New formulation of the Gompertz equation to describe the kinetics of untreated tumors
Autoři:
Antonio Rafael Selva Castañeda aff001; Erick Ramírez Torres aff003; Narciso Antonio Villar Goris aff004; Maraelys Morales González aff007; Juan Bory Reyes aff008; Victoriano Gustavo Sierra González aff009; María Schonbek aff010; Juan Ignacio Montijano aff001; Luis Enrique Bergues Cabrales aff001
Působiště autorů:
Departamento de Matemática Aplicada, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, Zaragoza, Spain
aff001; Departamento de Telecomunicaciones, Facultad de Ingeniería en Telecomunicaciones Informática y Biomédica, Universidad de Oriente, Santiago de Cuba, Cuba
aff002; Departamento de Biomédica, Facultad de Ingeniería en Telecomunicaciones Informática y Biomédica, Universidad de Oriente, Santiago de Cuba, Cuba
aff003; Universidad Autónoma de Santo Domingo, Santo Domingo, Dominican Republic
aff004; Universidad Católica Tecnológica del CIBAO, Ucateci, La Vega, Dominican Republic
aff005; Departamento de Ciencia e Innovación, Centro Nacional de Electromagnetismo Aplicado, Universidad de Oriente, Santiago de Cuba, Cuba
aff006; Departamento de Farmacia, Facultad de Ciencias Naturales y Exactas, Universidad de Oriente, Santiago de Cuba, Cuba
aff007; ESIME-Zacatenco, Instituto Politécnico Nacional, CD-MX, Mexico
aff008; Grupo de las Industrias Biotecnológica y Farmacéuticas (BioCubaFarma), La Habana, Cuba
aff009; Department of Mathematics, University of California Santa Cruz, Santa Cruz, CA, United States of America
aff010
Vyšlo v časopise:
PLoS ONE 14(11)
Kategorie:
Research Article
doi:
https://doi.org/10.1371/journal.pone.0224978
Souhrn
Background
Different equations have been used to describe and understand the growth kinetics of undisturbed malignant solid tumors. The aim of this paper is to propose a new formulation of the Gompertz equation in terms of different parameters of a malignant tumor: the intrinsic growth rate, the deceleration factor, the apoptosis rate, the number of cells corresponding to the tumor latency time, and the fractal dimensions of the tumor and its contour.
Methods
Furthermore, different formulations of the Gompertz equation are used to fit experimental data of the Ehrlich and fibrosarcoma Sa-37 tumors that grow in male BALB/c/Cenp mice. The parameters of each equation are obtained from these fittings.
Results
The new formulation of the Gompertz equation reveals that the initial number of cancerous cells in the conventional Gompertz equation is not a constant but a variable that depends nonlinearly on time and the tumor deceleration factor. In turn, this deceleration factor depends on the apoptosis rate of tumor cells and the fractal dimensions of the tumor and its irregular contour.
Conclusions
It is concluded that this new formulation has two parameters that are directly estimated from the experiment, describes well the growth kinetics of unperturbed Ehrlich and fibrosarcoma Sa-37 tumors, and confirms the fractal origin of the Gompertz formulation and the fractal property of tumors.
Klíčová slova:
Angiogenesis – Apoptosis – Fractals – Graphs – Histology – Interpolation – Malignant tumors – Fibrosarcoma
Zdroje
1. González MM, Joa JA, Cabrales LE, Pupo AE, Schneider B, Kondakci S et al., Is cancer a pure growth curve or does it follow a kinetics of dynamical structural transformation? BMC Cancer 2017; 17:174. doi: 10.1186/s12885-017-3159-y 28270135
2. Tjørve KMC, Tjørve E, The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the Unified-Richards family. Plos One 2017; 12:6.
3. Cabrales LE, Nava JJ, Aguilera AR, Joa JA, Ciria HM, González MM et al., Modified Gompertz equation for electrotherapy murine tumor growth kinetics: Predictions and new hypotheses. BMC Cancer 2010; 10:589. doi: 10.1186/1471-2407-10-589 21029411
4. Izquierdo-Kulich E, Regalado O, Nieto-Villar JM, Fractal origin of the Gompertz equation. Rev Cub Fis 2013; 30:26.
5. Mombach JCM, Lemke N, Bodmann BEJ, Idiart MAP, A mean-field theory of cellular growth. Europhys Lett 2002; 59:923–928.
