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Mathematical functions relevant to respiratory physiology


This book contains many examples of mathematical statements which relate respiratory variables under specified conditions. Appendix E is intended to refresh the memory of those readers whose knowledge of mathematics has been attenuated under the relentless pressure of new information acquired in the course of study of the biological sciences.

The most basic study of respiratory physiology requires familiarity with at least four types of mathematical relationship. These are:

  • a.

    The linear function.

  • b.

    The rectangular hyperbola or inverse function.

  • c.

    The parabola or squared function.

  • d.

    Exponential functions.

These four types of function will now be considered separately, with reference to examples drawn from this book.

Linear function

Examples

  • 1.

    Pressure gradient against flow rate with laminar flow (page 28). There is no constant factor, and the pressure gradient is zero when flow rate is zero.

  • 2.

    Respiratory minute volume against P co 2 (page 49). In this case there is a constant factor corresponding to a ‘negative’ respiratory minute volume when P co 2 is zero.

  • 3.

    Over a limited range, lung volume is proportional to inflating pressure (page 18). The slope of the line is then the compliance.

Mathematical statement

A linear function describes a change in one variable (dependent or y variable) that is directly proportional to another variable (independent or x variable). There may or may not be a constant factor which is equal to y when x is zero. Thus:


y = ax + b

where a is the slope of the line and b is the constant factor. In any one particular relationship a and b are assumed to be constant, but both may have different values under other circumstances. They are not therefore true constants (like π , for example), and are more precisely termed parameters, whereas y and x are variables.

Graphical representation

Figure E.1 shows a plot of a linear function following the convention that the independent variable ( x ) is plotted on the abscissa and the dependent variable ( y ) on the ordinate. Note that the relationship is a straight line, and simple regression analysis is based on the assumption that the relationship is of this type. If the slope ( a ) is positive, the line goes upwards and to the right. If the slope is negative, the line goes upwards and to the left.

• Fig. E.1, A linear function plotted on linear coordinates. Examples include pressure/flow rate relationships with laminar flow (see Fig. 3.2 ) and P co 2 /ventilation response curves (see Fig. 4.6 ).

The rectangular hyperbola or inverse function

Examples

  • 1.

    The ventilatory response to hypoxia (expressed in terms of P o 2 ) approximates to a rectangular hyperbola, asymptotic on the horizontal axis to the respiratory minute volume at high P o 2 and, on the vertical axis, to the P o 2 at which it is assumed ventilation increases towards infinity.

  • 2.

    The relationships of alveolar gas tensions to alveolar ventilation are conveniently described by rectangular hyperbolas (for carbon dioxide see page 130, and for oxygen see page 138). The curves are concave upwards for gases that are eliminated (e.g., carbon dioxide) and concave downwards for gases that are taken up from the lungs (e.g., oxygen). Curvature is governed by gas output (or uptake), and the asymptotes in each case are zero ventilation and partial pressure of the gas under consideration in the inspired gas.

  • 3.

    Airway resistance approximates to an inverse function of lung volume (page 31).

Mathematical statement

A rectangular hyperbola describes a relationship when the dependent variable y is inversely proportional to the independent variable x , thus:


y = a / x + b

The asymptote of x is its value when y is infinity, and the asymptote of y is its value when x is infinity. If b is zero, then the relationship may be simply represented as follows:


xy = a

Graphical representation

Figure E.2 A shows rectangular hyperbolas with and without constant factors. Changes in the value of a alter the curvature but not the asymptotes. Figure E.2 B shows the same relationships plotted on logarithmic coordinates. The relationship is now linear, but with a negative slope of unity, because if:


xy = a

• Fig. E.2, Rectangular hyperbolas plotted on ( A ) linear coordinates and ( B ) logarithmic coordinates. Examples include the relationships between alveolar gas tensions and alveolar ventilation (see Figs 9.9 and 10.2 ), P o 2 /ventilation response curves (see Fig. 4.8 ) and the relationship between airway resistance and lung volume (see Figs 3.5 and 21.12 ).

then:


log y =− log x + log a

The parabola or squared function

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