Advertisement

Advances in Difference Equations

, 2011:652789 | Cite as

An Effective Numerical Method and Its Utilization to Solution of Fractional Models Used in Bioengineering Applications

  • Ivo Petráš
Open Access
Research Article
Part of the following topical collections:
  1. Fractional Models and their Applications

Abstract

This paper deals with the fractional-order linear and nonlinear models used in bioengineering applications and an effective method for their numerical solution. The proposed method is based on the power series expansion of a generating function. Numerical solution is in the form of the difference equation, which can be simply applied in the Matlab/Simulink to simulate the dynamics of system. Several illustrative examples are presented, which can be widely used in bioengineering as well as in the other disciplines, where the fractional calculus is often used.

Keywords

Fractional Derivative Fractional Calculus Finite Impulse Response Fractional Differential Equation Power Series Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Recently, fractional calculus has played an increasing role in modeling complex phenomena in the fields of physics, chemistry, biology, and engineering (e.g., [1, 2, 3, 4]). The main characteristic of fractional derivatives, or more precisely derivatives of positive real order, is so called the "memory effect". It is well known that the state of many systems (biological, electrochemical, viscoelastic, etc.) at a given time depends on their configuration at previous times. The fractional derivative takes into account this history in its definition as a convolution with a function whose amplitude decays at earlier times as a power-law. Thus, the fractional derivative is natural to use when modeling biological systems in various bioengineering applications.

In this paper, we offer applications of fractional calculus in bioengineering, which are described by the fractional differential equations. Paper is organized as follows: basic definitions of fractional calculus, fractional-order systems and numerical method are presented first in Section 2. Three representative fractional-order models often used in bioengineering are described and numerically solved in Section 3. In Section 4 the questions of numerical analysis are discussed. Some conclusion remarks are mentioned in Section 5.

2. Preliminaries

2.1. Fractional Calculus

Fractional calculus is a topic in mathematics that is more than 300 years old. The idea of fractional calculus was suggested early in the development of regular (integer-order) calculus, with the first literature reference being associated with a letter, from Leibniz to L'Hospital in 1695. In this letter the half-order derivative was first mentioned.

There are several definitions of the fractional derivative/integral as a one common operator known as "differintegral" (see, e.g., [4, 5, 6]):

The Riemann-Liouville (RL) definition is given as

for Open image in new window and where Open image in new window is the Gamma function.

The Caputo's definition of fractional derivatives can be written as

for Open image in new window .

If we consider Open image in new window , where Open image in new window is a real constant and Open image in new window means the integer part, we can write the Grünwald-Letnikov (GL) definition as

where Open image in new window and Open image in new window are the bounds of operation for   Open image in new window . Usually, we assume lower boundary Open image in new window .

For many engineering applications the Laplace transform methods are often used. The Laplace transform of the RL, the GL, and Caputo's fractional derivative/integral, under zero initial conditions for order Open image in new window is given by [5]:
A function, which plays a very important role in the fractional calculus, was in fact introduced by Humbert and Agarwal [7]. It is a two-parameter function of the Mittag-Leffler type defined as [4]:

Note that fractional calculus holds many important and interesting properties, which were described for instance in [3, 4, 5].

2.2. Fractional-Order Systems

There are several possible interpretations of the fractional-order systems. Here are mentioned three of them.

A general fractional-order linear system can be described by a fractional differential equation of the form [4]:
where Open image in new window denotes the Riemann-Liouville, Caputo's or Grünwald-Letnikov fractional derivative depending on initial conditions and their physical meaning or by the corresponding transfer function of incommensurate real orders of the following form [4]:

where Open image in new window ,   Open image in new window are constants, and Open image in new window , Open image in new window are arbitrary real or rational numbers and without loss of generality they can be arranged as Open image in new window , and Open image in new window .

