Reirradiation at long time intervals in mouse kidney: a comparison between experimental results and the predictions of the F-type tissue model

The tolerance of a late-responding tissue to reirradiation after long time intervals has been analysed using the F-type tissue model. In this model the tissue is composed of identical cells, each of which is capable of extensive proliferation and of tissue-specific function. The model was adapted to calculate the response to two fractions of radiation given in a variable overall time. For two equal doses of radiation the repair of tissue damage after the first fraction could be detected theoretically by a change in the rate of cell depletion after retreatment and by an increase in the minimum cell number attained. For an 'experimental set-up', in which a constant first dose was followed by a range of retreatment doses in a variable overall time, the repair of tissue damage theoretically could be detected most sensitively by a shift of the dose-response curves to higher retreatment doses as the time interval between the two doses was increased. A prerequisite for a proper comparison of these dose-response curves was that the responses were evaluated at times after the first dose determined by the minimal latency times after high retreatment doses. From a comparison of these theoretical results with experimental findings for mouse kidneys it was concluded that no recovery of tissue function took place over a 6-month period. Instead it appeared that the kidneys had become more sensitive to irradiation over this period.     

EFFICIENT DESIGNS FOR STUDYING SYNERGISTIC DRUG COMBINATIONS

Distinguishing between pharmacologically additive and synergistic drug combinations requires experimental designs and statistical analyses that often require appreciable numbers of animals and much experimenter time. The current study employed a design in which individual dose-effect data from each drug were translated into theoretically additive total dose combinations, in a fixed drug proportion, in order to produce a composite additive dose-effect relation that could be compared with that of an actual mixture having the same proportion. Results from this approach, using a combination of intrathecal doses of morphine and clonidine, were virtually identical to those using isobolographic analysis of the same data set. Both analyses showed significant synergism for this combination and, in each method, it was not necessary to constrain the drug regression lines to parallelism. In contrast to the isobole approach, the use of the composite additive dose-effect relation also allows observation of the interaction over a range of effects while reducing the size of the data sets needed. © 1997 Elsevier Science Inc.     

Neurohypophyseal hormone-responsive renal adenylate cyclase. IV. A random-hit matrix model for coupline in a hormone-sensitive adenylate cyclase system

A "random-hit" matrix model is proposed to account for the dynamic and steady state relationship between occupation of bovine renal medullary membrane receptors by [Lys8]vasopressin (LVP) and neurohypophyseal hormones (NHH) and the associated activation of membrane-bound adenylate cyclase. The model was developed by systematic introduction of specific rules concerning receptor coupling into a general structural model which consists of two square matrices of identical size, one composed of homogeneous R ("receptor") units, the second of homogeneous C ("cyclase") units. R units are either occupied (RO) or unoccupied (RU); C units are either active (CA) or inactive (CI). Hormone molecules are envisioned to "collide" with R units randomly; collision with RU leads to "binding", and occupation is maintained for a characteristic mean occupancy time, TO. In this structure, each R unit has an "interaction field" which consists of the "twin" unit in the "C" matrix, and the 4 nearest neighbor C units surrounding the twin. Occupation of an R unit leads to activation of all CI units in the interaction field of that R; CA units in the interaction field are refractory. Thus binding at a given R may "recruit" a variable number of inactive neighboring C units (5, 4, 3, 2, 1, or 0). The model requires that there be individual coupling delays between the moment of binding at a given R and subsequent activation of CI units (mean coupling delay (Td) approximately 10% To). Activation of C units persists as long as the "parent" R is occupied and is maintained for an additional short time interval (Tp) after RO reverts to RU, corresponding to hormone dissociation from receptor. The model accounts for the following previously demonstrated relations between LVP occupation of receptors and adenylate cyclase activation in bovine renal medullary membranes: 1) the shape of the nonlinear steady state relation between normalized (percentage maximal) receptor occupation (O) and cyclase activation (A), uniformly observed in different membrane preparations: 2) variable hormone concentration-dependent trajectories of approach to the final steady state A:O value (A:Oss) which may be either monophasic or biphasic; 3) the loss of intrinsic adenylate cyclase activity observed in bovine membranes for a series of NHH analogs with progressively diminishing affinity for receptors. The model represents an explicit theory of coupling where a successive series of temporal events are quantitatively related to each other and privide major constraints to any interpretation of the molecular organization of receptors and adenylate cyclase units in membranes. The model excludes a number of mechanistic proposals and suggests a new hypothesis for membrane coupling with features which may be generally applicable to other hormone-sensitive adenylate cyclase systems.     

