Radiosensitivity of murine and human melanoma cells: a comparative study with different models

The in vitro radiosensitivity of one murine melanoma cell line (Cloudman S91 CCL 53.1) and three human melanoma cell strains (C8146C, C8161, and R83-4) were studied. Cells were irradiated by single dose X-rays and plated either in agar or on plastic. The survival curves were fitted by the single-hit multitarget, two-hit multitarget, and quadratic models. Multiple comparisons of the residual sum of squares suggested that the two-hit model was clearly inferior to the single-hit and quadratic models. No statistically significant difference was suggested for either the single-hit or quadratic models. Furthermore, on examination of the differences in correlations between the observed and predicted values, the residual plots (observed minus predicted over dose) failed to suggest a clear advantage of either the single-hit multitarget or the quadratic models. Either model could be recommended for analysis of in vitro radiation data.     

Survival of a mixture of cells of variable linear-quadratic sensitivity to radiation

The experimentally observed survival of a heterogeneous mixture of cells, each component of which obeys a different linear-quadratic survival response to ionizing radiation, is examined. It is shown that the survival relationship for the mixed population approaches a linear-quadratic form for low doses. The linear parameter of the low-dose relationship approached is equal to the average of the distribution of values of the linear parameter (alpha(i)) of the various components of the mixture. The quadratic parameter of the low-dose relationship approached is equal to the average of the distribution of values of the quadratic parameter (beta(i)) of the various components of the mixture minus one-half the variance of the distribution of the values of alpha(i). A numerical example of the survival expected for an exponentially growing population of Chinese hamster V79 cells is presented. From this it can be appreciated that the apparent value of the alpha and beta parameters obtained by fitting an experimentally obtained survival curve will depend on the range of doses over which survival is determined. The apparent value of beta is decreased at higher doses, producing a straightening of the survival curve to approach the exponential decrease in survival commonly observed for the terminal high-dose portion of survival curves. Apparent exponential survival at high doses is not inconsistent with linear-quadratic survival and may not indicate a multitarget or other mechanism of cell killing.     

Polynomials to model the growth of young bulls in performance tests

The use of polynomial functions to describe the average growth trajectory and covariance functions of Nellore and MA (21/32 Charolais+11/32 Nellore) young bulls in performance tests was studied. The average growth trajectories and additive genetic and permanent environmental covariance functions were fit with Legendre (linear through quintic) and quadratic B-spline (with two to four intervals) polynomials. In general, the Legendre and quadratic B-spline models that included more covariance parameters provided a better fit with the data. When comparing models with the same number of parameters, the quadratic B-spline provided a better fit than the Legendre polynomials. The quadratic B-spline with four intervals provided the best fit for the Nellore and MA groups. The fitting of random regression models with different types of polynomials (Legendre polynomials or B-spline) affected neither the genetic parameters estimates nor the ranking of the Nellore young bulls. However, fitting different type of polynomials affected the genetic parameters estimates and the ranking of the MA young bulls. Parsimonious Legendre or quadratic B-spline models could be used for genetic evaluation of body weight of Nellore young bulls in performance tests, whereas these parsimonious models were less efficient for animals of the MA genetic group owing to limited data at the extreme ages.     

Modeling of crossbred cattle growth, comparison between cubic and piecewise random regression models

Two analyses, cubic and piecewise random regression, were conducted to model growth of crossbred cattle from birth to about two years of age, investigating the ability of a piecewise procedure to fit growth traits without the complications of the cubic model. During a four-year period (1994-1997) of the Australian "Southern Crossbreeding Project", mature Hereford cows (N = 581) were mated to 97 sires of Angus, Belgian Blue, Hereford, Jersey, Limousin, South Devon, and Wagyu breeds, resulting in 1141 steers and heifers born over four years. Data included 13 (for steers) and eight (for heifers) live body weight measurements, made approximately every 50 days from birth until slaughter. The mixed model included fixed effects of sex, sire breed, age (linear, quadratic and cubic), and their interactions between sex and sire breed with age. Random effects were sire, dam, management (birth location, year, post-weaning groups), and permanent environmental effects and for each of these when possible, their interactions with linear, quadratic and cubic growth. In both models, body weights of all breeds increased over pre-weaning period, held fairly steady (slightly flattening) over the dry season then increased again towards the end of the feedlot period. The number of estimated parameters for the cubic model was 22 while for the piecewise model it was 32. It was concluded that the piecewise model was very similar to the cubic model in the fit to the data; with the piecewise model being marginally better. The piecewise model seems to fit the data better at the end of the growth period.     

