Matrix population models require the population to be divided into discrete stage classes. In many cases, especially when classes are defined by a continuous variable, such as length or mass, there are no natural breakpoints, and the division is artificial. We introduce the ''integral projection model,'' which eliminates the need for division into discrete classes, without requiring any additional biological assumptions. Like a traditional matrix model, the integral projection model provides estimates of the asymptotic growth rate, stable size distribution, reproductive values, and sensitivities of the growth rate to changes in vital rates. However, where the matrix model represents the size distributions, reproductive value, and sensitivities as step functions (constant within a stage class), the integral projection model yields smooth curves for each of these as a function of individual size. We describe a method for fitting the model to data, and we apply this method to data on an endangered plant species, northern monkshood (Aconitum noveboracense), with individuals classified by stem diameter. The matrix and integral models yield similar estimates of the asymptotic growth rate, but the reproductive values and sensitivities in the matrix model are sensitive to the choice of stage classes. The integral projection model avoids this problem and yields size-specific sensitivities that are not affected by stage duration. These general properties of the integral projection model will make it advantageous for other populations where there is no natural division of individuals into stage classes.

Matrix population models require the population to be divided into discrete stage classes. In many cases, especially when classes are defined by a continuous variable, such as length or mass, there are no natural breakpoints, and the division is artificial. We introduce the “integral projection model,” which eliminates the need for division into discrete classes, without requiring any additional biological assumptions. Like a traditional matrix model, the integral projection model provides estimates of the asymptotic growth rate, stable size distribution, reproductive values, and sensitivities of the growth rate to changes in vital rates. However, where the matrix model represents the size distributions, reproductive value, and sensitivities as step functions (constant within a stage class), the integral projection model yields smooth curves for each of these as a function of individual size. We describe a method for fitting the model to data, and we apply this method to data on an endangered plant species, northern monkshood (Aconitum noveboracense), with individuals classified by stem diameter. The matrix and integral models yield similar estimates of the asymptotic growth rate, but the reproductive values and sensitivities in the matrix model are sensitive to the choice of stage classes. The integral projection model avoids this problem and yields size‐specific sensitivities that are not affected by stage duration. These general properties of the integral projection model will make it advantageous for other populations where there is no natural division of individuals into stage classes.

Background Human exposures to bisphenol A (BPA) are widespread. The current study addresses uncertainties regarding human pharmacokinetics of BPA. Objective To reduce uncertainties about the metabolism and excretion of BPA in humans following oral administration. Methods We exposed six men and eight women to 100 μg/kg bw of deuterated BPA (d6-BPA) by oral administration and conducted blood and urine analysis over a three day period. The use of d6-BPA allowed administered d6-BPA to be distinguished from background native (unlabelled) BPA. We calculated the rate of oral absorption, serum elimination, half-life, area under the curve (AUC), urinary excretion, and metabolism to glucuronide and sulfate conjugates. Results Mean serum total (unconjugated and conjugated) d6-BPA Cmax of 1711 nM (390 ng/ml) was observed at Tmax of 1.1 ± 0.50 h. Unconjugated d6-BPA appeared in serum within 5–20 min of dosing with a mean Cmax of 6.5 nM (1.5 ng/ml) observed at Tmax of 1.3 ± 0.52 h. Detectable blood levels of unconjugated or total d6-BPA were observed at 48 hours in some subjects at concentrations near the LOD (0.001–0.002 ng/ml). The half-times for terminal elimination of total d6-BPA and unconjugated d6-BPA were 6.4 ± 2.0 h and 6.2 ± 2.6 h, respectively. Recovery of total administered d6-BPA in urine was 84–109%. Most subjects (10 of 14) excreted >90% as metabolites within 24 hours. Conclusions Using more sensitive methods, our study expands the findings of other human oral pharmacokinetic studies. Conjugation reactions are rapid and nearly complete with unconjugated BPA comprising less than 1% of the total d6-BPA in blood at all times. Elimination of conjugates into urine largely occurs within 24 hours.

Studies that utilize the rodent mammary gland (MG) as an endpoint for assessing the developmental toxicity of chemical exposures typically employ either basic dimensional measurements or developmental scoring of morphological characteristics as a means to quantify MG development. There are numerous means by which to report these developmental changes, leading to inconsistent translation across laboratories. The Sholl analysis is a method historically used for quantifying neuronal dendritic patterns. The present study describes the use of the Sholl analysis to quantify MG branching characteristics. Using this method, we were able to detect significant differences in branching density in MG of peripubertal female Sprague Dawley rats that had been exposed to vehicle or a potent estrogen. These data suggest the Sholl analysis can be an effective tool for quantitatively measuring an important characteristic of MG development and for examining associations between MG growth and density and adverse effects in the breast.