Pegasus Lectures, Inc. COPYRIGHT 2006 Volume I Companion Presentation Frank R. Miele Pegasus Lectures, Inc. Ultrasound Physics & Instrumentation 4 th Edition

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Pegasus Lectures, Inc. COPYRIGHT 2006 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

Pegasus Lectures, Inc. COPYRIGHT 2006 Mathematics: Level 1

Pegasus Lectures, Inc. COPYRIGHT 2006 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

Pegasus Lectures, Inc. COPYRIGHT 2006 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

Pegasus Lectures, Inc. COPYRIGHT 2006 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%  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.

Pegasus Lectures, Inc. COPYRIGHT 2006 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).

Pegasus Lectures, Inc. COPYRIGHT 2006 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).

Pegasus Lectures, Inc. COPYRIGHT 2006 Number Raised to a Power Raising a number to a power is a shorthand notation for multiplication. In the expression X A, 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.  2 3 = 2 x 2 x 2 = 8  2 5 = 2 x 2 x 2 x 2 x 2 = 32  5 2 = 5 x 5 = 25  5 5 = 5 x 5 x 5 x 5 x 5 = 3,125  (1/2) 3 = 1/2 x 1/2 x 1/2 = 1/8

Pegasus Lectures, Inc. COPYRIGHT 2006 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.

Pegasus Lectures, Inc. COPYRIGHT 2006 Exponential Notation Using powers of 10 to simplify large and small numbers  4,600,000,000 = 4.6 x 10 9  = 6.3 x  7,100 = 7.1 x 10 3  = 0.47 x 10 -9

Pegasus Lectures, Inc. COPYRIGHT 2006 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= ,000,000,000 M= ,000,000 k= ,000 h= da= d= c= m=  = n= 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.

Pegasus Lectures, Inc. COPYRIGHT 2006 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

Pegasus Lectures, Inc. COPYRIGHT 2006 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.

Pegasus Lectures, Inc. COPYRIGHT 2006 Linear Proportionality A proportional relationship between variables, in which, if one variable increases by x %, the other variable also increases by x %. y = x x y Increase by factor of 2

Pegasus Lectures, Inc. COPYRIGHT 2006 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

Pegasus Lectures, Inc. COPYRIGHT 2006 Inverse Relationships Fig. 2: Inverse Proportional Relationship (Pg 31) This is a graph of an inverse relationship. Notice that as x increases, y decreases.

Pegasus Lectures, Inc. COPYRIGHT 2006 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.

Pegasus Lectures, Inc. COPYRIGHT 2006 Distance Equation (Sound in the Body) 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.

Pegasus Lectures, Inc. COPYRIGHT 2006 Distance Equation 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. 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.

Pegasus Lectures, Inc. COPYRIGHT 2006 Distance Equation (Scaling for Depth) 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.

Pegasus Lectures, Inc. COPYRIGHT 2006 Distance Equation TimeDistanceImaging Depth 6.5  sec 1 cm0.5 cm 13  sec 2 cm1 cm 26  sec 4 cm2 cm 39  sec 6 cm3 cm 52  sec 8 cm4 cm 65  sec 10 cm5 cm 78  sec 12 cm6 cm 91  sec 14 cm7 cm 104  sec 16 cm8 cm 117  sec 18 cm9 cm 130  sec 20 cm10 cm 0 cm 1 cm 6.5  sec 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.

Pegasus Lectures, Inc. COPYRIGHT 2006 Time of Flight in the Body Fig. 3: Imaging 1 cm Requires 13  sec (Pg 39)

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