Four-parameter Logistic Model Research Articles

Comparisons of titer estimation methods for multiplexed pneumococcal opsonophagocytic killing assay

Titer estimation is one of the major components of immunoassay and vaccine development. A multiplexed in vitro opsonization assay (MOPA) is widely accepted to quantitate Streptococcus pneumococcal antibodies to serotype-specific pneumococcal capsular polysaccharide. Titer estimation of vaccine based on OPA is one important component of standardization of OPA, and the selected statistical method is a factor influencing the accuracy and precision of titer estimation. We evaluated five titer estimation methods for pneumococcal OPA in terms of precision and accuracy using three data sets generated by specifically designed experiments with both an eight-dilution and an eleven-dilution design. The bootstrap resampling technique was also used to determine the performance of the estimation. We concluded that the traditional direct method did not perform as well as the other four methods in terms of precision and accuracy of titer estimation. The Spearman–Kärber estimator might be biased upward for OPA titer estimation. The four-parameter logistic model (4PL) method is an alternative choice for OPA titer estimation. The eleven-dilution design provided more information than the eight-dilution design for titer estimation and enhanced precision of estimators. UAB opsotiter, computer software using the statistical language R and Microsoft Excel ®, was developed to implement OPA titer estimation.

A Four-Parameter Logistic Model for Estimating Titers of Functional Multiplexed Pneumococcal Opsonophagocytic Killing Assay

In vitro opsonophagocytosis killing assay (OPA) is widely accepted to quantitate Streptococcus pneumococcal antibodies to serotype-specific pneumococcal capsular polysaccharide (PS). A titer estimation method is needed for large scale data generated by OPA, and it is one component of OPA standardization. In order to improve the reliability of OPA results, we developed a nonlinear fitting method using the Levenberg–Marquardt algorithm with an option of a robust procedure to estimate titers of OPA data. Performance of the proposed method was evaluated by comparing precision and accuracy of titer estimation with traditional methods used in the literature by analyzing real experimental data sets. Goodness-of-fit to experimental data for the two model-based methods was also assessed. We conclude that the four-parameter logistic model is an alternative choice for titer estimation of OPA data. Computer software using the statistical language R and Microsoft Excel was developed to implement our calculation algorithm for OPA data.

Inactivation of Bacillus anthracis Spores by Liquid Biocides in the Presence of Food Residue

Biocide inactivation of Bacillus anthracis spores in the presence of food residues after a 10-min treatment time was investigated. Spores of nonvirulent Bacillus anthracis strains 7702, ANR-1, and 9131 were mixed with water, flour paste, whole milk, or egg yolk emulsion and dried onto stainless-steel carriers. The carriers were exposed to various concentrations of peroxyacetic acid, sodium hypochlorite (NaOCl), or hydrogen peroxide (H(2)O(2)) for 10 min at 10, 20, or 30 degrees C, after which time the survivors were quantified. The relationship between peroxyacetic acid concentration, H(2)O(2) concentration, and spore inactivation followed a sigmoid curve that was accurately described using a four-parameter logistic model. At 20 degrees C, the minimum concentrations of peroxyacetic acid, H(2)O(2), and NaOCl (as total available chlorine) predicted to inactivate 6 log(10) CFU of B. anthracis spores with no food residue present were 1.05, 23.0, and 0.78%, respectively. At 10 degrees C, sodium hypochlorite at 5% total available chlorine did not inactivate more than 4 log(10) CFU. The presence of the food residues had only a minimal effect on peroxyacetic acid and H(2)O(2) sporicidal efficacy, but the efficacy of sodium hypochlorite was markedly inhibited by whole-milk and egg yolk residues. Sodium hypochlorite at 5% total available chlorine provided no greater than a 2-log(10) CFU reduction when spores were in the presence of egg yolk residue. This research provides new information regarding the usefulness of peroxygen biocides for B. anthracis spore inactivation when food residue is present. This work also provides guidance for adjusting decontamination procedures for food-soiled and cold surfaces.

