Library Subscription: Guest
Critical Reviews™ in Biomedical Engineering

Published 6 issues per year

ISSN Print: 0278-940X

ISSN Online: 1943-619X

SJR: 0.262 SNIP: 0.372 CiteScore™:: 2.2 H-Index: 56

Indexed in

Fractional Calculus in Bioengineering, Part3

Volume 32, Issue 3&4, 2004, 183 pages
DOI: 10.1615/CritRevBiomedEng.v32.i34.10
Get accessGet access

ABSTRACT

Table of Contents
  • Fractional Calculus Applied to Distributed Systems Characterized by Partial Differential Equations
  • Electrochemical Applications of Fractional Calculus
  • Biomedical Applications of Fractional Calculus
  • Summary
  • Annotated Bibliography
  • Acknowledgments
Abstract:

Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub-threshold nerve propagation. By expanding the range of mathematical operations to include fractional calculus, we can develop new and potentially useful functional relationships for modeling complex biological systems in a direct and rigorous manner.
In Part 2 of this review (Crit Rev Biomed Eng 2004; 32(1):105-193), fractional calculus was applied to problems in nerve stimulation, dielectric relaxation, and viscoelastic materials by extending the governing differential equations to include fractional order terms. In this third and final installment, we consider distributed systems that represent shear stress in fluids, heat transfer in uniform one-dimensional media, and subthreshold nerve depolarization. Classic electrochemical analysis and impedance spectroscopy are also reviewed from the perspective of fractional calculus, and selected examples from recent studies in neuroscience, bioelectricity, and tissue biomechanics are analyzed to illustrate the vitality of the field.

CITED BY
  1. West Bruce, Complexity, Scaling, and Fractals in Biological Signals, in Wiley Encyclopedia of Biomedical Engineering, 2006. Crossref

  2. Dokoumetzidis Aristides, Macheras Panos, Fractional kinetics in drug absorption and disposition processes, Journal of Pharmacokinetics and Pharmacodynamics, 36, 2, 2009. Crossref

  3. Herzallah Mohamed A. E., Baleanu Dumitru, Fractional-Order Variational Calculus with Generalized Boundary Conditions, Advances in Difference Equations, 2011, 2011. Crossref

  4. Verotta Davide, Fractional dynamics pharmacokinetics–pharmacodynamic models, Journal of Pharmacokinetics and Pharmacodynamics, 37, 3, 2010. Crossref

  5. Rossikhin Yuriy A., Reflections on Two Parallel Ways in the Progress of Fractional Calculus in Mechanics of Solids, Applied Mechanics Reviews, 63, 1, 2010. Crossref

  6. Rida S. Z., Arafa A. A. M., New Method for Solving Linear Fractional Differential Equations, International Journal of Differential Equations, 2011, 2011. Crossref

  7. Rossikhin Yury A., Shitikova Marina V., The analysis of the impact response of a thin plate via fractional derivative standard linear solid model, Journal of Sound and Vibration, 330, 9, 2011. Crossref

  8. Goldstein R.J., Ibele W.E., Patankar S.V., Simon T.W., Kuehn T.H., Strykowski P.J., Tamma K.K., Heberlein J.V.R., Davidson J.H., Bischof J., Kulacki F.A., Kortshagen U., Garrick S., Srinivasan V., Ghosh K., Mittal R., Heat transfer—A review of 2004 literature, International Journal of Heat and Mass Transfer, 53, 21-22, 2010. Crossref

  9. Zhao Cunlu, Yang Chun, Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels, Applied Mathematics and Computation, 211, 2, 2009. Crossref

  10. Zhang C. P., Niu J., Lin Y. Z., Numerical Solutions for the Three-Point Boundary Value Problem of Nonlinear Fractional Differential Equations, Abstract and Applied Analysis, 2012, 2012. Crossref

  11. Agrawal S. K., Srivastava M., Das S., Synchronization between fractional-order Ravinovich–Fabrikant and Lotka–Volterra systems, Nonlinear Dynamics, 69, 4, 2012. Crossref

  12. Agrawal S.K., Srivastava M., Das S., Synchronization of fractional order chaotic systems using active control method, Chaos, Solitons & Fractals, 45, 6, 2012. Crossref

