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Applied Mathematics for Medics - Lecture Notes Tuesday 14th September 2021 Applied Mathematics for Medics Lecture Notes Gokulan Vethanayakam (gv286), Jie Lin Li (jll57), James Bibey (jb2373) Units and Symbols Units define numerical values in science. Unit conversion is one of the most important aspects of this, as some values (such as temperature) have multiple P peta- 1015 s second time T tera- 1012 m metre distance 9 G giga- 10 kg kilogram mass 6 M mega- 10 A ampere electrical current k kilo- 103 K kelvin temperature d deci- 10-1 mol mole amount of substance -2 c centi- 10 cd candela luminous intensity m milli- 10-3 L litre volume : 10 m 3 μ micro- 10-6 g gram mass: 10 kg -9 n nano- 10 ℃ celsius temperature : K - 273 p pico- 10-12 min minute time : 60 s f femto- 10-15 hr hour time : 3600 s day day time : 86400 s Converting between SI prefixes As a general rule to convert between SI prefixes, divide the prefix the value is currently in with the prefix that you want to convert the value to, to work out a scale factor. Multiply the scale factor by the value to get the new converted value. Express 95 nm in mm Scale Factor = 10 / 10 = 10 -6 95 nm = 95 . 10 mm 1Applied Mathematics for Medics - Lecture Notes Tuesday 14th September 2021 Take note when converting between areas and volumes, which involve powers to the units involved. The scale factor will need to have that power applied to it before in order for the conversion to work. Express 95 nm in mm 2 -6 2 -12 Scale Factor = (10 ) = 10 95 nm = 95 . 10-1mm 2 Greek Alphabet Letter Name Letter Name Letter Name Α α alpha Ι ι iota Ρ ρ rho Β β beta Κ κ kappa Σ σ/ς sigma Γ γ gamma Λ λ lambda Τ τ tau delta mu upsilon Δ δ Μ μ Υ υ epsilon nu phi Ε ε Ν ν Φ φ Ζ ζ zeta Ξ ξ xi Χ χ chi Η η eta Ο ο omicron Ψ ψ psi Θ θ theta Π π pi Ω ω omega Key Points ● Unit conversion is important to get right, and is a key source of calculation errors. ● milli- and micro- are easy to mix up due to sounding very similar. ● There are 7 SI base units, which can be used to derive all of the other SI units. 2Applied Mathematics for Medics - Lecture Notes Tuesday 14th September 2021 Rearranging Equations and Formulae Rearranging equations and formulae is a key skill in medical science. Formulae are used to help us to calculate useful values. These can be useful physiological values both theoretically and in hospital work, or help us in medical science to interpret experimental results. Simple Example - Alveolar Ventilation Formula V = (V . K) / P make K the subject A ECO2 ACO2 Multiply byACO2 : V A PACO2= VECO2. K Divide by ECO2 : K = (VA. ACO2 / ECO2 In this example, K is a constant useful for understanding ventilation. Moderate Example - Poiseuille’s Law 𝐿η 𝑅 ∝ 4 make r the subject 𝑟 4 4 Multiply by : 𝑟 . 𝑅 = 𝐿η 4 𝐿η Divide by𝑅 : 𝑟 = 𝑅 4𝐿η Hypercube root : 𝑟 = 𝑅 In this example, the ± sign should not be used, as r is radius, which can only be positive. This example also highlights the importance of consistent use of variable names - both capital R and lowercase r are in the formula, and it’s important to not mix them up. Complex Example - Alveolar Gas Formula 1 − 𝐼𝐶𝑂2 𝑃 = 𝑃 − 𝑃 .(𝐹 + ) make R the subject 𝐴𝑂2 𝐼𝑂2 𝐴𝐶𝑂2 𝐼𝑂2 𝑅 1 − 𝐼𝐶𝑂2 Subtract𝑃𝐼𝑂2 : 𝑃𝐴𝑂2 − 𝑃 𝐼𝑂2= − 𝑃 𝐴𝐶𝑂2. ( 𝐼𝑂2 + 𝑅 ) 1 −𝐼𝐶𝑂2 Multiply b− 1 : 𝑃 𝐼𝑂2− 𝑃 𝐴𝑂2 = 𝑃 𝐴𝐶𝑂2. ( 𝐼𝑂2 + 𝑅 ) 𝑃𝐼𝑂2 𝐴𝑂2 1 − 𝐼𝐶𝑂2 Divide by𝑃𝐴𝐶𝑂2 : 𝑃 = 𝐹 𝐼𝑂2 + 𝑅 𝐴𝐶𝑂2 𝑃𝐼𝑂2𝑃𝐴𝑂2 1 − 𝐼𝐶𝑂2 Subtract𝐹𝐼𝑂2 : 𝑃 − 𝐹 𝐼𝑂2 = 𝑅 𝐴𝐶𝑂2 𝑃𝐼𝑂2 𝐴𝑂2 Multiply by : 𝑅 .