Individual response of growing pigs to sulphur amino acid intake

Two N balance experiments were conducted to study the individual response of growing pigs to limiting amino acid (AA) intake. Series of fifteen diets with increasing concentration of sulphur amino acids (SAA, Expt 1) or methionine in the presence of excess cystine (Expt 2) were fed sequentially to nine pigs during a 15-day experimental period. The concentration of the AA under test ranged from 50% to 140% of the requirement while other essential AA were given in a 25% excess relative to the limiting AA. N retention was related to the limiting AA intake using rectilinear and curvilinear models. In Expt 1, the quadratic-plateau model fitted the individual data significantly better (p = 0.01) than the linear-plateau model. No difference was found between the two models in Expt. 2, presumably due to the sparing effect of excess cystine on methionine utilization. Exponential, saturation kinetics or four-parameter logistic models fitted to data for all pigs showed that their goodness of fit was similar to those of quadratic-plateau or linear-plateau models. Significant differences (p < 0.05) were found between individual plateau values for N retention within each experiment while the slopes of the regression lines did not significantly differ either in Expt 1 (p = 0.07) or Expt 2 (p = 0.45). There was a positive correlation between the slope and plateau values of the linear-plateau model in Expt 1 (r = 0.74, p = 0.02) but no significant correlation was found in Expt 2 (r = -0.48, p = 0.13). Marginal efficiencies of SAA and methionine utilization derived from the linear-plateau model were 0.43 and 0.65 respectively. Based on linear-plateau and quadratic-plateau models, daily requirements of SAA and methionine for a 50 kg pig were estimated to be 13.0 and 5.9 g and 14.3 and 6.1 g respectively.

Correlations between circulating insulin-like growth factor-I and thyroxine and cortisol hormone levels, and some biometrical traits in female brood stocks during the late stages of sex maturation and in juvenile Persian sturgeon (Acipenser persicus)

Relationships between insulin-like growth factor-I (IGF-I) and thyroxine (T4) and cortisol hormones were studied in female brood stocks of Persian sturgeon [Acipenser persicus; caught from both sea water (SW) and fresh water (FW)] during late stages of sex maturation. A number of biometrical traits were also studied that could represent the reproductive and/or growth states of brood stocks, and the possible relationship between IGF-I and growth was assessed in juvenile Persian sturgeons between 1 and 4 years of age. IGF-I, T4 and cortisol were measured in serum samples using commercially available kits. A four-parameter logistic model test was performed between the standard curves and the sample dilutions for each hormone. Parallelism, linearity and regression coefficients for the linearized standard curves and serial dilutions of samples were not significantly different (P < 0.05). Serum IGF-I levels in the juveniles were higher than those in the SW brood stocks, and cortisol levels in the former were lower than those in both brood stocks (P < 0.05). T4 levels in serum samples of juveniles were below the detectable level. IGF-I concentrations in juveniles were correlated with total weight, total length and fork length, but they did not increase significantly with increasing age from 1 to 4 years old (P < 0.05). Compared with SW brood stock, the FW brood stock was younger, had a smaller girth, smaller polarization index and higher ova diameter (P < 0.05). There were no differences between IGF-I and T4 levels in the two brood stocks, but cortisol levels were significantly higher in the FW brood stock. Percentage of fertilization was correlated with serum IGF-I in both brood stocks (P < 0.05). Our results support a role for IGF-I during the juvenile growth and reproductive physiology in female brood stocks of the Persian sturgeon.

L-Glutamate biosensor for estimation of the taste of tomato specimens

An amperometric biosensor has been developed for measurement of Umami, or the taste based on the amount of L-glutamate, in tomato foods. The biosensor is based on an enzyme-mediator system in which L-glutamate oxidase is used for biochemical oxidation of L-glutamate and a tetrafulvalene-tetracyanoquinodimethane (TTF-TCNQ) paste, prepared from the mixture of TTF-TCNQ salt, graphite powder, and silicone oil, serves as the mediator. The limit of detection, calculated by use of a four-parameter logistic model, was 0.05 mmol L(-1), and the limit of quantification was 0.15 mmol L(-1). The correlation coefficient (R2) was 0.990 and the relative standard deviation was no more than 1% (n=5). The response time (tau (95)) was 20-50 s, depending on concentration. The repeatability of the sensor was better than 5% (n=10). The sensor developed was stable for more than ten days.

Effect of Carbon Black on Devulcanized Ground Rubber Tire—Natural Rubber Vulcanizates: Cure Characteristics and Mechanical Properties

Ground rubber tire (GRT) powder is devulcanized using a counter-rotating twin screw extruder and then the product is mixed with 30 phr virgin natural rubber (NR) and HAF carbon black added in various proportions. The data obtained from a cure rheometer is used to fit in a four-parameter logistic model to correlate the variation of torque with time. The mechanical properties were evaluated, which suggested that optimum properties are obtained with a 15 phr HAF addition. The effects of a sulfur/accelerator (S/CBS) ratio study reveals that satisfactory properties are obtained at a S/CBS ratio of 1.5/0.75.