6. d’Onofrio A, Fractal growth of tumors and other cellular populations: linking the mechanistic to the phenomenological modeling and vice versa. Chaos, Solitons & Fractals 2009; 41:875–880.
7. Ribeiro FL, A Non-phenomenological Model of Competition and Cooperation to Explain Population Growth Behaviors. Bull Math Biol 2015; 77:409–433. doi: 10.1007/s11538-014-0059-z 25724311
8. Ribeiro FL, An attempt to unify some population growth models from first principles. Rev Bras Ensino Fis 2017; 39:e1311.
9. Ciria HMC, Quevedo MS, Cabrales LB, Bruzón RP, Salas ME, Pena OG et al., Antitumor effectiveness of different amounts of electrical charge in Ehrlich and fibrosarcoma Sa-37 tumors. BMC Cancer 2004; 4:87. doi: 10.1186/1471-2407-4-87 15566572
10. Cotran RS, Kumar V, Collins T, Patología Estructural y Funcional. Sexta Edición McGraw-Hill- Interamericana de España (S.A.U. Madrid); 1999. pp 277–347.
11. Cabrales LEB, The electrotherapy a new alternative for the treatment of the malignant tumors. Preclinical study. PhD thesis. Havana University, Biology Department, 2003.
12. Steel GG, Basic Clinical Radiobiology. Second Edition (Oxford University Press, Inc. New York); 1997. pp 1–30.
13. Yang WY, Cao W, Chung TS, Morris J, Applied Numerical Methods using MATLAB®. Wiley-Interscience (John Wiley & Sons, New Jersey); 2005. pp 117–156.
14. Cabrales LEB, Aguilera AR, Jiménez RP, Jarque MV, Ciria HMC, Reyes JB et al., Mathematical modeling of tumor growth in mice following low-level direct electric current. Math Simul Comp 2008; 78:112–120.
15. Waliszewski P, Konarski J, The Gompertzian curve reveals fractal properties of tumor growth. Chaos, Solitons and Fractals 2003; 16:665–674.
16. Molski M, Biological growth in the fractal space-time with temporal fractal dimension. Chaotic Model Simul 2012; 1:169–175.
17. Shim EB, Kim YS, Deisboeck TS, Analyzing the dynamic relationship between tumor growth and angiogenesis in a two dimensional finite element model; 2007. Preprint. Available from: arXiv:q-bio/0703015v1 (q-bio.TO). Preprint, posted February 10, 2016.
18. Sokolov I, Fractals: a possible new path to diagnose and cure cancer? Future Oncology 2015; 11: 3049–3051. doi: 10.2217/fon.15.211 26466999
19. Breki CM, Dimitrakopoulou-Starauss A, Hassel J, Theoharis T, Sachpekidis C, Pan L et al., Fractal and multifractal analysis of PET/CT images of metastatic melanoma before and after treatment with ipilimumab. EJNMMI Research 2016; 6:61. doi: 10.1186/s13550-016-0216-5 27473846
20. Tavakol ME, Lucas C, Sadri S, NG EYK, Analysis of breast thermography using fractal dimension to establish possible difference between malignant and benign patterns. J Healthc Eng 2010; 1: 27–43.
21. Baish JW, Jain RK, Fractals and cancer. Cancer Research 2000; 60:3683–3688. 10919633
22. Hanahan D, Weinberg RA, Hallmarks of Cancer: The Next Generation. Cell 2011; 144:646–674. doi: 10.1016/j.cell.2011.02.013 21376230
23. Stępień R, Stępień P, Analysis of contours of tumor masses in mammograms by Higuchi’s fractal dimension. Biocybern Biomed Eng 2010; 30:49–56.
24. Gazit Y, Berk DA, Leunig M, Baxter LT, Jain RK, Scale-invariant behavior and vascular network formation in normal and tumor tissue. Phys Rev Lett 1995; 75:2428–2431. doi: 10.1103/PhysRevLett.75.2428 10059301
25. Ribeiro FL, dos Santos RV, Mata AS, Fractal dimension and universality in avascular tumor growth; 2016. Phys Rev E 2017; 95:1–9.
