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After reviewing the evidence on the relation of vertical transmission of HIV to stage of infection in the mother, I developed a stochastic model of transmission in which the probability of transmission per week is proportional to the virus load in the mother. The virus load in different stages of the infection is measured by viral RNA levels or tissue culture infectious virus levels in plasma. The constant of proportionality is assumed to be different for transmission during pregnancy, during parturition, and during breast-feeding. Using data on transmission from mothers who are in the primary stage of infection, I estimated the constant of proportionality and calculated the probability of transmission during pregnancy as a function of the time pregnancy starts in relation to the stage of the infection. For breast-feeding, I calculated the conditional probability of transmission by breast-feeding for 20 weeks, dependent on the infant escaping infection during pregnancy and parturition. As might be expected, the probabilities of transmission are highest if the mother is in the primary stage of infection or in late stages of the disease and is quite low when the mother is in the asymptomatic stage of the infection.

PIP: Based on a review of the literature on maternal-child HIV transmission, the author presents a stochastic model in which the probability of HIV transmission per week during pregnancy, parturition, and breast feeding is conceptualized as proportional to the mother's viral load at that stage. The model combines data on culturable virus levels and RNA copies in plasma as measures of maternal viral load. Through use of transmission data on women in the primary stage of infection, the probability of HIV transmission during pregnancy was calculated as a function of the time pregnancy starts in relation to the stage of infection. Also calculated was the conditional probability of transmission by breast feeding for 20 weeks, dependent on the infant escaping infection during pregnancy and delivery. Predicted transmission probabilities were quite low during the asymptomatic stage of maternal infection and highest when the mother was in the primary stage of infection or the late stages of the disease. RNA data emerged as a better basis for the calculations than culturable virus. The considerable variation by country in reported rates of maternal-child HIV transmission could be explained by changes in the mix of women in the primary, asymptomatic, and later stages of infection.

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The combination of two factors gives early HIV infection an especially strong influence on transmission dynamics: (a) increased transmission probabilities and (b) increased transmission potential of partners infected during this period. Most attention has been focused on the first factor because it fits the way we usually think about risk factors affecting individuals. The second factor acts not on individuals, but across chains of transmission. It is missed by models with constant partnership formation rates over an individual's life or with random mixing. It cannot be assessed from available data collected from individuals. Its assessment requires data from both individuals in a partnership. We demonstrate that this second effect can be so strong that early infection can dominate transmission dynamics even when transmission probabilities are only modestly increased. This second effect is not directly parameterized in our models but arises from two realistic types of temporal variation in partnership formation: (a) Partnership formation rates vary by age with preferential partnership formation in one's own age group, and (b) individuals of any age can experience transient periods of high-risk partnership formation. In a model with only the age-related effect, early infection is observed to dominate transmission dynamics when 20% of transmissible virus is allocated to the first 6 weeks of infection, 7% to middle infection, and 73% to late infection. This domination occurs both early in the course of an epidemic and later when endemic infection levels have been reached. When the second effect is added, early infection is seen to dominate transmission in a model allocating 10% of transmissible virus to the first 6 months, 40% to middle infection, and 50% to late infection. In this model, transmission probabilities during early infection are only 4.17 times those of middle infection and half those of late-stage infection.

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The gamma and Erlang density functions describe a large class of lagged, right-skewed distributions. The Erlang distribution has been shown to be the analytic solution for a chain of compartments with identical rate constants. This relationship makes it useful for the analysis of first-pass pulmonary drug uptake data following intravenous bolus administration and the incorporation of this analysis into an overall systemic drug disposition model. However, others have shown that one Erlang density function characterizes the residence time distribution of solutes in single tissues with significant systematic error. We propose a model of two Erlang density functions in parallel that does characterize well the arterial appearance of indocyanine green, antipyrine, and alfentanil administered simultaneously by right atrial bolus injection. We derive the equations that permit calculation of the higher order moments of a system consisting of two parallel Erlang density functions and use the results of these calculations from the data for all three indicators to estimate pulmonary capillary blood volume and mean transit time in the dog.

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In pooled data methods such as naive pooled data methods and NONMEM, the number of sample points per individual may be less than the number of unknown parameters so that the values of the parameters are not estimable in individuals. However, for the moments of the distributions of the parameters to be estimable, the basic parameters must be identifiable.