Comprehensive Algorithm for Quantitative Real-Time Polymerase Chain Reaction

Quantitative real-time polymerase chain reactions (qRT-PCR) have become the method of choice for rapid, sensitive, quantitative comparison of RNA transcript abundance. Useful data from this method depend on fitting data to theoretical curves that allow computation of mRNA levels. Calculating accurate mRNA levels requires important parameters such as reaction efficiency and the fractional cycle number at threshold (CT) to be used; however, many algorithms currently in use estimate these important parameters. Here we describe an objective method for quantifying qRT-PCR results using calculations based on the kinetics of individual PCR reactions without the need of the standard curve, independent of any assumptions or subjective judgments which allow direct calculation of efficiency and CT. We use a four-parameter logistic model to fit the raw fluorescence data as a function of PCR cycles to identify the exponential phase of the reaction. Next, we use a three-parameter simple exponent model to fit the exponential phase using an iterative nonlinear regression algorithm. Within the exponential portion of the curve, our technique automatically identifies candidate regression values using the P-value of regression and then uses a weighted average to compute a final efficiency for quantification. For CT determination, we chose the first positive second derivative maximum from the logistic model. This algorithm provides an objective and noise-resistant method for quantification of qRT-PCR results that is independent of the specific equipment used to perform PCR reactions.

Modeling the concentration–response function of the herbicide dinoseb on Daphnia magna (survival time, reproduction) and Pseudokirchneriella subcapitata (growth rate)

Models describing dose–response relationships are becoming increasingly popular in ecotoxicology. They allow simple and thorough evaluations of toxicity test results, including inter- and extrapolations to concentrations or exposure times other than those tested. Simple parametric regression models are of particular interest because their parameters may be attributed mechanistic meanings and they can be applied without sophisticated mathematical and computational support. We recently proposed a four-parameter logistic regression model to fit the survival data of Daphnia magna under dinoseb stress. The model parameters are the maximum survival time, the minimum time required for an individual to die, effect concentration, EC 50, and a curve shape parameter. This model has now been applied to compare the lethality and reproduction toxicity of D. magna and the growth inhibition of Pseudokirchneriella subcapitata under dinoseb stress. It can be fitted adequately to all the measured data and the parameters can be attributed biological meanings in any of the three endpoints. A comparison of the modeled concentration–response functions of all three endpoints for dinoseb toxicity shows that the range of ECs with respect to both D. magna and algae is steep (a decrease of between 0.1 and 0.6 mg/L). The survival and reproduction of D. magna exhibit similar characteristic concentration–response functions and toxicities. The statistical no-effect concentration (SNEC) is 0.14 (survival) and 0.11 (reproduction) mg/L, respectively. On the other hand, algae seem to be less sensitive to dinoseb than D. magna (SNEC: 0.48 mg/L). However, further investigations of individual algae may lead to a more suitable comparison. We speculate that the four parameters of the model function can be related to specific properties of chemicals and organisms. Characterization of these properties would allow simple and appropriate estimation of the toxic effects of these chemicals.

Defining the linear portion of a sigmoid‐shaped curve: bend points

AbstractFormulae are provided that define the ‘bend points’, the beginning and end of the essentially linear dose–response region, for the four‐parameter logistic model. The formulae are expressed in both response and dose units. The derivation of the formulae is shown in order to illustrate the general nature of the methodology. Examples are given that describe how the formulae may be used while planning and conducting bioassays. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Mapping of antibody responses to the protective antigen of Bacillus anthracis by flow cytometric analysis.

Knowledge of the target and functional capability of the antibody response against an antigen provides more specific and relevant information about protective immunity than measuring the total amount of antibody produced against an antigen. Using flow cytometry, a competitive assay has been created for measuring the antibody response against important epitopes of an antigen. Monoclonal antibodies (mAbs) against the protective antigen (PA) component of Bacillus anthracis are available that neutralize the activity of lethal toxin (LeTx). Flow cytometric analysis revealed that these mAbs bind PA conjugated to polystyrene latex microspheres. Unlabeled mAbs against PA competed with fluorescent mAbs that bind the homologous epitope but not with fluorescent mAbs that bind heterologous epitopes. Four-parameter logistic models were developed for measuring the antibody response against two epitopes of PA. Sera from anthrax-vaccinated rabbits inhibited the binding of fluorescent mAbs to either epitope; the degree of inhibition correlated with the dilution of sera. The response in the rabbit sera to either epitope on PA depended on the dose of vaccine administered to the rabbits. No inhibition was seen with sera from control animals. With no species-specific components, this assay could be adapted readily for comparing responses between species.

