AIMS Mathematics, 2017, 2(4): 692-705. doi: 10.3934/Math.2017.4.692

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Fractional calculus and the ESR test

1 Department of Applied Mathematics, Imecc–Unicamp, Sérgio Buarque de Holanda, 651 13083–859, Campinas, SP, Brazil
2 Department of Medical Genetics, School of Medical Sciences–Unicamp, 13083–887, Campinas, SP, Brazil

We consider the partial differential equation of a mathematical model proposed by Sharmaet al. [1] to describe the concentration of nutrients in blood, a factor which influences erythrocytesedimentation rate. Introducing in it a fractional derivative in the Caputo sense, we create a new, timefractionalmathematical model which contains, as a particular case, the original model. We obtainan analytic solution of this time-fractional partial differential equation in terms of Mittag-Leffler andWright functions and to show that our model is more realistic than the Sharma model.
  Article Metrics


1. G. C. Sharma, M. Jain, R. N. Saral, A mathematical model for concentration of blood affecting erythrocyte sedimentation, Comput. Biol. Med. 26 (1996), 1–7.    

2. E. Kucharz, The forgotten contribution of dr. Edmund Faustyn Biernacki (1866-1911) to the discovery of the erythrocyte sedimentation rate, J. Lab. Clin. Med. 112 (1988), 279–280.

3. E. J. Kucharz, Edmund Biernacki and erythrocyte sedimentation rate, J. Lab. Clin. Med. 329 (1987), 696.    

4. A. Grzybowski, J. J. Sak, Who discovered the erythrocyte sedimentation rate? J. Rheumatol. 38 (2011), 1521–1522.    

5. E. Biernacki, Die spontane blutsedimentirung als eine wissenschaftliche und praktischklinische untersuchungsmethode? Deut. Med. Wochenschr., Georg Thieme Verlag, Stuttgart, 23 (1897), 769–772.    

6. E. Biernacki, Samoistna sedymentacja krwi, jako naukowa i praktyczno-kliniczna metoda badania (spontaneous sedimentation of red blood cells in clinical practice), Gazeta Lekarska, 36 (1897), 962–968.

7. A. Westergren, Studies of the suspension stability of the blood in pulmonary tuberculosis1, Acta Medica Scandinavica, Wiley Online Library, 54 (2009), 247–282.    

8. A. Westergren, The technique of the red cell sedimentation reaction, Am. Rev. Tuberc, 14 (1926), 94–101.

9. R. Fahraeus, The suspension-stability of blood, Acta Med. Scand. 55 (1921), 1–228.

10. R. Fahraeus, The suspension stability of the blood, Physiol. rev. 9 (1929), 241–274.

11. M. Brigden, The erythrocyte sedimentation rate: still a helpful test when used judiciously, Postgrad. Med. Taylor and Francis, 103 (1998), 257–274.    

12. N. Van den Broek, E. Letsky, Pregnancy and the erythrocyte sedimentation rate, Brit. J. Obstet. Gynaec. Elsevier, 108 (2001), 1164–1167.

13. J. S. Olshaker, D. A. Jerrard, Pregnancy and the erythrocyte sedimentation rate, J. Emerg. Med. Elsevier, 108 (1997), 869–874.

14. S. E. Bedell, B. T. Bush, Erythrocyte sedimentation rate. from folklore to facts, Am. J. Med. Elsevier, 78 (1985), 1001–1009.    

15. M. Morris, F. Davey, Basic examination of blood, Clinical diagnosis and management by laboratory methods, WB Saunders Company Philadelphia, 20 (2001), 479–519.

16. P. Chaturani, S. Narasimbham, R. Puniyani, et al. A comparative study of erythrocyte sedimentation rate of hypertension and normal controls, In: Physiol. Fluid Dynamics II: Tata McGraw Hill New Delhi, 20 (1987), 265–280.

17. I. Talstad, P. Scheie, H. Dalen, J. Roli, Influence of plasma proteins on erythrocyte morphology and sedimentation, Scand. J. Haematol. Wiley Online Library, 31 (1983), 478–484.    

18. S. Nayha, Normal variation in erythrocyte sedimentation rate in males over 50 years old, Scand. J. Prim. Health, Taylor and Francis, 5 (1987), 5–8.    