The fractional-order linear time-invariant system can also be represented by the following state-space model:

where Open image in new window , Open image in new window , and Open image in new window are the state, input and output vectors of the system and Open image in new window , Open image in new window , Open image in new window , and Open image in new window are the fractional orders. If Open image in new window , system (2.8) is called a commensurate-order system, otherwise it is an incommensurate-order system.

In this paper, we will also consider the general incommensurate fractional-order nonlinear system represented as follows:
where Open image in new window are nonlinear functions and Open image in new window are initial conditions. The vector representation of (2.9) is:

where Open image in new window for Open image in new window , Open image in new window and Open image in new window .

The equilibrium points of system (2.10) are calculated via solving the following equation

and we suppose that Open image in new window is an equilibrium point of the fractional-order nonlinear system (2.10).

2.3. Discrete Time Approximation of Fractional Calculus: Numerical Method

In general, if a function Open image in new window is approximated by a grid function, Open image in new window , where Open image in new window is the grid size, the approximation for its fractional derivative of order Open image in new window can be expressed as [8]:

where Open image in new window is the backward shift operator and Open image in new window is a generating function. This generating function and its expansion determine both the form of the approximation and the coefficients [9]. In this way, the discretization of continuous fractional-order differentiator/integrator Open image in new window Open image in new window can be expressed as Open image in new window . It is known that the forward difference rule is not suitable for applications to causal problems [8, 9].

As a generating function, Open image in new window can be used in generally the following formula [10]:

where Open image in new window and Open image in new window are denoted the gain and phase tuning parameters, respectively, and Open image in new window is sampling period. For example, when Open image in new window and Open image in new window , the generating function (2.13) becomes the forward Euler, the Tustin, the Al-Alaoui, the backward Euler, the implicit Adams rules, respectively. In this sense the generating formula can be tuned more precisely.

The expansion of the generating functions can be done by power series expansion (PSE). It is very important to note that PSE scheme leads to approximations in the form of polynomials of degree Open image in new window , that is, the discretized fractional order derivative is in the form of finite impulse response (FIR) filters, which have only zeros [11].

In this paper, for directly discretizing Open image in new window , Open image in new window , we will concentrate on the FIR form of discretization where as a generating function we will adopt a backward Euler rule. The mentioned operator, obtained from (2.13) for Open image in new window , raised to power Open image in new window , has the form
Then, the resulting transfer function, approximating the fractional-order operators, can be obtained by applying the relationship [12]:

where Open image in new window is the Open image in new window transform of the output sequence Open image in new window , Open image in new window is the Open image in new window transform of the input sequence Open image in new window , and Open image in new window denotes the expression, which results from the power series expansion of the function Open image in new window .

Doing so gives [13]:

where Open image in new window denotes the discrete equivalent of the fractional-order operator, considered as processes, and Open image in new window is the polynomial with degree Open image in new window of variable Open image in new window .

By using the short memory principle [4], the discrete equivalent of the fractional-order integrodifferential operator, Open image in new window , is given by
where Open image in new window is the memory length and Open image in new window are binomial coefficients Open image in new window where [14]
For practical numerical calculation of the fractional derivative and integral we can derive the formula from relation (2.17), where the sampling period Open image in new window is in numerical evaluation replaced by the time step of calculation Open image in new window , then we get

where Open image in new window for Open image in new window or Open image in new window for Open image in new window in the relation (2.19). By using a relation (2.14) we obtained a first-order approximation Open image in new window of the fractional derivative of order Open image in new window . Another possibility for the approximation is use, the trapezoidal rule, that is, the use of the generating function (2.13) for Open image in new window and then the PSE, which is convergent of order 2. Other forms of generation functions for higher-order approximation of the fractional order derivative Open image in new window are presented in [9].

Obviously, for this simplification we pay a penalty in the form of some inaccuracy. If Open image in new window , we can easily establish the following estimate for determining the memory length Open image in new window , providing the required accuracy Open image in new window [4]:

An evaluation of the short memory effect and convergence relation of the error between short and long memory were clearly described and also proved in [4].