The effect of misonidazole as a hypoxic radiosensitizer at low dose

Using an automated low dose survival assay, the radiosensitizing effectiveness of misonidazole at low radiation dose (0-6 Gy) was measured in cultured mammalian cells. Also measured was its effectiveness at high doses of radiation (0-35 Gy) using the conventional survival assay. In both cases, several concentrations of the drug from 0 to 5 mM were used. The data at low doses were analyzed by a two-parameter mathematical equation with linear and quadratic dose terms, S = e-alpha D-beta D2, which proved to be a good fit to the experimental data at all misonidazole concentrations. It is shown that whereas the coefficient of the quadratic dose term, beta, increases significantly with increasing misonidazole concentration, the drug does not significantly affect the coefficient of the linear term, alpha. The enhancement ratio (ER) of misonidazole is shown to be decreased at lower doses. The clinical implications of this result are discussed.     

Computer programs for the analysis of cellular survival data

Four programs have been written to enable radiobiologists to build a computer data base of cellular dose-survival data, calculate cell survival with a correction for cell multiplicity at the time of irradiation, fit various survival models to the data by iteratively weighted least squares, and calculate the ratio of survival levels corresponding to specified doses or the ratio of doses that produce specified survival levels (e.g., oxygen enhancement ratio or relative biological effectiveness). The programs make plots of survival curves and data, and they calculate standard errors and confidence intervals of the fitted survival curve parameters and ratios. The programs calculate survival curves for the linear-quadratic, repair-saturation, single-hit multitarget, linear-multitarget, and repair-misrepair models of cell survival and have been designed to accommodate the addition of other survival models in the future. The programs can be used to compare the accuracy with which different models fit the data, determine if a difference in fit is statistically significant, and show how the estimated value of a survival curve parameter, such as the extrapolation number or the final slope, varies with the survival model. The repair of radiation-induced damage is analyzed in a novel way using these programs.     

Modeling the synergistic effect of high pressure and heat on inactivation kinetics of Listeria innocua: a preliminary study

The survival curves of Listeria innocua CDW47 by high hydrostatic pressure were obtained at four pressure levels (138, 207, 276, 345 MPa) and four temperatures (25, 35, 45, 50 degrees C) in peptone solution. Tailing was observed in the survival curves. Elevated temperatures and pressures substantially promoted the inactivation of L. innocua. A linear and two non-linear (Weibull and log-logistic) models were fitted to these data and the goodness of fit of these models were compared. Regression coefficients (R2), root mean square (RMSE), accuracy factor (Af) values and residual plots suggested that linear model, although it produced good fits for some pressure-temperature combinations, was not as appropriate as non-linear models to represent the data. The residual and correlation plots strongly suggested that among the non linear models studied the log-logistic model produced better fit to the data than the Weibull model. Such pressure-temperature inactivation models form the engineering basis for design, evaluation and optimization of high hydrostatic pressure processes as a new preservation technique.     

Updated Models for the Effects of Temperature,Initial pH,NaCl, and NaNO2 on the Aerobic and Anaerobic Growth of Listeria monocytogenes

