++
The construction of a curve or straight line by plotting observed or experimental data on a graph is an important method of visualizing relationships between variables. By general custom, the values of the independent variable (x) are placed on the horizontal line in a plane, or on the abscissa (x axis), whereas the values of the dependent variable are placed on the vertical line in the plane, or on the ordinate (y axis), as demonstrated in Fig. 2-3. The values are usually arranged so that they increase linearly or logarithmically from left to right and from bottom to top.
++
In pharmacokinetics, time is the independent variable and is plotted on the abscissa (x axis), whereas drug concentration is the dependent variable and is plotted on the ordinate (y axis).
++
Two types of graphs or graph paper are usually used in pharmacokinetics. These are Cartesian or rectangular coordinate (Fig. 2-4) and semilog graph or graph paper (Fig. 2-5).
++
Semilog paper is available with one, two, three, or more cycles per sheet, each cycle representing a 10-fold increase in the numbers, or a single log10 unit. This paper allows placement of the data at logarithmic intervals so that the numbers need not be converted to their corresponding log values prior to plotting on the graph. Similarly, when plotted using a computer program, one to three logarithmic cycles are generally plotted on a graph.
++
Fitting a curve to the points on a graph implies that there is some sort of relationship between the variables x and y, such as dose of drug versus pharmacologic effect (eg, lowering of blood pressure). Moreover, when using curve fitting, the relationship is not confined to isolated points but is a continuous function of x and y. In many cases, a hypothesis is made concerning the relationship between the variables x and y. Then, an empirical equation is formed that best describes the hypothesis. This empirical equation must satisfactorily fit the experimental or observed data. If the relationship between x and y is linearly related, then the relationship between the two can be expressed as a straight line.
++
Physiologic variables are not always linearly related. However, the data may be arranged or transformed to express the relationship between the variables as a straight line. Straight lines are very useful for accurately predicting values for which there are no experimental observations. The general equation of a straight line is
++
++
where m = slope and b = y intercept. Equation 2.10 could yield any one of the graphs shown in Fig. 2-6, depending on the value of m. The absolute magnitude of m gives some idea of the steepness of the curve. For example, as the value of m approaches 0, the line becomes more horizontal. As the absolute value of m becomes larger, the line slopes farther upward or downward, depending on whether m is positive or negative, respectively. For example, the equation
++
++
indicates a slope of –15 and a y intercept at +7. The negative sign indicates that the curve is sloping downward from left to right, and the positive nature of the y intercept says that the line intercepts the y axis at +7, above the x axis.
++
+++
Determination of the Slope
+++
Slope of a Straight Line on a Rectangular Coordinate Graph
++
The value of the slope may be determined from any two points on the curve (Fig. 2-7). The slope of the curve is equal to Δy/Δx, as shown in the following equation:
++
++
The slope of the line plotted in Fig. 2-7 is
++
++
Because the y intercept is equal to 3.5, the equation for the curve by substitution into Equation 2.10 is
++
++
+++
Slope of a Straight Line on a Semilog Graph
++
When using semilog paper, the y values are plotted on a logarithmic scale without performing actual logarithmic conversions, whereas the corresponding x values are plotted on a linear scale. However, to determine the slope of a straight line on semilog paper graph, the y values must be converted to logarithms, as shown in the following equation:
++
++
The slope value is often used to calculate k, a constant that determines the rate of drug decline:
++
++
Very often an empirical equation is calculated to show the relationship between two variables. Experimentally, data may be obtained that suggest a linear relationship between an independent variable x and a dependent variable y. The straight line that characterizes the relationship between the two variables is called a regression line. In many cases, the experimental data may have some error and therefore show a certain amount of scatter or deviations from linearity. The least-squares method is a useful procedure for obtaining the line of best fit through a set of data points by minimizing the deviation between the experimental and the theoretical line. In using this method, it is often assumed, for simplicity, that there is a linear relationship between the variables. If a linear line deviates substantially from the data, it may suggest the need for a nonlinear regression model, although several variables (multiple linear regression) may be involved. Nonlinear regression models are complex mathematical procedures that are best performed with a computer program (see Appendix B).
++
When the equation of a linear model is examined, the dependent variables can be expressed as the sum of products of the independent variables and parameters. In nonlinear models, at least one of the parameters appears as other than a coefficient. For example,
++
Linear model: y = ax, y = ax + bx + cx2, y = ax + bx1 + cx2
++
Nonlinear model: y = ax/(b + cx), y = 10e–3x (a, b, and c are parameters and x and x1 are variables)
++
The second nonlinear example as written is nonlinear, but may be transformed to a linear equation by taking the natural log on both sides:
++
+++
Problems of Fitting Points to a Graph
++
When x and y data points are plotted on a graph, a relationship between the x and y variables is sought. Linear relationships are useful for predicting values for the dependent variable y, given values for the independent variable x.
++
The linear regression calculation using the least-squares method is used for calculation of a straight line through a given set of points. However, it is important to realize that, when using this method, one has already assumed that the data points are related linearly. Indeed, for three points, this linear relationship may not always be true. As shown in Fig. 2-8, Riggs (1963) calculated three different curves that fit the data accurately. Generally, one should consider the law of parsimony, which broadly means “keep it simple”; that is, if a choice between two hypotheses is available, choose the more simple relationship.
++
++
If a linear relationship exists between the x and y variables, one must be careful as to the estimated value for the dependent variable y, assuming a value for the independent variable x. Interpolation, which means filling the gap between the observed data on a graph, is usually safe and assumes that the trend between the observed data points is consistent and predictable. In contrast, the process of extrapolation means predicting new data beyond the observed data, and assumes that the same trend obtained between two data points will extend in either direction beyond the last observed data points. The use of extrapolation may be erroneous if the regression line no longer follows the same trend beyond the measured points.
++
+++
Determination of Order
++
Graphical representation of experimental data provides a visual relationship between the x values (generally time) and the y axis (generally drug concentrations). Much can be learned by inspecting the line that connects the data points on a graph. The relationship between the x and y data will determine the order of the process, data quality, basic kinetics, number of outliers, and provide the basis for an underlying pharmacokinetic model. To determine the order of reaction, first plot the data on a rectangular graph. If the data appear to be a curve rather than a straight line, the reaction rate for the data is non-zero order. In this case, plot the data on a semilog graph. If the data now appear to form a straight line with good correlation using linear regression, then the data likely follow first-order kinetics. This simple graph interpretation is true for one-compartment, IV bolus (Chapter 3). Curves that deviate from this format are discussed in other chapters in terms of route of administration and pharmacokinetic model.