Nonlinear Dynamics in the Progression of Atherosclerotic Fatty Streaks: Morphometric Analysis.

Atherosclerotic lesions are heterogeneous in terms of their cellular and lipid composition. While heterogeneity can be the result of stochastic noise, an alternate hypothesis is that the differences observed among individual lesions arise from deterministic chaos. Five New Zealand white rabbits were fed a diet containing 0.15% cholesterol for 6 months. Segments of the aorta were fixed in formalin, stained en bloc with Nile red (NR) and filipin (F), and en face fluorescence microscopy was used to map the distribution of lipids in fatty streaks (FS). The smallest lesions detected stained only with filipin. Larger lesions stained with both Nile red and filipin and two distinct regions of Nile red staining, NR-orange (rich in polar lipids) and NR-yellow (rich in neutral lipids) were observed. Digital overlays revealed a "nested" arrangement of F, NR-orange, and NR-yellow. The lesions also showed marked heterogeneity in their lipid composition. Thus, although initially similar, as FS increased in size, their composition became divergent, suggesting that the ultimate composition of a FS was highly sensitive to its initial composition. Sensitivity to initial condition is one of the hallmarks of deterministic systems. To determine if FS were self-similar, another hallmark of deterministic chaos, the borders of the different regions defined by NR and F staining were subjected to fractal analysis. For each lesion, the borders of the F, NR-orange, and NR-yellow regions were found to be fractal. Return maps were constructed for the differently stained regions. Analysis of the entire 104-lesion data set showed that although the data could be described by a four-parameter logistic model, the population was not chaotic. However, return maps drawn for the maxima of the NR-orange stained regions demonstrated chaos. Taken together, the data suggest that deterministic chaos plays a role in the evolution of atheromatous disease but, in common with most biologic systems, as the lesions progress, chaotic behavior is dampened.

Quantitation of immunoglobulin to hepatitis E virus by enzyme immunoassay.

We developed a quantitative enzyme immunoassay (EIA) for antibody to hepatitis E virus (HEV) by using truncated HEV capsid protein expressed in the baculovirus system to improve seroepidemiology, to contribute to hepatitis E diagnosis, and to enable vaccine evaluations. Five antigen lots were characterized; we used a reference antiserum to standardize antigen potency. We defined Walter Reed antibody units (WR U) with a reference antiserum by using the four-parameter logistic model, established other reference pools as assay standards, and determined the conversion factor: 1 WR U/ml = 0.125 World Health Organization unit (WHO U) per ml. The EIA performed consistently; median intra- and inter-test coefficients of variation were 9 and 12%, respectively. The accurate minimum detection limit with serum diluted 1:1,000 was 5.6 WR U/ml; the test could detect reliably a fourfold antibody change. In six people followed from health to onset of hepatitis E, the geometric mean antibody level rose from 7.1 WR U/ml to 1,924.6 WR U/ml. We used the presence of 56- and 180-kDa bands by Western blotting as a confirmatory test and to define true-negative and -positive serum specimens. A receiver-operating characteristics plot identified 30 WR U/ml as an optimum cut-point (sensitivity, 86%; specificity, 89%). The EIA detected antibody more sensitively than a commercially available test. The EIA was transferred to another laboratory, where four operators matched reference laboratory results for a panel of unknowns. Quantitation of antibody to HEV and confirmation of its specificity by Western blotting make HEV serology more meaningful.

EFFECTS OF DINOSEB ON THE LIFE CYCLE OF DAPHNIA MAGNA: MODELING SURVIVAL TIME AND A PROPOSAL FOR AN ALTERNATIVE TO THE NO-OBSERVED-EFFECT CONCENTRATION