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A review of the data on infectivity per contact for transmission of the HIV suggests that the infectivity may be on the order of 0.1-0.3 per anal intercourse in the period of the initial infection, 10(-4) to 10(-3) in the long asymptomatic period, and 10(-3) to 10(-2) in the period leading into AIDS. The pattern of high contagiousness during the primary infection followed by a large drop in infectiousness may explain the pattern of epidemic spread seen in male homosexual cohorts in the early years of the epidemic. Simulations of cohorts of homosexual males, using that range of parameter values, indicate the following: (a) The initial fast rise and then more or less rapid flattening of the incidence curve of seropositives is primarily due to rapid initial spread, yielding a group of infecteds all of whom pass into the low infectivity asymptomatic period at close to the same time. All this occurs only if the basic reproduction number for the primary infection is > 1. (b) The behavioral changes that have been reported all started after the incidence of new infections began to fall, too late to have a major effect on the initial rise. The behavioral changes had a major effect in slowing down the subsequent rise in the number of seropositives. (c) High activity groups play an important role in the early rapid rise of the epidemic. However, it is not likely that the rapid decrease in rate of growth of seropositives is solely due to saturation of these very high activity groups. Although the evidence for this interpretation of the role of the primary infection is not conclusive, its implications for prevention and for vaccine trials are so markedly different from those of other interpretations that we consider it to be an important hypothesis for further testing.

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This study was designed to define the effect of human aging on hypoglycemia counterregulatory mechanisms. A hyperinsulinemic (2 mU.kg-1.min-1) glucose clamp procedure was used to control glucose and insulin levels during stepwise lowering of plasma glucose. Counterregulatory hormones, hepatic glucose production (HGP), glucose utilization, and symptoms of hypoglycemia were studied in 13 healthy young [age 24 +/- 1 (SE) yr] and 11 healthy old (age 65 +/- 1 yr) nondiabetic volunteers on two occasions: 1) at matched euglycemia and 70 and 60 mg/dl (study 1) and 2) at matched euglycemia and 60 and 50 mg/dl (study 2). The old had consistently lower epinephrine (P < 0.005), glucagon (P < 0.02), cortisol (P < 0.05), and pancreatic polypeptide (P < 0.02) responses at the 60-mg/dl glucose step in study 1. However, these differences were no longer detectable at the more severe hypoglycemic stimulus of 50 mg/dl in study 2. A consistent increase in HGP occurred in both groups only at the 50-mg/dl glucose step (study 2) and was not different between young and old. There were also no differences in symptom responses between young and old. In summary, we found that elderly individuals have a subtle impairment of the glucose counterregulatory response during moderate hypoglycemia, but this impairment is no longer detectable during more severe hypoglycemia.

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We compare the stochastic and deterministic versions of an SI model with recruitment, background deaths, and deaths due to the disease. For the stochastic version, analysis of the mean number of susceptibles, mx, and infecteds, m(y), and of the means conditioned on nonextinction of the infection, m*x and m*y, shows that (1) if R0 < or = 1, the disease dies out monotonically for the deterministic and stochastic models, and (2) if R0 > 1, the disease dies out early with a probability close to (1/R0)a, where a is the number of infecteds introduced, or m(y) rises to a peak and then dies out slowly. For small populations, N, the peak is an obvious maximum. If N > or = 100, the peak in m(y) is hidden in a long, nearly stationary plateau and m*y is close to the deterministic endemic level for a large range of parameter values. The analytical results are illustrated with simulations. The results for the SI model are motivated by and compared with the corresponding results for the closed SIS model.

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I present a review and synthesis of the basic theory, steady state, and non-steady state for the calculation of metabolite production rates for systems that have a central well-mixed compartment that is the site of tracer input and sampling. The theory is then applied to the calculation of glucose production. If the only inputs are into the central compartment, an experimental design that involves varying tracer infusion rates to maintain constant specific activity in the central compartment and the same constant specific activity in the peripheral compartments allows calculation of the endogenous production. That holds even if the models are unidentifiable. The correct equation and Steele's pool fraction approximation reduce to the same result for this experimental design. However, that does not justify the use of Steele's equation when there are deviations from the exact experimental design. When the specific activity in the central compartment is not constant, model-dependent correction terms to Steele's equation are needed.

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We compare threshold results for the deterministic and stochastic versions of the homogeneous SI model with recruitment, death due to the disease, a background death rate, and transmission rate beta cXY/N. If an infective is introduced into a population of susceptibles, the basic reproduction number, R0, plays a fundamental role for both, though the threshold results differ somewhat. For the deterministic model, no epidemic can occur if R0 less than or equal to 1 and an epidemic occurs if R0 greater than 1. For the stochastic model we find that on average, no epidemic will occur if R0 less than or equal to 1. If R0 greater than 1, there is a finite probability, but less than 1, that an epidemic will develop and eventuate in an endemic quasi-equilibrium. However, there is also a finite probability of extinction of the infection, and the probability of extinction decreases as R0 increases above 1.