( 𝑃 − 𝐹 𝐼𝑂2) = 1 − 𝐹 𝐼𝐶𝑂2 𝐴𝐶𝑂2 1 − 𝐼𝐶𝑂2 Divide by bracketed portio: 𝑅 = 𝑃 − 𝑃 ( 𝐼𝑃2 𝐴𝑂− 𝐹 ) 𝐴𝐶𝑂2 𝐼𝑂2 3Applied Mathematics for Medics - Lecture Notes Tuesday 14th September 2021 In this example, the fraction ends up being quite complex! This highlights the importance of rearranging formulae before inserting values. Key Points ● We can calculate specific values from other known values by rearranging formulae. ● Rearranging formulae before inserting values generally leads to fewer errors. ● Rearranging equalities depends on performing the same function to both sides. ● The ± sign should always be considered when roots have been used. Remember that some types of value (such as radius) cannot be negative, whilst others (such as pressure) can be. ● If the same letter is used in a formula as both a capital letter and a lowercase letter, it is important to not mix them up. Proportions (∝) Equations with ∝ can be converted to = by finding the constant Example - Ohm’s Law V ∝ I V = I x constant V = I x R Find the current (I) through a resistor of resistance R = 2 Ω if the voltage across the resistor is 6 V. V = 6 V R = 2Ω I = V / R 6 / 2 = 3 A Equations with = can be converted to ∝ by dropping any constants. Example - y = kx + c y = k x + c x and y are variables k and c are constants What is y proportional to? y ∝ x Let k = 2 and c = 3 4Applied Mathematics for Medics - Lecture Notes Tuesday 14th September 2021 For y = kx For y = kx + c y ∝ x If x = 1 If x = 1 y= 2 x 1 y = (2 x 1) + 3 y= 2 y = 5 If x = 2 If x =2 y = 2 x 2 y = (2x2) + 3 y = 4 y = 7 Not y ∝ x + c Key Points ● Values in equations and formulae can be generally split into variables (which we can find the value of given certain inputs) and constants. ● By finding the constant and replacing ∝ with =, we can calculate the values of unknown variables. ● Equations with = can be converted to ∝ by dropping any constants. Types of Graphs There are lots of types of graphs and charts, which are chosen based on the data type they represent. These include: ● Line graphs ● Scatter graphs ● Bar graphs ● Histograms ● Pie charts Data can be split into various types: ● Quantitative data is data that takes a numerical value. ○ Discrete data takes specific cardinal values along the number line. ■ Days of hospitalisation after appendix surgery. ○ Continuous data can be anywhere along a numerical spectrum, potentially with infinite precision if it were possible to measure this. ■ Birth weight at gestational age. ● Qualitative data is data that can be split into separate described categories. ○ Nominal data can be split into categories that have no logical order. ■ Blood groups. ○ Ordinal data can be split into categories that follow a logical order. ■ Colour palette. ○ Boolean, or binary, data takes one of two values, such as true or false. ■ Biological sex. 5Applied Mathematics for Medics - Lecture Notes Tuesday 14th September 2021 Key Points ● The type of graph is chosen based on the type(s) of data it represents. Some graph choices are inappropriate for some data. ● Quantitative data can generally be either discrete (taking specific values) or continuous (taking a value along a potentially infinite number line). ● Qualitative data can generally be either nominal (uncategorised), ordinal (where the categories follow a logical pattern), or Boolean / binary (usually true or false), Handling Graphs Graphs are really helpful in medical science. They help us to observe trends, and can be used to identify normal physiology and states of illness. y = mx + c The formula y = mx + c describes the general formula of a straight line graph. x and y describe paired coordinate values, m is the gradient of the line (change in y as x changes), and the constant c. In the graph to the right, the red line has a gradient of 0.5 and a constant c of 2, whilst the blue line has a gradient of -1 and a constant c of 5. ● The y-intercept (when x = 0) is equal to the constant c. We can imagine that if c is 0 and x is 0, that y will always be 0 no matter the multiplicative effect of m. ● The