# Modeling Vessels in Human Cardiovascular System and Calculating Pressure Wastes Using Lumped Method

M Mirzaee, O Ghasemalizadeh, B Firoozabadi

###### Keywords

arterioles, cardiovascular systems, electrical analogy lumped method, large arteries, pressure wastes, pulmonary

###### Citation

M Mirzaee, O Ghasemalizadeh, B Firoozabadi. *Modeling Vessels in Human Cardiovascular System and Calculating Pressure Wastes Using Lumped Method*. The Internet Journal of Bioengineering. 2008 Volume 3 Number 2.

###### Abstract

One of the most effective ways to model human cardiovascular system is lumped method (Electrical analogy). In this research Lumped method as an effective way is used for making equivalent circuit of body vessels. This complicated circuit includes equivalent segment for arteries, pulmonary, atrium, left and right ventricles with their equivalent circuits. Furthermore, in our complex circuit some additional points are considered in respect to previous works to improve this method. Some of the most important considerations are exact modeling of right and left ventricles as the main pressure suppliers, division of ascending aorta into 27 segments, and adding peristaltic motions of vessels in descending arteries as a further pressure supplier respect to ventricles. So the accuracy of our modeling increased so much. At the end wastes of each artery are considered and their relevant graphs are brought.

### Introduction

To analyze cardiovascular system and effects of diseases on it different ways are usable such as lumped model, one or multi-dimensional modeling and experimental methods. In this context lumped method were used with the goal of providing better understanding and simulation of the blood flow in the human cardiovascular system. Such modeling has been done before but not by this power. The first computer models describing the arterial system such as ascending aorta and carotids were introduced a multi-branched model of the arterial tree in a usable form for digital computers. By this method different physiological conditions became considerable. Later more detailed models were applied to reach more accurate results _{1} . To analyze the human cardiovascular system mathematically, more simplified model should be considered to decrease the difficulty of investigating. A pulsatile-flow model of the left and right ventricles (as suppliers) and 2-segment aorta were constructed and the changes in flow behavior investigated. Later lumped (electrical analogy) model was developed to analyze cardiovascular systems easily with suitable accuracy. An electrical model which focused on the vessel properties was made by Young and his team _{2} . One-dimensional axisymmetric Navier-Stokes equations for time dependent blood flow in a rigid vessel had been used to derive lumped models relating flow and pressure _{3} . The effects of external factors on vessels wall were investigated in blood flow _{4} . A complicated non-linear computer model for pressure changes and flow propagation in the human arterial system was drived _{5} . The model had 55 arterial compartments and was based on one-dimensional flow equations to simulate effects of hydrodynamic parameters on blood flow. A computer model of stenosis was introduced and analyzed with a related software _{6} . Later a simple model of lumped method was presented for human body _{7} .

This paper describes modeling of the whole human cardiovascular system using an extensive equivalent electronic circuit (lumped method). The model consists of about 150 RLC segments representing the arterial and cardiac systems that would be explained later. Respect to previous researches in electrical analogy, our modeling has more advantages and considerations. Also, body vessels equivalent compartments in the circuit is more detailed, especially ascending aorta which is the start point of blood flow after the left ventricle has 27 segments to cross flow completely and without wasting. So, more exact results for different vessels are reachable. Furthermore, adding suppliers to model waveform movement of descending vessels made our modeling more and more accurate and citable. At the end pressure wastes along different vessels are considered and their graphs with citable answers are shown.

Also to model each part of body its equivalent circuit is used.

Here is a 36-vessel body tree which is utilized in this research to model human cardiovascular system vessels. Also information about these vessels is shown in table 1.

### Modeling Bases

To model human cardiovascular system we chose different equivalent electrical elements to express different mechanical properties of vessels, blood and heart.

In our modeling process atriums, ventricles, every blood vessel, set of all capillaries, arterioles and veins have been presented by some compartments consisting of a resistor, an inducer and a capacitor.

The number of compartments would be chosen by the purpose and the required accuracy .So more compartments would be used for main arteries.

Voltage, current, charge, resistance and capacitance, inductor in the electronic circuit are respectively equivalent to blood pressure, blood flow, volume, resistance, compliance and flow inertia in the cardiovascular system. Ground potential (reference for voltage measurements) is assumed to be zero as usual. The relation between mechanical properties of cardiovascular system and their equivalent electrical elements are as follow:

0.01ml/Pa = 1 µF (compliance - capacitance)
1 Pa.s ^{2} /ml = 1 µH (inertia - inductor)
1 Pa.s/ml = 1 kΩ (resistance - resistance)
1mmHg = 1 volt (pressure - voltage)
133416 ml = 1A (volume - charge)

Following equations is taken to introduce needed electrical elements for simulation _{8} .

