Elsevier

Applied Soft Computing

Volume 52, March 2017, Pages 605-629
Applied Soft Computing

Design of bio-inspired heuristic technique integrated with interior-point algorithm to analyze the dynamics of heartbeat model

https://doi.org/10.1016/j.asoc.2016.10.009Get rights and content

Highlights

  • Design of biological inspired heuristics to analyze the dynamics of heartbeat model.

  • The strength of ANNs, GA, and IPAs is exploited to solve the Heartbeat dynamic system.

  • Design scheme is tested effectively on variants of problems by taking different values of parameters in the system.

  • Comparison from reference solution established the correctness of the proposed scheme.

  • Results of performance indices validate consistent accuracy and convergence of the scheme.

Abstract

In this study, bio-inspired computing is presented for finding an approximate solution of governing system represents the dynamics of the HeartBeat Model (HBM) using feed-forward Artificial Neural Networks (ANNs), optimized with Genetic Algorithms (GAs) hybridized with Interiort-Point Algorithm (IPA). The modeling of the system is performed with ANNs by defining an unsupervised error function and optimization of unknown weights are carried out with GA-IPA; in which, GAs is used as an effective global search method and IPA for rapid local convergence. Design scheme is applied to study the dynamics of HBM by taking different values for perturbation factor, tension factor in the muscle fiber and the length of the muscle fiber in the diastolic state. A large number of simulations are performed for the proposed scheme to determine its effectiveness and reliability through different performance indices based on mean absolute deviation, Nash-Sutcliffe efficiency, and Thiel’s inequality coefficient.

Introduction

Bio-inspired heuristics based on universal approximation strength of Artificial Neural Networks (ANNs) is exploited to address many applications arising in different fields of applied science and technology, for instance, few potential examples in the field of bioinformatics are parameter indentification to predict the relation between chemical and electrical dynamics of Honeys [1], optimization of pulse-taking depth by the width of artery [2], robust and accurate segmentation of abdominal organs [3], crop classification of polarimetric synthetic aperture radar (SAR) images [4], patient classification and outcome prediction in End Stage Kidney Disease (ESKD) [5], solution of difficult fruit classification problem [6] and reliable indentification of oculomotor behavior of humanbeing [7] etc. Recently, Stochastic Numerical Solvers (SNSs) are developed based on artificial intelligence algorithms using feed forward ANNs optimized with local and global search methodologies for solving a variety of differential equations involving both integer and fractional derivatives [8], [9], [10], [11], [12], [13]. Few significant applications of SNSs based on these artificial intelligence techniques are oscillatory problems of nonlinear Van-der Pol type equations [14], fuel ignition dynamics of combustion theory [15], nonlinear Boundary Value Problems (BVPs) of 2-dimensional Bratu’s type equations [16], system on singular BVPs [17], nonlinear BVPs of functional differential equation of Pantograph type [18], BVPs of second order differential equations [19], nonlinear algebraic and transcendental equations of single variable [20], Initial Value Problems (IVPs) of nonlinear Painlevé type equations [21], inverse Kinematics problems [22], BVP of nonlinear Toresch’s type equations [23], thermodynamic studies of the spherical gas cloud model [24] BVPs of convergent and divergent magnetodyrodynamics (MHD) flow based on nonlinear Jeffery-Hamel type equations [25], [26], nanofluidic problems [27], nonlinear singular BVPs of Lane–Emden type equations [28], nonlinear Riccati fractional order systems [11], Legendre neural networks for ODEs [29], nonlinear Navier Stokes problems [30] and thin film flow of third grade fluids [31]. Beside the well established worth of SNSs, these solvers looks promising to solve the mathematical problems arising in bioinformatics like heart dynamics [32], [33], tumor growth [34], HIV infection model [35], heat conduction model of human head [36] etc. Keeping in view of these facts authors are motivated to make explorations and exploitations in SNSs for designing alternate, accurate and reliable computing platform for Bioinformatics problems especially arising in the study of heartbeat dynamics.

Aim of this study is to design alternate stochastic numerical solver for a reliable solution of governing mathematical relations representing the nonlinear control hypothesis to the heartbeat dynamics models. Numerical and analytical studies have been carried out by the research community for mathematical models of heart dynamics [37], [38], [39], [40], [41], however, all these procedures based on well established deterministic methodologies, while stochastic solvers are not applied to analyze the dynamics of the systems. Therefore, in this study ANNs optimized with GAs hybrid with IPA based stochastic numerical solver is designed to analyze the dynamics of HBM. Validation of the proposed results is made through comparative studies from reference numerical solutions based on Adams method for a number of scenarios for the system by taking variations in the values of the perturbation factor of the system, length of muscle fiber in the diastolic state and tension in the muscle fiber. Accuracy and convergence of the proposed scheme are evaluated through the results of statistical analysis based on a large number of independent runs of the algorithms.

The rest of the paper is organized as follows: in section two, a brief description of system model for heartbest cycle is presented. In Section three, the design methodology is presented in terms of ANNs modeling, formulation of a fitness function, and procedure adopted for learning of design parameters of ANNs. In Section four, results of numerical experimentations for the proposed scheme are presented in different scenarios of HBM. In section five, comparative study on the basis of results of statistical analysis is presented through different performance measures. Conclusion and future research directions are given in the last Section.

