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Analytic Considerations and Axiomatic Approaches to the Concept Cell Death and Cell Survival Functions in Biology and Cancer Treatment

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GeNeDis 2014

Part of the book series: Advances in Experimental Medicine and Biology ((AEMB,volume 822))

Abstract

This study contains a discussion on the connection between current mathematical and biological modeling systems in response to the main research need for the development of a new mathematical theory for study of cell survival after medical treatment and cell biological behavior in general. This is a discussion of suggested future research directions and relations with interdisciplinary science. In an effort to establish the foundations for a possible framework that may be adopted to study and analyze the process of cell survival during treatment, we investigate the organic connection among an axiomatic system foundation, a predator–prey rate equation, and information theoretic signal processing. A new set theoretic approach is also introduced through the definition of cell survival units or cell survival units indicating the use of “proper classes” according to the Zermelo–Fraenkel set theory and the axiom of choice, as the mathematics appropriate for the development of biological theory of cell survival.

*Author contributed equally with all other contributors.

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Notes

  1. 1.

    Center for Cell Decision Processes (MIT), National Centers for System Biology.

    http://www.systemscenters.org/centers/center-for-cell-decision-processes/.

  2. 2.

    Kelly LaFleur, Russell’s Paradox, Department of Mathematics University of Nebraska-Lincoln, July 2011.

  3. 3.

    Cantor [8] and Schroder [4], cited in van Heijenoort [47], copied from Simmons [43].

  4. 4.

    Colyvan Mark, “Vagueness and Truth”, in H. Dyke (ed.), From Truth to Reality: New Essays in Logic and Metaphysics, Routledge, 2009,pp. 29–40.

  5. 5.

    http://vladimirmatveev.ru/mainprinciples.html.

  6. 6.

    Healthy Cells vs. Cancer Cells, A.P. John Institute for Cancer Research https://www.apjohncancerinstitute.org/frequently-asked-questions/healthy-cells-vc-cancer-cells.

References

  1. Aaron MR, Simon MK (1966) Approximation of error probability in a regenerative repeater with quantized feedback. Bell Syst Tech J 45:1845

    Article  Google Scholar 

  2. Anand P, Kunnumakara AB, Sundaram C, Harikumar KB, Tharakan ST, Lai OS, Sung BY, Aggarwal BB (2008) Cancer is a preventable disease that requires major lifestyle changes. Pharm Res 25:2097–2116

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  3. Arditi R, Berryman AA (1991) The biological-control paradox. Trends Ecol Evol 6:32–32

    Article  CAS  PubMed  Google Scholar 

  4. Arditi R, Ginzburg LR (1989) Coupling in predator prey dynamics—ratio-dependence. J Theor Biol 139:311–326

    Article  Google Scholar 

  5. Balazsi G, Van Oudenaarden A, Collins JJ (2011) Cellular decision making and biological noise: from microbes to mammals. Cell 144:910–925

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  6. Bartlett MS (1957) On theoretical models for competitive and predatory biological systems. Biometrika 44:27–42

    Article  Google Scholar 

  7. Burns RC, Lawson AD (1964) Quantized probability circuit design principles applied to linear circuits. IEEE Trans Reliab R 13:16

    Article  Google Scholar 

  8. Cantor G (1932) Gesammelte Abhandlungen mathematischen und philosophischen inhalts (Ernst Zermelo ed.). Springer, Berlin, Includes excerpts of his correspondence with Dedekind (1899)

    Google Scholar 

  9. Champagnat N (2006) A microscopic interpretation for adaptive dynamics trait substitution sequence models. Stochastic Processes Appl 116:1127–1160

    Article  Google Scholar 

  10. Chen WW, Niepel M, Sorger PK (2010) Classic and contemporary approaches to modeling biochemical reactions. Genes Dev 24:1861–1875

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  11. Chesson P (1978) Predator–prey theory and variability. Annu Rev Ecol Systemat 9:323–347

    Article  Google Scholar 

  12. Colyvan M (2009) Vagueness and truth. In: Dyke H (ed) From truth to reality: new essays in logic and metaphysics. Routledge, New York, NY

    Google Scholar 

  13. Ferriere R, Bronstein JL, Rinaldi S, Law R, Gauduchon M (2002) Cheating and the evolutionary stability of mutualisms. Proc Biol Sci 269:773–780

