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