Unit 1B: Basic Concepts of Math & Physics Used in Ultrasound

Relationships

• Fundamental to understanding ultrasound physics.

• Unrelated: Two entities are not associated.

  • Example: Hair color and shoe size, height and eye color.

• Related or proportional: Two entities are affiliated, but the specific relationship isn't defined.

  • Example: Taxes and income, Santa Claus and Christmas, exam score and studying.

• Directly related or directly proportional: When one entity increases, the other also increases.

  • Example: Experience and age, skill and practice, quality of wine and age.

• Inversely related or inversely proportional: When one entity increases, the other decreases.

  • Example: Golf score and skill, car's mileage and weight, grades and partying time.

• If c increases, f increases: $\uparrow = \uparrow$

• If f decreases, f increases: $\downarrow = \uparrow$

Basic Physical Quantities & Units of Measurement

• Base quantities:

  • Length (l)

  • Mass (m)

  • Time (t)

• Secondary quantities (derived quantities): Combinations of base quantities.

  • Area: 2-dimensional (squared) = $m^2$, $cm^2$, $ft^2$

  • Volume (V): 3-dimensional (cubed) = $cm^3$, $ft^3$, cc

  • Density: Mass divided by volume ($m/V$)

  • Velocity: Length divided by time ($v = l/t$)

Systems of Units

• Various systems represent base quantities.

• The International System of Units (SI) is an extension of the MKS system.

• MKS/SI:

  • Length: Meter (m)

  • Mass: Kilogram (kg)

  • Time: Second (s)

• CGS:

  • Length: Centimeter (cm)

  • Mass: Gram (g)

  • Time: Second (s)

• British:

  • Length: Foot (ft)

  • Mass: Slug

  • Time: Second (s)

Standard of Length

• Measure of distance or circumference.

• Base unit: meter (m).

• Subdivided into centimeters (cm) and millimeters (mm).

  • $1 m = 100 cm = 1000 mm$

  • $1 cm = 10 mm$

• Relationship between British and metric systems:

  • $1 in = 2.54 cm = 25.4 mm$

• Other units: feet, yards.

Standard of Mass

• Quantity of matter within a given object.

• Base unit: kilogram (kg).

  • Defined as the mass of 1000 $cm^3$ of water at 0° Celsius.

• Subdivided into grams (g).

  • $1 kg = 1000 g$

• Mass ≠ weight.

  • Kilogram is a unit of mass.

  • Newton and pound (British unit) are units of weight.

Standard of Time

• Measure of duration of events.

• Base unit: second (s).

• Subdivided into milliseconds (ms) and microseconds ($\mu$s).

  • $1 s = 1,000 ms = 1,000,000 \mu s$

• Other units: minute, year, decade, century.

Volume

• Amount of 3-dimensional space occupied by a substance.

• Cubic units: $cm^3$, cc, liters (L).

• Measuring volume of irregularly shaped object:

  • Immerse in full tank of water.

  • Measure missing water.

  • Volume of missing water = volume of object.

• In the body, volume is calculated by multiplying 3 dimensions (length x width x height).

  • Units are cubed ($cm^3$).

Expression of Measurements

• Every measurement has two parts:

  • Unit

  • Magnitude (numerical value)

• Numerical values require corresponding units.

• Example: "How old is Judy?" "6" is incomplete. It could be 6 days, months, years, or decades.

Unit Conversions

• Converting from one unit to another.

• Examples:

  • 4 quarters in 1 dollar

  • 365 days in 1 year

• Total doesn't change; only the unit.

Mathematical Terms

• Coefficient: Multiplication factor (number in front of a variable).

  • Example: 2x

• Reciprocal: Relationship between two expressions that, when multiplied, equal one.
  $\frac{1}{2} * 2 = 1$

Solving Equations

• Equations express relationships between variables.

• Variables denoted by letters.

• Solution found by manipulating the equation to solve for the variable (when only one variable is present).

• Techniques to isolate the unknown:

  • Adding/subtracting the same term from both sides does not change the value.

  • Multiplying/dividing both sides by the same factor does not change the value (division by nonzero real number).

• Whatever is done to one side must be done to the other.