    Article  PubMed Central  PubMed  Google Scholar 

  14. Freedman HI, Waltman P (1984) Persistence in models of 3 interacting predator–prey populations. Math Biosci 68:213–231

    Article  Google Scholar 

  15. Goldsmith A (1943) Quantized probability. Phys Rev 64:376

    Article  Google Scholar 

  16. Graf S, Luschgy H, Pages G (2012) The local quantization behavior of absolutely continuous probabilities. Ann Probab 40:1795–1828

    Article  Google Scholar 

  17. Gray RM, Neuhoff DL (1998) Quantization. IEEE Trans Inform Theor 44:2325–2383

    Article  Google Scholar 

  18. Hâajek P, Pudlâak P (1993) Metamathematics of first-order arithmetic. Springer, Berlin

    Book  Google Scholar 

  19. Harrison GW (1979) Global stability of predator–prey interactions. J Math Biol 8:159–171

    Article  Google Scholar 

  20. Hoge H (1976) Calculation of error probability in regenerative systems with quantized feedback. Aeu Int J Electron Comm 30:130–133

    Google Scholar 

  21. Hsu SB (1981) On a resource based ecological competition model with interference. J Math Biol 12:45–52

    Article  Google Scholar 

  22. Kalyanasundaram S, Chong EKP, Shroff NB (2004) Markov decision processes with uncertain transition rates: sensitivity and max hyphen min control. Asian J Contr 6:253–269

    Article  Google Scholar 

  23. Kleinbaum DG (2005) Survival analysis; a self-learning text. Springer, New York, NY

    Google Scholar 

  24. Kossak R, Schmerl JH (2006) The structure of models of Peano arithmetic. Oxford logic guides 50. Clarendon, Oxford

    Book  Google Scholar 

  25. Kuipers B (1989) Qualitative reasoning—modeling and simulation with incomplete knowledge. Automatica 25:571–585

    Article  Google Scholar 

  26. Leslie PH (1958) A stochastic model for studying the properties of certain biological systems by numerical methods. Biometrika 45:16–31

    Article  Google Scholar 

  27. Ling G (2007) Nano-protoplasm: the ultimate unit of life. Physiol Chem Phys Med NMR 39:111–234

    CAS  PubMed  Google Scholar 

  28. Ling GN (1965) Physical state of water in living cell and model systems. Ann N Y Acad Sci 125:401

    Article  CAS  PubMed  Google Scholar 

  29. Ling GN (1988) A physical theory of the living state—application to water and solute distribution. Scanning Microsc 2:899–913

    CAS  PubMed  Google Scholar 

  30. Ling GN (1994) The new cell physiology—an outline, presented against its full historical background, beginning from the beginning. Physiol Chem Phys Med NMR 26:121–203

    CAS  PubMed  Google Scholar 

  31. link G, Ebrary INC (2004) One hundred years of Russell’s paradox mathematics, logic, philosophy. De Gruyter series in logic and its applications 6. Walter de Gruyter, New York

    Google Scholar 

  32. Lomsadze YM (1962) Relativistically invariant formulation of theory of quantized probability amplitude field. Nucl Phys 37:147

    Article  CAS  Google Scholar 

  33. Lomsadze YM, Krivsky IY, Khimitch IV (1962) A consistently relativistically invariant formulation of the quantised probability-amplitude field theory. Phys Lett 2:80–81

    Article  Google Scholar 

  34. Lomsadze YM, Krivsky IY, Khimitch IV (1962) A possible experimental determination of the universal constant-I in the quantised probability amplitude field theory. Phys Lett 2:218–219

    Article  Google Scholar 

  35. Maddika S, Ande SR, Panigrahi S, Paranjothy T, Weglarczyk K, Zuse A, Eshraghi M, Manda KD, Wiechec E, LOS M (2007) Cell survival, cell death and cell cycle pathways are interconnected: implications for cancer therapy. Drug Resist Updat 10:13–29

    Article  CAS  PubMed  Google Scholar 

  36. Masutani K (1993) Effects of survival thresholds upon one-dimensional dynamics of single-species populations. Bull Math Biol 55:1–13

    Article  Google Scholar 

  37. Mitra D, Mukherjee D, Roy AB, Ray S (1992) Permanent coexistence in a resource-based competition system. Ecol Model 60:77–85

    Article  Google Scholar 

  38. Ohta M, Hatakeyama K, Nishimura M (1979) Unified study of the multivariate joint probability function of the state variables with quantized levels for a stochastic environmental system with discrete data—theory and experiment. J Sound Vib 66:75–89