Phase diagrams for impure lipid systems. Application to lipid/anaesthetic mixtures

A physical model is presented to describe theoretically the temperature-dependent interactions of lipid bilayers with small molecules such as anaesthetics. Based on an earlier model, a triangular lattice in which each site is occupied by a single lipid chain is constructed and the small (anaesthetic) molecules are assumed to occupy interstitial sites in the centre of each lattice triangle. The phase characteristics of such lipid/anaesthetic mixtures are described in terms of the interaction parameters between lipid-lipid, lipid-anaesthetic and anaesthetic-anaesthetic molecules. Depending on the chemical nature of the interacting species the following three models are formulated: Model I. An interstitial model in which the only perturbation is in the head-group region of the bilayer and direct interactions between neighbouring anaesthetic molecules are taken into account. Model II. Here, only hydrophobic interactions between anaesthetics and lipids are considered. Model III. Both van der Waals' and coulombic interactions are taken into account. Phase diagrams for the three models are obtained by numerical calculation over a wide range of interaction parameters. It is shown that in all three models, lateral phase separation takes place due to the presence of anaesthetics. The heat of transition, however, is found to be virtually independent of the anaesthetic concentration.     

Multivariate slope ratio assay with repeated measurements

A method is given for analyzing a slope ratio assay in which a test drug is compared with a standard drug, two or more response variates being measured on each subject at each of several successively increased drug doses. The method requires all subjects to receive the same number of doses, all subjects on the same drug to receive the same doses, the ratio of corresponding doses of the two drugs to be constant over the successive increases, and response variables to be measured only once on each subject at each dose with no missing data allowed. The technique is also applicable when doses are randomly assigned, provided there is no carry-over effect between doses. For each of the J response variates, the relative potency of the test drug with respect to the standard is defined and estimated in the usual way; a 100(1-alpha)% confidence region is then obtained for the vector of the J relative potencies. A procedure is given for testing the equality of some or all of the J relative potencies; an estimator of a common relative potency is obtained by a standard multivariate least squares method. A common relative potency is of interest because the multiple outcome variables are often different indicators of a general physiologic response. The procedures in the paper are illustrated by a simple example concerning the effects of two anesthetics on children.     

Data analysis problems in the area of pharmacokinetics research.

Of interest in compartment analysis is the estimation of kinetic first order rate constants and quantification of the amount of drug present in each compartment at every point in time. From the data pertaining to only one compartment, it is not always possible to estimate all the first order rate parameters of linear and non-linear compartment systems. When multiple varying doses are administered at different intervals of time, the equations quantifying the maximum and minimum amount of drug accumulated in each interval may be used. To obtain estimates of kinetic parameters, statistical adjustments may be used in the kinetic equations.     

Mimicry of food intake: the dynamic interplay between eating companions

Numerous studies have shown that people adjust their intake directly to that of their eating companions; they eat more when others eat more, and less when others inhibit intake. A potential explanation for this modeling effect is that both eating companions'' food intake becomes synchronized through processes of behavioral mimicry. No study, however, has tested whether behavioral mimicry can partially account for this modeling effect. To capture behavioral mimicry, real-time observations of dyads of young females having an evening meal were conducted. It was assessed whether mimicry depended on the time of the interaction and on the person who took the bite. A total of 70 young female dyads took part in the study, from which the total number of bites (N = 3,888) was used as unit of analyses. For each dyad, the total number of bites and the exact time at which each person took a bite were coded. Behavioral mimicry was operationalized as a bite taken within a fixed 5-second interval after the other person had taken a bite, whereas non-mimicked bites were defined as bites taken outside the 5-second interval. It was found that both women mimicked each other''s eating behavior. They were more likely to take a bite of their meal in congruence with their eating companion rather than eating at their own pace. This behavioral mimicry was found to be more prominent at the beginning than at the end of the interaction. This study suggests that behavioral mimicry may partially account for social modeling of food intake.     

