Tihomir Ivanov

Original Research Articles in Peer-Reviewed Journals from Past BIOMATH Conferences

The four past Biomath conferences in Sofia (1995, 2011, 2012, 2013) resulted in a total number of 116 original research articles, as follows: 4 artricles in J.UCS, 59 articles in Elsevier journal CAMWA, 14 articles in B&BE and 39 articles in journal BIOMATH. A list of titles of these articles follows.

Stability Analysis of an Inflation of Internally-Pressurized Hyperelastic Spherical Membranes Connected to Aneurysm Progression

In this study, we consider a nonlinear second-order ordinary differential equation to describe the inflation of a thin-walled hyperelastic spherical membrane, subjected to an internal distention pressure. We examine the stability of the equilibria of the basic model using three different forms of strain energy functions (SEFs), representing the mechanical properties of elastomers and soft tissues. It is shown that the mechanical stability or instability of the membrane material is associated with the monotonicity or non-monotonicity of the pressure-stretch relation. We define two types of instabilities according to the specific constitutive relation. We prove analytically that a stable inflation of the membrane can retain or can change to an unstable one depending on the specific analytical form of SEF and its material parameters. We derive conditions for the stability/instability of the equilibria of the model with the different SEFs and relate the identified unstable equilibrium states to development and rupture of aneurysms.

Initial Calibration of MEMS Accelerometers, Used for Measuring Inclination and Toolface

In the present work, we consider calibrating MEMS accelerometers for the purpose of determining orientation in space. We propose a new objective function whose minimization gives an estimate of the calibration coefficients. The latter takes into account the specifics of measuring toolface and inclination in seek of better accuracy, when the device is used for this purpose. To the best of our knowledge, such an objective function has not been mentioned in the literature. The calibration algorithm is described in detail because, even though, some of the steps are standard from the point of view of a numerical analyst, this could be helpful for an engineer or an applied scientist, looking to make a concrete implementation for applied purposes. On the basis of numerical experiments with sensor data, we compare the accuracy of the proposed algorithm with a classical method. We show that the proposed one has an advantage when sensors are to be used for orientation purposes.

Analysis of a Bioreactor Model with Microbial Growth Phases and Spatial Dispersal

We consider a batch mode bioreactor model proposed by Alt and Markov (2012). The model is developed using the fact that the bacterial growth undergoes four phases: lag, log, stationary and death phase. First, we modify the model by introducing additional (the so-called transport) terms to describe continuously stirred bioreactor dynamics. For this model we compute the equilibrium points and study their asymptotic stability. Some basic properties of the solutions like uniform boundedness and uniform persistence are also established. Then we extend the model by adding diffusion terms to the equations. The latter reaction-diffusion equations are studied numerically. Thereby, solutions in the form of travelling waves are found.

Computational Study of a Bioreactor Model with Microbial Growth Phases and Spatial Dispersal

We consider a batch mode bioreactor model proposed in [1]. The model is developed using the fact that the bacterial growth undergoes several phases: lag, log, stationary and death phase [1], [2]. First we modify the model by introducing additional (the so-called transport) terms to describe continuously stirred bioreactor dynamics. Then we extend the model by adding diffusion terms to the equations [3]. The latter reaction diffusion equations are studied numerically. Thereby, solutions in the form of travelling waves are found. References [1] R. Alt, S. Markov, Theoretical and Computational Studies of Some Bioreactor Models, Computers and Mathematics with Applications 643, 2012, 350{360. [2] S. Markov, On the Mathematical Modelling of Microbial Growth: Some Computational Aspects, Serdica Journal of Computing 5 2, 2011, 153{168. [3] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Third Edition, Springer, 2003