Mathematical Models for Tumor Growth and the Reduction of Overtreatment

Background To improve our understanding of the natural course of head and neck paragangliomas (HNPGL) and ultimately differentiate between cases that benefit from early treatment and those that are best left untreated, we studied the growth dynamics of 77 HNPGL managed with primary observation. Methods Using digitally available magnetic resonance images, tumor volume was estimated at three time points. Subsequently, nonlinear least squares regression was used to fit seven mathematical models to the observed growth data. Goodness of fit was assessed with the coefficient of determination (R 2) and root-mean-squared error. The models were compared with Kruskal–Wallis one-way analysis of variance and subsequent post-hoc tests. In addition, the credibility of predictions (age at onset of neoplastic growth and estimated volume at age 90) was evaluated. Results Equations generating sigmoidal-shaped growth curves (Gompertz, logistic, Spratt and Bertalanffy) provided a good fit (median R 2: 0.996–1.00) and better described the observed data compared with the linear, exponential, and Mendelsohn equations (p < 0.001). Although there was no statistically significant difference between the sigmoidal-shaped growth curves regarding the goodness of fit, a realistic age at onset and estimated volume at age 90 were most often predicted by the Bertalanffy model. Conclusions Growth of HNPGL is best described by decelerating tumor growth laws, with a preference for the Bertalanffy model. To the best of our knowledge, this is the first time that this often-neglected model has been successfully fitted to clinically obtained growth data.

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Mathematical models to describe the growth curves of white-egg layers

Semina Ciências Agrárias ◽

10.5433/1679-0359.2018v39n3p1327 ◽

2018 ◽

Vol 39 (3) ◽

pp. 1327

Author(s):

Cleber Franklin Santos de Oliveira ◽

João Marcos Novais Tavares ◽

Gerusa Da Silva Salles Corrêa ◽

Bruno Serpa Vieira ◽

Silvana Alves Pedrozo Vitalino Barbosa ◽

...

Keyword(s):

Mathematical Models ◽

Mean Squared Error ◽

Computer Software ◽

Logistic Models ◽

Growth Curves ◽

Gompertz Model ◽

Information Criterion ◽

Coefficient Of Determination ◽

Population Densities

The aim of this study was to compare mathematical models describing growth curves of white-egg layers at different population densities. To fit the models, 4,000 growing white-egg layers were utilized. The experimental design was completely randomized, with population densities of 71, 68, 65, 62, and 59 birds per cage in the starter phase and 19, 17, 15, 13, and 11 birds per cage in the grower phase, with 10 replicates each. Birds were weighed weekly to determine the average body weight and the weight gain. Gompertz and Logistic models were utilized to estimate their growth. The data analysis was carried out using the PROC NLMIXED procedure of the SAS® statistical computer software to estimate the parameters of the equation because mixed models were employed. The mean squared error, the coefficient of determination, and Akaike’s information criterion were used to evaluate the quality of fit of the models. The studied models converged for the description of the growth of the birds at the different densities studied, showing that they were appropriate for estimating the growth of white-egg layers housed at different population densities. The Gompertz model showed a better fit than the Logistic model.

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Isotonic regression analysis of Guzerá cattle growth curves

Revista CERES ◽

10.1590/0034-737x201865010004 ◽

2018 ◽

Vol 65 (1) ◽

pp. 24-27 ◽

Cited By ~ 1

Author(s):

Adriano Rodrigues ◽

Lucas Monteiro Chaves ◽

Fabyano Fonseca Silva ◽

Idalmo Pereira Garcia ◽

Darlene Ana Souza Duarte ◽

...