Data from several studies were appended to aerobic and anaerobic data sets that had been previously used to develop response surface models describing the growth kinetics of Listeria monocytogenes (Buchanan and Phillips, 1990). The expanded data sets included 709 aerobic and 358 anaerobic growth curves fitted with the Gompertz equation, and representing 189 and 150 unique combinations of four variables (temperature, pH, NaCl, NaNO₂), respectively. Response surface models were developed for (1) the Gompertz B and M terms and (2) lag phase durations (LPD) and generation times (GT). In addition to modeling NaCl as a variable, a second set of response surface models was developed by substituting calculated water activity as a variable. A number of data transformations were evaluated in an attempt to better utilize no-growth data. Full quadratic models of the natural logarithm transformation of the data (no-growth data excluded) predicted values that fit the observed data well. The assignment of GT=50[emsp4 ]h and LPD=600[emsp4 ]h (the approximate maximum duration of experiments) for the variable combinations that did not support growth proved to be the most effective means of making use of the no-growth data. However, this approach did not offer any clear advantage over quadratic models where the no-growth data were excluded. Error matrices were developed for the LPD and GT models to provide 95 % confidence intervals. The agreement between observed and predicted growth kinetics was excellent considering the number and ranges of the variables encompassed in the models. The models provide reasonable predictions of the growth of L. monocytogenes in foods. The full quadratic models of LPD and GT without inclusion of the no-growth data were selected for inclusion in the USDA Pathogen Modeling Program, release 5.1.     

Micronucleus induction in Vicia faba roots. Part 1. Absence of dose-rate, fractionation, and oxygen effect at low doses of low LET radiations

Micronucleus indication in Vicia faba roots has been evaluated after irradiation with 60Co gamma-rays. The dependence of the damage on dose, dose rate, fractionation, and oxygen has been studied. The best fit to the experimental data in the dose region between 7 and 190 cGy is represented, for single-dose exposures, by a linear + quadratic relationship. In the low-dose region, between 7 and 20 cGy, where the linear dose dependence is dominant, no dose-rate, fractionation, or oxygen effect could be observed. These effects were, however, present in the high-dose region, where the quadratic dependence is dominant.     

A continuous method to compute model parameters for soft biological materials

Developing appropriate mathematical models for biological soft tissues such as ligaments, tendons, and menisci is challenging. Stress-strain behavior of these tissues is known to be continuous and characterized by an exponential toe region followed by a linear elastic region. The conventional curve-fitting technique applies a linear curve to the elastic region followed by a separate exponential curve to the toe region. However, this technique does not enforce continuity at the transition between the two regions leading to inaccuracies in the material model. In this work, a Continuous Method is developed to fit both the exponential and linear regions simultaneously, which ensures continuity between regions. Using both methods, three cases were evaluated: idealized data generated mathematically, noisy idealized data produced by adding random noise to the idealized data, and measured data obtained experimentally. In all three cases, the Continuous Method performed superiorly to the conventional technique, producing smaller errors between the model and data and also eliminating discontinuities at the transition between regions. Improved material models may lead to better predictions of nonlinear biological tissues' behavior resulting in improved the accuracy for a large array of models and computational analyses used to predict clinical outcomes.     

Population models for the health effects of ionizing radiation

In this paper we review recently-developed extension frailty, quadratic hazard, stochastic process, microsimulation, and linear latent structure models, which have the potential to describe the health effects of human populations exposed to ionizing radiation. We discuss the most common situations for which such models are appropriate. We also provide examples of how to estimate the parameters of these models from datasets of various designs. Carcinogenesis models are reviewed in context of application to epidemiologic data of population exposed to ionizing radiation. We also discuss the ways of how to generalize stochastic process and correlated frailty models for longitudinal and family analyses in radiation epidemiology.     

Estimating the proportion cured of cancer: Some practical advice for users

BackgroundCure models can provide improved possibilities for inference if used appropriately, but there is potential for misleading results if care is not taken. In this study, we compared five commonly used approaches for modelling cure in a relative survival framework and provide some practical advice on the use of these approaches.Patients and methodsData for colon, female breast, and ovarian cancers were used to illustrate these approaches. The proportion cured was estimated for each of these three cancers within each of three age groups. We then graphically assessed the assumption of cure and the model fit, by comparing the predicted relative survival from the cure models to empirical life table estimates.ResultsWhere both cure and distributional assumptions are appropriate (e.g., for colon or ovarian cancer patients aged <75 years), all five approaches led to similar estimates of the proportion cured. The estimates varied slightly when cure was a reasonable assumption but the distributional assumption was not (e.g., for colon cancer patients ≥75 years). Greater variability in the estimates was observed when the cure assumption was not supported by the data (breast cancer).ConclusionsIf the data suggest cure is not a reasonable assumption then we advise against fitting cure models. In the scenarios where cure was reasonable, we found that flexible parametric cure models performed at least as well, or better, than the other modelling approaches. We recommend that, regardless of the model used, the underlying assumptions for cure and model fit should always be graphically assessed.     

