Environmental toxicology is the study of the fate of toxic chemicals once they are released into the environment. It considers the effects on entire ecosystems by first examining the consequences of those chemicals on cells, organisms, populations, and communities. Ecotoxicology covers a wide range of disciplines, including genetics, chemistry, physiology, and, of course, mathematics. Math can be found in almost all aspects of ecotoxicology, from equations for populations growth rate to standards set for the benefit of all scientists. It is difficult to classify the different types of mathematics used in ecotoxicology, since they are all interrelated.

One of the most fundamental aspects of the study of environmental toxicology is an understanding of the dose-response relationship on the level of individual organisms. A dose-response relationship is the connection between exposure and a specified endpoint. An endpoint is a physical change in the cells or biochemistry of the organisms being studied. Endpoints can vary based upon the chemical that is introduced. An Effective Dose (ED) is the amount of a chemical that brings about an endpoint specified by the researcher. A Lethal Dose (LD) is the dose that causes death to the organism. This is the most common endpoint to study, as it is the easiest to observe and often has the greatest effect on an ecosystem. Variations on ED and LD are Effective Concentration (EC) and Lethal Concentration (LC). A concentration refers to the buildup of chemicals in a system, through many successive doses. These terms are all used in conjunction with a number value between 1 and 100, the most common example being LD₅₀. An LD₅₀ is the dose that is required to cause the death of 50% of a test group on a regular basis. This means that this is the median lethal dose, or one that has a fifty percent chance of causing death. It, as well as measures of ED₅₀, EC₅₀, and LC₅₀, are used to compare toxic chemicals and help researchers to understand the mechanisms that chemicals use to cause toxicity. (Walker, 108)

The values that are obtained for lethal doses are usually represented graphically. In graphical analysis, the percentage of animals that die in each group following the administration of the dose is plotted. There are three distinct but similar types of graphs used to determine the LD₅₀: cumulative, distributive, and probit. A cumulative graph begins with the amount of chemical needed to produce any mortality. The dose directly before this is referred to as No Observable Effect Level (NOEL). The cumulative graph plots the percentage of organisms responding against the amount of the dose. (See graph) Those organisms that did not respond are not shown on the graph. The graph will be an S-shaped curve, where the LD₅₀ can be easily determined by observing which dose caused the death of 50% of the test group. (Walker, 122)

The next type of graph is a Gaussian distribution. This distribution is the ideal type expected of a dose-response relationship. Again, this graph plots percentage response against dosage. However, this graph produces a bell-shaped curve because it takes into account the actual amount of organisms that died upon the administration of each individual dose. It more clearly shows the relative sensitivity of the organisms and is used to calculate the standard deviation from the mean. This graph shows the percentages of organisms that are resistant to a given dose. The LD₅₀ is again the dose causing mortality to 50% of the test group, but the Gaussian distribution shows the percentages of organisms that survived the LD₅₀, and so on. The LD₅₀ is the mean of the statistical distribution. The standard deviation (SD) is a measurement of how close the collected data are to the mean. The mean ± one standard deviation includes 68.8% of the test organisms. The mean ± 2SD = 95.5%, while the mean ± 3SD = 99.7% of the organisms. This means that the dose that is three standard deviations greater (+3) than the mean dose will cause the death of 99.7% of the organisms. Also, the dose that is three standard deviations less (–3) than the mean will only cause the death of 0.3% of the organisms.

The final type of graphical representation is the probit, or probability unit, graph. The percent response in converted to units of deviation, called Normal Equivalent Deviates (NED). One NED equals one standard deviation. To avoid negative numbers, the standard deviation is added to 5. In this way, 50% response, or 0 standard deviations, is equal to 5 probits. +2 standard deviation equals 7 probits, and –2 standard deviation equals 3 probits. The probit units are plotted against a log scale of the dosage. (See graph) The LD₅₀ is determined by observing the dosage that is found at the probit unit of 5. The plot is the only form of dose-response that yields a straight line throughout. (Walker, 122)

