Lifetime analyses frequently apply a parametric functional description from measured data of the Kaplan-Meier non-parametric estimate (KM) of the survival probability. The cumulative Weibull distribution function (WF) is the primary choice to parametrize the KM. but some others (e.g. Gompertz, logistic functions) are also widely applied. We show that the cumulative two-parametric Weibull function meets all requirements. The Weibull function is the consequence of the general self-organizing behavior of the survival, and consequently shows self-similar death-rate as a function of the time. The ontogenic universality as well as the universality of tumor-growth fits to WF. WF parametrization needs two independent parameters, which could be obtained from the median and mean values of KM estimate, which makes an easy parametric approximation of the KM plot. The entropy of the distribution and the other entropy descriptions are supporting the parametrization validity well. The goal is to find the most appropriate mining of the inherent information in KM-plots. The two-parameter WF fits to the non-parametric KM survival curve in a real study of 1180 cancer patients offering satisfactory description of the clinical results. Two of the 3 characteristic parameters of the KM plot (namely the points of median, mean or inflection) are enough to reconstruct the parametric fit, which gives support of the comparison of survival curves of different patient’s groups.
Szentgyorgyi, A. (2019) Life Is Nothing But an Electron Looking for a Place to Rest. https://www.goodreads.com/work/quotes/46106688-the-big-picture-on-the-origins -of-life-meaning-and-the-universe-itsel
Kauffman, S.A. (1993) The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, New York, Oxford. https://doi.org/10.1007/978-94-015-8054-0_8
Haken, H. (1987) Self-Organization and Information. Physica Scripta, 35, 247-254. https://doi.org/10.1088/0031-8949/35/3/006
Sornette, D. (2000) Chaos, Fractals, Self-Organization and Disorder: Concepts and Tools. Springer Verlag, Berlin, Los Angeles.
Wolfram, S. (2002) A New Kind of Science. Wolfram Media Inc., Champaign.
Camazine, S., Deneubourg, J.L., Franks, N.R., et al. (2003) Self-Organization in Biological Systems. Princeton Studies in Complexity, Princeton Univ. Press, Princeton, Oxford.
Bozorgmehr, J.E.H. (2014) The Role of Self-Organization in Developmental Evolution. Theory in Biosciences, 133, 145-163. https://doi.org/10.1007/s12064-014-0200-4
Johnson, B.R. and Lam, S.K. (2010) Self-Organization, Natural Selection, and Evolution: Cellular Hardware and Genetic Software. Bioscience, 60, 879-885. https://doi.org/10.1525/bio.2010.60.11.4
Glancy, J., Sone, J.V. and Wilson, S.P. (2016) How Self-Organization Can Guide Evolution. Royal Society Open Science, 3, Article ID: 160553. https://doi.org/10.1098/rsos.160553
Kurakin, A. (2011) The Self-Organizing Fractal Theory as a Universal Discovery Method: The Phenomenon of Life. Theoretical Biology and Medical Modelling, 8, 4. https://doi.org/10.1186/1742-4682-8-4
Scheffer, M. and Nes, V.E.H. (2006) Self-Organized Similarity, the Evolutionary Emergence of Groups of Similar Species. PNAS, 103, 6230-6235. https://doi.org/10.1073/pnas.0508024103
Deering, W. and West, B.J. (1992) Fractal Physiology. IEEE Engineering in Medicine and Biology, 11, 40-46. https://doi.org/10.1109/51.139035
West, B.J. (1990) Fractal Physiology and Chaos in Medicine. World Scientific, Singapore, London.
Bassingthwaighte, J.B., Leibovitch, L.S. and West, B.J. (1994) Fractal Physiology. Oxford Univ. Press, New York, Oxford. https://doi.org/10.1007/978-1-4614-7572-9
Li, W. (1989) Spatial 1/f Spectra in Open Dynamical Systems. Europhysics Letters, 10, 395-400. https://doi.org/10.1209/0295-5075/10/5/001
Schlesinger, M.S. (1987) Fractal Time and 1/f Noise in Complex Systems. Annals of the New York Academy of Sciences, 504, 214-228. https://doi.org/10.1111/j.1749-6632.1987.tb48734.x
Bak, P., Tang, C. and Wieserfeld, K. (1988) Self-Organized Criticality. Physical Review A, 38, 364. https://doi.org/10.1103/PhysRevA.38.364
Musha, T. and Sawada, Y. (1994) Physics of the Living State. IOS Press, Amsterdam.
von Bertalanffy, L. and Pirozynski, W.J. (1953) Tissue Respiration, Growth, and Basal Metabolism. Biology Bulletin, 105, 240-256. https://doi.org/10.2307/1538640
von Bertalanffy, L. (1968) General System Theory: Foundations, Development, Applications. George Braziller, New York.