x-intercept (when y = 0) is equal to -c/m. Calculating the Gradient of Straight Lines The gradient m can be calculated based on a pair of coordinates on the line. The formula: m = (y2- y1) / 2x -1x ) describes the change in y as x changes for the pairs of coordinates (x1,1 ) and (2 2y ). Calculating the gradient can be very useful. The gradient describes the relationship between 2 variables, which could be alcohol consumption and liver damage, or the change in the voltage of a cell membrane over time as an action potential occurs, among various relationships. Exponential Curves A curve is considered to be exponential when the rate at which the graph rises/falls is proportional to the x-value. All exponential curves can be modelled with the equation: bx y = a . e , whereby a and b are constants. 6Applied Mathematics for Medics - Lecture Notes Tuesday 14th September 2021 We shall group these curves into 3 rough classes: ● Positive exponential: Whereby a > 0 and b > 0 ○ Example: Growth of bacteria (exponential phase) ● Negative exponential (Exponential Decay): Whereby b < 0 ○ Example: Plasma concentration of a single drug infusion given intravenously ● Inverse exponential: Whereby a < 0 ○ Example: Plasma concentration of continuous drug infusions given intravenously Sigmoidal curves are also present within medical and biological sciences. These are often harder to model with an equation, but have a characteristic S-shape curve. You may have come across this profile with the haemoglobin dissociation curve profile. Another example that you will come across within pre-clinical studies are dose-response curves. Calculating the Gradient of Exponential Curves Many relationships in biomedical science are exponential, from drug half-life to renal filtration tests. Logarithms can be used to convert an exponential relationship into a straight line relationship. A logarithm of the form log (b) = x is analogous to a = b. In other words, it a reverses the exponential function. Calculating the Gradient of Irregular Curves Curves, especially those with no clear algebraic equation, are difficult to model. A tangent is a straight line that touches a curve at a single point. Because it only touches the curve once, it has the same gradient as the section of the curve it touches. We can draw a tangent onto a curve. As we can easily calculate the gradient of a straight line, we can approximate the gradient of the curve. Calculus involves two components: differentiation and integration. We can take the equation for a curve, and differentiate it to find a new equation that describes the gradient at any point along the curve. As computers can estimate an algebraic equation for many curves, they can also work out the gradient. Key Points ● The general formula y = mx + c is helpful when we are interpreting graphs, as it can be used to define relationships in data sets. 7Applied Mathematics for Medics - Lecture Notes Tuesday 14th September 2021 ● Based on y = mx + c, we can calculate the gradient using 2 known coordinate pairs, and the x-intercept and the y-intercept using the constants m and c. ● We can convert many data sets to fit the relationship y = mx + c. We may use indices, roots, logarithms, or other functions. bx ● Exponential curves can be modelled using “y = a . e ”, with the values of a and b determining whether it is positive, negative or inverse in profile ● Sigmoidal curves also exist but are harder to model ● Logarithms can be used to convert exponential relationships into linear relationships. ● Tangents and calculus can be useful for modelling the gradient of irregular curves. Concentrations Concentration refers to the amount of substance in a given -3 volume of solution, with standard units of mol dm . Every bodily fluid contains dissolved substances. Some of these are intrinsic to function, and others are waste. Example - Serial Dilution You are doing an experiment. You prepare a 1 M standard solution of a blue dye. You perform 4 serial dilutions, each diluting by a factor of 5. What is the concentration of blue dye in each of the diluted solutions? If we dilute by a factor of 5 each time (5 times more dilute), that means that each solution has 1/5 (or 0.2) of the original concentration. This is our dilution factor. ● Undiluted : 1 mol dm -3 -3 -3 ● Dilution 1 : 1 . 