Math 121 - Calculus for Biology I Spring Semester, 2009 Allometric Modeling
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Allometric Modeling

1. Cumulative AIDS cases
2. Link to Nonlinear Least Squares
3. Allometric or Power Law Model
4. Review of Exponents and Logarithms
5. Graphing Exponentials and Logarithms
6. Worked Examples
7. Finding Allometric Models
8. Log-Log Graphs
9. References

Cumulative AIDS cases

• AIDS has had a significant impact on both personal behavior and public policy
• The new protease inhibitors have significantly improved the quality of life for those who are HIV positive; however, this has come at a substantial cost to society
• The new drugs are extremely expensive, are difficult to take because of the complex scheduling requirements to be effective, and have many strong side effects (besides not always working for a particular person or strain of the HIV virus)
• In turn, there are a number of people who are now avoiding safe sex practices as they no longer fear the "Death Sentence" that used to be associated with an HIV infection

Below is a figure illustrating the HIV virus.

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	[click for enlarged and detailed image] illustration by Russell Kightley Media, all rights reserved http://www.rkm.com.au

• Society needs to know the extent of this disease from both an economic and sociological perspective
• We need to know what is the expected case load in the future
• This is an extremely complex modeling problem
• Below is a table of cumulative cases of AIDS between 1981 and 1992 [1]
• An animated .gif shows the spread of the disease (through mortality statistics) over a similar period of time

Year	Cumulative AIDS Cases (thousands)
1981	97
1982	709
1983	2,698
1984	6,928
1985	15,242
1986	29,944
1987	52,902
1988	83,903
1989	120,612
1990	161,711
1991	206,247
1992	257,085

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	click here to view mpeg

• The data is clearly not linear
• There are general methods for finding the least squares best fit to nonlinear data
• These techniques are very complicated an often difficult to implement
• A hyperlink provides an applet for finding the best nonlinear least squares fit to the data for cumulative AIDS cases, which is different from the technique we'll show below.

Allometric or Power Law Model

• Using a least squares fit to nonlinear data can be extremely difficult
• A few standard nonlinear models used in biological applications are easily analyzed
• The technique developed in this section is known as the Power Law of Modeling
• It is also referred to as Allometric Modeling
• Allometric models are used in modeling complex biological phenomena
  • Used when the actual mechanisms underlying the model behavior are complex
  • There is a need to make some predictions

Allometric Models

• Allometric models assume a relationship between two sets of data, x and y, that satisfy a power law of the form

• A and r are parameters that are chosen to best fit the data in some sense
• This model assumes that when x = 0, then y = 0
• This method provides its best predictive capabilities when examining a situation that lies between the given data points

Example

• For example, if the number of species of herptofauna on Carribean islands is determined for a collection of islands with varying areas
  • This model would give a reasonable estimate for the expected number of species on another Carribean island with an area that lies between the collected data
  • It would not be appropriate for extending to a large continent as the area is significantly beyond the range of the collected data
  • It wouldn't even be appropriate for another island such as Iceland, which lies in a different type of climate and has a different geography.

• Allometric models are found by taking the logarithms of the data
• Determine if the log-log graph of the data produces roughly on a straight line
• If this is the case, then a power law relationship makes a reasonable model

Below is an applet showing the linear least squares fit to the logarithms of the data for cumulative aids cases, and the graph to the right shows the modeling relationship with a normal scale. The allometric model has x be time in years since 1980 and y be the cumulative AIDS cases.

• The minimum least squares for the log of the data gives J(A,r) = 0.10
• The best slope is r = 3.27
• The best intercept is ln(A) = 4.42
• This gives the best fit power law for this model as

• The graph of the power law provides a reasonable fit to the data
• The fit is weakest at the end where we'd like to use the model to predict the cumulative AIDS cases for the next year
• The model predicts 366,990 cases in 1993, which is clearly too high from the given data
• The analysis gives some indication of the rate of growth for this disease
  • Gives a first approximation for improved models
  • Could be applied to expected spread of another disease with similar infectivity as HIV
• This modeling technique is valuable for analysis of other data sets and can provide insight into the underlying biology of a problem
• A better fit to the data is shown in the nonlinear least squares appendix that can be viewed through the hyperlink
• More examples are in the computer labs.