Labra, F.A., Marquet, P.A. and Bozinovic, F. (2007) Scaling Metabolic Rate Fluctuations. PNAS, 104, 10900-10903. https://doi.org/10.1073/pnas.0704108104
Goldberger, A.L., Amaral, L.A., Hausdorff, J.M., et al. (2002) Fractal Dynamics in Physiology: Alterations with Disease and Aging. PNAS Colloquium, 99, 2466-2472. https://doi.org/10.1073/pnas.012579499
Szendro, P., Vincze, G. and Szasz, A. (2001) Pink Noise Behaviour of the Bio-Systems. European Biophysics Journal, 30, 227-231. https://doi.org/10.1007/s002490100143
Calder III, W.A. (1984) Size, Function and Life History. Dover Publications Inc., Mineola, New York.
West, G.B. and Born, J.H. (2000) Scaling in Biology. Oxford University Press, Oxford.
West, G.B., Woodruf, W.H. and Born, J.H. (2002) Allometric Scaling of Metabolic Rate from Molecules and Mitochondria to Cells and Mammals. Proceedings of the National Academy of Sciences of the United States of America, 99, 2473-2478. https://doi.org/10.1073/pnas.012579799
West, G.B. and Brown, J.H. (2005) The Origin of Allometric Scaling Laws in Biology from Genomes to Ecosystems: Towards a Quantitative Unifying Theory of Biological Structure and Organization. Journal of Experimental Biology, 208, 1575-1592. https://doi.org/10.1242/jeb.01589
Lane, N. (2006) Mitochondria: Key to Complexity. In: Martin, W., Ed., Origins of Mitochondria and Hydrogenosomes, Springer, Heidelberg, Chapter 2. https://doi.org/10.1007/978-3-540-38502-8_2
Brown, J.H., West, G.B. and Enquis, B.J. (2005) Yes, West, Brown and Enquist’s Model of Allometric Scaling Is Both Mathematically Correct and Biologically Relevant. Functional Ecology, 19, 735-738. https://doi.org/10.1111/j.1365-2435.2005.01022.x
Glazier, D.S. (2014) Metabolic Scaling in Complex Living Systems. Systems, 2, 451-540. https://doi.org/10.3390/systems2040451
Kolmogorov, N.N. (1937) On the Statistical Theory of the Crystallization of Metals. Bull Acad Sci UssR Math Ser, 1, 355-359.
Joshnson, W.A. and Mehl, P.A. (1939) Reaction Kinetics in Processes of Nucleation and Growth. Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers, 135, 416.
Avrami, M.A. (1939) Kinetics of Phase Change, Parts: I-II-III. The Journal of Chemical Physics, 7, 1103.
Levine, L.E., Narayan, K.L. and Kelton, K.F. (1997) Finite Size Corrections for the Johnson-Mehl-Avrami-Kolmogorov Equation. Journal of Materials Research, 12, 124-131. https://doi.org/10.1557/JMR.1997.0020
Fanfoni, M., Persichetti, L. and Tomellini, M. (2012) Order and Randomness in Kolmogorov-Johnson-Mehl-Avrami-Type Phase Transitions. Journal of Physics: Condensed Matter, 24, Article ID: 355002. https://doi.org/10.1088/0953-8984/24/35/355002
Cope, F.W. (1977) Detection of Phase Transitions and Cooperative Interactions by Avrami Analysis of Sigmoid Biological Time Curves for Muscle, Nerve, Growth, Firefly, and Infrared Phosphorescence, of Green Leaves, Melanin and Cytochrome C. Physiological Chemistry and Physics, 9, 443-459.
Cope, F.W. (1977) Solid State Physical Replacement of Hodgkin-Huxley Theory. Phase Transformation Kinetics of Axonal Potassium Conductance. Physiological Chemistry and Physics, 9, 155-160.
