scholarly journals Population Modeling of Tumor Growth Curves, the Reduced Gompertz Model and Prediction of the Age of a Tumor

Author(s):  
Cristina Vaghi ◽  
Anne Rodallec ◽  
Raphaelle Fanciullino ◽  
Joseph Ciccolini ◽  
Jonathan Mochel ◽  
...  
2020 ◽  
Vol 16 (2) ◽  
pp. e1007178 ◽  
Author(s):  
Cristina Vaghi ◽  
Anne Rodallec ◽  
Raphaëlle Fanciullino ◽  
Joseph Ciccolini ◽  
Jonathan P. Mochel ◽  
...  

2019 ◽  
Author(s):  
C. Vaghi ◽  
A. Rodallec ◽  
R. Fanciullino ◽  
J. Ciccolini ◽  
J. Mochel ◽  
...  

AbstractTumor growth curves are classically modeled by ordinary differential equations. In analyzing the Gompertz model several studies have reported a striking correlation between the two parameters of the model.We analyzed tumor growth kinetics within the statistical framework of nonlinear mixed-effects (population approach). This allowed for the simultaneous modeling of tumor dynamics and interanimal variability. Experimental data comprised three animal models of breast and lung cancers, with 843 measurements in 94 animals. Candidate models of tumor growth included the Exponential, Logistic and Gompertz. The Exponential and – more notably – Logistic models failed to describe the experimental data whereas the Gompertz model generated very good fits. The population-level correlation between the Gompertz parameters was further confirmed in our analysis (R2 > 0.96 in all groups). Combining this structural correlation with rigorous population parameter estimation, we propose a novel reduced Gompertz function consisting of a single individual parameter. Leveraging the population approach using bayesian inference, we estimated the time of tumor initiation using three late measurement timepoints. The reduced Gompertz model was found to exhibit the best results, with drastic improvements when using bayesian inference as compared to likelihood maximization alone, for both accuracy and precision. Specifically, mean accuracy was 12.1% versus 74.1% and mean precision was 15.2 days versus 186 days, for the breast cancer cell line.These results offer promising clinical perspectives for the personalized prediction of tumor age from limited data at diagnosis. In turn, such predictions could be helpful for assessing the extent of invisible metastasis at the time of diagnosis.Author summaryMathematical models for tumor growth kinetics have been widely used since several decades but mostly fitted to individual or average growth curves. Here we compared three classical models (Exponential, Logistic and Gompertz) using a population approach, which accounts for inter-animal variability. The Exponential and the Logistic models failed to fit the experimental data while the Gompertz model showed excellent descriptive power. Moreover, the strong correlation between the two parameters of the Gompertz equation motivated a simplification of the model, the reduced Gompertz model, with a single individual parameter and equal descriptive power. Combining the mixed-effects approach with Bayesian inference, we predicted the age of individual tumors with only few late measurements. Thanks to its simplicity, the reduced Gompertz model showed superior predictive power. Although our method remains to be extended to clinical data, these results are promising for the personalized estimation of the age of a tumor from limited measurements at diagnosis. Such predictions could contribute to the development of computational models for metastasis.


2020 ◽  
Vol 42 (2) ◽  
Author(s):  
Édipo Menezes da Silva ◽  
Maraísa Hellen Tadeu ◽  
Victor Ferreira da Silva ◽  
Rafael Pio ◽  
Tales Jesus Fernandes ◽  
...  

Abstract Blackberry is a small fruit with several properties beneficial to human health and its cultivation is an alternative for small producers due to its fast and high financial return. Studying the growth of fruits over time is extremely important to understand their development, helping in the most appropriate crop management, avoiding post-harvest losses, which is one of the aggravating factors of blackberry cultivation, being a short shelf life fruit. Thus, growth curves are highlighted in this type of study and modeling through statistical models helps understanding how such growth occurs. Data from this study were obtained from an experiment conducted at the Federal University of Lavras in 2015. The aim of this study was to adjust nonlinear, double Logistic and double Gompertz models to describe the diameter growth of four blackberry cultivars (‘Brazos’, ‘Choctaw’, ‘Guarani’ and ‘Tupy’). Estimations of parameters were obtained using the least squares method and the Gauss-Newton algorithm, with the “nls” and “glns” functions of the R statistical software. The comparison of adjustments was made by the Akaike information criterion (AICc), residual standard deviation (RSD) and adjusted determination coefficient (R2 aj). The models satisfactorily described data, choosing the Logistic double model for ‘Brazos’ and ‘Guarani’ cultivars and the double Gompertz model for ‘Tupy’ and ‘Choctaw’ cultivars.


2018 ◽  
Vol 80 (01) ◽  
pp. 072-078 ◽  
Author(s):  
Berdine Heesterman ◽  
John-Melle Bokhorst ◽  
Lisa de Pont ◽  
Berit Verbist ◽  
Jean-Pierre Bayley ◽  
...  

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.


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 ◽  
...  

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.


1988 ◽  
Vol 74 (3) ◽  
pp. 269-274 ◽  
Author(s):  
Romano Demicheli ◽  
Roberto Foroni ◽  
Fernando C. Giuliani ◽  
Giuseppina Savi

The influence of tumor growth kinetics on response to doxorubicin treatment of C3H mammary carcinoma was investigated. Gompertzian growth curves were obtained for the tumor mass of each mouse by a computerized best fit program. The response was assessed by evaluating: a) the total clonogenic cell reduction as a fraction of the initial tumor volume or the tumor volume that should result at the end of treatment in a free growth condition, and b) the partial clonogenic cell reduction at each drug administration, assuming a first order cell kill hypothesis. Slowly growing tumors at each dose level showed a significantly poorer response than rapidly growing tumors. Each response index exhibited a linear correlation with the specific instantaneous growth rate at the time of treatment. Data also suggested a dose-response dependence.


2007 ◽  
Vol 5 (4) ◽  
pp. 69
Author(s):  
C. Murayama ◽  
M. Isozaki ◽  
A. Shintani ◽  
A. Kamijo ◽  
Y. Maeda ◽  
...  

Blood ◽  
2005 ◽  
Vol 106 (11) ◽  
pp. 3452-3452
Author(s):  
Anton C. Martens ◽  
Henk Rozemuller ◽  
Ellen van der Spek ◽  
Lijnie Bogers-Boer ◽  
Niels van de Donk ◽  
...  

Abstract Preclinical testing of new therapeutical strategies for the treatment of multiple myeloma (MM) requires animal models that closely resemble human disease and allow quantitative evaluation of the applied therapy. Models that meet both requirements have thus far not been described. Here we present a novel in vivo MM model by engraftment of MM U266 or RPMI-8226/S cells, both of human origin, into RAG2γc double knock-out mice. These mice are totally immune deficient because they lack T-, B and NK cells and the mice easily accept human cells (van Rijn et al., Blood 2003, Rozemuller et al., 2004). After intravenous injection of 2x106 MM cells engraftment and outgrowth occurred in all mice but was limited to the bone marrow compartment only. Flow cytometry (FCM) confirmed the presence of human CD45/38/138 positive MM cells in femur, spine, tibia and sternum bone specimens. Infiltration into other organs was not observed. In a next step MM cells were stably transduced using a retroviral vector encoding both the Green Fluorescent Protein (GFP) and firefly Luciferase (fLuc) marker genes. Technical advances in recent years in optical imaging by Bioluminescence Imaging (BLI) techniques allow visualization and quantification of bioluminescent light by detecting photons that are transmitted through mammalian tissue. When luciferase converts the substrate luciferin, photons are emitted that can be registered by using sensitive CCCD cameras. The absolute number of photons that are produced correlates, in our application, with local tumor mass. Mice were injected i.v. with 2x106 GFP-fLuc transduced MM cells (U266 or RPMI8226/S) and imaged weekly using BLI. Within 2 weeks after injection significant BLI signals were detectable. Per mouse 5-10 foci showed luciferase activity, predominantly in the pelvic region, skull, limbs, sternum, ribs and the spine. This low frequency of engraftment is in line with earlier reports on RPMI8226/S (Mitsiades et al., Cancer Res 2003). At 9 weeks the first mice developed hind leg paralysis which could be attributed to tumor associated spinal lesions. After 12 weeks the last mouse was sacrificed. BLI revealed that the intensity of light production at the various sites of tumor growth within individual mice as well as between mice showed a similar increase. This reflects an increase in tumor mass. Quantitative analysis of subsequent BLI images allowed construction of tumor growth curves of the total tumor mass per mouse as well as for the individual foci of MM growth in individual mice. We typically observed exponential growth, with growth curves running parallel with an average population doubling time of approximately 4–5 days. The range in which tumor growth could be monitored (and as a consequence also the response to treatment) spans 3–4 decades. In contrast with previously reported murine models for human MM where -next to bone marrow homing- also extra-skeletal tumors were observed our model almost exclusively shows homing of MM cells to the BM and is therefore more consistent with the clinical manifestation in myeloma patients. The major advantage of the model is the option for quantitative evaluation of the effect of a given treatment on the tumorload. Currently we are studying the efficacy of newly developed geranyl-geranyl-transferase inhibitors (GGTI). In conclusion, we have developed a novel in vivo model to study the characteristics of homing and outgrowth of MM and for quantitative evaluation of the efficacy of the therapeutic intervention applied.


2016 ◽  
Vol 16 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Monika Michalczuk ◽  
Krzysztof Damaziak ◽  
Antoni Goryl

Abstract The study analyzed the growth of medium-growing chickens of the CCGP experimental line, using Gompertz, Logistic, and Richards models as well as body gain curves. The birds were reared until 9 wk of age. To fit BW values to the applied models, determination coefficients (R2,Ř2) and standard error of the mean (±SE) were calculated for 487 male and 493 female chickens. The comparison of results obtained demonstrated the Gompertz model to be the most precise equation to describe the growth of both sexes of CCGP chickens, though in all examined models the determination coefficients were approximating 99%. According to the Gompertz model, the chickens may reach the maximum BW at the age of 16 wk (5900 g - males and 4000 g - females), whereas the maximum daily BW gain - on day 47 (69.0 g) in males and on day 41 (50.0 g) in females. Values achieved in the Logistic model were the most diverging from the values obtained with other models, whereas the Richards model may be successfully applied to estimate BW of chickens. Females were reaching the maximum BW gains earlier, but the curve of their BW gain was proceeding with two peaks (at ca. 4 wk - 313.09 g/wk and at 6 wk - 327.59 g/wk), which was probably due to partial growth deceleration as a result of allowing the birds to use free ranges on day 14. In the case of males, the maximum BW gain (727.35 g/wk) was reached at 6.2 wk.


2010 ◽  
Vol 43 ◽  
pp. 736-739
Author(s):  
Kui Hua Zhan ◽  
Lun Bai ◽  
Hong Qin Dai

We divide a solid tumor in vascular phase into a vascular layer, an oxygen-diffusion layer, an oxygen-resistant layer and a necrotic layer outside-to-inside, and have constructed a layered model for tumor growth. The actual growth curves of the tumors have been obtained by using different invasion speeds of the vascular layer. The simulative calculation results indicate that the angiogenesis rate is positively correlated with malignant degree of the tumors, if the tumors enter vascular phase, survival time of tumor patients is about 1-4 years, but the tumor patients without angiogenesis can live with tumor.


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