RESONANT FREQUENCIES OF SMALL ARTERIAL SEGMENTS AS DETERMINING FACTORS FOR ESTIMATION OF TERMINAL SEGMENTS IN ELECTROMECHANICAL ARTERIAL TREE MODEL RESONANT FREQUENCIES OF SMALL ARTERIAL SEGMENTS AS DETERMINING FACTORS FOR ESTIMATION OF TERMINAL SEGMENTS IN ELECTROMECHANICAL ARTERIAL TREE MODEL

divide the modeled arterial tree to the discrete arterial segments with defined length. We used for our simulation of arterial system behavior the model consisting of interconnection of single vessel segments (see Fig. 1). A model of the nonsymmetrical arterial tree was analyzed. The arterial branching model was based on the principle of the electromechanical analogy. The nonsymmetrical arterial tree was divided to 7 arterial generations, starting from larger arteries towards small arteries and arterioles. For each generation the resonant frequency and the transfer function were calculated and there are presented the pressure simulation results for each arterial generation. Terminal segments were determined regarding to calculation of the resonant frequencies and the transfer functions and simulation results.


Introduction
The arteries deliver a blood containing oxygen to the tissues of whole human body. Blood delivery starts at heart output connected to aortic arterial segments and continues towards the periphery. On the peripheral side of the arterial system small arteries play more than one role such as blood delivering pipelines. They serve for regulation of systemic vascular resistance by changing their radii which are modulated by innervated smooth muscles [1].

Arterial Branching
In this contribution we adopted the results of works which deal with arterial branching questions [2 -6]. In most cases, the arterial branching consists of mother vessel which bifurcates into two daughter vessels. We can define the daughter vessel radii by Eq. (1) and Eq.
where rd 1 are daughter vessel radii, rp is radius of the mother vessel and a , b are the coefficients modulating the daughter vessel radii [2] and [3]. More information about derivation of the concrete values of the a and b coefficients can be found in the [3] where author defines asymmetry and area ratio of the mother and daughter vessels. Olufsen in [6] combines generalized "Murray's law" [4] and [5] with Zamir's [3] definitions of bifurcating vessel ratios.

Model of Arterial System Based on Electromechanical Analogy
The properties of the arterial system can be transformed by using appropriate electromechanical analogies to system consisting of discrete electrical elements such as resistor, inductor or capacitor [7 -10]. A blood pressure is equivalent to electrical voltage and a blood flow can be modeled as electrical current. These equivalences can be used in the case when we divide the modeled arterial tree to the discrete arterial segments with defined length. We used for our simulation of arterial system behavior the model consisting of interconnection of single vessel segments (see Fig. 1).
are approximately equal to arteriole radii. We attempted to solve this problem in another way.

Transfer Function of Single Arterial Segments
At first, we made an analysis of transfer function of arterial segments placed in the single generations (see Fig. 2). Pressure or voltage transfer function of the single segments is defined by: where U2 o is output harmonic voltage of the simplified arterial segment (see Fig. 3) and U1 o is input harmonic voltage of this segment. According to the theory of electromechanical analogy the voltage is interconnected with the pressure in physiological segment described by its mechanical properties.
where ZL o is longitudinal impedance of arterial segment in Fig. 1 and YT o is transversal admittance of this segment.
Both parts of this basic arterial segment model can be expressed by (8) and (9): The meaning of the elements, which are described by (3), (4) and (5), is listed in Table 1 [7] and [8].
Elements and parameters of the vessel segments Table 1 Element

Small arteries branching implementation
Before implementation of small artery models to existing model of arterial system involving the large arteries, it is needed to determine a finishing radius of last arterial generation. Radii of arterioles vary from 5 to 30 µm [1]. Therefore the new generation of small arteries should be finished when their radii Resonant frequencies of the arterial segments placed in the different generations Table 2 α-side β-side

Determination of outflow condition
As written in the text above, we used the calculation of the transfer functions for assessment of the properties of the single arterial segments. They can be used for determination of outflow conditions of the segments placed in the last generation of small arteries, concretely in our case in the 7 th generation. We can list several approaches how to determine an element which finishes arterial tree. We can start with "windkessel" models and continue with simple resistive elements which represent resistive nature of capillary bed [11] and [12].
In our contribution we focus on determining outflow condition in the following way: in the real world the arterial tree consists of approximately 20 generations. It follows from that there are 2 20 elements needed for modeling the last generation [2] and [11]. By using the calculation of the transfer functions we can estimate "cut-off" generation which means that we can determine when we can stop the generating of new elements and by that way it is possible to minimize computing costs of computer simulation. In our work we set the threshold for stopping generating of the new segments by observing the resonant properties of the single segments on the alpha and beta sides of the particular arterial generations.
As defined by (1) and (2) the radius of the new segment descends much slower on the alpha side than on the beta side of the modeled arterial tree. It follows from Fig. 4 which characterizes transfer functions of arterial tree segments (see Fig. 2) on the alpha side. The change in resonant frequency is much evident in single generations on the beta side of arterial bed (Fig. 5).
The meaning of the elements used in (8), (9) is listed in Table 1.
From the point of view of the electrical circuit theory, we can express impedance of the unloaded segment such as the serial connection of its longitudinal impedance (8) and reciprocal of its transversal admittance (9): The total impedance will be helpful at analysis of transfer characteristics of the single arterial segments placed in different generations. Concretely, there are resonant frequencies which are indicators of the arterial segment properties regarding to inductive and capacitive properties of the equivalent electrical circuit (Figs 1 -3). The resonant frequency of the corresponding segment (see two-port circuit in Fig. 3) is determined by equaling of the imaginary part of the total impedance to zero: Having expressed the single elements and edited the Eq.
The values of the resonant frequencies of the arterial segments placed in the different generation are listed in Table 2.

Conclusion
We estimated the criteria for stopping of creation of the new elements in the next generation by evaluating the resonant peaks in single transfer functions. The segment placed on the beta side of modeled arterial tree has resonant frequency approximately equal to 10 4 rad•s -1 . It is negligible from the point of view of possible heart rate frequencies which can exist in arterial system. Therefore, we can assume that the segments placed in the seventh generation have resistive character in the range of the possible heart rate frequencies and with them connected upper harmonic components which form the basic pulse wave propagating through arteries and they can be finished by simple resistive elements -resistors with values comparable to total resistance of the previous segment.
The modeling of the vascular system by using the electromechanical analogies can lead to better understanding of processes which occur in arteries. Also it could be possible to evaluate the degree of pathological changes in arterial system by using appropriate measurement methods (e.g. photoplethysmography) [13] and by reverse comparing of measured and simulated data. In this way the malformation such as arterial stenosis or aneurysm could be deeply studied. We performed the simulations of the model of arterial tree consisting of small arteries towards arterioles. This model was implemented to the existing model of arterial system based on electromechanical analogy. Comparing to the previous arterial tree model [7 -10], where the arterial peripheral segments were ended by terminal segments at the level of large arteries, the arterial branching continues to the seventh arterial generation in our implementation. Input signal of this branching extension corresponds to the mother vessel curve (blue line -see Fig. 6). The radii of the segments placed on the left side of the modeled arterial tree are around 45 µm (see Fig. 7) which is value close to arteriolar tree border (arterioles have resistive character and serve for peripheral vascular resistance regulation [1]). The simulated pressure values (see Fig. 6) in the seventh generation correspond to pressure values measured in arteriolar tree [1].