19. International Committee for Standardization in Haematology, for Standardization in. Reference method for the erythrocyte sedimentation rate (ESR) test on human blood, Brit. J. Haematol. 24 (1973), 671–673.

20. International Committee for Standardization in Haematology, Recommendations for measurement of erythrocyte sedimentation rate, J. Clin. Pathol. 46 (1993), 198–203.

21. K. V. Boroviczeny, L. Bottiger, B. Bull, et al. Recommendation for measurement of erythrocyte sedimentation rate of human blood: International Committee for Standardization in Haematology, Am. J. Clin. Pathol. The Oxford University Press, 68 (1977), 505–507.

22. B. S. Bull, G. Brecher, An evaluation of the relative merits of the wintrobe and westergren sedimentation methods, including hematocrit correction, Am. J. Clin. Pathol. 62 (1974), 502–510.    

23. J. Whelan, C. R. Huang, A. L. Copley, Concentration profiles in erythrocyte sedimentation in human whole blood, Biorheology, 7 (1971), 205–212.    

24. C. R. Huang, J. Whelan, H. H. Wang, et al. A mathematical model of sedimentation analysis applied to human whole blood, Biorheology, 8 (1971), 157–163.    

25. W. K. Sartory, Prediction of concentration profiles during erythrocyte sedimentation by a hindered settling model, Biorheology, 11 (1974), 253–264.    

26. A. Reuben, A. Shannon, Some problems in the mathematical modelling of erythrocyte sedimentation, Math. Med. Biol. IMA, 11 (1990) 145–156.    

27. E. N. Lightfoot, Transport Phenomena in Living Systems, Wiley, New York, 1996.

28. E. K. Lenzi, L. C. Malacarne, R. S. Mendes, et al. Anomalous diffusion, nonlinear fractional fokker-planck equation and solutions, Physica A, 319 (2003), 245–252.    

29. F. Mainardi, G. Pagnini, The Wright functions as solution of the time-fractional diffusion equation, Appl. Math. Comput. 141 (2003), 51–62.    

30. R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), 1–77.    

31. J. Vanterler da C. Sousa, Erythrocyte sedimentation: A fractional model, Phd Thesis, Imecc-Unicamp, Campinas, 2017.

32. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, 198, 1999.

33. M. Caputo, F. Mainardi, Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento, 1 (1971), 161–178.    

34. R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Springer, Wien and New York, 54 (2008), 223–276.

35. L. Debnath, D. Bhatta, Integral Transforms and Their Applications, Second Edition, Chapman & Hall/CRC, Taylor and Francis Group, Boca Raton, 2007.

36. P. Dyke, An Introduction to Laplace Transforms and Fourier Series, Springer, New York, 2001.    

37. M. El-Shahed, A. Salem, An extension of wright function and its properties, J. Math. Hindawi Publishing Corporation, 2015.    

38. R. Figueiredo Camargo, E. Capelas de Oliveira, J. Vaz Jr., On anomalous diffusion and the fractional generalized langevin equation for a harmonic oscillator, J. Math. Phys. 50 (2009), 123518.    

39. J. M. Harris, J. L. Hirst, M. J. Mossinghoff, Combinatorics and Graph Theory, Springer, New York, 2008.

40. R. Gorenflo, A. A. Kilbas, F. Mainardi, et al. Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014.

41. W. N. Schirmer, H. de M. Lisboa, R. d. F. P. M. Moreira, et al. Modeling of adsorption of volatile organic compounds on the carbon nanotubes cup-stacked using the model linear driving force (in portuguese), Acta Scientiarum. Technology, 32 (2010), 159–166.

42. J. Grote, R. Susskind, P. Vaupel, Oxygen diffusivity in tumor tissue ds-carcinosarcoma under temperature conditions within the range of 2040 c, Pflgers Archiv, Springer, 72 (1977), 37–42.    

43. D. Aksnes, J. Egge, A theoretical model for nutrient uptake in phytoplankton, Marine Ecology Progress Series, Oldendorf, 70 (1991), 65–72.    

44. F. Silva Costa, E. Capelas de Oliveira, J. Vaz Jr, On a class of inverse Laplace transform, not published.

Copyright Info: © 2017, J. Vanterler da C. Sousa, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved