Transcript Ultrasound Physics Volume I

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%
numerator
Fractions 
denominator
 1/5 = 0.2 = 20%
 1/50 = 0.02 = 2%
 14/100 = 0.14 = 14%
 28/200 = 14/100 = 0.14 = 14%
This is a good time to have your students build up tables of fractions and decimal equivalents (without calculators). By doing
this from their heads, the students will start to improve their abilities to recognize patterns. (Such as 1/5 = 0.2, 1/50 = 0.02,
and 1/500 = 0.002). This skill is helpful for converting between periods and frequencies, dealing with percentage change
(such as percent stenosis), and for understanding relative rates of change between related variables.
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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
It is a good idea to point out that with respect to reciprocals, both physical units and variables behave the same way as
constants. Note that 1/(1 MHz) can be rewritten in a simplified form as 1 sec since 1/1 = 1, 1/M = , and 1/Hz = sec. This
last step can be seen since Hz is the same as 1/sec. Therefore, 1/Hz is the same as 1/1/sec which equals sec (two
reciprocals of a value cancel each other out).
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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
To make variables a little less intimidating, it is useful to have the students create their own equations beginning with
variable definition. For example, you could ask the students to write an equation that expresses how much money they
would earn if they are paid $20 per hour. In this case, they would need to create a variable which represents the number
of hours worked, and a variable which represents how much money they earn. For example: Let S = (salary) money
earned and t = time (hours worked). The equation would then be S= $20/hr * t (hrs).
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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
Note that a positive exponent expresses the idea of multiplication so a negative exponent expresses the idea of division.
Another way to express division is to multiply by a reciprocal.
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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

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
Note: it is very important for the students to make a distinction between upper (capital) and lower case letters. (M = mega
and m = milli). Also, it is valuable to stress the fact that the table is based on reciprocals.
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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
Proportionality is a way of expressing a relative relationship, whereas equations express absolute relationships. In
other words, relationships express the rate of change between variables, but not the actual value of the variables.
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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
1
y 
 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)
m
Distance  1540
 time
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
Distance  rate  time
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
1 cm
1  102 m
so
time =

 6.5  10 6sec  6.5 sec
m
m
1540
1540
sec
sec

So it takes 6.5 sec to travel 1 cm or:
13 sec to image a structure at 1 cm because of the roundtrip effect.
At this point it is very important that the students start developing very specific language skills. Normal language tends to be
relatively sloppy so that students often have difficulty distinguishing between a roundtrip versus distance measure. For
example, stating that sound travels a distance of 10 cm implies the same required time as stating that sound is used to image a
structure at 5 cm (5 cm into the patient and 5 cm out of the patient). Attention paid to “language” now will show benefits later
on.
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Distance Equation (Scaling for Depth)
Distance  rate  time
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 sec to travel 1 cm, it takes 65 sec to travel 10 cm.
• Since it takes 13 sec to image a structure at 1 cm, it takes 130 sec to
image a structure at 10 cm.
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Distance Equation
6.5  sec
6.5  sec
0 cm
1 cm
Again notice that this table indicates the
linear relationship between time and
distance as well as the fact that caution
must always be exercised as to whether
the question is asking a one-way
measurement or a roundtrip measurement.
Time
Distance
Imaging Depth
6.5 sec
1 cm
0.5 cm
13  sec
2 cm
1 cm
26  sec
4 cm
2 cm
39  sec
6 cm
3 cm
52  sec
8 cm
4 cm
65  sec
10 cm
5 cm
78  sec
12 cm
6 cm
91  sec
14 cm
7 cm
104  sec
16 cm
8 cm
117  sec
18 cm
9 cm
130  sec
20 cm
10 cm
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Time of Flight in the Body
Fig. 3: Imaging 1 cm Requires 13 sec (Pg 39)
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Notes