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Commonly used measures of effect, such as risk ratios and odds ratios, may be quite biased when used to assess the effect of factors that alter transmission risks given exposure to infected individuals. This is demonstrated in a simulation model involving a higher-risk behavior and a lower-risk behavior affecting the sexual transmission of human immunodeficiency virus. The bias arises because population contact patterns between higher-risk and lower-risk persons change their relative probabilities of exposure to an infected individual as an epidemic progresses. The assessment of contact patterns is thus central to risk assessment for contagious diseases. A new formulation of selective mixing presented here, together with a structured mixing specification of the social settings of contact, provides a theoretic framework for the investigation of contact pattern determinants.

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In experiments on biological systems one often cannot measure all state variables (compartments). Given a particular experiment of that type, a basic kinetic parameter may have no effect on the observations; such a parameter is an insensible parameter for that experiment. A parameter may influence the observations and not be uniquely determinable; such a parameter is nonidentifiable for that experiment. Only identifiable parameters can be estimated uniquely, by that experiment. I review the basic theory to check identifiability for a nominal value of a parameter (local identifiability), and present some examples of problems that may arise in estimation.

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A model for the spread of human immunodeficiency virus (HIV) in a population of male homosexuals is presented. The population is divided into five groups on the basis of degree of sexual activity. Within each group, the individuals are classified as 1) susceptible; 2) infective; or 3) removed because of a lack of sexual activity associated with advanced acquired immunodeficiency disease (AIDS). The infective individuals are further subdivided into four stages of infection. Analyses of the model address two questions with regard to the spread of HIV: (1) What is the effect of level of sexual activity on an individual's risk for infection, and (2) What is the effect that assumptions about mixing between groups have on both individual risk and transmission throughout a population? Results from analyses using a number of different parameter estimates show that increased levels of sexual activity increase the likelihood that an individual will become infected. In addition, the initial spread of the disease is markedly affected by variation in the amount of contact among individuals from different subpopulations. The steady-state incidence of the disease is not markedly affected by variation in the contact patterns, but the size of the steady-state population and therefore the proportion of infected individuals in the population does vary significantly with changes in the degree of mixing among subpopulations. These results show clearly the sensitivity of model outcomes to variation in the patterns of contact among individuals and the need for better data on such interactions to aid in understanding and predicting the spread of HIV.

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For biological systems one often cannot set up experiments to measure all of the state variables. If only a subset of the state variables can be measured, it is possible that some of the system parameters cannot influence the measured state variables or that they do so in combinations that do not define the parameters' effects separately. Such parameters are unidentifiable and are in theory unestimable. Given a model of the system, linear or nonlinear, and initial estimates of the values of all parameters, we exhibit a simple theory and describe a program for checking the local identifiability of the parameters at the initial estimates for given experiments on the model. The program, IDENT, is available from the authors.

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Kinetic analysis and integrated system modeling have contributed significantly to understanding the physiology and pathophysiology of metabolic systems in humans and animals. Many experimental biologists are aware of the usefulness of these techniques and recognize that kinetic modeling requires special expertise. The Resource Facility for Kinetic Analysis (RFKA) provides this expertise through: (1) development and application of modeling technology for biomedical problems, and (2) development of computer-based kinetic modeling methodologies concentrating on the computer program Simulation, Analysis, and Modeling (SAAM) and its conversational version, CONversational SAAM (CONSAM). The RFKA offers consultation to the biomedical community in the use of modeling to analyze kinetic data and trains individuals in using this technology for biomedical research. Early versions of SAAM were widely applied in solving dosimetry problems; many users, however, are not familiar with recent improvements to the software. The purpose of this paper is to acquaint biomedical researchers in the dosimetry field with RFKA, which, together with the joint National Cancer Institute-National Heart, Lung and Blood Institute project, is overseeing SAAM development and applications. In addition, RFKA provides many service activities to the SAAM user community that are relevant to solving dosimetry problems.

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The functional significance of high affinity agonist binding to receptors that interact with guanine nucleotide regulatory proteins has remained controversial. Preincubation of human platelet membranes with the full alpha 2-agonist UK 14,304 in the absence of GTP increases the potency of the agonist to inhibit adenylate cyclase in a pre-steady state (15-sec) assay. The EC50 after preincubation (6 +/- 1 nM) is within a factor of 2 of the high affinity Kd for [3H]UK 14,304 binding determined under identical conditions (2.7 +/- 0.1 nM). In contrast, in the usual steady state measurements (15 min) or in pre-steady state measurements without agonist preincubation, the EC50 values (74 +/- 1 and 207 +/- 8 nM, respectively) are near the low affinity Kd for [3H]UK 14,304 binding. Reduction of the GTP concentration in steady state adenylate cyclase assays also decreases the EC50 for UK 14,304 from 40 +/- 5 nM at 10 microM GTP to 14 +/- 5 nM with no added GTP. Both sets of experimental observations are accommodated by a complete kinetic model of inhibition in which the high affinity ternary complex of drug, receptor, and G protein leads to the response. Explicit rate parameters are included for agonist binding, receptor-G protein interactions, GTP binding, and hydrolysis. Despite the functional role of the high affinity state of the alpha 2-receptor in this model, the steady state EC50 for agonist-mediated inhibition correlates best with the Kd of low affinity agonist binding in the presence of high levels of GTP. Under conditions in which formation of the high affinity ternary complex is favored, the EC50 for responses approaches the high affinity Kd.