Risk assessment is in urgent need of more accurate toxic effect endpoints than those currently in use, especially for low concentrations. Often such endpoints are estimated by analysis of variance, linear interpolation, or smoothing. As these statistical methods are not always satisfactory, some authors have proposed to describe the entire dose-response curves by fully formalized parametric regression models whose parameters have toxicological meaning. These models allow a better evaluation of pollutant effects, including inter- and extrapolation to any other than the measured effect values. Following this line, a four-parameter logistic regression model (standard model) was fitted to survival data of Daphnia magna under pesticide (dinoseb) stress. The heterogeneity of the variance was taken into account with a both-sides logarithmic transformation. Besides the standard model, a hormesis and a threshold model were tested too. These two others models have been described in the literature and might better represent the dose-response function we are looking for. All three models showed a good fit to our data, and the statistics gave no hints as to which model is the most appropriate. As no evidence was seen for hormesis or for the existence of a threshold concentration, we used the simplest, namely, the standard model, for most of our calculations. Model calculations allow the quantification of the effects on individuals' longevity as well as on mean survival time of the population. We used them to define a no-effect value, the statistical-no-effect concentration (SNEC). The SNEC is based on the confidence bands of the modeled regression and represents the highest value for which an effect is statistically not different from the control. The SNEC is an alternative to classical endpoints, like the no-observed-effect concentration (NOEC) or the low-effect concentrations (e.g., EC10, EC5, EC1).

Nonlinear Dynamics in the Progression of Atherosclerotic Fatty Streaks: Morphometric Analysis

AbstractAtherosclerotic lesions are heterogeneous in terms of their cellular and lipid composition. While heterogeneity can be the result of stochastic noise, an alternate hypothesis is that the differences observed among individual lesions arise from deterministic chaos. Five New Zealand white rabbits were fed a diet containing 0.15% cholesterol for 6 months. Segments of the aorta were fixed in formalin, stained en bloc with Nile red (NR) and filipin (F), and en face fluorescence microscopy was used to map the distribution of lipids in fatty streaks (FS). The smallest lesions detected stained only with filipin. Larger lesions stained with both Nile red and filipin and two distinct regions of Nile red staining, NR-orange (rich in polar lipids) and NR-yellow (rich in neutral lipids) were observed. Digital overlays revealed a “nested” arrangement of F, NR-orange, and NR-yellow. The lesions also showed marked heterogeneity in their lipid composition. Thus, although initially similar, as FS increased in size, their composition became divergent, suggesting that the ultimate composition of a FS was highly sensitive to its initial composition. Sensitivity to initial condition is one of the hallmarks of deterministic systems. To determine if FS were self-similar, another hallmark of deterministic chaos, the borders of the different regions defined by NR and F staining were subjected to fractal analysis. For each lesion, the borders of the F, NR-orange, and NR-yellow regions were found to be fractal. Return maps were constructed for the differently stained regions. Analysis of the entire 104-lesion data set showed that although the data could be described by a four-parameter logistic model, the population was not chaotic. However, return maps drawn for the maxima of the NR-orange stained regions demonstrated chaos. Taken together, the data suggest that deterministic chaos plays a role in the evolution of atheromatous disease but, in common with most biologic systems, as the lesions progress, chaotic behavior is dampened.

Calibration and LOD/LOQ Estimation of a Chemiluminescent Hybridization Assay for Residual DNA in Recombinant Protein Drugs Expressed in E. coli Using a Four-Parameter Logistic Model

Calibration is the process of fitting a model based on reference data points (x, y), then using the model to estimate an unknown x based on a new measured response, y. In DNA assay, x is the concentration, and y is the measured signal volume. A four-parameter logistic model was used frequently for calibration of immunoassay when the response is optical density for enzyme-linked immunosorbent assay (ELISA) or adjusted radioactivity count for radioimmunoassay (RIA). Here, it is shown that the same model or a linearized version of the curve are equally useful for the calibration of a chemiluminescent hybridization assay for residual DNA in recombinant protein drugs and calculation of performance measures of the assay.

New analytic framework for understanding sympathetic baroreflex control of arterial pressure.

The sympathetic baroreflex is an important feedback system in stabilization of arterial pressure. This system can be decomposed into the controlling element (mechanoneural arc) and the controlled element (neuromechanical arc). We hypothesized that the intersection of the two operational curves representing their respective functions on an equilibrium diagram should define the operating point of the arterial baroreflex. Both carotid sinuses were isolated in 16 halothane-anesthetized rats. The vagi and aortic depressor nerves were cut bilaterally. Carotid sinus pressure (CSP) was sequentially altered in 10-mmHg increments from 80 to 160 mmHg while sympathetic efferent nerve activity (SNA) and systemic arterial pressure (SAP) were recorded simultaneously under various hemorrhagic conditions. The mechanoneural arc was characterized by the response of SNA to CSP and the neuromechanical arc by the response of SAP to SNA. We parametrically analyzed the relationship between input and output for each arc using a four-parameter logistic equation model. In baseline states, the two arcs intersected each other at the point at which the instantaneous gain of each arc attained its maximum. Severe hemorrhage lowered the gain and offset of the neuromechanical arc and moved the operating point, whereas the mechanoneural arc remained unchanged. The operating points measured under the closed-loop conditions were indistinguishable from those estimated from the intersections of the two arc curves on the equilibrium diagram. The average root mean square errors of estimate for arterial pressure and SNA were 2 and 3%, respectively. Such an analytic approach could explain a mechanism for the determination of the operating point of the sympathetic baroreflex system and thus helps us integratively understand its function.