For general numerical solution of the fractional differential equation, let us consider the following initial value problem
with initial conditions Open image in new window , where Open image in new window . Using approximation (2.19), we obtain the numerical solution, which can be expressed as

where Open image in new window . For the memory term expressed by sum, a "short memory" principle can be used or without using "short memory" principle, we put Open image in new window for all Open image in new window in (2.22).

3. Fractional-Order Models in Bioengineering Applications

There are many fractional-order models, which were already used in bioengineering applications as for example [3, 4, 15]: model of neuron, bioelectrode model, model of respiratory mechanics, compartmental model of pharmacokinetics, and so forth, In this section we mention and describe only three of them, namely model of the cells, nuclear magnetic resonance (NMR) model, and Lotka-Volterra (parasite-host or predator-prey) model.

3.1. Fractional-Order Viscoelastic Models of Cells

Cells have an essential biological roles and often change shape, attach and detach from surface, and sometimes divide. Such activities require the deformation in response to local stress. The rheological behavior of these cells can be modeled with the following fractional differential equation [3]:

where Open image in new window is stress, Open image in new window is strain, Open image in new window is the static elastic modulus, Open image in new window is fractional relaxation time constant, and Open image in new window is the viscosity.

If we apply the Laplace transform to system (3.1), assuming that the initial conditions are all zeros, we obtain
As it was mentioned in [3], the parameter Open image in new window can be neglected. For a step function Open image in new window in applied stress, Open image in new window , the creep response can be written as
The inverse Laplace transform of this expression can be written by using a Laplace transform of the Mittag-Leffler function [4]:
and we obtain an analytical solution in the form
For numerical solution of the fractional differential equation (3.1) for Open image in new window , we can use relations (2.18) and (2.19). The resulting difference equation has the form

where Open image in new window for Open image in new window , where Open image in new window and Open image in new window is time step of calculation, and Open image in new window is obtained from initial condition, for example, Open image in new window for zero initial condition.

Let us assume the following model parameters: Open image in new window , zero initial condition, Open image in new window  sec, Open image in new window , and Open image in new window .

Comparison of the analytical solution (3.5) and the numerical solution (3.6) of the fractional differential equation (3.1) for the parameters Open image in new window , Open image in new window , zero initial condition, Open image in new window  sec, Open image in new window , and Open image in new window is depicted in Figure 1.
Figure 1

Comparison of analytical and numerical solutions of fractional-order viscoelastic models of cell (3. 3) for simulation time 5 sec, step Open image in new window , and Open image in new window in (3.6).

As we can observe in Figure 1, the numerical solution fits the analytical solution and we can say that both solutions are consistent.

3.2. Fractional-Order Bloch Equations in NMR

In physics and bioengineering, specifically in NMR or magnetic resonance imaging, the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization Open image in new window as a function of time when relaxation times are Open image in new window (spin-lattice) and Open image in new window (spin-spin). The physical basis for Open image in new window relaxation involves the protons losing their energy to the surrounding lattice, hence the name spin-lattice relaxation. Open image in new window involves the loss of phase coherence between the protons processing in the transverse plane. Different tissues in the body have different values of Open image in new window and Open image in new window . The values depend on the strength of the magnetic field.

Now, we consider the fractional-order Bloch equations, where integer-order derivatives are replaced by fractional-order ones. Mathematical description of the fractional-order system with Caputo's derivatives is expressed as [16]

where Open image in new window , Open image in new window , and Open image in new window are the derivative orders. Here, Open image in new window , Open image in new window , and Open image in new window have the units of Open image in new window to maintain a consistent set of units for the magnetization.

Numerical solution of Bloch equations (3.7) was obtained by using the relationship (2.22), which leads to solution in the form [17]:

where Open image in new window is the simulation time, Open image in new window , for Open image in new window , and ( Open image in new window , Open image in new window , Open image in new window ) is the start point (initial conditions).