BCG-Mediated Bladder Cancer Immunotherapy: Identifying Determinants of Treatment Response Using a Calibrated Mathematical Model

Intravesical Bacillus Calmette Guérin (BCG) immunotherapy is considered the standard of care for treatment of non-muscle invasive bladder cancer; however the treatment parameters were established empirically. In order to evaluate potential optimization of clinical parameters of BCG induction therapy, we constructed and queried a new mathematical model. Specifically, we assessed the impact of (1) duration between resection and the first instillation; (2) BCG dose; (3) indwelling time; and (4) treatment interval of induction therapy – using cure rate as the primary endpoint. Based on available clinical and in vitro experimental data, we constructed and parameterized a stochastic mathematical model describing the interactions between BCG, the immune system, the bladder mucosa and tumor cells. The primary endpoint of the model was the probability of tumor extinction following BCG induction therapy in patients with high risk for tumor recurrence. We theoretically demonstrate that extending the duration between the resection and the first BCG instillation negatively influences treatment outcome. Simulations of higher BCG doses and longer indwelling times both improved the probability of tumor extinction. A remarkable finding was that an inter-instillation interval two times longer than the seven-day interval used in the current standard of care would substantially improve treatment outcome. We provide insight into relevant clinical questions using a novel mathematical model of BCG immunotherapy. Our model predicts an altered regimen that may decrease side effects of treatment while improving response to therapy.     

Estimation of Bayesian confidence intervals for the product of conditional survival probabilities

A model is derived to estimate the survival probability of a time interval when censorings occur. The time interval is divided into partial intervals in order to obtain the conditional survival probabilities, each of which is a parameter of a Binomial distributed random variable. To allow for the dependence between the events in the different intervals these parameters are transformed. Corresponding a priori density functions are formulated regarding both the Bayesian uniform distribution and the special model. The a posteriori density function is derived for the product of the conditional survival probabilities, and formulae for the BAYE sian confidence interval and the expectation are given. Lower and upper bounds for the confidence interval and the expectation are derived. Some examples are given to compare the results with other methods.     

Effects of repeated doses of MDMA ("ecstasy") on cell-mediated immune response in humans.

Cell-mediated immune response after the administration of two repeated doses of 100 mg 3,4-methylenedioxymethamphetamine (MDMA) at 4-hour and 24-hour intervals was evaluated in two randomised, double-blind and cross-over clinical trials conducted in healthy male MDMA consumers. MDMA produced a time-dependent decrease in the CD4/CD8 T-cell ratio due to a decrease in the number of CD4 T-helper cells, a decrease in the functional responsiveness of lymphocytes to mitogenic stimulation, and a simultaneous increase in natural killer cells. In case of two 100 mg MDMA doses given 4 hour apart, immune alterations produced by the first dose were strengthened by the second one. At 24 hours after treatment, statistically significant residual effects were observed for all the altered immune parameters after the administration of two MDMA doses if compared to single dose and placebo. In the second clinical trial, the second 100 mg MDMA dose given 24 hours after the first dose produced immunological changes significantly greater than those induced by the initial drug administration and which seemed to show a delayed onset. Significant residual effects were observed for all the immune parameters as late as 48 hours after the second dose. These results show that repeated administration of MDMA with both a short and a long time interval between doses extends the critical period following MDMA administration, already observed after a single dose, in which immunocompetence is severely compromised.     

A mathematical model of the chemotherapeutic treatment of acute myeloblastic leukemia.

Based on our previous mathematical model of the acute myeloblastic leukemic (AML) state in man, we superimpose a chemotherapeutic drug treatment regimen. Our calculations suggest that small changes in the protocol can have significant effects on the result of treatment. Thus, the optimal period between drug doses is the S-phase interval of the leukemic cells--about 20h--and the greater the number of doses administered in a given course treatment, the longer the rest interval should be before the next course is administered. For a patient with a "slow" growing AML cell population, remission can be achieved with one or two courses of treatment, and further suppression of the leukemic population can be achieved with continued courses of treatment. However, for patients with a "fast" growing AML cell population, a similar aggressive treatment regimen succeeds in achieving remission status only at the cost of very great toxic effects on the normal neutrophil population and its precursors.     

Quantification of long-term doxorubicin response dynamics in breast cancer cell lines to direct treatment schedules

While acquired chemoresistance is recognized as a key challenge to treating many types of cancer, the dynamics with which drug sensitivity changes after exposure are poorly characterized. Most chemotherapeutic regimens call for repeated dosing at regular intervals, and if drug sensitivity changes on a similar time scale then the treatment interval could be optimized to improve treatment performance. Theoretical work suggests that such optimal schedules exist, but experimental confirmation has been obstructed by the difficulty of deconvolving the simultaneous processes of death, adaptation, and regrowth taking place in cancer cell populations. Here we present a method of optimizing drug schedules in vitro through iterative application of experimentally calibrated models, and demonstrate its ability to characterize dynamic changes in sensitivity to the chemotherapeutic doxorubicin in three breast cancer cell lines subjected to treatment schedules varying in concentration, interval between pulse treatments, and number of sequential pulse treatments. Cell populations are monitored longitudinally through automated imaging for 600–800 hours, and this data is used to calibrate a family of cancer growth models, each consisting of a system of ordinary differential equations, derived from the bi-exponential model which characterizes resistant and sensitive subpopulations. We identify a model incorporating both a period of growth arrest in surviving cells and a delay in the death of chemosensitive cells which outperforms the original bi-exponential growth model in Akaike Information Criterion based model selection, and use the calibrated model to quantify the performance of each drug schedule. We find that the inter-treatment interval is a key variable in determining the performance of sequential dosing schedules and identify an optimal retreatment time for each cell line which extends regrowth time by 40%-239%, demonstrating that the time scale of changes in chemosensitivity following doxorubicin exposure allows optimization of drug scheduling by varying this inter-treatment interval.     

A mathematical model for the temporal pattern of a population structure,with particular reference to the flour beetle

Equations are derived to represent the time course of the population numbers of the various stages of the flour beetle. The assumption of constant duration of the life stages and the absence of delayed effects leads to equations from which the various population numbers can be calculated in terms of the parameters of the system. Formulae are given for the estimation of many of the latter from observations on population numbers. Calculations show that the principal features of the observed changes in population structure can be accounted for on the basis of a simple model in which it is further assumed that each interaction is proportional to the product of the numbers of the two given interacting stages. A more detailed analysis may require secondary interaction coefficients. Suggestions for estimation of such coefficients are given.     

Tumor escape from immune elimination

A simple kinetic model is proposed for the interaction between a tumor and the immune system. Special attention is given to the phenomenon of “sneaking through” and the associated phenomena of tumor regression and recurrence. Sneaking through refers to the situations in which small antigenic tumors grow progressively, medium-sized tumors are rejected and large ones break through again. The combination of two factors is proposed as being essential for explaining this behavior: (a) The dependence of the immune response on antigen dose. (b) The negative intervention of immunosuppressors, or inhibitory factors.The immune response is described here as a repeated antigen-dependent stimulation of lymphoid cells to proliferate, with a parallel process of antigen-dependent differentiation to a terminal phase. For too small or too large antigen doses the growth in the number of cells is counter-balanced by loss from the proliferative pool through natural decay or by enhanced differentiation, respectively. When this is combined with the blocking of resting and proliferating precursor cells by factors originating from tumor cells, computer simulations demonstrate that the model is able to account for the observed patterns of tumor behavior. The model allows for a discussion of the significance of various biological parameters and is amenable for testing.     

Drug recirculation model with multiple cycles occurring at unequal time intervals

A pharmacokinetic model for enterohepatic recycling has been developed to take into account multiple recirculations likely to occur at various times after intravenous or subcutaneous injection, after infusion, or after a single oral administration of a drug. The times when the gall bladder empties, the duration of infusion and the number of recirculations may be arbitrarily chosen (for simulations) or computed (for optimization) to express the concentration in the central compartment at any time. Without a new theoretical calculation, the area under the concentration curve may be obtained as a function of the model parameters. As an example, the model is applied to an experimental case of four recirculations after oral administration and to a new drug data fitting.     

Probabilistic secretion of quanta from nerve terminals at synaptic sites on muscle cells: non-uniformity, autoinhibition and the binomial hypothesis

A model of the secretion of a quantum at a release site is proposed in which, following the influx of calcium ions, synaptic vesicles are made available for release by the activation of kappa phosphorylation steps with rate alpha. At any time during this process the vesicles may become unavailable for secretion at rate gamma. On completion of the kappa phosphorylation steps the vesicles participate in the formation of a fusion pore with the terminal membrane to give exocytosis at rate delta. Changes in alpha, delta and kappa are shown to produce characteristic changes in the number and timecourse of quantal secretions following a nerve impulse, which are similar to those observed following drug treatments that are thought to act selectively on each of these processes. The number of quanta secreted from nerve terminals that consist of many release sites does not fluctuate much during a low frequency train of impulses: the variance is small compared with the mean level, so secretion follows binomial rather than Poisson statistics. A theory is derived that shows that variations in the probability of secretion amongst these release sites of any particular kind fails to reduce the variance of the total secretion from the terminal; Poisson rather than binomial statistics then still apply. The theory shows that an interaction between release sites is required to reduce this variance and such an effect is provided if secretion at a site inhibits secretion at nearby sites. Simulations show that incorporating this process of autoinhibition into the model reproduces the experimental observations on the effects of calcium ions on the binomial parameters p and n as well as on the relative constancy of p during facilitation and depression of quantal secretion. Methods for estimating the timecourse of changes in the probability of secretion at release sites following an impulse, by using either the time of occurrence of first, second, third or later quantal latencies, are given. These procedures show that current methods for estimating the time-dependent probability changes are inadequate for detecting interaction between release sites, such as autoinhibition, unless this is relatively large. Therefore, estimates from third quantal latencies are used.     

Modelling the dynamics of the electron transport rate measured by PAM fluorimetry during Rapid Light Curve experiments

We propose a dynamic model specifically designed to simulate changes in the photosynthetic electron transport rate, which is calculated from fluorescence measurements when plants are exposed, for a short time, to a series of increasing photon flux densities. This model simulates the dynamics of the effective yield of photochemical energy conversion from the maximum and natural chlorophyll fluorescence yields, taking into account a cumulative effect of successive irradiations on photosystems. To estimate a characteristic time of this effect on photosystems, two series of experiments were performed on two benthic diatom culture concentrations. For each concentration, two different series of irradiations were applied. Simplified formulations of the model were established based on the observed fluorescence curves. The simplified versions of the model streamlined the parameters estimation procedure. For the most simplified version of the model (only 4 parameters) the order of magnitude of the characteristic time of the residual effect of irradiation was about 38 s (within a confidence interval between 20 and 252 s). The model and an appropriate calibration procedure may be used to assess the physiological condition of plants experiencing short time-scale irradiance changes in experimental or field conditions.     

Estimating local interaction from spatiotemporal forest data, and Monte Carlo bias correction

We point out a general problem in fitting continuous time spatially explicit models to a temporal sequence of spatial data observed at discrete times. To illustrate the problem, we examined the continuous time Markov model for forest gap dynamics. A forest is assumed to be apportioned into discrete cells (or sites) arranged in a regular square lattice. Each site is characterized as either a gap or a non-gap site according to the vegetation height of trees. The model incorporates the influence of neighboring sites on transition rate: transition rate from a non-gap to a gap site increases linearly with the number of neighbors that are currently in the gap state, and vice versa. We fitted the model to the spatiotemporal data of canopy height observed at the permanent plot in Barro Colorado Island (BCI). When we used the approximate maximum likelihood method to estimate the parameters of the model, the estimated transition rates included a large bias-in particular, the strength of interaction between nearby sites was underestimated. This bias originated from the assumption that each transition between two observation times is independent. The interaction between sites at local scale creates a long chain of transitions within a single census interval, which violates the independence of each transition. We show that a computer-intensive method, called Monte Carlo bias correction (MCBC), is very effective in removing the bias included in the estimate. The global and local gap densities measuring spatial aggregation of gap sites were computed from simulated and real gap dynamics to assess the model. When the approximate likelihood estimates were applied to the model, the predicted local gap density was clearly lower than the observed one. The use of MCBC estimates, suggesting a strong interaction between sites, improved this discrepancy.     

Collective animal behavior from Bayesian estimation and probability matching

Animals living in groups make movement decisions that depend, among other factors, on social interactions with other group members. Our present understanding of social rules in animal collectives is mainly based on empirical fits to observations, with less emphasis in obtaining first-principles approaches that allow their derivation. Here we show that patterns of collective decisions can be derived from the basic ability of animals to make probabilistic estimations in the presence of uncertainty. We build a decision-making model with two stages: Bayesian estimation and probabilistic matching. In the first stage, each animal makes a Bayesian estimation of which behavior is best to perform taking into account personal information about the environment and social information collected by observing the behaviors of other animals. In the probability matching stage, each animal chooses a behavior with a probability equal to the Bayesian-estimated probability that this behavior is the most appropriate one. This model derives very simple rules of interaction in animal collectives that depend only on two types of reliability parameters, one that each animal assigns to the other animals and another given by the quality of the non-social information. We test our model by obtaining theoretically a rich set of observed collective patterns of decisions in three-spined sticklebacks, Gasterosteus aculeatus, a shoaling fish species. The quantitative link shown between probabilistic estimation and collective rules of behavior allows a better contact with other fields such as foraging, mate selection, neurobiology and psychology, and gives predictions for experiments directly testing the relationship between estimation and collective behavior.