Zhang, H. (2005) Reconstructing DNA Replication Kinetics from Small DNA Fragments. Ms. Thesis, Simon Fraser University, Burnaby. https://doi.org/10.1103/PhysRevE.73.051903
Cope, F.W. (1977) The Kinetics of Biological Phase Transitions Manifested by Sigmoid Time Curves: A Review of Approaches. Physiological Chemistry and Physics, 8, 519-527.
Cope, F.W. (1980) Avrami Analysis of Electrical Switching in Hydrated Melanin Suggest Dependence on a Phase Transition. Physiological Chemistry and Physics, 12, 537-538.
Suckjoon, J. and Bechhoefer, J. (2005) Nucleation and Growth in One Dimension. II. Application to DNA Replication Kinetics. Physical Review E, 71, Article ID: 011909. https://doi.org/10.1103/PhysRevE.71.011909
Zhang, H.Y. (2005) Reconstructing DNA Replication Kinetics from Small DNA Fragments. Simon Fraser University, Burnaby.
Szasz, O., Szigeti, G.P. and Szasz, A. (2017) On the Self-Similarity in Biological Processes. Open Journal of Biophysics, 7, 183-196. https://doi.org/10.4236/ojbiphy.2017.74014
Binney, J.J., Dowrick, N.J., Fisher, A.J., et al. (1992) The Theory of Critical Phenomena, an Introduction to the Renormalization Group. Oxford Science Publications, Oxford.
Augis, J.A. and Bennett, J.E. (1978) Calculation of the Avrami Parameters for Heterogeneous Solid State Reactions Using a Modification of the Kissinger Method. Journal of Thermal Analysis, 13, 285-291. https://doi.org/10.1007/BF01912301
Jena, A.K. and Chaturvedi, M.C. (1992) Phase Transformations in Materials. Prentice Hall, Upper Saddle River, 247.
Aloev, V.Z., Kozlov, G.V. and Zaikov, G.E. (2004) Relationship between the Exponent of the Kolmogorov-Avrami Equation and the Fractal Dimension in the Crystallisation of Uniaxially Stretched Crosslinked Polychloroprene. Kauchuk I Rezina, No. 3, 38-39. https://doi.org/10.1177/0307174X0403101210
Hassan, M.K. and Kurths, J. (2004) Scale Invariant Fractal and Slow Dynamics in Nucleation and Growth Processes. https://arxiv.org/pdf/cond-mat/0407715.pdf
Cope, F.W. (1977) Detection of Phase Transitions and Cooperative Interactions by Avrami Analysis of Sigmoid Biological Time Curves for Muscle, Nerve, Growth, Firefly, and Infrared Phosphorescence, of Green Leaves, Melanin and Cytochrome C. Physiological Chemistry and Physics, 9, 443-459.
Cope, F.W. (1977) Solid State Physical Replacement of Hodgkin-Huxley Theory. Phase Transformation Kinetics of Axonal Potassium Conductance. Physiological Chemistry and Physics, 9, 155-160.
Cope, F.W. (1980) Avrami Analysis of Electrical Switching in Hydrated Melanin Suggest Dependence on a Phase Transition. Physiological Chemistry and Physics, 12, 537-538.
Izquierdo-Kulich, E. and Nieto-Villar, J.M. (2013) Morphogenesis and Complexity of the Tumor Patterns. In: Rubio, R.G., Ryazantsev, Y.S., Starov, V.M., Huang, G.-X., Chetverikov, A.P., Arena, P., Nepomnyashchy, A.A., Ferrus, A. and Morozov, E.G., Eds., Without Bounds: A Scientific Canvas of Nonlinearity and Complex Dynamics. Understanding Complex Systems, Springer-Verlag, Berlin Heidelberg, 657-691. https://doi.org/10.1007/978-3-642-34070-3_48
Davies, P.C.W., Demetrius, L. and Tuszynski, J.A. (2011) Cancer as a Dynamical Phase Transition. Theoretical Biology and Medical Modelling, 8, 30. https://doi.org/10.1186/1742-4682-8-30
Enderling, H., Hahnfeldt, P., Hlatky, L. and Almog, N. (2012) Systems Biology of Tumor Dormancy: Linking Biology and Mathematics on Multiple Scales to Improve Cancer Therapy. Cancer Research, 72, 2172-2175. https://doi.org/10.1158/0008-5472.CAN-11-3269
Szasz, O., Szigeti, G.P., Szasz, A. and Benyo, Z. (2018) Role of Electrical Forces in Angiogenesis. Open Journal of Biophysics, 8, 49-67. https://doi.org/10.4236/ojbiphy.2018.82005
Szasz, O., Vincze, Gy., Szigeti, Gy.P., Benyo, Z. and Szasz, A. (2018) An Allometric Approach of Tumor-Angiogenesis. Medical Hypothesis, 116, 74-78. https://doi.org/10.1016/j.mehy.2018.03.015
Szasz, O., Szigeti, G.P. and Szasz, A. (2018) The Intrinsic Self-Time of Biosystems. Open Journal of Biophysics, 9, 131-145.