Solving Equations - Examples

• Example 1:

  • $2x + 4 = 10$

  • Subtract 4 from both sides: $2x + 4 - 4 = 10 - 4$

  • $2x = 6$

  • Divide both sides by 2: $\frac{2x}{2} = \frac{6}{2}$

  • $x = 3$

  • Test: $2(3) + 4 = 10$

• Example 2:

  • $3x + 1 = 4x - 6$

  • Add 6 to both sides: $3x + 1 + 6 = 4x - 6 + 6$

  • $3x + 7 = 4x$

  • Subtract 3x from both sides: $3x - 3x + 7 = 4x - 3x$

  • $7 = x$

  • Test: $3(7) + 1 = 4(7) - 6$

Exponents

• Product of identical factors represented in exponential notation:

  • $3 \times 3 \times 3 \times 3 = 3^4$

  • $a \times a \times a \times a = a^4$

• 3 (or a) is the base.

• Superscript 4 is the power.

  • Indicates how many times the base is multiplied by itself.

Exponents - Rules

• Bases must be the same to combine exponents.

• Multiplication: $a^m a^n = a^{m+n}$

• Division: $\frac{a^m}{a^n} = a^{m-n}$

Exponents - Special Cases

• Any base raised to the power zero (0) is equal to 1.

  • $10^0 = 1$

  • $2^0 = 1$

  • $1000^0 = 1$

• Other operations:

  • $(a^m)^n = a^{m*n}$

• Fractional exponents:

  • $a^{1/n}$ ($n^{th}$ root of a)

  • If n = 2, then $a^{1/n}$ represents the square root of a $\sqrt{a}$

  • If n = 3, then $a^{1/n}$ represents the cube root of a $\sqrt[3]{a}$

Exponents - Negative Integers

• Bases raised to negative powers express small numbers.

• Negative power is equal to its reciprocal:

  • $a^{-n} = \frac{1}{a^n}$

• Example:

  • If a = 2 and n = -3, then $2^{-3} = \frac{1}{2^3} = \frac{1}{2 \times 2 \times 2} = \frac{1}{8} = 0.125$

• Result is positive with value between 0 and 1.

Scientific Notation (Power of 10 Notation)

• Shorthand method for very large or very small numbers.

• Positive exponent: number > 1.

  • $4^2 = 16$

  • $2^3 = 8$

• Negative exponent: number < 1.

  • $4^{-2} = \frac{1}{4^2} = \frac{1}{16} = 0.0625$

  • $2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125$

• Exponent of zero: number = 1

  • $4^0 = 1$

  • $2^0 = 1$

Scientific Notation - Examples

• Shift decimal point so that resulting number is between 1 and 10.

• Multiply by the appropriate power of 10 (number of places moved).

• Examples:

  • 1,000,000 = $1.0 \times 10^6$

  • 500 = $5.0 \times 10^2$

  • 0.79 = $7.9 \times 10^{-1}$

  • 0.000124 = $1.24 \times 10^{-4}$

  • 1742 = $1.742 \times 10^3$

Scientific Prefixes - Metric System

• Powers of Ten - Prefix - Symbol - Meaning

  • $10^9$ - Giga - G - Billion

  • $10^6$ - Mega - M - Million

  • $10^3$ - Kilo - k - Thousand

  • $10^2$ - Hecto - h - Hundred

  • $10^1$ - Deka - da - Ten

  • $10^{-1}$ - Deci - d - Tenth

  • $10^{-2}$ - Centi - c - Hundredth

  • $10^{-3}$ - Milli - m - Thousandth

  • $10^{-6}$ - Micro - $\mu$ - Millionth

  • $10^{-9}$ - Nano - n - Billionth

Complementary Metric Units

• Pairs that "belong together."

• Ultrasound: frequency in MHz, period in $\mu$s.

  • Millions of cycles and millionths of seconds.

  • If frequency (f) = 1 MHz, then period (T) = 1 $\mu$s

• Numbers Prefixes Symbols

  • Billions & billionths Giga & nano G & n

  • Millions & millionths Mega & micro M & $\mu$

  • Thousands & thousandths Kilo & milli k & m

  • Hundreds & hundredths Hecto & centi h & c

  • Tens & tenths Deca & deci da & d

Conversion Math

• Set up an equation like converting moles in chemistry.

• Multiply by the ratio of the units you are converting with the current unit on the bottom.
  Ex:

• How many cm are there in 85 meters?

  • $85 m * \frac{100 cm}{1 m} = 8,500 cm$

• How many how many km are in 30 meters?

  • $30 m * \frac{1 km}{1000 m} = 0.03 km$

Conversion Ladder

• Conversion Ladder, King Henry Doesn't Usually Drink Chocolate Milk

• Tera = T

• Giga = G

• Mega = M

• Kilo = k

• Hecto = h

• Deka = da

• Multiply by 10, Shift Decimal Point to Right

• UNIT

• Divide by 10, Shift Decimal Point to Left

• Deci = d

• Centi = c

• Milli = m

• Micro = $\mu$

• Nano = n

• Pico = p

Logarithms

• $\log_{10} 1,000,000 = 6$ Since $10^6=1,000,000$

Logarithms

• Logarithms are the inverse function of an exponent.

• A number x can be represented in the form $x = b^y$ (x > 0), where b = base & y = exponent.

• Taking the logarithm of a number finds the exponent for which the number would be obtained if the base were raised to that exponent.

• The logarithm to the base b of x is an exponent: $\log_b x = y$

• Common logarithms (base 10): $x = 10^y$; $\log_{10} x = y$

Logarithms with base 10

• Example:
  $\log_{10} 10000$ = 4

Logarithms with base 10 - Table

Logarithms - Properties

• Logarithms have properties similar to exponents.

• Logarithm of a number raised to the nth power is the product of n and the logarithm of the number: $\log(x^n) = n \log (x)$

• Example: $\log(2^3) = 3 \log 2$

• $\log 8 = 3(0.3010)$

• $0.9030 = 0.9030$

Binary Numbers

• Computers operate on the binary number system (two digits: 0 and 1).

• Computers convert alphabetic characters, decimal values, and logic functions into binary values.

• Computation is handled by adjusting on/off switches in the computer.

• 0 = off

• 1 = on

Binary Number - Powers

Binary Number - Equivalents

Binary Numbers - Conversion to Decimal

• Convert the binary number 1011010 to decimal form.

• Build a table with columns representing powers of 2 (rightmost is $2^0$).

• Line 1: Powers of twos ($2^0, 2^1, 2^2, 2^3,… 2^n$).

• Line 2: Decimal values of powers of twos (1, 2, 4, 8,…).

• Line 3: Input binary number (rightmost digit in the column of $2^0$).

• Line 4: Decimal value:

  • If column value is zero, the decimal value will be zero.

  • If column value is 1, value corresponds to decimal value of that power of 2.

• Add all columns; the sum represents the decimal number.

• Example: 1011010 = 2 + 8 + 16 + 64 = 90

Binary Numbers - Conversion to Binary

• Express the number 193 in binary form.

• Start dividing the number by 2.

• If division is perfect, report 0 in the column of remainders.

• If division is not perfect, report 1 in the column of remainders.

• Finish division with 1 ÷ 2, which gives 1 in the column of remainders.

• Read the column of remainders from bottom to top.

• The answer is 11000001.

Binary Numbers in Computer Language

• A single binary digit (0 or 1) is called a bit.

• Computers use as many bits as necessary to express a decimal number.

• Encoding is translating from ordinary characters to computer-compatible characters or binary digits.

• The 26 characters of the alphabet and other special characters are usually encoded by 8 bits.

Binary Numbers in Computer Language - Bytes

• Bits are grouped into bunches of 8 called bytes (8 bits = 1 byte).

• Depending on the computer configuration, 2 bytes usually constitute a word.

• Computer capacity is expressed by the number of bytes that can be accommodated by computer memory.

• One kbyte is equal to 8 x 1024 bits.

• Note: In computer use, kilo is not metric; it represents $2^{10}$.

Geometry

• Geometry is the study of shapes, sizes, and positions of objects.

• Ultrasound uses geometric graphs to represent the relationships of two variables (typically velocity & time).