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We used compartmental analysis to analyze the kinetics of distribution and metabolism of norepinephrine (NE) and to determine whether the increase in plasma norepinephrine concentration (PNE) during sodium restriction in humans is due to sympathetic nervous system (SNS) activation. [3H]-NE infusion and postinfusion decay were measured in young subjects in the supine position and during 60 min of standing during normal sodium (NS) diet and after 7 days of 10 meq/day sodium-restricted (SR) diet. The mean supine PNE was greater during SR diet compared with NS diet (154 +/- 9 vs. 185 +/- 12 pg/ml, P = 0.02, n = 10). During both NS and SR diets, upright PNE increased (163 +/- 4 vs. 359 +/- 38 pg/ml and 182 +/- 8 vs. 401 +/- 26 pg/ml, respectively, multivariate one-way analysis of variance, P less than 0.001, alpha = 0.05). The increases of PNE with both SR diet and upright posture were accompanied by a fall in NE metabolic clearance rate (MCR1). During SR diet this was due to a fall in the volume of distribution of NE (6.1 +/- 0.4 vs. 5.0 +/- 0.4 liters, P = 0.003, n = 10). In contrast to the effect of upright posture to increase NE release into the extra-vascular compartment (NE2), during SR diet there was no change in NE2 (1.63 +/- 0.09 vs. 1.62 +/- 0.1 micrograms.min-1.m-2, P = 0.97, n = 10). Thus the increase in PNE during SR diet in humans can be explained by a fall in the volume of distribution of NE, resulting in a decrease in MCR1.(ABSTRACT TRUNCATED AT 250 WORDS)

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The present study was undertaken to quantify more precisely and to begin to address the problem of heterogeneity of the kinetics of distribution and metabolism of norepinephrine (NE) in humans, by using compartmental analysis. Steady-state NE specific activity in arterialized plasma during [3H]NE infusion and postinfusion plasma disappearance of [3H]NE were measured in eight healthy subjects in the supine and upright positions. Two exponentials were clearly identified in the plasma [3H]NE disappearance curves of each subject studied in the supine (r = 0.94-1.00, all P less than 0.01) and upright (r = 0.90-0.98, all P less than 0.01) positions. A two-compartment model was the minimal model necessary to simultaneously describe the kinetics of NE in the supine and upright positions. The NE input rate into the extravascular compartment 2, estimated with the minimal model, increased with upright posture (1.87 +/- 0.08 vs. 3.25 +/- 0.2 micrograms/min per m2, P less than 0.001). Upright posture was associated with a fall in the volume of distribution of NE in compartment 1 (7.5 +/- 0.6 vs. 4.7 +/- 0.3 liters, P less than 0.001), and as a result of that, there was a fall in the metabolic clearance rate of NE from compartment 1 (1.80 +/- 0.11 vs. 1.21 +/- 0.08 liters/min per m2, P less than 0.001). We conclude that a two-compartment model is the minimal model that can accurately describe the kinetics of distribution and metabolism of NE in humans.

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The observations in an experiment define a set of observational parameters that are functions of the basic kinetic parameters of the model of the system. The problem of identifiability is concerned with whether the observational parameters uniquely specify the basic kinetic parameters. As such, it depends only on the functional relation between the two levels of parameters and not on errors of observation and the estimation procedure. It should be checked before doing the experiment. Given initial estimates of the basic kinetic parameters, identifiability can be checked, in a local sense, from data generated by simulating the experiment on the model.

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We have examined the role of myoglobin to facilitate O2 diffusion to active mitochondria in skeletal muscle by constructing computer-simulation experiments. Steady-state mitochondrial O2 consumption under different conditions of supply partial pressure of O2 (PO2) in a system with and without myoglobin were examined for a one-dimensional slab of tissue. O2 consumption by mitochondria was saturable with the mitochondria located in bands at uniform intervals throughout the tissue. Under these conditions, myoglobin provides a measurable increase in O2 transport for supply PO2 below 10 Torr and diffusion lengths expected for skeletal muscle fibers. We conclude that under circumstances where hypoxia lowers PO2 below 10 Torr that myoglobin begins to provide a measurable increase in O2 delivery to mitochondria.

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