González, M.M., Joa, J.A.G., Cabrales, L.E.B., Pupo, A.E.B., Schneider, B., Kondakci, S., Ciria, H.M.C., Reyes, J.B., Jarque, M.V., O’Farril Mateus, M.A., Tamara Rubio González, T.R., Brooks, S.C.A., Cáceres, J.L.H. and González, G.V.S. (2017) Is Cancer a Pure Growth Curve or Does It Follow a Kinetics of Dynamical Structural Transformation? BMC Cancer, 17, 174. https://doi.org/10.1186/s12885-017-3159-y
Izquierdo-Kulich, E.E., Alonso-Becerra and Nieto-Villar, J.M. (2011) Entropy Production Rate for Avascular Tumor Growth. Journal of Modern Physics, 2, 615. https://doi.org/10.4236/jmp.2011.226071
Betancourt-Mar, J.A., Cocho, G., Mansilla, R. and Nieto-Villar, J.M. (2017) What Can Be Learned from a Phase Transitions in Tumor Growth? Insights in Biomedicine, 2, 1.
Agus, D.B., Alexander, J.F., Arap, W., Ashili, S., Aslan, J.E., Austin, R.H., Backman, V., Bethel, K.J., et al. (2013) A Physical Sciences Network Characterization of Non-Tumorigenic and Metastatic Cells. Scientific Reports, 3, Article No. 1449. https://doi.org/10.1038/srep01449
Stehlik, M., Mrkvicka, T., Filus, J. and Filus, I. (2012) Recent Developments on Testing in Cancer Risk: A Fractal and Stochastic Geometry. Journal of Reliability and Statistical Studies, 5, 83-95.
Guiot, C., Degiorgis, P.G., Delsanto, P.P., Gabriele, P. and Deisboeck, T.S. (2003) Does Tumor Growth Follow a “Universal Law”? Journal of Theoretical Biology, 225, 147-151. https://doi.org/10.1016/S0022-5193(03)00221-2
Bose, P., Brockton, N.T., Guggisberg, K., Nakoneshny, S.C., Kornaga, E., Klimowicz, A.C., Tambasco, M. and Dort, J.C. (2015) Fractal Analysis of Nuclear Histology Integrates Tumor and Stromal Features into a Single Prognostic Factor of the Oral Cancer Microenvironment. BMC Cancer, 15, 409. https://doi.org/10.1186/s12885-015-1380-0
Sole, R.V. (2003) Phase Transitions in Unstable Cancer Cell Populations. The European Physical Journal B, 35, 117-123. https://doi.org/10.1140/epjb/e2003-00262-8
Loeb, L.A., Loeb, K.R. and Anderson, J.P. (2003) Multiple Mutations and Cancer. PNAS, 100, 776-781. https://doi.org/10.1073/pnas.0334858100
Bielas, J.H., Loeb, K.R., Rubin, B.P., True, L.D. and Loeb, L.A. (2006) Human Cancers Express a Mutator Phenotype. PNAS, 103, 18238-18242. https://doi.org/10.1073/pnas.0607057103
Neumann, J. (1928) Zur Theorie der Gesellschaftsspiele. Mathematische Annalen, 100, 295-320. (English Translation: Tucker, A.W. and Luce, R.D. (1959) On the Theory of Games of Strategy. Contributions to the Theory of Games, 4, 13-42. https://doi.org/10.1007/BF01448847
Ben-David, S., Borodin, A., Karp, R., Tardos, G. and Wigderson, A. (1994) On the Power of Randomization in On-Line Algorithms. Algorithmica, 11, 2-14. https://doi.org/10.1007/BF01294260
Si, T. (2008) Game Theory and Topological Phase Transition.
Menon, S.N., Sasidevan, V. and Sinha, S. (2008) Emergence of Cooperation as a Non-Equilibrium Transition in Noisy Spatial Games. Frontiers in Physics, 6, 34. https://doi.org/10.3389/fphy.2018.00034
Kaplan, E.L. and Meier, P. (1958) Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53, 457-481. https://doi.org/10.1080/01621459.1958.10501452
Etikan, I., Abubakar, S. and Alkassim, R. (2017) The Kaplan Meier Estimate in Survival Analysis. Biometrics & Biostatistics International Journal, 5, Article ID: 00128. https://doi.org/10.15406/bbij.2017.05.00128
Efron, B. (1967) The Two-Sample Problem with Censored Data. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 4, 831-852.
Gill, R.D. (1980) Censoring and Stochastic Integrals. Mathematical Centre Tract No. 124, Mathematisch Centrum, Amsterdam. https://doi.org/10.1111/j.1467-9574.1980.tb00692.x
Birnbaum, Z.W. and Saunders, S.C. (1969) A New Family of Life Distributions. Journal of Applied Probability, 6, 319-327. https://doi.org/10.2307/3212003
Camazine, S., Deneubourg, J.-L., Franks, N.R., Sneyd, J., Theraulaz, G. and Bonabeau, E. (2003) Self-Organization in Biological Systems. Princeton Studies in Complexity, Princeton Univ. Press, Princeton, Oxford.
May, K.A. and Solomon, J.A. (2013) Four Theorems on the Psychometric Function. PLoS ONE, 8, e74815. https://doi.org/10.1371/journal.pone.0074815
Weibull, W. (1939) The Statistical Theory of the Strength of Materials. Ingeniors Vetenskaps Academy Handlingar, 151-153, 45-55.
Weibull, W. (1951) The Statistical Distribution Function of Wide Applicability. Journal of Applied Mechanics, 8, 293-297.
Brown, W.K. and Wohletz, K.H. (1995) Derivation of the Weibull Distribution Based on Physical Principles and Its Connection to the Rosin-Rammler and Lognormal Distributions. Journal of Applied Physics, 78, 2758-2764. https://doi.org/10.1063/1.360073
Batdorf, S.B. (1978) Fundamentals of the Statistical Theory of Fracture. In: Bradt, R.C., Hasselman, D.P.H. and Lange, F.F., Eds., Fracture Mechanics of Ceramics, Vol. 3, Plenum Press, New York, 1.
Gompertz, B. (1825) On the Nature of the Function Expressive of the Law of Human Mortality and on a New Mode of Determining the Value of Life Contingencies. Philosophical Transactions of the Royal Society of London, 115, 513-585. https://doi.org/10.1098/rstl.1825.0026
Gavrilov, L.A. and Gavrilova, N.S. (2001) The Reliability Theory of Aging and Longevity. Journal of Theoretical Biology, 213, 527-545. https://doi.org/10.1006/jtbi.2001.2430
Wilson, D.L. (1994) The Analysis of Survival (Mortality) Data: Fitting Gompertz, Weibull, and Logistic Functions. Mechanisms of Ageing and Development, 74, 15-33. https://doi.org/10.1016/0047-6374(94)90095-7
Pham, H. (2008) Mortality Modeling Perspectives. In: Recent Advances in Reliability and Quality in Design, Springer Series in Reliability Engineering, Springer-Verlag, London, 509-516. https://doi.org/10.1007/978-1-84800-113-8_25
Waliszewski, P. and Konarski, J. (2003) The Gompertzian Curve Reveals Fractal Properties of Tumor Growth. Chaos, Solitons and Fractals, 16, 665-674. https://doi.org/10.1016/S0960-0779(02)00469-1
Bru, A., Albertos, S., Subiza, J.L., García-Asenjo, J.L. and Bru, I. (2003) The Universal Dynamics of Tumor Growth. Biophysical Journal, 85, 2948-2961. https://doi.org/10.1016/S0006-3495(03)74715-8
Demicheli, R., Biganzoli, E., Boracchi, P., Greco, M., Hrushesky, W.J.M. and Retsky, M.W. (2006) Allometric Scaling Law Questions the Traditional Mechanical Model for Axillary Lymph Node Involvement in Breast Cancer. Journal of Clinical Oncology, 24, 4391-4396. https://doi.org/10.1200/JCO.2006.05.5988
Oguntunde, P.E., Balogun, O.S., Okagbue, H.I. and Bishop, S.A. (2015) The Weibull-Exponential Distribution: Its Properties and Applications. Journal of Applied Sciences, 15, 1305-1311. https://doi.org/10.3923/jas.2015.1305.1311
El-Bassiouny, A.H., El-Damcese, M.A., Mustafa, A. and Eliwa, M.S. (2017) Exponentiated Generalized Weibull-Gompertz Distribution with Application in Survival Analysis. Journal of Statistics Applications & Probability, 6, 7-16. https://doi.org/10.18576/jsap/060102
Ricklefs, R.E. and Scheuerlein, A. (2002) Biological Implications of the Weibull and Gompertz Models of Aging. Journal of Gerontology: Biological Sciences, 57A, B69-B76. https://doi.org/10.1093/gerona/57.2.B69
Waliszewski, P. and Konarski, J. (2005) A Mystery of the Gompertz Function. In: Losa, G.A., Merlini, D., Nonnenmacher, T.F. and Weibel, E.R., Eds., Fractals in Biology and Medicine, Birkhäuser, Basel, 277-286. https://doi.org/10.1007/3-7643-7412-8_27
Watts, D.J. and Strogatz, S.H. (1998) Collective Dynamics of “Small-World” Networks. Nature, 393, 440-442. https://doi.org/10.1038/30918
Nijhout, H.F. (2011) Dependence of Morphometric Allometries on the Growth Kinetics of Body Parts. Journal of Theoretical Biology, 288, 35-43. https://doi.org/10.1016/j.jtbi.2011.08.008
Nijhout, H.F. and German, R.Z. (2012) Developmental Causes of Allometry: New Models and Implications for Phenotypic Plasticity and Evolution. Integrative and Comparative Biology, 52, 43-52. https://doi.org/10.1093/icb/ics068
West, G.B., Brown, J.H. and Enquist, B.J. (2001) A General Model for Ontogenetic Growth. Nature, 413, 628-631. https://doi.org/10.1038/35098076
Hajian-Tilaki, K.O., Hanley, J.A., Joseph, L. and Collet, J.-P. (1977) A Comparison of Parametric and Nonparametric Approaches to ROC Analysis of Quantitative Diagnostic Tests. Medical Decision Making, 17, 94-102. https://doi.org/10.1177/0272989X9701700111
Bex, P.J., Metha, A.B. and Makous, W. (1998) Psychophysical Evidence for a Functional Hierarchy of Motion Processing Mechanisms. Journal of the Optical Society of America A, 15, 769-777. https://doi.org/10.1364/JOSAA.15.000769
Pelli, D.G. and Farell, B. (1995) Psychophysical Methods. In: Bass, M., Van Stryland, E.W., Williams, D.R. and Wolfe, W.L., Eds., Handbook of Optics, 2nd Edition, McGraw-Hill, New York, I (29.21-29.13).
Wang, H., Wang, Z., Li, X., et al. (2011) A Robust Approach Based on Weibull Distribution for Clustering Gene Expression Data. Algorithms for Molecular Biology, 6, 14. https://doi.org/10.1186/1748-7188-6-14
Piantanell, L. (1986) A Mathematical Model of Survival Kinetics. I. Theoretical Basis. Archives of Gerontology and Geriatrics, 5, 107-118. https://doi.org/10.1016/0167-4943(86)90014-2
Economos, E.C. (1982) Rate of Aging, Rate of Dying and the Mechanism of Mortality. Archives of Gerontology and Geriatrics, 1, 3-27. https://doi.org/10.1016/0167-4943(82)90003-6
Weon, B.M. and Je, J.H. (2010) Predicting Human Lifespan Limits. Scientific Research, 2, 984-989. https://doi.org/10.4236/ns.2010.29120
Weon, B.M. and Je, J.H. (2011) Plasticity and Rectangularity in Survival Curves. Scientific Reports, 1, Article No. 104. https://doi.org/10.1038/srep00104
Pugno, N.M. (2007) A Statistical Analogy between Collapse of Solids and Death of Living Organisms: Proposal for a “Law of Life”. Medical Hypotheses, 69, 441-447. https://doi.org/10.1016/j.mehy.2006.10.067
Pugno, N.M. (2005) On the Statistical Law of Life. Department of Structural Engineering, Politecnico di Torino, Torino.