While LD₅₀ is one value that is used for comparison among chemicals and for testing their toxicity, there are many other values that are used for setting standards of comparison. These values are used by researchers to communicate and standardize methods of analysis. One example of this is the Bioconcentration Factor, BCF. The chemical entering an organism from the water can be seen as an ideal model—the chemical moving from a water environment to a lipid, or fatty, environment. The BCF is found by dividing the concentration of the compound in the organism by the concentration of the compound in the environment. If the concentration of the pollutant in the organism increases faster than the concentration of the pollutant in the environment, then the compound is bioaccumulating, or accumulating in the organism. For aquatic environments, there is a simple laboratory method for determining the potential of a chemical to accumulate in organisms. This is the evaluation of the Octanol-Water Coefficient, K_ow. Uptake of a chemical by an aquatic organism can often be seen as simply passive diffusion of the chemical into the organism. The chemical, in order to diffuse, must have an affinity for the lipid barrier it must cross to get into the organism. It also must have a similar affinity for the water environment on the inside of the organism. To determine K_ow, known amounts of water and octanol are placed in a separatory funnel. They will not mix, because octanol is hydrophobic, and analogous to the lipid membranes of the cells of the aquatic organisms. The chemical is added to the funnel, and the water and octanol are both tested to determine in which solvent the chemical accumulated. K_ow is determined by dividing the concentration of the compound in the octanol by the concentration in the water. A K_ow close to 1 indicates that the chemical can easily move across barriers into organisms. (See graph) A value much smaller than 1 shows high water solubility, and a value much higher than 1 shows high lipid solubility. It must be kept in mind that the K_ow value is a model, and that in a natural environment chemicals react with each other, are metabolized by the organism, and many other effects. However, K_ow is one mathematical representation that is helpful for comparison of chemicals and for making important estimates. (Walker, 72)

Besides K_ow, there is another mathematical equation that determines the hazard of the release of a certain chemical into the environment. This is Henry’s Law Constant (H) and it is a measure of the volatilization of a chemical from water. This is the ability of the chemical to become a vapor, and come out of the soil or water on its own. It is also a measure of the ability of the chemical to move around in the soil, especially in the air trapped within it. The constant can be calculated as follows:

H = (V_p)

(S)

Where V_p is the vapor pressure of the chemical in atmospheres, and S is the solubility in moles per cubic meter. The reciprocal of the product of H and the LC₅₀ for the chemical will give an F factor, which will be an indication of the toxic potential of the chemical in a water environment. A high F value indicates a high toxic potential when found in water, where a low F value indicates little toxic potential. If a chemical vaporizes easily from water, it is less likely to be available to organisms there. However, if the chemical is found in soil, more volatility can lead to more toxic potential. If a chemical is volatile, it can come off of soil particles and move around in the air in the soil. The less the chemical adsorbs to the soil particles, the more it is available to organisms, and therefore, the more potential it has for toxicity. (Schuurmann, 359)

Along with graphical representations and the use of constants, there are many other mathematical applications in ecotoxicology. Many of these focus around equations used in the study of the toxic chemicals. An example of an equation useful to researchers is the Henderson-Hasselbach equation. This equation deals with pH, which is the log of the concentration of H⁺ ions that dissociate from a compound in water. An acid is a chemical that donates H⁺, and has a low pH, and a base is one that accepts H⁺, and has a high pH. An acid is represented HA à H⁺ and A^- where the H⁺ dissociates. A base can be seen as H⁺ + B^- à BH, where the base accepts the H⁺ ion. Only the nonionized form of the compound can passively diffuse across membranes. The pKa is the pH where the chemical concentration is 50% ionized and 50% nonionized. The Henderson-Hasselbach equation is used to determine the rate of diffusion of a chemical across membranes, especially across stomach lining into the blood steam when a chemical is ingested. It should be kept in mind that the pH of the stomach is 1 (very acidic), and the pH of the blood plasma is 7.4 (weakly basic). The equation is as follows:

pH = pKa + log [basic form]

[acidic form]

For a weak acid the concentration of the basic form over the concentration of the acidic form denotes [H⁺] + [A^-] / [AH]. For a weak base, the formula is [B] + [H⁺] / [BH⁺]. The consequences of this equation can easily be seen in the examination of an example. Aspirin has a pKa of 3, and is weakly acidic. To determine the potential for aspirin to remain in the blood once it has been transported there, the values need to be inserted into the equation:

7 = 3 + log [B]

[A]

where 7 is the pH of the blood and 3 is the pKa of the aspirin. In this way, it is found that [B] / [A] is equal to 10,000 / 1. This means that the ratio of the acidic form (AH) to the basic form (H⁺ + A^-) is 1 to 10,000. Since passive diffusion can only occur in the form of AH, once the aspirin is absorbed into the blood, it will not readily diffuse out of it. A similar example can be used to determine the potential for the aspirin to be absorbed by the stomach lining, with a pH of about 1:

1 = 3 + log [B]

[A]

From this equation, it can be seen that [B] / [A] is 100 / 1. This means that the ratio of the nonionized form to the ionized form is 100 to 1. Since more of the molecules of aspirin when in the stomach are in the neutral form, they are likely to be absorbed. (Schuurmann, 510)

Thus far, only the effects of chemicals on individual organisms or test groups of organisms have been considered. However, mathematics in ecotoxicology can also be related to the study of entire populations. This can be seen in studies of Population Growth Rate (PGR). Population growth rate is defined as the population increase per unit time, divided by the number of individuals in the population. A PGR of zero indicates that a population is neither increasing nor decreasing. A positive PGR is a sign that the population is increasing, and a negative PGR shows that a population is decreasing. This can be understood mathematically if population size, N(t), is plotted as a function of time, t. dN/dt represents the population increase per unit time, in units of animals per unit time. This is put on a per animal basis by dividing by the number of individuals in a population. This means that PGR = (1 / N) (dN / dt). For example, if there are 1000 individuals in a population which is increasing by 1 individual per year, the PGR is 0.001 per animal per year. If population growth rate is constant, then N(t) = N(0)e^rt, which shows that a constant PGR leads to an exponential population increase. A similar factor that is sometimes considered is net reproductive rate, l . It is defined as l = e^r, which is equivalent to r = log_e l . Net reproductive rate is the factor by which a population is multiplied each year. If a population doubles, then l = 2. In this case, r = log_e 2, so population growth rate equals 0.693. (Walker, 214)

This population growth rate depends on many factors. Among these are the birth and death rates of individuals, the growth rates of individuals, and the timing of breeding attempts. The calculation of Population Growth Rate can take into account all of these factors by using the Euler-Lotka Equation:

1 = ½ n₁ l₁ e^-rt₁ + ½ n₂ l₂ e^-rt₂ + ½ n₃ l₃ e^-rt₃ + ….

Where t₁ represents the age of the first breeding, t₂ is the age of the second breeding, and so on. The n values represent the number of offspring produced by each female at each breeding attempt. The probability of a female surviving from birth to each of the times t is shown in the l values. PGR in the equation is r, and can be derived when the value is found which balances the right side of the equation at 1. In this equation, it is assumed that the proportion of females in a population is constant. Another way of mathematically analyzing the life histories of populations is for researchers to record the number of organisms surviving and the number of offspring produced at regular intervals. Then, this information is tabulated in a matrix, called a population projection matrix, which is then analyzed using matrix algebra. (Hoffman, 688)

Mathematics can be used in a variety of ways in the study of environmental toxicology. Its use can be applied to all aspects of the discipline, ranging from the effects of chemicals on individual organisms to the effects on entire populations. Different forms of graphical representations are often utilized to analyze toxicological data. Also, standards are often set using mathematical derivations, and important equations are also used to make necessary calculations. In this way, mathematics is a fundamental aspect of ecotoxicology.

Hoffman, David, et. al. Handbook of Ecotoxicology. Boca Raton: Lewis Publishers. 1995.

Schuurmann, Gerrit and Markert, Bernd. Ecotoxicology: Ecological Fundamentals, Chemical Exposure, and Biological Effects. New York: John Wiley and Sons, Inc. 1998.

Walker, C. H., et. al. Principles of Toxicology. Bristol, PA: Taylor and Francis Ltd. 1997.