Development of a radioimmunoassay for the determination of zolpidem in biological samples.

The development of a specific and sensitive radioimmunoassay for the detection of zolpidem and its metabolites in urine and serum samples is described. The assay can be used to pre-screen forensic and emergency samples. 6-Methyl-2-(4-methylphenyl)imidazo[1,2-a]-pyridine-3-acetic acid was prepared as a hapten and was coupled directly and indirectly to bovine serum albumin. The immunization of rabbits with the hapten-bovine serum albumin conjugates resulted in the production of highly specific antibodies, showing no significant cross-reactivities towards existing drugs. The four-parameter logistic model was used to process the calibration data into a fitting curve (r2 = 0.9764). Intra- and inter-assay relative standard deviations were < 6.30 and < 12.28%, respectively. The limit of quantification was 0.1 ng ml-1. Using this assay, zolpidem levels were determined in urine and serum, and could be easily detected up to 48 h after oral intake of 10 mg of zolpidem.

Supercritical fluid extraction-immunoassay for the rapid screening of cocaine in hair

Supercritical fluid extraction-radioimmunoassay (SFE-RIA) was evaluated as a rapid screening tool for the detection of cocaine residues in human hair. SFE was performed using carbon dioxide modified with triethylamine (TEA) and water, with off-line collection of extracted cocaine in methanol. Extracts were analyzed for the presence of cocaine using a commercially available solid-phase RIA kit. In order to develop a suitable RIA calibration method for reliable measurement of cocaine in SF extracts, calibration data was compared for methanolic standards both with and without the presence of the SF modifier. Methanol had only a minor impact on immunoassay performance, producing an 11% decrease in maximum binding counts. In contrast, the presence of TEA/H2O profoundly degraded assay performance, producing a 60% suppression in assay counts. To preserve RIA sensitivity, SF extracts were evaporated under nitrogen to remove the modifier and reconstituted in methanol for RIA analysis. Incorporation of the evaporation step permitted the use of modifier-free cocaine calibrators in methanol. Calibration data was at to a four-parameter logistic model. SFE-RIA analysis of a series of drug-free hair samples established an RIA cut-off value for distinguishing between a negative and presumptive positive cocaine sample at an SF extract concentration of 1.2 ng/mL, or a hair concentration of 0.07 ng/mg. The robustness of the SFE-RIA method was demonstrated by the analysis of a variety of hair samples from both drug users and non-users. The quantitative SFE-RIA findings correlated well with the values obtained by an acid incubation/GC-MS method. © 1996 John Wiley & Sons, Inc.

Predicting Biomass of Beaver Food from Willow Stem Diameters

Beaver (Castor canadensis Kuhl) and willow (Salix spp.) are important components of riparian restoration on degraded western rangelands. Land managers need quantitative information to evaluate carrying capacity and potential habitat quality for beavers in riparian-willow systems. Our objectives were to determine the best model to predict biomass components of coyote willow (S. exigua Nuttall) from basal stem diameters and compare model predictions to diameter class averages. The study was conducted in a shrub-steppe ecosystem of northwestern Colorado. We estimated oven-dried weights of annual and total beaver food and total live biomass by diameter class from a sample of 160 willow stems. Several variants of a logistic function were fit with nonlinear least squares regression to select a model that best predicted mean biomass by stem diameter. A four-parameter logistic model provided the best fit for all 3 stem components. Predicted biomass estimates of beaver food and total live biomass had smaller standard errors than sample means for all 10 stem diameter class midpoints. Percentage of stem weight that was beaver food varied from 93.6% for the smallest stems to 12.2% for the largest. We concluded that the logistic model provided reliable estimates of beaver food biomass and could be used with food consumption rates and stem density data to evaluate carrying capacity for beaver or test assumptions in the beaver habitat suitability index model.