  13. Badr Abdallah Ali, El-Hoety Hanan Salem, Monte-Carlo Galerkin Approximation of Fractional Stochastic Integro-Differential Equation, Mathematical Problems in Engineering, 2012, 2012. Crossref

  14. Kiryakova Virginia, Luchko Yuri, Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators, Open Physics, 11, 10, 2013. Crossref

  15. Chen Hua, Chen Wen, Zhang Binwu, Cao Haitao, Robust Synchronization of Incommensurate Fractional-Order Chaotic Systems via Second-Order Sliding Mode Technique, Journal of Applied Mathematics, 2013, 2013. Crossref

  16. Liu Yuji, Nieto Juan J., Otero-Zarraquiños Óscar, Existence Results for a Coupled System of Nonlinear Singular Fractional Differential Equations with Impulse Effects, Mathematical Problems in Engineering, 2013, 2013. Crossref

  17. Pu Yi-Fei, Zhou Ji-Liu, Siarry Patrick, Zhang Ni, Liu Yi-Guang, Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach-Based Multiscale Denoising Model for Texture Image, Abstract and Applied Analysis, 2013, 2013. Crossref

  18. Das Saptarshi, Maharatna Koushik, Fractional dynamical model for the generation of ECG like signals from filtered coupled Van-der Pol oscillators, Computer Methods and Programs in Biomedicine, 112, 3, 2013. Crossref

  19. Popović Jovan K., Pilipović Stevan, Atanacković Teodor M., Two compartmental fractional derivative model with fractional derivatives of different order, Communications in Nonlinear Science and Numerical Simulation, 18, 9, 2013. Crossref

  20. Radwan Ahmed G., Fouda M. E., Optimization of Fractional-Order RLC Filters, Circuits, Systems, and Signal Processing, 32, 5, 2013. Crossref

  21. Luchko Yuri, Fractional wave equation and damped waves, Journal of Mathematical Physics, 54, 3, 2013. Crossref

  22. Chen Jiejie, Zeng Zhigang, Jiang Ping, Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks, Neural Networks, 51, 2014. Crossref

  23. Popović Jovan K., Spasić Dragan T., Tošić Jela, Kolarović Jovanka L., Malti Rachid, Mitić Igor M., Pilipović Stevan, Atanacković Teodor M., Fractional model for pharmacokinetics of high dose methotrexate in children with acute lymphoblastic leukaemia, Communications in Nonlinear Science and Numerical Simulation, 22, 1-3, 2015. Crossref

  24. Mescia Luciano, Bia Pietro, Caratelli Diego, Fractional Derivative Based FDTD Modeling of Transient Wave Propagation in Havriliak–Negami Media, IEEE Transactions on Microwave Theory and Techniques, 62, 9, 2014. Crossref

  25. Ding Zhixia, Shen Yi, Wang Leimin, Global Mittag-Leffler synchronization of fractional-order neural networks with discontinuous activations, Neural Networks, 73, 2016. Crossref

  26. Ding Dongsheng, Qi Donglian, Wang Qiao, Fractional-order integral state space modeling and quasi state analysis via block operational matrix scheme, The 26th Chinese Control and Decision Conference (2014 CCDC), 2014. Crossref

  27. El-Sayed Ahmed M. A., Rida Saad Z., Arafa Anas A. M., On Solutions of Generalized Bacterial Chemotaxis Model in a Semi-Solid Medium, Applied Mathematics, 02, 12, 2011. Crossref

  28. Ding Zhixia, Shen Yi, Projective synchronization of nonidentical fractional-order neural networks based on sliding mode controller, Neural Networks, 76, 2016. Crossref

  29. Yadav Vijay K., Srikanth N., Das S., Dual function projective synchronization of fractional order complex chaotic systems, Optik, 127, 22, 2016. Crossref

  30. Wu Huaiqin, Wang Lifei, Wang Yu, Niu Peifeng, Fang Bolin, Global Mittag-Leffler projective synchronization for fractional-order neural networks: an LMI-based approach, Advances in Difference Equations, 2016, 1, 2016. Crossref

