Mathematical relationships By Dr. Ahmed Mostafa Assist. Prof. of anesthesia & I.C.U

Graphs show relationships between the depicted variables (Visual expression). The x-axis (horizontal) is sometimes described as the abscissa: on which the independent variable is plotted. The y-axis (vertical) is sometimes described as the ordinate: on which the dependent variable is plotted.

Linear relationships -It is represented by a straight line. -It occurs when the rate of change between the two variables occurs by equal amount. -As: 1.Laminar flow in a uniform. 2.Expansion of mercury in a thermometer.

Linear relationships -The value of y is equal to the value of x at every point. -The slope when drawn correctly should be at 45º if the scales on both axes are the same.

Linear relationships -The line will be a straight line passing through zero. -Slope ἀ proportionate to k. -If we wished to obtain the sum of all y’s for their respective x’s up to X, this would be given as the area of triangle XYO, otherwise known as the area under the line k, and the process of obtaining this area is called integration.

Linear relationships -When x is 0, y must be 0 + b. -The slope of the graph is given by the multiplier a. For example, when the equation states that y=2x, then y will be 4 when x is 2, and 8 when x is 4, etc. The slope of the line will, therefore, be twice as steep as that of the line given by y=1x.

Non linear relationships Logarithms: -The power (x) to which a base must be raised in order to produce the number given as for the equation x =log base (number). -The base can be any number; common numbers are 10, 2 and e ( ). Log 10 (100) is, therefore, the power to which 10 must be raised to produce the number 100; for 10 2 = 100, therefore, the answer is x = 2.

Non linear relationships -Rules of logarithms: 1- Multiplication becomes addition. Log(x y) = log (x) + log (y) 2- Division becomes subtraction. Log (x/y) = log (x) – log (y) 3- Reciprocal becomes negative. Log (1/x) = - log (x)

Non linear relationships -Rules of logarithms: 4- Power becomes multiplication. Log (x n ) = n.log (x) 5- Any log of its own base is one. Log 10 (10) 6- Any log of 1 is zero because n 0 always = 1. Log 10 (1) = 0

Non linear relationships Euler’s number: -Represents the numerical value and is the base of natural logarithms. -Represented by the symbol ‘e’.

Non linear relationships Exponential: - A function whereby the x variable becomes the exponent of the equation ( y=e x ). - We are normally used to x being represented in equations as the base unit (i.e. y = x 2 ). In the exponential function, it becomes the exponent (y=e x ), which conveys some very particular properties.

Non linear relationships Types of exponential curves: -At negative values of x, the slope is shallow but the gradient increases sharply when x is positive. -The curve intercepts the y axis at 1 because any number to the power 0 (as in e 0 ) equals 1. -Most importantly, the value of y at any point equals the slope of the graph at that point.

Non linear relationships Types of exponential curves: -The curve crosses y axis at value of a. -It tends towards infinity as value of t increases. - This is clearly not a sustainable physiological process but could be seen in the early stages of bacterial replication where y equals number of bacteria and multiplication of some cancer cells

Non linear relationships Types of exponential curves: -The x axis is again an asymptote and the line crosses the y axis at 1. -This time the curve climbs to infinity as x becomes more negative. -This is because -x is now becoming more positive. -The curve is simply a mirror image, around the y axis, of the positive exponential curve seen above.

Non linear relationships Types of exponential curves: -The curve crosses the y axis at a value of a. -It declines exponentially as t increases. -The line is asymptotic to the x axis. -This curve is seen in physiological processes such as drug elimination, lung volume during passive expiration and decay of radioactive isotope.

Non linear relationships Types of exponential curves: -The curve passes through the origin and has an asymptote that crosses the y axis at a value of a. -Although y increases with time, the curve is actually a negative exponential. -This is because the rate of increase in y is decreasing exponentially as t increases. -This curve may be seen clinically as a wash- in curve or that of lung volume during positive pressure ventilation using pressure controlled ventilation and uptake of inhalational anesthetic.

Duration of exponential function In exponential process, although the quantity decrease (in wash out curve), it never actually reaches zero. Consequently the total length of time taken by the exponential process is infinite and the total time cannot be used to measure the duration of the exponential process. Therefore, alternative values are used to describe the rate of any exponential process:

Duration of exponential function 1.Half-life: The time taken for the value of an exponential function to decrease by half is the half-life and is represented by the symbol t ½. An exponential process is said to be complete after five half-lives. At this point, % of the process has occurred.

Duration of exponential function

2. Time constant: The time it would have taken for a negative exponential process to complete, were the initial rate of change to be maintained throughout. It was given the symbol t. Or, The time taken for the value of an exponential to fall to 37%of its previous value. Or, The time taken for the value of an exponential function change by a factor of e 1. Or, The reciprocal of the rate constant. The time constant = 1/e = 1/ = 37%

Duration of exponential function An exponential process is said to be complete after three time constants. At this point 94.9% of the process has occurred. Uses of time constant: - Describe the expiration: If the expiratory time constant is 0.3 second, then the values of the tidal volume which are expired and those remaining in the lungs are as follows: a.After 0.3 s (1t), 63% of TV is expired and 37% remains in the lungs. b.After 0.6 s (2t), 86.5% of TV is expired and 13.5% remains in the lungs. c.After 0.9 s (3t), 95% of TV is expired and 5% remains in the lungs.

Duration of exponential function

3. Rate constant: The reciprocal of the time constant. Given the symbol k. Or, A marker of the rate of change of an exponential process. The rate constant acts as a modifier to the exponent as in the equation y=ekt (e.g. in a savings account, k would be the interest rate; as k increases, more money is earned in the same period of time and the exponential curve is steeper).

Duration of exponential function

Thank you Dr. Ahmed Mostafa