Stehlik, M., Hermann, P. and Nicolis, O. (2016) Fractal Based Cancer Modelling. REVSTAT—Statistical Journal, 14, 139-155.
Liu, S., Wang, Y., Xu, K., Wang, Z., Fan, X., Zhang, C., Li, S., Qiu, X. and Jiang, T. (2017) Relationship between Necrotic Patterns in Glioblastoma and Patient Survival: Fractal Dimension and Lacunarity Analyses Using Magnetic Resonance Imaging. Scientific Reports, 7, Article No. 8302. https://doi.org/10.1038/s41598-017-08862-6
Stehlik, M., Wartner, F. and Minarova, M. (2013) Fractal Analysis for Cancer Research: Case Study and Simulation of Fractals. Pliska Studia Mathematica Bulgarica, 22, 195-206.
Weston, C.L., Douglas, C., Craft, A.W., Lewis, I.J. and Machin, D. (2004) Establishing Long-Term Survival and Cure in Young Patients with Ewing’s Sarcoma. British Journal of Cancer, 91, 225-232. https://doi.org/10.1038/sj.bjc.6601955
Zhang, Z.-H. (2016) Parametric Regression Model for Survival Data: Weibull Regression Model as an Example. Annals of Translational Medicine, 4, 484. https://doi.org/10.21037/atm.2016.08.45
Jones, G. and Rocke, D.M. (2002) Multivariate Survival Analysis with Doubly-Censored Data: Application to the Assessment of Accutane Treatment for Fibrodysplasia Ossificans Progressive. Statistics in Medicine, 21, 2547-2562. https://doi.org/10.1002/sim.1123
Pourhoseingholi, M.A., Pourhoseingholi, A., Vahedi, M., Moghimi Dehkordi, B., Safaee, A., Ashtari, S. and Zali, M.R. (2011) Alternative for the Cox Regression Model: Using Parametric Models to Analyze the Survival of Cancer Patients. Iranian Journal of Cancer Prevention, 4, 1-9.
Pourhoseingholi, M., Hajizadeh, E., Moghimi Dehkordi, B., Safaee, A., Abadi, A. and Zali, M.R. (2007) Comparing Cox Regression and Parametric Models for Survival of Patients with Gastric Carcinoma. Asian Pacific Journal of Cancer Prevention: APJCP, 8, 412-416.
Hoyle, M.W. and Henley, W. (2011) Improved Curve Fits to Summary Survival Data: Application to Economic Evaluation of Health Technologies. BMC Medical Research Methodology, 11, 139. https://doi.org/10.1186/1471-2288-11-139
Barriga, G.D.C., Louzada-Neto, F., Ortega, E.M.M. and Cancho, V. (2010) A Bivariate Regression Model for Matched Paired Survival Data: Local Influence and Residual Analysis. Statistical Methods and Applications, 19, 477-495. https://doi.org/10.1007/s10260-010-0140-1
Andersson, T. (2007) Analyzing Time Trends in Cancer Patient Survival Using Cure Fraction Models. U.U.D.M. Project Report 4, Department of Mathematics, Uppsala University, Uppsala.
Fuller, A.F. and Griffiths, C.M. (1983) Gynecologic Oncology. M. Nijhoff, The Hague. https://doi.org/10.1007/978-1-4613-3852-9
Lambert, P.C., Thompson, J.R., Weston, C.L. and Dickman, P.W. (2007) Estimating and Modeling the Cure Fraction in Population-Based Cancer Survival Analysis. Biostatistics, 8, 576-594. https://doi.org/10.1093/biostatistics/kxl030
Cover, T.M. and Thomas, J.A. (2005) Elements of Information Theory. Wiley, Hoboken. https://doi.org/10.1002/047174882X
Szasz, A., Dani, A., Varkonyi, A. and Magyar, T. (2005) Retrospective Analysis of 1180 Oncological Patients Treated by Electro-Hyperthermia in Hungary. Strahlentherapy Onkologie (Radiation Oncology), 181, 121-122.
Szasz, A., Szasz, N. and Szasz, O. (2010) Oncothermia: Principles and Practices. Springer Science, Heidelberg, Ch. 4, 295-302. https://doi.org/10.1007/978-90-481-9498-8