Comparison of the proposed numerical solution (3.8) with an analytical solution has been done in [17] and obtained results show a good consistency of both solutions. In aforementioned work the Matlab function and the Matlab/Simulink model for solution of the fractional-order Bloch equations (3.7) have also been created, which can be widely used for simulations with various parameters Open image in new window , Open image in new window , Open image in new window , and Open image in new window for desired simulation time Open image in new window and initial conditions ( Open image in new window , Open image in new window , Open image in new window ).

Let us consider the following parameters for tissue—gray matter of brain—for a magnetic field strength of 1.5 T from [18]: Open image in new window , Open image in new window , Open image in new window , equilibrium Open image in new window , orders Open image in new window , and Open image in new window , respectively.

Numerical solution (state space trajectory) of the fractional-order Bloch equations (3.7) with parameters: Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and initial conditions Open image in new window , Open image in new window , Open image in new window obtained by equations (3.8) for Open image in new window , Open image in new window , and Open image in new window is depicted in Figure 2. State space trajectory for the integer-order Bloch equations with the same parameters is depicted in Figure 3.
Figure 2

Numerical solutions of fractional-order ( Open image in new window ) Bloch (3. 7) in state space for simulation time 1 sec Open image in new window , and Open image in new window in (3.8).

Figure 3

Numerical solutions of integer-order ( Open image in new window ) Bloch equations (3.7) in state space for simulation time Open image in new window sec, Open image in new window , and Open image in new window in (3.8).

We can observe in both figures that fractional orders in the Bloch equations provide expanded model with different behavior for describing a more general NMR, which can find applications in complex materials exhibiting memory.

3.3. Fractional-Order Lotka-Volterra System

The fractional-order Lotka-Volterra (or fractional-order predator-prey model or parasite-host) system was proposed and described as [19]:

where Open image in new window are prey and predator densities, respectively, and all constants Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window are positive. For Open image in new window and Open image in new window we obtain a well-known model proposed by Alfred Lotka in 1910 and independently by Vito Volterra in 1926.

The stability analysis and numerical solutions of such kind of system have been already studied in [19]. There are two equilibria, when the system (3.9) is solved for Open image in new window and Open image in new window . The above system of equations yields to Open image in new window and Open image in new window if Open image in new window . The stability of the equilibrium point Open image in new window is of importance. If it were stable, nonzero populations might be attracted towards it. However, as the fixed point at the origin is a saddle point, and hence unstable, we find that the extinction of both species is difficult in the model. The second fixed point Open image in new window is not hyperbolic, so no conclusions can be drawn from the linear analysis. However, the system admits a constant of motion and the level curves are closed trajectories surrounding the fixed point. Consequently, the levels of the predator and prey populations cycle and oscillate around this fixed point.

Numerical solution of the fractional-order Lotka-Volterra system (3.9) is given by using a relation (2.22) as

where Open image in new window is the simulation time, Open image in new window , for Open image in new window , and ( Open image in new window , Open image in new window ) is the start point (initial conditions).

Let us assume the following parameters of system (3.9): Open image in new window and orders Open image in new window and Open image in new window , respectively.

Numerical solution (state plane trajectory) of the fractional-order Lotka-Volterra equations (3.9) with parameters: Open image in new window , and initial conditions Open image in new window , Open image in new window obtained by equations (3.10) for Open image in new window , Open image in new window , and Open image in new window is depicted in Figure 4. State plane trajectory for the integer-order Lotka-Volterra equations with the same parameters is depicted in Figure 5.
Figure 4

Numerical solutions of fractional-order ( Open image in new window ) Lotka-Volterra equations (3.9) in state plane for simulation time 60 sec, Open image in new window , and Open image in new window in (3.10).

Figure 5

Numerical solutions of integer-order ( Open image in new window ) Lotka-Volterra equations (3.9) in state plane for simulation time 60 sec, Open image in new window , and Open image in new window in (3.10).

According to knowledge of author, there is no exact analytical solution of the fractional-order Lotka-Volterra equations, which could be compared with the numerical solution. The only possibility is to compare proposed numerical method with an approximate solution obtained via different numerical methods as for example homotopy perturbation method, variational iteration method, and so on.