Review of Exponents and Logarithms

Here are several properties of exponents that you should remember from algebra.

• For solving equations with exponents, the inverse function of the exponent is needed, the logarithm

• then the inverse equation that solves for x is given by

• The a in the above expression is called the base of the logarithm.

• Properties of logarithms

• The only property that requires the base of the logarithm for Property 5
• All other properties are independent of which base is used
• The two most common logarithms that are used are log₁₀ and log_e
• The natural logarithm, often denoted log or ln, is the one most commonly used (and is the default on your calculator)
• We will soon see the importance of the natural base e
• Unless otherwise specified, use the natural logarithm. (Note that Excel defaults to log₁₀.)

Graphing Exponentials and Logarithms

• The exponential function, e^x, and the natural logarithm, ln(x), are inverse functions
• This section shows the graphs of these functions
• The study of e^x arises very naturally in Calculus applications
• For graphing purposes, e is an irrational number between 2 and 3, more precisely, e = 2.71828....
• The domain of e^x is all of x, becoming extremely small very fast for x < 0 (a horizontal asymptote of y = 0) and growing very fast for x > 0
• Its range is y > 0
• Similarly, the graph of y = e^-x has the same y-intercept of 1, but its the mirror reflection through the y-axis of y = e^x

• Since ln(x) is the inverse function of e^x, the graph of this function mirrors the graph of e^x through the line y = x
• The domain of ln(x) is x > 0, while its range is all values of y
• As y = ln(x) becomes undefined at x = 0, there is a vertical asymptote at x = 0

• The exponential function plays a role in many applications, so it is very important to understand this function
• Several examples are illustrated in the hyperlinked  Worked Examples for Exponentials and Logarithms section.

Finding Allometric Models

• The Allometric model for two sets of data, x and y satisfies a power law of the form

• We want to choose the parameters A and r that best fit the data
• Take the logarithm of both sides, then use the properties of logarithms to simplify the equation

• From this formula, the logarithm of the data, ln(x) and ln(y), when graphed, fits a straight line
• Take X = ln(x), Y = ln(y), and a = ln(A), then

Y = a + rX

• This is a line with a slope of r and a Y-intercept of ln(A)

AIDS Example

• Below is a table that includes both the data and the logarithms of the data.

Below shows a graph of the logs of the data (year-1980 and cumulative AIDS cases) along with the best straight line fit.

• This plot shows that when the logarithms of the data for the cumulative AIDS cases are plotted against the logarithms of the time since 1980, then these logarithmic data lie fairly close to a straight line
• The data are flattening for the later years suggesting a diminished rate of increase
• The least squares best fit of the straight line to the logarithms of the data give a slope of r = 3.274 and intercept of a = ln(A) = 4.415, which gives A = 82.70
• Thus, an allometric or power law model is a reasonable description of the data.

Log-Log Graphs

• The log-log plot allows direct plotting of the data
• The user graphs the data directly onto a graph with logarithmic scales on the axes
• If the data falls on a straight line, one suspects an allometric or power law model

• Below we show a plot of the original data on cumulative AIDS cases against the date - 1980 on a graph with logarithmic scaled axes.

Example: Weight and Pulse

• Smaller animals have a higher pulse than larger animals
• Assume an allometric model (A Lab will justify this)
• Suppose a 17 g (or .017 kg) mouse has a pulse of 500 beats/min and a 68 kg human has a pulse of 65
• Use these data to form an allometric model
• Predict the pulse for a 1.34 kg rabbit.

Solution:

• The power law gives

• Logarithms give:

• This is a straight line in ln(P) and ln(w) with slope of k and intercept of ln(A)
• From the data,

• The slope k is given by:

• This slope with one of the points is used to find ln(A):

• If we use the first equation with a 1.34 kg rabbit, then it gives P = 171.

References:

[1] E. K. Yeargers, R. W. Shonkwiler, and J. V. Herod, 1996, An Introduction to the Mathematics of Biology: with Computer Algebra Models, Birkhäser, Boston.