    Article  Google Scholar 

  39. Ohta M, Yamaguchi S, Katho Y (1986) Multivariate probability expression of quantized state variables for a nonstationary system and its application to an acoustic environment. J Sound Vib 104:465–482

    Article  Google Scholar 

  40. Ohta M, Yamaguchi S, Mitani Y (1984) A precise noise probability estimation from roughly observed data with quantized levels. J Acoust Soc Am 76:122–127

    Article  Google Scholar 

  41. Perthame B, Gauduchon M (2010) Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations. Math Med Biol 27:195–210

    Article  PubMed  Google Scholar 

  42. Priest G (2002) Beyond the limits of thought. Clarendon/Oxford University Press, Oxford

    Book  Google Scholar 

  43. Quine WV (1951) On the consistency of new foundations. Proc Natl Acad Sci U S A 37:538–540

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  44. Schröder E (1890–1905) Vorlesungen über die algebra der logik, 3 vols. B.G. Teubner, Leipzig, Reprints: (1966) Chelsea; (2000) Thoemmes Press

    Google Scholar 

  45. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:623–656

    Article  Google Scholar 

  46. Simmons K (2000) Sets, classes and extensions: a singularity approach to Russell’s paradox. Phil Stud 100:109–149

    Article  Google Scholar 

  47. Specker EP (1953) The Axiom of Choice in Quine New Foundations for Mathematical Logic. Proc Natl Acad Sci U S A 39:972–975

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  48. Takahashi K, Yamanaka S (2006) Induction of pluripotent stem cells from mouse embryonic and adult fibroblast cultures by defined factors. Cell 126:663–676

    Article  CAS  PubMed  Google Scholar 

  49. TYSON JJ, BAUMANN WT, CHEN C (2011) Dynamic modelling of oestrogen signalling and cell fate in breast cancer cells. Nat Rev Cancer 11(7):523–532

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  50. van Heijenoort J (1967) From Frege to Gödel; a source book in mathematical logic, 1879–1931. Harvard University Press, Cambridge

    Google Scholar 

  51. von Neumann J (1997) An axiomatization of set theory. Aut Aut 280:107–123

    Google Scholar 

  52. Vopenka P (1979) Mathematics in the alternative set theory. B. G. Teubner, Leipzig

    Google Scholar 

  53. Wheatley N (2011) A sorites paradox in the conventional definition of amount of substance. Metrologia 48:L17–L21

    Article  Google Scholar 

  54. Yu RC, Rappaport SM (1997) A lung retention model based on Michaelis-Menten-like kinetics. Environ Health Perspect 105:496–503

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  55. Zehna PW, Johnson RL (1962) Elements of set theory. College mathematics series. Allyn and Bacon, Boston, MA

    Google Scholar 

  56. Zhou HX, Rivas GN, Minton AP (2008) Macromolecular crowding and confinement: biochemical, biophysical, and potential physiological consequences. Annu Rev Biophys 37:375–397

    Article  PubMed Central  CAS  PubMed  Google Scholar 

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Author notes

    Authors and Affiliations

    1. Departments of Mathematics and Biomedical Physics, East Carolina University, 124 Austin Building, East Fifth Street, Greenville, NC, 27858-4353, USA

      Ioannis Gkigkitzis

    2. Physics and Computer Science, Wilfrid Laurier University, Science Building, Room N2078 75 University Ave. W, Waterloo, ON, N2L 3C5, Canada

      Ioannis Haranas

    3. Department of Computer Science, SIU Southern Illinois University, Carbondale, Illinois, 62902, USA

      Carlos Austerlitz

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    1. Ioannis Gkigkitzis
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    Correspondence to Ioannis Haranas .

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    1. Department of Informatics, Ionian University, Corfu, Greece

      Panayiotis Vlamos

    2. Department of Informatics, Ionian University, Corfu, Greece

      Athanasios Alexiou

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    Gkigkitzis, I., Haranas, I., Austerlitz, C. (2015). Analytic Considerations and Axiomatic Approaches to the Concept Cell Death and Cell Survival Functions in Biology and Cancer Treatment. In: Vlamos, P., Alexiou, A. (eds) GeNeDis 2014. Advances in Experimental Medicine and Biology, vol 822. Springer, Cham. https://doi.org/10.1007/978-3-319-08927-0_11

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