4. Discussion

The proposed numerical method is also known as Euler method which is based on the Grünwald-Letnikov definition of the fractional derivative and can be used for numerical solution of the fractional differential equation even if the fractional-order derivative in differential equation is Caputo's or Riemann-Liouville type. It is based on the fact that for a wide class of functions, all three definitions of the fractional derivatives are equivalent [4].

Sometimes the Euler method is not accurate enough; it only has order one. We have to do a numerical analysis, which consists of not only the design of numerical methods, but also analysis of three main concepts.
  1. (i)

    Consistency and order. Tell us how well it approximates the solution, we can say, method is consistent if it has an order greater than 0. The method used in this article has order 1 and therefore it is consistent. Order is determined by generating function. Consistency is a necessary condition for convergence, but not sufficient.

     
  2. (ii)
    Convergence. It means whether the method approximates the solution, in other words, a numerical method is said to be convergent if the numerical solution Open image in new window approaches the exact solution Open image in new window as the time step size Open image in new window goes to 0. The method described in this article is convergent because the following condition is satisfied:

    for Open image in new window . For instance, we can observe a good result in comparison of exact solution and numerical solution shown in Figure 1. The time step was Open image in new window .

     
  3. (iii)

    Stability and stiffness. It says whether errors are damped out. For some differential equations, application of standard methods exhibit instability in the solutions, though other methods may produce stable solutions. This behavior in the equation is described as stiffness. Method described in article provides a stable solution.

     

The numerical method (2.22) proposed for the initial value problem (2.21) holds all three above-mentioned conditions and can be used for solution of linear and nonlinear fractional differential equations. Based on performed experiments, we can consider what is the optimal choice of time step Open image in new window in order to get maximum accuracy in the approximated solution for minimum computational cost. We have used the time steps Open image in new window , Open image in new window , and 0.0005. Numerical solutions show than we may accept the results obtained in this way. The size of the time step also depends on desired relative error in the solution.

5. Conclusions

In this paper, we presented an effective numerical method and its application to solution of linear and nonlinear models of fractional order used in bioengineering applications. For some of them, Matlab functions [15, 17, 20] were also published. Here, three illustrative examples have been presented as well. It is worth to note that some other methods are also appropriate for solution of such kind of problem, for example predictor-corrector method [19], Podlubny's matrix approach [21, 22], quadrature formula approach [23], multistep method [24], and frequency (Oustaloup's) method [6], but it has some restrictions, especially for the fractional nonlinear models [25]. In further work, it is necessary to improve this method with proper mathematical analysis and exact determination of the time step size Open image in new window .

Notes

Acknowledgment

This work was supported in part by the Slovak Grant Agency for Science under Grants VEGA: 1/0390/10, 1/0497/11, 1/0746/11, Grants APVV-0040-07 and SK-PL-0052-09.