Keyword(s):

Goodness Of Fit ◽

Mean Squared Error ◽

Least Squares Method ◽

Age Groups ◽

Growth Curves ◽

Generalized Least Squares ◽

Isotonic Regression ◽

Coefficient Of Determination ◽

Linear Regression Models ◽

Generalized Least Squares Method

ABSTRACT The objective of this study was to apply data transformation via isotonic regression in growth curves studies of Guzerá cattle whose data presented disturbances characterized by decreased body weight in certain age groups. Weight-age data were collected on newly weaned Guzerá males according to the methodology of weight gain tests (WGT) defined by the Brazilian Association of Zebu Breeders (ABCZ). The Logistic, Von Bertalanffy and Gompertz models were fitted to weight-age data using the generalized least squares method for non-linear regression models through the Gauss-Newton algorithm. The proposed transformation based on isotonic regression theory proved to be efficient; and the Logistic model was the best to describe the growth of animals, with a high percentage of convergence (100%) and goodness of fit assessed by the mean squared error (MSE) and the coefficient of determination (R2).

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A sinusoidal equation as alternative to conventional growth functions to describe the evolution of growth in quail

Spanish Journal of Agricultural Research ◽

10.5424/sjar/2019173-14973 ◽

2019 ◽

Vol 17 (3) ◽

pp. e0606

Author(s):

Hassan Darmani-Kuhi ◽

James France ◽

Secundino López ◽

Navid Ghavi Hossein-Zadeh

Keyword(s):

Goodness Of Fit ◽

Mean Squared Error ◽

Growth Curves ◽

Growth Dynamics ◽

Information Criteria ◽

Growth Patterns ◽

Growth Data ◽

Suitable Alternative ◽

Growth Functions ◽

Age Related

Aim of study: The aim of the present study was to introduce a sinusoidal equation into poultry science by applying it to temporal growth data from quail.Material and methods: To examine the performance of the sinusoidal equation in describing the growth patterns of quail, four conventional growth functions (Gompertz, logistic, López and Richards) were used as reference in this study. Comparison of models was carried out by analysing model behaviour when fitting the curves using nonlinear regression and assessing statistical performance. Maximum log-likelihood estimation, mean squared error, Akaike and Bayesian information criteria were used to evaluate the general goodness-of-fit of each model to the different data profiles.Main results: The selected sinusoidal equation precisely describes the growth dynamics of quail. Comparison of the growth functions in terms of the goodness-of-fit criteria revealed that the sinusoidal equation was one of the most appropriate functions to describe the age-related changes of bodyweight in quail.Research highlights: To the best of our knowledge there are no studies available on the use of sinusoidal equations to describe the evolution of growth in quail. The sinusoidal equation used in this study represents a suitable alternative to conventional growth functions to describe the growth curves for a range of strains/lines of male and female Japanese quail.

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Reference range of fetal myocardial area by three-dimensional ultrasonography and its applicability in fetuses of pre-gestational diabetic women

Journal of Perinatal Medicine ◽

10.1515/jpm-2018-0342 ◽

2019 ◽

Vol 47 (4) ◽

pp. 422-428 ◽

Cited By ~ 1

Author(s):

José Francisco Melo Júnior ◽

Nathalie Jeanne Bravo-valenzuela ◽

Luciano Marcondes Machado Nardozza ◽

Alberto Borges Peixoto ◽

Rosiane Mattar ◽

...

Keyword(s):

Pregnant Women ◽

Goodness Of Fit ◽

Reference Range ◽

Polynomial Regression ◽

Three Dimensional ◽

Image Correlation ◽

Coefficient Of Determination ◽

Cross Sectional ◽

Myocardial Area ◽

Significant Difference

Abstract Objective To determine the reference range for the myocardial area in healthy fetuses using three-dimensional (3D) ultrasonography and validate these results in fetuses of pregnant women with pre-gestational diabetes mellitus (DM). Methods This cross-sectional retrospective study included 168 healthy pregnant women between gestational weeks 20 and 33+6 days. The myocardial area was measured using spatio-temporal image correlation (STIC) in the four-chamber view. Polynomial regression models were used, and the goodness of fit of the models were evaluated by the coefficient of determination (R2). Intra- and inter-observer reproducibility was determined using the concordance correlation coefficient (CCC). Validation was performed in 30 pregnant women with pre-gestational DM. Results There was a strong correlation (R2=0.71, P<0.0001) between myocardial area and gestational age. There was good intra- and inter-observer reproducibility, with a CCC of 0.86 and 0.83, respectively. However, there was no significant difference in the mean myocardial area between healthy fetuses and fetuses of women with pre-gestational DM (0.11 cm2, P=0.55). Conclusion The reference range was determined for the myocardial area in fetuses, and there was no significant difference in this variable between healthy fetuses and the fetuses of women with pre-gestational DM.