  31. Dokoumetzidis Aris, Fractional Pharmacokinetics, in Modeling in Biopharmaceutics, Pharmacokinetics and Pharmacodynamics, 30, 2016. Crossref

  32. Wu Huaiqin, Wang Lifei, Niu Peifeng, Wang Yu, Global projective synchronization in finite time of nonidentical fractional-order neural networks based on sliding mode control strategy, Neurocomputing, 235, 2017. Crossref

  33. Magin R.L., Ovadia M., Modeling the Cardiac Tissue Electrode Interface Using Fractional Calculus, Journal of Vibration and Control, 14, 9-10, 2008. Crossref

  34. Yadava Vijay K., Das Subir, Cafagna Donato, Nonlinear synchronization of fractional-order Lu and Qi chaotic systems, 2016 IEEE International Conference on Electronics, Circuits and Systems (ICECS), 2016. Crossref

  35. Vats Mukti, Mishra Sumit, Baghini Mahdieh, Chauhan Deepak, Srivastava Rohit, De Abhijit, Near Infrared Fluorescence Imaging in Nano-Therapeutics and Photo-Thermal Evaluation, International Journal of Molecular Sciences, 18, 5, 2017. Crossref

  36. Arshad Sadia, Baleanu Dumitru, Huang Jianfei, Tang Yifa, Al Qurashi Maysaa Mohamed, Dynamical analysis of fractional order model of immunogenic tumors, Advances in Mechanical Engineering, 8, 7, 2016. Crossref

  37. Yadav Vijay K., Kumar Rakesh, Leung A.Y.T., Das Subir, Dual phase and dual anti-phase synchronization of fractional order chaotic systems in real and complex variables with uncertainties, Chinese Journal of Physics, 57, 2019. Crossref

  38. Ren Jiapeng, Wu Huaiqin, Global Synchronization in the Finite Time for Variable-Order Fractional Neural Networks with Discontinuous Activations, Optical Memory and Neural Networks, 27, 2, 2018. Crossref

  39. Chen Wenping, Lü Shujuan, Chen Hu, Liu Haiyu, Optimal error estimate of the Legendre spectral approximation for space-fractional reaction–advection–diffusion equation, Advances in Difference Equations, 2018, 1, 2018. Crossref

  40. Hu Han-Ping, Wang Jia-Kun, Xie Fei-Long, Dynamics Analysis of a New Fractional-Order Hopfield Neural Network with Delay and Its Generalized Projective Synchronization, Entropy, 21, 1, 2018. Crossref

  41. Ullah Malik Zaka, Al-Aidarous Eman S, Baleanu Dumitru, New Aspects of Immunogenic Tumors Within Different Fractional Operators, Journal of Computational and Nonlinear Dynamics, 14, 4, 2019. Crossref

  42. Fernandez Arran, Baleanu Dumitru, Differintegration with Respect to Functions in Fractional Models Involving Mittag-Leffler Functions, SSRN Electronic Journal , 2018. Crossref

  43. Meng Fanqi, Zeng Xiaoqin, Wang Zuolei, Impulsive anti-synchronization control for fractional-order chaotic circuit with memristor, Indian Journal of Physics, 93, 9, 2019. Crossref

  44. Yadav Vijay K., Das S., Combination synchronization of fractional order n-chaotic systems using active backstepping design, Nonlinear Engineering, 8, 1, 2019. Crossref

  45. West Bruce J., Control from an Allometric Perspective, in Progress in Motor Control, 629, 2009. Crossref

  46. Radwan Ahmed G., Fouda Mohammed E., Introduction, in On the Mathematical Modeling of Memristor, Memcapacitor, and Meminductor, 26, 2015. Crossref

  47. Ullah Malik Zaka, Alzahrani Abdullah K., Baleanu Dumitru, An efficient numerical technique for a new fractional tuberculosis model with nonsingular derivative operator, Journal of Taibah University for Science, 13, 1, 2019. Crossref

  48. Patnaik Sansit, Hollkamp John P., Semperlotti Fabio, Applications of variable-order fractional operators: a review, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 476, 2234, 2020. Crossref