0.2 = 0.2 mol dm = 200 mmol dm ● Dilution 2 : 1 . 0.2 = 0.04 mol dm = 40 mmol dm -3 3 -3 -3 ● Dilution 3: 1 . 0.2 = 0.008 mol dm = 8 mmol dm ● Dilution 4 : 1 . 0.2 = 0.0016 mol dm = 1.6 mmol dm -3 Understanding serial dilutions is important for spectrophotometry when preparing standard solutions for a calibration curve. This is a key tool both in research and in biochemical tests. -1 Example - 1.5 g kg Protein Infusion You know that 1.5 g of protein is needed per day for every kg of body mass (for a sedentary patient in hospital). A patient weighs 70 kg. One type of intravenous solution available has 50 g of protein per L of fluid. What is the minimum volume of solution needed (in ml)? Begin by working out the protein needed for a 70 kg patient: 1.5 g kg . 70 kg = 105 g -1 Then work out the volume needed: 105 g / 50 g L = 2.1 L = 2100 ml 8Applied Mathematics for Medics - Lecture Notes Tuesday 14th September 2021 Example - 5% m/v Glucose Solution A 500 ml solution of dextrose (glucose) states that it has a glucose concentration of 5% mass for volume (m/v). Calculate the moles of glucose present in the solution. Firstly, we must recall the formula for calculating amount of substance: moles = mass / M.r For this, we need two values: ● M rs calculated by knowing that the molecular formula of glucose is C H6O12 a6d that the A r of the constituent elements are approximately 12, 1, and 16 respectively. (6 . 12) + (12 . 1) + (6 . 16) = 180. ● Mass is calculated by multiplying 500 ml by 5%, giving 25 g of glucose. We insert these values into our formula: moles = 25 / 180 ≈ 0.139 moles Key Points ● Every bodily fluid dissolves important substances. We can measure their concentration to give us information on health outcomes, and what interventions may be needed. ● By knowing the concentrations of solutions in the lab, we can perform better research and biochemical testing. ● Serial dilution involves diluting solutions that themselves have already been diluted. 9Applied Mathematics for Medics - Lecture Notes Tuesday 14th September 2021 Summary ● Unit conversion is a common and avoidable error when performing calculations. ● Knowing common symbols in formulae speeds up calculations. ● Rearrange equations and formulae before solving with numbers to avoid errors. ● Proportions can be converted to equations by finding constants. ● Graph choice is based on the data type. ● Graphs may be modelled mathematically to understand them better. ● Calculations involving concentration are common and worth practicing. References Greek letter table taken from English Wikipedia “Greek Alphabet” (edited) - CC BY-SA 3.0 https://en.wikipedia.org/wiki/Greek_alphabet Some examples taken from or inspired by “Essential Maths for Medics and Vets” by J A Koenig - CC BY-SA Image Sources (Furfur, 9 May 2016) “File:AB0-Blutgruppen in Deutschland.svg” - Wikimedia Commons https://commons.wikimedia.org/wiki/File:AB0-Blutgruppen_in_Deutschland.svg (Jim.belk, 14 November 2010) “File:Linear Function Graph.svg” https://commons.wikimedia.org/wiki/File:Linear_Function_Graph.svg (Cristian Quinzacara, 27 April 2021) “File:Tangent line to a curve.svg” https://commons.wikimedia.org/wiki/File:Tangent_line_to_a_curve.svg 10