References

  1. 1.
    Caponetto R, Dongola G, Fortuna L, Petráš I: Fractional Order Systems: Modeling and Control Applications. World Scientific, Singapore; 2010.Google Scholar
  2. 2.
    Hilfer R: Applications of Fractional Calculus in Physics. World Scientific, River Edge, NJ, USA; 2000:viii+463.CrossRefMATHGoogle Scholar
  3. 3.
    Magin RL: Fractional Calculus in Bioengineering. Begell House; 2006.Google Scholar
  4. 4.
    Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar
  5. 5.
    Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York, NY, USA; 1974:xiii+234.MATHGoogle Scholar
  6. 6.
    Oustaloup A: La Dérivation non Entière. Hermes, Paris, France; 1995.MATHGoogle Scholar
  7. 7.
    Humbert P, Agarwal RP: Sur la fonction de Mittag-Leffler et quelques-unes de ses généralisations. Bulletin des Sciences Mathématiques 1953, 77: 180-185.MathSciNetMATHGoogle Scholar
  8. 8.
    Gorenflo R: Fractional calculus: some numerical methods. In Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures. Volume 378. International Centre for Mechanical Sciences, Udine, Italy; 1996:277-290.Google Scholar
  9. 9.
    Lubich Ch: Discretized fractional calculus. SIAM Journal on Mathematical Analysis 1986,17(3):704-719. 10.1137/0517050MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Barbosa RS, Tenreiro Machado JA, Silva MF: Time domain design of fractional differintegrators using least-squares. Signal Processing 2006,86(10):2567-2581. 10.1016/j.sigpro.2006.02.005CrossRefMATHGoogle Scholar
  11. 11.
    Dorčák L', Petráš I, Terpák J, Zborovjan M: Comparison of the methods for discrete approximation of the fractional-order operator. Acta Montanistica Slovaca 2003,8(4):236-239.Google Scholar
  12. 12.
    Vinagre BM, Podlubny I, Hernández A, Feliu V: Some approximations of fractional order operators used in control theory and applications. Fractional Calculus & Applied Analysis 2000,3(3):231-248.MathSciNetMATHGoogle Scholar
  13. 13.
    Vinagre BM, Chen YQ, Petráš I: Two direct Tustin discretization methods for fractional-order differentiator/integrator. Journal of the Franklin Institute 2003,340(5):349-362. 10.1016/j.jfranklin.2003.08.001MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dorčák L': Numerical Models for the Simulation of the Fractional-Order Control Systems. The Academy of Science, Institute for Experimental Physics, Košice, Slovakia; 1994.Google Scholar
  15. 15.
    Petráš I, Magin RL: Numerical solution of two compartmental biological system model. In Proceedings of the 4th IFAC Workshop on Fractional Differentiation and Its Applications, October 2010, Badajoz, Spain. University of Extremadura;Google Scholar
  16. 16.
    Magin R, Feng X, Baleanu D: Solving the fractional order Bloch equation. Concepts in Magnetic Resonance A 2009,34(1):16-23.CrossRefGoogle Scholar
  17. 17.
    Petráš I: Modeling and numerical analysis of fractional-order Bloch equations. Computers and Mathematics with Applications 2011,61(2):341-356. 10.1016/j.camwa.2010.11.009MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Webb A: Introduction to Biomedical Imaging. John Wiley & Sons, New York, NY, USA; 2003.Google Scholar
  19. 19.
    Ahmed E, El-Sayed AMA, El-Saka HAA: Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. Journal of Mathematical Analysis and Applications 2007,325(1):542-553. 10.1016/j.jmaa.2006.01.087MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Petráš I: Fractional Order Chaotic Systems (Matlab toolbox). MathWorks, Inc., FileExchange, 2010, http://www.mathworks.com/matlabcentral/fileexchange/27336
  21. 21.
    Podlubny I: Matrix approach to discrete fractional calculus. Fractional Calculus & Applied Analysis 2000,3(4):359-386.MathSciNetMATHGoogle Scholar
  22. 22.
    Podlubny I, Chechkin A, Skovranek T, Chen Y, Vinagre Jara BM: Matrix approach to discrete fractional calculus. II. Partial fractional differential equations. Journal of Computational Physics 2009,228(8):3137-3153. 10.1016/j.jcp.2009.01.014MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Diethelm K: An algorithm for the numerical solution of differential equations of fractional order. Electronic Transactions on Numerical Analysis 1997, 5: 1-6.MathSciNetMATHGoogle Scholar
  24. 24.
    Lubich Ch: Fractional linear multistep methods for Abel-Volterra integral equations of the first kind. IMA Journal of Numerical Analysis 1987,7(1):97-106. 10.1093/imanum/7.1.97MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Tavazoei MS, Haeri M: Limitations of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Analysis: Theory, Methods & Applications 2008,69(4):1299-1320. 10.1016/j.na.2007.06.030MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Ivo Petráš. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Institute of Control and Informatization of Production Processes, Faculty of BERGTechnical University of KošiceKošiceSlovakia

Personalised recommendations