Chapter four: CARDIAC ULTRASOUND TUTORIAL SERIES

Terminlogy of the Physics of Ultrasound (continued)


by


Dr z. Gooding


Physics versus Mathematics:

This chapter is an extension of the previous, to stress the importance that a thorough mastery of the "Fundamentals of Physics", reenforced by a basic knowledge of Mathematics, will help understand the Physics of Ultrasound.

To illustrate the above statement, let's take a look at the following example and see how it can serve our purpose. If someone drops a ball from the top of Sears Tower, one can measure the "time" it takes the ball to hit the ground and also compute the "distance" covered, which is
basically the height of the Tower. With the knowledge that a basic law of physics known as Law of Gravity is involved, the height (h) of the tower can be expressed mathematically as follows:

h = 1/2 g t x t

where h is the height of the tower expressed in meters, g the local acceleration of gravity (9.8 meters per second x second) and t the time of fall in seconds.

Note that g and t are known, and t is squared (t x t).
Similar laws that mathematically describe the fundamental nature of ultrasound will be explained. But first a basic math review is necessary.

It is recommended that the candidates for the ultrasound boards should become familiar and understand each type of math problem reviewed below such as:

1. Fractions (addition, multiplication, and division);












2. Decimals;












3. Significant figures;












4. Scientific notations;












5. Rules of exponents;












6. Binary system;












7. Graphing;












8. Units of measurements.

Fractions:

By definition a fraction (x/y) has 2 members a "numerator" (x) and a "denominator (y).

Rule 1: both addition and subtraction of fractions require a common denominator.

1/5 + 3/4 =====> (1/5 x 4) + (3/4 x 5) = (4/20) + (15/20) = 19/2

3/4 - 1/5 =====> (3/4 x 5) - (1/5 x 4) = 15/20 - (4/20) = 11/20

Rule 2: in a multiplication , numerators are multiplied together and denominators together.

1/5 x 3/4 = 1 x 3/ 5 x 4 = 3/20

Rule3: in a division, invert the second term and then proceed as in multiplication;

3/4 divided by 1/5 =====> 3/4 x 5/1 = 15/4

Decimals:

Rule1: When the denominator of a fraction is a power of 10, that fraction can be readily
converted into decimals:

5/10 = 0.5

5/100 = 0.05

5/1000 = 0.005

Rule2: If the denominator is not a power of 10, the decimal can be obtained by
performing the division

5/9 = 0.55

Significant Figures:

Consider the entries below:

45.684 (3 decimals) + 7.7 (1 decimal) + 6.15 (2decimals) = 59.534 = 59.5

• In both addition and subtraction, the total must be rounded to the same number of digits as the entry with the least number of decimal places. Therefore the total is 59.5.
  



• In multiplication and division round to 2 digits.

45.68 x 7.7 = 351.736 =====> 351.74

45.8/7.7 = 5.93247 =====> 5.93

Algebra:

As a branch of Mathematics (exact science), the goal of Algebra is to solve for "unknown quantities" (aka variables) represented by alphabetic letters (x, y, z) in an equation.

An equation is defined by its "sides" or "members". In the following example: x = 2y, x is the first member and 2y is the second member.

Algebra also has rules, used to solve for the variables encountered in diagnostic ultrasound equations.

Rule1: when a variable is multiplied by a number, both members of the equation must be divided by that number.

6x = 12 =====> 6x/6 = 12/6 =====> x = 2

Rule2: when numbers are added to a variable, subtract that number from both members of the equation.

x + 6 = 20 =====> x + 6 - 6 = 20 - 6 =====> x = 14

Rule3: when the equation is in the form of a fration, cross multiply and solve for the variable.

y/5 = 10/25 =====> y x 25 = 5 x 10 =====> y = 2

At this point the candidates are invited to take a quiz in order to practice working with equations and hence solving for a variety of variables similar to the ones given in ultrasound board exams.