Response of potato tuber yield to stem injury by European corn borer (Lepidoptera: Crambidae) in the Mid-Atlantic United States.

The response of potato, Solanum tuberosum L., tuber yield to stem injury by European corn borer, Ostrinia nubilalis (Hübner), larvae was investigated in the Mid-Atlantic region of the United States for 3 yr. This response was described for 'Superior', 'Atlantic', and 'Snowden' potato, which are early-season, midseason, and late-season maturing cultivars, respectively. To model the yield/injury relationship, a range of corn borer injury levels was established in the field by augmenting the natural infestation with varying densities of laboratory-reared larvae. Linear and nonlinear regression analyses (plateau and second-order polynomial models) were used to describe the relationship between yield of U.S. No. 1 grade tubers and the percentage of stems injured by corn borer larvae. The maturity of the cultivar did not affect the response of potato yield to stem injury. In nine of 14 experiments, potato tolerated high levels of corn borer injury (55-90% of stems injured) without yield loss, suggesting that control of corn borer may not be necessary. Yet, in one of five Superior tests, in two of four Atlantic tests and in two of five Snowden tests, corn borer injury significantly reduced yield. Of the five data sets in which corn borer injury reduced yield, the plateau model fit two data sets and the quadratic model fit one data set. In two of the three cases, these models accounted for nearly identical amounts of total variation in yield as that accounted for by the linear model. The linear model fit four of the five data sets, but the R2 values were low for three of the four tests (0.10, 0.18, and 0.31). The parameter or parameters that interact with corn borer injury to cause tuber yield reduction should be identified before economic injury levels and thresholds are developed.     

One-step analysis of seed storage data and the longevity of Arabidopsis thaliana seeds

Seeds of two ecotypes of Arabidopsis thaliana, NW20 and N1601, were aged over a range of saturated salt solutions at temperatures between 6 degrees C and 55 degrees C. For each ecotype, the results from 37 storage experiments were summarized using the Ellis and Roberts viability equations and a modified version of these equations which allows for a proportion of 'non-respondents'. For both models, two approaches were taken in order to model the effect of moisture content (MC) and temperature on seed longevity. The first, a two-step approach, involved fitting individual survival curves and then multiple regression analysis of the fitted parameters on moisture content and temperature. For the second approach, the full viability models were fitted in one step, including the multiple regression for the effects of MC and temperature within the generalized linear model used to describe each survival curve. This one-step approach takes into account the full variability of the data and provides the best predictions of seed longevity based on the original assumptions of the Ellis and Roberts viability equations. As a consequence of taking into account all the variation, this one-step approach is more sensitive and thus more likely to detect changes due to reducing the number of parameters in the model as being significant. Whilst both approaches indicated that seeds from the two Arabidopsis ecotypes have the same response to MC and temperature, parameter values did differ between the approaches, with the one-step approach providing the better fit. The best model for these two ecotypes, from the one-step approach, confirmed a quadratic relationship between temperature and longevity, but the magnitude of the non-linearity is not as large as indicated by the universal value for the quadratic term.     

Generalized models vs. classification tree analysis: Predicting spatial distributions of plant species at different scales

Abstract. Statistical models of the realized niche of species are increasingly used, but systematic comparisons of alternative methods are still limited. In particular, only few studies have explored the effect of scale in model outputs. In this paper, we investigate the predictive ability of three statistical methods (generalized linear models, generalized additive models and classification tree analysis) using species distribution data at three scales: fine (Catalonia), intermediate (Portugal) and coarse (Europe). Four Mediterranean tree species were modelled for comparison. Variables selected by models were relatively consistent across scales and the predictive accuracy of models varied only slightly. However, there were slight differences in the performance of methods. Classification tree analysis had a lower accuracy than the generalized methods, especially at finer scales. The performance of generalized linear models also increased with scale. At the fine scale GLM with linear terms showed better accuracy than GLM with quadratic and polynomial terms. This is probably because distributions at finer scales represent a linear sub‐sample of entire realized niches of species. In contrast to GLM, the performance of GAM was constant across scales being more data‐oriented. The predictive accuracy of GAM was always at least equal to other techniques, suggesting that this modelling approach is more robust to variations of scale because it can deal with any response shape.     