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A COMPARISON OF NONLINEAR MODELS FOR DESCRIBING WEIGHT-AGE DATA IN ANGLO-NUBIAN DOES

Revista Caatinga ◽

10.1590/1983-21252019v32n125rc ◽

2019 ◽

Vol 32 (1) ◽

pp. 251-258 ◽

Cited By ~ 2

Author(s):

Francisco Arthur Arré ◽

José Elivalto Guimarães Campelo ◽

José Lindenberg Rocha Sarmento ◽

Luiz Antônio Silva Figueiredo Filho ◽

Diego Helcias Cavalcante

Keyword(s):

Goodness Of Fit ◽

Mean Squared Error ◽

Nonlinear Models ◽

Age Groups ◽

Coefficient Of Determination ◽

Group Model ◽

Age Group ◽

Von Bertalanffy ◽

Absolute Deviation ◽

Model Interaction

ABSTRACT The objective of this study was to determine the optimum age at last weighing and compare the goodness of fit of nonlinear models used to fit longitudinal weight-age data to describe the growth pattern of Anglo-Nubian does. Weights of 104 animals from birth to 60 months of age were grouped into 10 age groups at six-month intervals. In each age group, parameters A (asymptotic weight), B (integration constant), and K (maturity index) were estimated using the Brody, Gompertz, logistic, and von Bertalanffy models. Data were analyzed using analysis of variance in a factorial design (10 age groups × 4 nonlinear models). The age group × model interaction was not significant. Mean estimates of A, B, and K were significantly different between age groups up to 30 months (p < 0.05), indicating that the estimated curve is affected by weights taken before this age independent of the model. The values of mean squared error (MSE), mean absolute deviation (MAD), coefficient of determination (R2) and Rate of convergence (RC) at each age group up to 30 months were compared to determine the goodness of fit of nonlinear models. The ranking of fit was logistic, Brody, von Bertalanffy, and Gompertz. The logistic and Brody models respectively estimated the smallest and largest asymptotic weight. Longitudinal weight records taken until 30 months of age are most appropriate for estimating the growth of Anglo-Nubian does using nonlinear models.

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Method to generate growth and degrowth models obtained from differential equations applied to agrarian sciences

Semina Ciências Agrárias ◽

10.5433/1679-0359.2018v39n6p2659 ◽

2018 ◽

Vol 39 (6) ◽

pp. 2659 ◽

Cited By ~ 3

Author(s):

André Luiz Pinto dos Santos ◽

Guilherme Rocha Moreira ◽

Cicero Carlos Ramos de Brito ◽

Frank Gomes-Silva ◽

Maria Lindomárcia Leonardo da Costa ◽

...

Keyword(s):

Differential Equations ◽

Goodness Of Fit ◽

Least Squares Method ◽

Growth Curves ◽

Information Criterion ◽

Coefficient Of Determination ◽

Absolute Deviation ◽

Marquardt Algorithm ◽

Maturation Rate ◽

Biological Interpretation

This study aims to propose a method to generate growth and degrowth models using differential equations as well as to present a model based on the method proposed, compare it with the classic linear mathematical models Logistic, Von Bertalanffy, Brody, Gompertz, and Richards, and identify the one that best represents the mean growth curve. To that end, data on Undefined Breed (UB) goats and Santa Inês sheep from the works of Cavalcante et al. (2013) and Sarmento et al. (2006a), respectively, were used. Goodness-of-fit was measured using residual mean squares (RMS), Akaike information criterion (AIC), Bayesian information criterion (BIC), mean absolute deviation (MAD), and adjusted coefficient of determination . The models’ parameters (?, weight at adulthood; ?, an integration constant; ?, shape parameter with no biological interpretation; k, maturation rate; and m, inflection point) were estimated by the least squares method using Levenberg-Marquardt algorithm on the software IBM SPSS Statistics 1.0. It was observed that the proposed model was superior to the others to study the growth curves of goats and sheep according to the methodology and conditions under which the present study was carried out.