Section snippets

Mathematical model of heartbeat dynamics

The dynamics of Heartbeat Model (HBM) are composed of two states in a single cycle; diastole, i.e., relaxed state, and systole, i.e., contracted state. The basic characteristic for on which the mathematical model of heartbeat is developed are given below:

  • a)

    the system express an equilibrium state related to diastole;

  • b)

    The model has a threshold to produce the electrochemail wave come from the pacemaker in order to ccontract the heart into systole;

  • c)

    The model return rapidly to the equilibrium state.

Proposed design methodology

Proposed design scheme for heartbeat dynamics model consists of two parts; firstly, the design of unsupervised ANN model for governing mathematical system given in Eq. (1) while in the second phase, optimization of weights for these networks with the help of soft computing techniques based on GAs and IPAs. Necessary definitions or expression of performance measure are also introduced in this section for comparison of the results.

Numerical experimentations

The results of numerical simulations for heartbeat dynamics model based on second order nonlinear ODE which are transformed into first order system IVP of nonlinear ODEs is presented. Different problems based on values of perturbation factor, tension factor in the muscle fiber T and typical length factor of muscle xd are considered.

Based on referenes [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], ANN hybrid with global and local

Comparative studies through different performance metrics

Comparative study through different performance indices is presented in order to determine the level of the accuracy and percentage of convergence of the proposed design methodology.

The reliability and effectiveness of the design scheme is evaluated based on the values of fitness, MAD, TIC, and ENSE for hundred independent runs and results are shown in Fig. 9, Fig. 10, Fig. 11, Fig. 12, for values of fitness, MAD, TIC and ENSE, respectively, against number of runs of the algorithms. It is seen

Conclusions

Design of an effective computational intelligent algorithm for solving governing mathematical relations based on nonlinear second order differential equation represented dynamics of heart models using neural networks optimized with genetic algorithms and interior-point method. Proposed results are found in good agreement with the reference Adam numerical solutions for each case on the basis of statistical analysis through different performance measures based on MAD, TIC and ENSE values as well

Acknowledgment

The authors would like to express their appreciation to theUnited Arab Emirates University Research Affairs for the financial support of grant No. COS/IRG-09/15.

References (61)

  • M.A.Z. Raja

    Stochastic numerical techniques for solving Troesch’s problem

    Inf. Sci.

    (2014)
  • M.A.Z. Raja et al.

    Numerical treatment for nonlinear MHD Jeffery-Hamel problem using neural networks optimized with interior point algorithm

    Neurocomputing

    (2014)
  • M.A.Z. Raja et al.

    Stochastic numerical solver for nanofluidic problems containing multi-walled carbon nanotubes

    Appl. Soft Comput.

    (2016)
  • S. Mall et al.

    Chebyshev neural network based model for solving Lane–Emden type equations

    Appl. Math. Comput.

    (2014)
  • S. Mall et al.

    Application of Legendre neural network for solving ordinary differential equations

    Appl. Soft Comput.

    (2016)
  • M.A.Z. Raja et al.

    Stochastic numerical treatment for thin film flow of third grade fluid using unsupervised neural networks

    J. Taiwan Inst. Chem. Eng.

    (2015)
  • M. Suchorsky et al.

    Three oscillator model of the heartbeat generator?

    Commun. Nonlinear Sci. Numer. Simul.

    (2009)
  • S.-R.F.S.M. Gois et al.

    An analysis of heart rhythm dynamics using a three-coupled oscillator model

    Chaos, Solitons Fractals

    (2009)
  • M.A.Z. Raja et al.

    Design of bio-inspired computational intelligence technique for solving steady thin film flow of Johnson-Segalman fluid on vertical cylinder for drainage problem

    Tiawan Inst. Chem. Eng.

    (2016)
  • M. Martins et al.

    Hybridization between multi-objective genetic algorithm and support vector machine for feature selection in walker-assisted gait

    Comput. Methods Programs Biomed.

    (2014)
  • M.A.Z. Raja et al.

    Reliable numerical treatment of nonlinear singular Flierl-Petviashivili equations for unbounded domain using ANN, GAs, and SQP

    Appl. Soft Comput.

    (2016)
  • G.G. Wang et al.

    Chaotic krill herd algorithm

    Inf. Sci.

    (2014)
  • G.G. Wang et al.

    An effective krill herd algorithm with migration operator in biogeography-based optimization

    Appl. Math. Modell.

    (2014)
  • L. Guo et al.

    A new improved krill herd algorithm for global numerical optimization

    Neurocomputing

    (2014)
  • Y. Zhang et al.

    Crop classification by forward neural network with adaptive chaotic particle swarm optimization

    Sensors

    (2011)
  • M. Al-Smadi et al.

    A computational method for two-point boundary value problems of fourth-order mixed integrodifferential equations

    Math. Prob. Eng.

    (2013)
  • M.A.Z. Raja et al.

    Comparison of three unsupervised neural network models for first Painleve´ Transcendent

    Neural Comput. Appl.

    (2015)
  • O. bu-Arqub et al.

    Application of continuous genetic algorithm for nonlinear system of second-order boundary value problems

    Appl. Math.

    (2014)
  • S. Mall et al.

    Regression based neural network training for the solution of ordinary differential equations

    Int. J. Math. Model. Numer. Optim.

    (2013)
  • S. Mall et al.

    Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems

    Neural Comput. Appl.

    (2014)
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