An Effective Numerical Method and Its Utilization to Solution of Fractional Models Used in Bioengineering Applications
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.
KeywordsFractional Derivative Fractional Calculus Finite Impulse Response Fractional Differential Equation Power Series Expansion
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.
2.1. 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.
for Open image in new window .
2.2. Fractional-Order Systems
There are several possible interpretations of the fractional-order systems. Here are mentioned three of them.
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 .
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.
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
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 . 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].
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 .
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 .
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 .
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 .
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 .
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
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.
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.
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.
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.
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  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 : 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.
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
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 . 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.
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).
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.
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 .
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.
- (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:(4.1)
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.
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 , Podlubny's matrix approach [21, 22], quadrature formula approach , multistep method , and frequency (Oustaloup's) method , but it has some restrictions, especially for the fractional nonlinear models . 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 .
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.
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