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A hydrological model skill score and revised R-squared

Hydrology Research ◽

10.2166/nh.2021.071 ◽

2021 ◽

Author(s):

Charles Onyutha

Keyword(s):

Mathematical Modeling ◽

Goodness Of Fit ◽

Hydrological Model ◽

Mean Squared Error ◽

Skill Score ◽

Coefficient Of Determination ◽

Model Skill ◽

Squared Error ◽

Quality Coefficient ◽

Correlation Term

Abstract Despite the advances in methods of statistical and mathematical modeling, there is considerable lack of focus on improving how to judge models’ quality. Coefficient of determination (R2) is arguably the most widely applied ‘goodness-of-fit’ metric in modelling and prediction of environmental systems. However, known issues of R2 are that it: (i) can be low and high for an accurate and imperfect model, respectively; (ii) yields the same value when we regress observed on modelled series and vice versa; and (iii) does not quantify a model's bias (B). A new model skill score E and revised R-squared (RRS) are presented to combine correlation, term B and capacity to capture variability. Differences between E and RRS lie in the forms of correlation and the term B used for each metric. Acceptability of E and RRS was demonstrated through comparison of results from a large number of hydrological simulations. By applying E and RRS, the modeller can diagnostically identify and expose systematic issues behind model optimizations based on other ‘goodness-of-fits’ such as Nash–Sutcliffe efficiency (NSE) and mean squared error. Unlike NSE, which varies from −∞ to 1, E and RRS occur over the range 0–1. MATLAB codes for computing E and RRS are provided.

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Classical Mathematical Models for Prediction of Response to Chemotherapy and Immunotherapy

10.1101/2021.10.23.465549 ◽

2021 ◽

Author(s):

Narmin Ghaffari Laleh ◽

Chiara Maria Lavinia Loeffler ◽

Julia Grajek ◽

Katerina Stankova ◽

Alexander T. Pearson ◽

...

Keyword(s):

Tumor Growth ◽

Mathematical Models ◽

Early Treatment ◽

Goodness Of Fit ◽

Tumor Volume ◽

Human Tumor ◽

Gompertz Model ◽

Text Book ◽

Response To Chemotherapy ◽

Volume Measurements

Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, even the simplest text-book models have been barely validated in real world-data of human patients. In this study, we fitted a range of differential equation models to tumor volume measurements of patients undergoing chemo-therapy or cancer immunotherapy for solid tumors. We used a large dataset of 1472 patients with three or more measurements per target lesion, of which 652 patients had six or more data points. We show that the early treatment response shows only moderate correlation with the final treatment re-sponse, demonstrating the need for nuanced models. We then perform a head-to-head comparison of six classical models which are widely used in the field: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model. Several models provide a good fit to tumor volume measurements, with the Gompertz model providing the best balance between goodness of fit and number of parameters. Similarly, when fitting to early treatment data, the general Bertalanffy and Gompertz models yield the lowest mean absolute error to forecasted data, indicating that these models could potentially be effective at predicting treatment outcome. In summary, we pro-vide a quantitative benchmark for classical textbook models and state-of-the art models of human tumor growth. We publicly release an anonymized version of our original data, providing the first benchmark set of human tumor growth data for evaluation of mathematical models.

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Choice of non-linear models to determine the growth curve of meat-type quail

Ciência Rural ◽

10.1590/0103-8478cr20190990 ◽

2021 ◽

Vol 51 (2) ◽

Author(s):

Marta Jeidjane Borges Ribeiro ◽

Fabyano Fonseca Silva ◽

Maíse dos Santos Macário ◽

José Aparecido Santos de Jesus ◽

Claudson Oliveira Brito ◽

...