  49. Ben Makhlouf A., Boucenna D., Hammami M. A., Existence and Stability Results for Generalized Fractional Differential Equations, Acta Mathematica Scientia, 40, 1, 2020. Crossref

  50. Balatif O., Boujallal L., Labzai A., Rachik M., Stability Analysis of a Fractional-Order Model for Abstinence Behavior of Registration on the Electoral Lists, International Journal of Differential Equations, 2020, 2020. Crossref

  51. Birs Isabela, Muresan Cristina, A non-Newtonian impedance measurement experimental framework: modeling and control inside blood-like environments—fractional-order modeling and control of a targeted drug delivery prototype with impedance measurement capabilities, in Automated Drug Delivery in Anesthesia, 2020. Crossref

  52. Yousef A. M., Salman S. M., Backward Bifurcation in a Fractional-Order SIRS Epidemic Model with a Nonlinear Incidence Rate, International Journal of Nonlinear Sciences and Numerical Simulation, 17, 7-8, 2016. Crossref

  53. Ali Farhad, Ahmad Zubair, Arif Muhammad, Khan Ilyas, Nisar Kottakkaran Sooppy, A Time Fractional Model of Generalized Couette Flow of Couple Stress Nanofluid With Heat and Mass Transfer: Applications in Engine Oil, IEEE Access, 8, 2020. Crossref

  54. Boucenna Djalal, Ben Makhlouf Abdellatif, El‐hady El‐sayed, Hammami Mohamed Ali, Ulam‐Hyers‐Rassias stability for generalized fractional differential equations, Mathematical Methods in the Applied Sciences, 44, 13, 2021. Crossref

  55. Jiang Lei, Lai Li, Yu Tao, Luo Maokang, Collective behaviors of two coupled harmonic oscillators driven by different frequency fluctuations with fractional damping, Journal of Statistical Mechanics: Theory and Experiment, 2021, 6, 2021. Crossref

  56. Liu Lu, Zhang Shuo, Xue Dingyu, Chen Yang Quan, General robustness analysis and robust fractional‐order PD controller design for fractional‐order plants, IET Control Theory & Applications, 12, 12, 2018. Crossref

  57. Liu Weizhen, Jiang Minghui, Fei Kaifang, Dissipativity Analysis of Memristor-Based Fractional-Order Hybrid BAM Neural Networks with Time Delays, International Journal of Nonlinear Sciences and Numerical Simulation, 20, 7-8, 2019. Crossref

  58. Magin R.L., Ovadia M., MODELING THE CARDIAC TISSUE ELECTRODE INTERFACE USING FRACTIONAL CALCULUS, IFAC Proceedings Volumes, 39, 11, 2006. Crossref

  59. Khan Hassan, Kumam Poom, Khan Qasim, Khan Shahbaz, Hajira , Arshad Muhammad, Sitthithakerngkiet Kanokwan, The Solution Comparison of Time-Fractional Non-Linear Dynamical Systems by Using Different Techniques, Frontiers in Physics, 10, 2022. Crossref

  60. JAN MUHAMMAD NAEEM, ZAMAN GUL, AHMAD IMTIAZ, ALI NIGAR, NISAR KOTTAKKARAN SOOPPY, ABDEL-ATY ABDEL-HALEEM, ZAKARYA M., EXISTENCE THEORY TO A CLASS OF FRACTIONAL ORDER HYBRID DIFFERENTIAL EQUATIONS, Fractals, 30, 01, 2022. Crossref

  61. Masood Saadia, Naeem Muhammad, Ullah Roman, Mustafa Saima, Bariq Abdul, Rihan Fathalla A., Analysis of the Fractional-Order Delay Differential Equations by the Numerical Method, Complexity, 2022, 2022. Crossref

  62. Owolabi Kolade M., Robust synchronization of chaotic fractional-order systems with shifted Chebyshev spectral collocation method, Journal of Applied Analysis, 27, 2, 2021. Crossref

  63. Chen Boshan, Chen Jiejie, Global O(t−α) stability and global asymptotical periodicity for a non-autonomous fractional-order neural networks with time-varying delays, Neural Networks, 73, 2016. Crossref

Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections Prices and Subscription Policies Begell House Contact Us Language English 中文 Русский Português German French Spain