Modeling study on photosynthetic-light response curves of a C4 plant,maize

Aims A light response curve can reflect a plant’s ability to utilize light, which is also a key tool in determining the relationship between photosynthetic capacity and environmental factors; however the model accuracies concerning the light response curve remain elusive. The objectives of this study were to compare and assess the model accuracies related to a light response curve and the effects of drought. Methods A field rain shelter was used to control the soil water conditions. To obtain photosynthesis parameters from the light response curve and the drought effects, the relevant models (including the rectangular model, non-rectangular hyperbolic model, modified rectangular hyperbolic model, exponential model, quadratic function model, and a newly modified model) were applied to fit the light response curves. The validity of each model was tested by analyzing the differences between the fitted values obtained by the models and the measured values. Important findings The newly modified model has been proved to performing relatively better in accurately describing the light response curve patterns, and credibly obtaining the crucial photosynthetic parameters such as the maximum net photosynthetic rate, light saturation point, light compensation point, and dark respiration rate, especially under high radiation conditions.     

Use of linear and non-linear functions to describe the growth of young sport- and race-horses born in Normandy

The objectives of this study were to establish standards for growth and to model the evolution of wither height (WH) between birth and adult age in different breeds of sport- and race-horses. Therefore, 398 foals, then yearlings of three different breeds, were measured regularly between birth and 18 months of age. Linear and non-linear functions were compared for describing the growth in each breed group. The monomolecular, Gompertz, logistic and cubic models correctly estimated WH in the three breeds during the first 2 years (R2 = 0.99, s.e. 3.9 to 4.5) and better than the cubic and quadratic models (R2 = 0.93, s.e. = 4.7 to 5.3). The logarithmic and power model seemed better in the last part of the growth period (2 to 6 years, R2 = 0.85, s.e. = 5.6 to 5.9). The linear model did not fit with data on most of the growth period. Comparison of the growth in the three breeds using these models confirmed that race-horses had an intense growth in their first months whereas sport-horses had a more regular growth prolonged in their first years of life.     

A mathematical model of brain tumour response to radiotherapy and chemotherapy considering radiobiological aspects

Glioblastoma is the most frequent and malignant brain tumour. For many years, the conventional treatment has been maximal surgical resection followed by radiotherapy (RT), with a median survival time of less than 10 months. Previously, the use of adjuvant chemotherapy (given after RT) has failed to demonstrate a statistically significant survival advantage. Recently, a randomized phase III trial has confirmed the benefit of temozolomide (TMZ) and has defined a new standard of care for the treatment of patients with high-grade brain tumours. The results showed an increase of 2.5 months in median survival, and 16.1% in 2 year survival, for patients receiving RT with TMZ compared with RT alone. It is not clear whether the major benefit of TMZ comes from either concomitant administration of TMZ with RT, or from six cycles of adjuvant TMZ, or both.The objectives were to develop our original model, which addressed survival after RT, to construct a new module to assess the potential role of TMZ from clinical data, and to explore its synergistic contribution in addition to radiation. The model has been extended to include radiobiological parameters. The addition of the linear quadratic equation to describe cellular response to treatment has enabled us to quantify the effects of radiation and TMZ in radiobiological terms.The results indicate that the model achieves an excellent fit to the clinical data, with the assumption that TMZ given concomitantly with RT synergistically increases radiosensitivity. The alternative, that the effect of TMZ is due only to direct cell killing, does not fit the clinical data so well. The addition of concomitant TMZ appears to change the radiobiological parameters. This aspect of our results suggests possible treatment developments.Our observations need further evaluations in real clinical trials, may suggest treatment strategies for new trials, and inform their design.