Keyword(s):

Goodness Of Fit ◽

Linear Models ◽

Growth Curves ◽

Information Criterion ◽

Metabolizable Energy ◽

Coefficient Of Determination ◽

Parameter Estimates ◽

Coturnix Coturnix ◽

Non Linear ◽

Meat Type

ABSTRACT: The objective of this study was to compare non-linear models fitted to the growth curves of quail to determine which model best describes their growth and check the similarity between models by analyzing parameter estimates.Weight and age data of meat-type European quail (Coturnix coturnix coturnix) of three lines were used, from an experiment in a 2 × 4 factorial arrangement in a completely randomized design, consisting of two metabolizable energy levels, four crude protein levels and six replicates. The non-linear Brody, Von Bertalanffy, Richards, Logistic and Gompertz models were used. To choose the best model, the Adjusted Coefficient of Determination, Convergence Rate, Residual Mean Square, Durbin-Watson Test, Akaike Information Criterion and Bayesian Information Criterion were applied as goodness-of-fit indicators. Cluster analysis was performed to check the similarity between models based on the mean parameter estimates. Among the studied models, Richards’ was the most suitable to describe the growth curves. The Logistic and Richards models were considered similar in the analysis with no distinction of lines as well as in the analyses of Lines 1, 2 and 3.

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Pre- and postnatal growth phenomena of Burchell's Zebra Equus Burchelli Antiquorum

Koedoe ◽

10.4102/koedoe.v18i1.915 ◽

1975 ◽

Vol 18 (1) ◽

Cited By ~ 4

Author(s):

G.L. Smuts

Keyword(s):

Linear Regression ◽

Body Mass ◽

Growth Curves ◽

Mass Range ◽

Postnatal Growth ◽

Regression Equations ◽

Growth Data ◽

Body Dimension ◽

Equus Burchelli ◽

Significant Difference

Between 1969 and 1972 growth data were collected from 175 zebra Equus burchelli antiquorum and 138 zebra embryos and foetuses from the Central District of the Kruger National Park, Republic of South Africa. Statistical analysis of data indicated no significant difference between body mass of adult stallions (range == 267,3 to 373,3 kg; mean = 318,5 kg; n = 57) and adult non-pregnant mares (range = 272,6 to 386,9 kg; mean = 321,6 kg; n = 51) (t = 0,587). The heaviest zebra had a body mass of 429,4 kilogram. This was a pregnant mare carrying a 35,2 kg foetus. Von Bertalanffy growth curves indicated that shoulder heights in young zebra may reach the adult range by one year of age, the adult body mass range is, however, only attained after three years of age. These curves also showed that age classification of free roaming zebra is only reliable up to the age of about two years, after which individual variation is too great. Stallions were significantly taller at the shoulder than mares (mean = 1,8 cm) (t = 2,032) and neck thickness was the only body dimension showing visible sexual dimorphism in adults. Here the stallion had a neck girth on average 8,1 cm greater than the mare. Regression equations for estimating body mass from body dimensions were calculated by using a standard logarithmic transformation and fitting a linear regression by the method of least squares and also by undertaking standard straight line linear regression analyses. Exponential curves obtained by the first method indicated that growth was not isometric (not linear) and that the ratios of any of the dimensions of length to body mass were con- stantly changing, i.e. growth is allometric. Marked allometric growth differences existed between the two sexes except in the case of the heart girth-body mass relationship. Comparison of growth data from E. b. antiquorum with that of E. b. boehmi from Tanzania (Sachs 1967), indicates that E. b. antiquorum is considerably larger. Body masses differ by an average of 70 kg and 102 kg for stallions and mares respectively. Average birth mass for zebra was 33,7 kg. The largest foetus had a body mass of 39,0 kilogram. Foetal growth curves are provided. The first signs of body stripes occur at between 250 and 270 days of pregnancy (gestation period = 375 days).

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