26. Zhong JT, Yu J, Wang HJ, Shi Y, Zhao TS, He BX et al., Effects of endoplasmic reticulum stress on the autophagy, apoptosis, and chemotherapy resistance of human breast cancer cells by regulating the PI3K/AKT/mTOR signaling pathway. Tumor Biol 2017; 39:1010428317697562.
27. Win TT, Jaafar H, Yusuf Y, Relationship of angiogenic and apoptotic activities in soft-tissue sarcoma. South Asian J Cancer 2014; 3:171–174. doi: 10.4103/2278-330X.136799 25136525
28. Huang D, Lan H, Liu F, Wang S, Chen X, Jin K, et al., Anti-angiogenesis or pro-angiogenesis for cancer treatment: focus on drug distribution. Int J Clin Exp Med 2015; 8:8369–8376. 26309490
29. Nyberg P, Xie L, Kalluri R., Endogenous inhibitors of angiogenesis. Cancer Res 2005; 65:3967–3979. doi: 10.1158/0008-5472.CAN-04-2427 15899784
30. Falconer K, Fractal geometry. Mathematical foundations and applications. Chapter 2, Second edition (John Wiley & Sons, Ltd., Chichester, England); 2003. pp 33.
31. Kremheller J, Vuong AT, Yoshihara L, Wall WA, Schrefler BA, A monolithic multiphase porous medium framework for (A-) vascular tumor growth. Comput Methods Appl Mech Eng 2018; 340:657–683.
32. Verma A, Pitchumani R, Fractal description of microstructures and properties of dynamically evolving porous media. Int J Heat Mass Transf 2017; 81:51–55.
33. Grizzi F, Fractal geometry as a tool for investigating benign and malignant breast mammography lesions. Fractal Geometry and Nonlinear Anal in Med and Biol 2015; 1:16–18.
34. Rangayyan RM, Nguyen TM, Fractal analysis of contours of breast masses in mammograms. J Digit Imaging 2007; 20:223–237. doi: 10.1007/s10278-006-0860-9 17021926
35. Pardoll DM, The blockade of immune checkpoints in cancer immunotherapy. Nat Rev Cancer 2012; 12:252–264. doi: 10.1038/nrc3239 22437870
36. Hahnfeldt P, Panigrahy D, Folkman J, Hlatky L, Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res 1999; 59:4770–4775. 10519381
37. Perthame B, Some mathematical models of tumor growth; 2015. Universite Pierre et Marie Curie, Paris (June 2014), 23–32. Available from: https://www.ljll.math.upmc.fr/perthame/cours_M2.pdf.
38. Enderling H, Chaplain MAJ, Mathematical modeling of tumor growth and treatment. Curr Pharm Des 2014; 20:4934–4940. doi: 10.2174/1381612819666131125150434 24283955
39. Izquierdo-Kulich E, de Quesada MA, Pérez-Amor CM, Texeira ML, Nieto-Villar JM, The dynamics of tumor growth and cells pattern morphology. Math Biosci Eng 2009; 6:547–559. 19566125
Článek vyšel v časopise
PLOS One
2019 Číslo 11
- S diagnostikou Parkinsonovy nemoci může nově pomoci AI nástroj pro hodnocení mrkacího reflexu
- Proč při poslechu některé muziky prostě musíme tančit?
- Je libo čepici místo mozkového implantátu?
- Chůze do schodů pomáhá prodloužit život a vyhnout se srdečním chorobám
- Pomůže v budoucnu s triáží na pohotovostech umělá inteligence?
Nejčtenější v tomto čísle
- A daily diary study on maladaptive daydreaming, mind wandering, and sleep disturbances: Examining within-person and between-persons relations
- A 3’ UTR SNP rs885863, a cis-eQTL for the circadian gene VIPR2 and lincRNA 689, is associated with opioid addiction
- A substitution mutation in a conserved domain of mammalian acetate-dependent acetyl CoA synthetase 2 results in destabilized protein and impaired HIF-2 signaling
- Molecular validation of clinical Pantoea isolates identified by MALDI-TOF
Zvyšte si kvalifikaci online z pohodlí domova
Všechny kurzy