• Trigonometry is the study of the relationships between the angles and sides of triangles.

• https://study.com/academy/lesson/graph-terminology-axis-range-scale.html

Geometry - Angles

• Measuring angles:

  • One complete revolution = 360°

  • A straight line = 180°

  • A right angle = 90°

Trigonometry - Triangles

• A triangle is a three-sided geometric figure.

• All angles of a triangle, when added together, must equal 180°.

• If one of the angles is equal to 90°, the triangle is a right triangle.

• https://thirdspacelearning.com/gcse-maths/geometry-and-measure/angles-in-a-triangle/

Trigonometry - Right Triangles

• Side opposite the right angle is the hypotenuse.

  • Always the longest side of a right triangle.

• An opposite side and an adjacent side are named with respect to the angle of interest.

Trigonometry - Functions

• Trigonometric function: a function of an angle expressed as a ratio of two sides of a right triangle.

• For a particular angle, the ratio of any two sides is constant regardless of the size of the triangle.

• Ratios:

  • Sine (sin)

  • Cosine (cos)

  • Tangent (tan)

• Trigonometric functions are equations with three variables; if any two variables are known, the third can be solved.

• https://www.skillsyouneed.com/num/trigonometry.html

Trigonometry - Application

• 
  
  

• In ultrasound physics, sin and cos are commonly used.
  
  

  
  

  Angle	Sin	Cos
  0°	0.000	1.000
  30°	0.500	0.866
  45°	0.707	0.707
  60°	0.866	0.500
  90°	1.000	0.000
  Newton’s Laws of Motion

  • Mechanics is the study of motion.

  • In 1686, Sir Isaac Newton presented three fundamental laws of motion.

  • Diagnostic medical ultrasound uses high-frequency mechanical waves transmitted through and reflected from body tissues.

  • Doppler ultrasound uses the motion of blood in the heart or vessels.

  • Mechanics is important to understanding the physics of ultrasound.

  Newton’s Laws of Motion - Velocity and Acceleration

  • Motion described by velocity and acceleration.

  • Velocity (speed): how fast something is moving.

    • Example: Speedometer in miles per hour (or kilometers per hour).

    • In ultrasound: meters per second (m/s), centimeters per second (cm/s), or millimeters per microsecond (mm/$\mu$s).

  Newton’s Laws of Motion - Equations

  • Velocity (v): rate of change of position with time.

    • $v = \frac{d}{t}$ (v = velocity, d = distance, t = time)

  • Acceleration (a): rate of change in velocity per unit of time.

    • $a = \frac{v}{t}$

  • If velocity is constant, acceleration is zero.

  • Constant acceleration of 2 m/$s^2$: velocity increases by 2 m/s each second.

  Newton’s First Law (Law of Inertia)

  • A body remains at rest or continues moving with constant velocity in a straight line unless acted on by an external force.

  • If no force acts on an object, there is no acceleration.

  • Inertia: property of matter that resists a change in its state of motion.

  Newton’s Second Law

  • Force: A push or a pull on an object.

    • $F = m \times a$ (F = Force, m = mass, a = acceleration)

  • Units:

    • SI: Newton (N); 1 N = 1 kg m/$s^2$

    • CGS: dyne; 1 dyne = 1 g cm/$s^2$

  Newton’s Third Law

  • To every action, there is an equal and opposite reaction.

  • If a heavy block is pushed, the block pushes back with the same force.

  • If a very large force is exerted, the block accelerates away.

  Work & Power

  • Work (W): force applied times the distance.

    • $W = F \times d$

    • Units: SI - Joules (J); CGS – ergs; 1 J = $10^7$ ergs

  • Power (P): rate of doing work.

    • $P = \frac{W}{time}$ or $P = \frac{Energy}{time}$

    • Units: Watt (W), mW (milliwatts)

  Energy

  • Energy (E): the ability to do work.

    • Units: SI - Joules (J); CGS – ergs

  • Forms of energy: mechanical, chemical, electrical, thermal, and nuclear.

  • Law of Conservation of Energy:

    • Energy can be transformed but not created or destroyed; total energy is constant.

    • Example: Electrical energy converted to ultrasound energy in a transducer.

