 6.3 x 10-6
 7,100 = 7.1 x 103
 0.00000000047 = 0.47 x 10-9
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Metric Abbreviations
Think about how much easier the metric system is than the English system; all
you have to do is move the decimal point by the number of places specified by
the exponent.
G
M
k
h
da
d
c
m
m
n
= 109
= 106
= 103
= 102
= 101
= 10-1
= 10-2
= 10-3
= 10-6
= 10-9
1,000,000,000
1,000,000
1,000
100
10
0.1
0.01
0.001
0.000001
0.000000001
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Direct Relationships
Fig. 1: Linear Proportional Relationship (Pg 30)
This is a graph of the equation y = 3x. Notice that as x increases, y also
increases. This type of relationship in which both variables change in the same
direction is called a direct (proportional) relationship
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Proportionality
Proportionality is a relationship between variables in which one variable
increases, the other variable also increases.
The symbol for proportionality is 
y  x  if x increases, y increases
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Linear Proportionality
Increase by factor of 2
A proportional relationship between variables, in which, if one variable
increases by x %, the other variable also increases by x %.
y
y=x
5
4
3
2
1
1
2
3
4
5
x
Increase by factor of 2
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Inverse Proportionality
Inverse proportionality is a relationship between variables in which if
one variable increases, the other variable decreases.
For inverse proportionality we still use the same symbol () but write the related
variable in its reciprocal form.
For example, to state y is inversely proportional to x we would write: y  1/x
y 
1
 if x increases , y decreases
x
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Inverse Relationships
Fig. 2: Inverse Proportional Relationship (Pg 31)
This is a graph of an inverse relationship. Notice that as x increases, y
decreases.
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Distance Equation (General)
By multiplying a velocity (rate) by time, the distance is calculated. This
equation is well known to most people since it is commonly employed to
determine how long it will take to drive between two locations.
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Distance Equation (Sound in the Body)
D istance  1540
m
 tim e
sec
The speed of sound in the body is much faster than we can drive a car.
(1540 m/sec is approximately 1 miles per second.) As a result, the time
to travel distances on the order of cm’s in the body will be much less than
1 second.
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Distance Equation
D ista n ce  ra te  tim e
We will begin by calculating the time it takes for sound to travel 1 cm in the body
(assuming a propagation velocity of 1540 m/sec). Since we want to solve for
time, we must rewrite the equation in the form time = distance/rate.

distance
 time
rate
2
so
1 cm
1  10 m
6
time =

 6.5  10 sec  6.5m sec
m
m
1540
1540
sec
sec
So it takes 6.5 msec to travel 1 cm or:
13 msec to image a structure at 1 cm because of the roundtrip effect.
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Distance Equation (Scaling for Depth)
D ista n ce  ra te  tim e
Since the travel time is linearly proportional to the distance, we can calculate
the time to travel 1 cm and then scale the answer by the actual travel distance.
Examples:
• Since it takes 6.5 msec to travel 1 cm, it takes 65 msec to travel 10 cm.
• Since it takes 13 msec to image a structure at 1 cm, it takes 130 msec to
image a structure at 10 cm.
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Distance Equation
6.5 m sec
1 cm
6.5 m sec
0 cm
Time
Distance
Imaging Depth
6.5 msec
1 cm
0.5 cm
13 m sec
2 cm
1 cm
26 m sec
4 cm
2 cm
39 m sec
6 cm
3 cm
52 m sec
8 cm
4 cm
65 m sec
10 cm
5 cm
78 m sec
12 cm
6 cm
91 m sec
14 cm
7 cm
104 m sec
16 cm
8 cm
117 m sec
18 cm
9 cm
130 m sec
20 cm
10 cm
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Time of Flight in the Body
Fig. 3: Imaging 1 cm Requires 13 msec (Pg 39)
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