Wk 1 Mathematics in the Physics of Ultrasound

Week 1: Mathematics in the Physics of Ultrasound

  • Instructor: Deanna Armstrong

  • Course: MEDS 12001, CQUniversity, Australia

Learning Objectives

  • Understand the importance of mathematics in diagnostic ultrasound.

  • Familiarize with metric abbreviations/prefixes and conversions related to diagnostic ultrasound.

  • Review relative mathematics, focusing on:

    • Linear Relationships

    • Non-linear Relationships

    • Inverse Relationships

Metric Abbreviations

Name

Prefix

Symbol

Number (Power of 10)

Giga

G

109=1,000,000,00010^9 = 1,000,000,000

Mega

M

106=1,000,00010^6 = 1,000,000

Kilo

k

103=1,00010^3 = 1,000

Hecto

h

102=10010^2 = 100

Deka

da

101=1010^1 = 10

Base Unit

i.e. meter, sec, Hertz

100=110^0 = 1

Deci

d

10−1=0.110^{-1} = 0.1

Centi

c

10−2=0.0110^{-2} = 0.01

Milli

m

10−3=0.00110^{-3} = 0.001

Micro

μ

10−6=0.00000110^{-6} = 0.000001

Nano

n

10−9=0.00000000110^{-9} = 0.000000001

Reciprocal Units

  • Reciprocal Relationships

    • Giga: 1/n=G1 / n = G where n is in nano

    • Micro: 1/M=μ1 / M = μ where M is in mega

    • Kilo: 1/m=k1 / m = k where m is in milli

Period and Frequency Calculations

  • Period Calculation Example

    • Given: 5 MHz transducer

    • Formula: T=1fT = \frac{1}{f}

    • Calculation:

    • f=5MHz=5×106Hzf = 5MHz = 5 \times 10^6 Hz

    • T=15×106=0.2μsT = \frac{1}{5 \times 10^6} = 0.2 \mu s

  • Frequency Calculation Example

    • Given: Period = 0.1 millisecond

    • Formula: f=1Tf = \frac{1}{T}

    • Calculation:

    • T=0.1 msec=0.1×10−3T = 0.1 \text{ msec} = 0.1 \times 10^{-3}

    • f=10.1×10−3=10KHzf = \frac{1}{0.1 \times 10^{-3}} = 10 KHz

Unit Conversions

  • Example Conversion:

    • 12 doughnuts = 1 dozen

    • 300 cm = 3 m

    • 0.003 GHz = 3 MHz

    • 3200 μs = 3.2 ms

  • Converting 9,700,000 μs:

    • Equation: 9700000μs=9.7s=9.7Ksec9700000μs = 9.7s = 9.7 Ksec

    • Also, 1540m/sec=1.54Km/sec1540 m/sec = 1.54 Km/sec

Wavelength and Frequency Relationships

  • Wavelength Calculation:

    • Given: Wavelength extλ=0.1extmmext{λ} = 0.1 ext{ mm} in soft tissue

    • Formula: f=vλf = \frac{v}{λ}

    • Calculation:

    • v=1540m/sv = 1540 m/s

    • f=15400.0001=15.4MHzf = \frac{1540}{0.0001} = 15.4 MHz

    • Additional:

    • f=cλf = \frac{c}{λ} with $c = $1550 m/s

Linear Proportional Relationships

  • Definition: Ratio between two variables is constant:

    • yx=k\frac{y}{x} = k or y=kxy = kx

    • Example:

    • 1 egg requires 2 cups of milk:

      • 1E=2M1E = 2M

    • If you double the number of eggs, you double the milk.

Newton's 2nd Law of Motion

  • Equation: F=maF = ma

    • Represents a linear relationship where force is proportional to acceleration with constant mass.

Proportionality Concepts

  • Proportionality notation: y∝xy ∝ x

  • In Inverse Proportional Relationships:

    • y∝1xy ∝ \frac{1}{x} where one variable increases as another decreases.

Non-Linear Proportionality

  • Illustrated by:

    • Attenuation and Absorption of Sound

    • Rayleigh Scattering

    • Echo Intensity Transmission/Reflection

    .