  Mechanical Energy

  • Two forms of mechanical energy:

    • Kinetic energy (KE): associated with motion.

      • $KE = \frac{1}{2} m \times v^2$

      • m = mass (kg), v = velocity (m/s)

    • Potential energy (PE): stored energy of position or configuration.

      • Gravitational PE: $PE = m \times g \times h$

      • m = mass (kg), g = acceleration of gravity on Earth (9.8 m/$s^2$), h = height (meters)

  Other Forms of Energy

  • Chemical Energy: released by a chemical reaction.

    • Energy provided by our bodies through chemical reactions involving food.

  • Electrical Energy: work done when an electron moves through an electric potential.

  • Thermal Energy: energy of motion at the atomic level.

    • Kinetic energy of atoms, measured by temperature.

  • Nuclear Energy: energy in the nucleus of an atom.

    • Controlled release in nuclear power plants.

    • Uncontrolled release - atomic bomb.

  Heat

  • Definition: Random disordered motion of molecules.

  • The more rapid and disordered the motion, the more heat the body contains.

  • The unit of heat:

    • Calorie = the heat necessary to raise the temperature of 1 g of water by 1° C

    • The currently accepted value for the mechanical equivalent of heat is 1 calorie = 4.186 Joules

  Heat Transfer

  • Heat is transferred from one place to another in three ways:

    • Conduction: molecular motion from high to low temperature

      • Hot and cold object in contact -> equalization of temperatures.

    • Convection: transfer of “hot” molecules in a gas or liquid.

      • Radiator/forced-air furnace - warms a room by convection.

    • Radiation: depends on the temperature of the object.

      • The reddish glow emitted by hot objects is evidence of heat transfer by radiation.

  Temperature

  • Measured with a thermometer, calibrated at the freezing and boiling points of water.

  • Three scales:

    • Fahrenheit (F)

    • Celsius (C)

    • Kelvin (K)

  Temperature - Equations

  
  

  • 
    
    

  • These scales are interrelated by the following equations:
    $T<em>C = \frac{5}{9} (T</em>F - 32)$
    $T<em>F = \frac{9}{5} T</em>C + 32$
    $T<em>K = T</em>C + 273$

  • 
    
    

  • Reference Points
    
    

    
    

    	C	F	K
    Absolute zero	-273°	-120°	0°
    Freezing water	0°	32°	273°
    Boiling water	100°	212°	373°
    Percentages

    • A number ratio expressed as a fraction of 100.

    • Used for expressing proportions of a whole or part of a total.

    • Unitless.

    • Example:

      • What percent of 16 is 4?

      • $\frac{4}{16} = \frac{1}{4} = \frac{?}{100}$

      • $\frac{25}{100} = 25\%$

      • 1 ÷ 4 = 0.25

      • Move the decimal over two places to the right to get the percentage

      • 025 = 25%

    Percentages - Application

    • Because of the Law of Conservation of Energy, it is important to understand how to work with percentages.

    • Percentages can be used to calculate numerical values.

    • Example:

      • How much is 25% of 16?

      • Move the decimal place to the left 2 places to get a decimal that can be used as a multiplier

      • 25% = 0.25

      • 16 × 0.25 = 4

    Percentages - The Whole

    • Since the total is 100%, the whole can be referred to as 1 (by moving the decimal 2 places to the left).

    Relative Change & Difference

    • Percentages describe an increase or decrease in the total.

    • The value corresponding to the total must be known.

    • Increase of 50%: you still have the whole plus another half.

    • Calculated by multiplying by 1.5 (150%).

    Relative Change & Difference - Decrease

    • If there is a decrease of 25%, you only have part of the whole.

    • Calculated by multiplying by the decimal of the percentage, then subtracting that value from the whole.

    • Example:

    • $1 \times 0.25 = 0.25$

    • $1 - 0.25 = 0.75 75\%$

    Application to US Physics

    • Clearly indicate the units of measure.

    • Pay attention to quantities that are directly or inversely proportional.

    • Certain quantities do not have any connection and are unrelated.

    • Be consistent with the system of units used to express physical quantities. Use SI.

    • Pay attention to conversions.