Blood vessel resistance (R), depending on blood viscosity and vessel diameter, is simulated by resistors:

Where µ is blood viscosity, l and A are in respect length and cross section area of each artery segment. The quantity of µ is 0.0035 (kg/m.s).

This simulation has considered because blood viscosity will cause resistance against Blood flow crossing.

The blood inertia (L) is simulated by inductors:

Where ρ is blood density and its quantity would be 1050 (kg/m ^{3} ).

Reason of this consideration is variability of flow acceleration in pulsatile blood flow, so an inductor can model inertia of blood flow very clearly.

The vessel compliance (C) is considered using capacitors:

In these formulas r, E, and h are in respect artery radius, Elasticity module and thickness of arteries.

h is wall thickness that this variable parameter would be changed by the type of artery and also diseases or abnormities in cardiovascular system. The wall thickness quantities in normal conditions are obtained from Physiological text _{9} using simplified equation 4:

Different heart shutters are modeled by using appropriate diodes, because shutters like diodes cross the flow in one direction. Considering this fact is important because in some parts of cardiovascular system, inverse current movement will cause to great danger to health. So to reach a good model, choosing appropriate diodes would be counted as an inseparable part. Type of diodes is visible in circuit at figure 3.

For the reason of this simulation, it should be noted that by passing blood thorough vessels, the vessels would be expanded or contracted, so they can keep blood or release it, and this is exactly like what a capacitor does. By these statements each vessel is modeled by some compartments, which includes one resistance, one capacitor, and one inductor. The next step is to introduce a model to put these elements together. Below model is chosen to make the circuit _{8} .

Quantities of Compartment's elements are easily achievable by using equations 1, 2, and 3. Computed values of circuit Elements are shown in table 1. Also, it shall be noted that substituting these quantities in their relevant elements should be done with adequate precision and in a special manner.

In this table n is number of each artery segments.

### Circuit Description

### Heart Equivalent circuit

### Heart and Ascending Aorta

The complex circuit for heart is shown in figure 3.

In this circuit an exact model of ventricles pressure has been used as the supply of power. The simulated and exact pressure graphs have been compared in Fig8 for right ventricle and Fig9 for left ventricle.

From the figures it is obvious that left and right ventricles pressure will change, in turn between 120-11 volt (mmHg) and 29-7 volt (mmHg) which is in complete agreement with physiological texts _{9} .

Also the right atrium and ventricle are modeled by two capacitors 216.45 µF and 150 µF. Also the left atrium and ventricle are represented by two capacitors 101 µF and 25 µF _{8} .

As said before we should use appropriate diodes to model shutters. Diodes used for tricuspid, pulmonary, mitral and aortic shutters are in respect 120NQ045, QSCH5545/- 55C, SD41 and SD41.

It should be noted that capillaries are so small but have an important role in cardiovascular system which without them the circulation of blood would not be completed. In our circuit marker “Vcc3” plays role of capillaries which connect arteries to veins. If these parts of circuit don't put correctly in their places, the modeling and its equivalent circuit wouldn't accompany real and exact answers.

### Main Arteries Circuits

Complex circuit of upper and downer body is shown in figures 4, 5, 6, and 7.

Ascending aorta would be subdivided into 27 segments which elements quantities are shown in table 1. This work is done because it is the most effective artery in whole cardiovascular system.

Also number of segments of other arteries would be chosen by their importance.

As said before voltage suppliers would be considered to model motion of vessels muscles. This should just be added to important descending arteries from aorta such as thoracic, abdominal aorta, iliac and femoral arteries.

In these to cases elements could be determined by using equation 1, 2 and 3. Also the calculated quantities are brought in table 1.

It should be added that VCC(x), allude to co potential points.

It should be noted that by using more than 80 segments to model whole arteries there is no leakage of current in the system and arteries pressure graph are quite acceptable comparing to the input pressure. It means blood flow after exiting from left ventricle and crossing from arteries and without decreasing of its quantity, would enter veins from capillaries. By power of right ventricle it will pump to pulmonary and will pour to left ventricle again so the cycle would be completed.

Also model is capable of showing blood and vessels properties in different points. For example the pressure (voltage) and volume (charge) graphs can be obtained from the different points of the circuit easily. But in this circuit we would quietly focus on pressure graphs that are productive in clinical researches.

Human Cardiovascular system equivalent circuit which its elements quantities have been earned from modeling principles is shown at figures 3-7. In each part of circuit increasing the number of compartments would increase our answers accuracy respect to real values and it is because, this work will increase the number of capacitors, consequently, their values would decrease and this will lead to lesser leakage of current.

### Confirmation part

To verify the simulation the focus is just on ventricles and ascending aorta.

Fig8 shows the simulated pressure graph of right ventricle varying between 29–7 mmHg (volt). The real graph of right ventricle confirms our simulation results which are in complete accordance with physiological data of reference _{9} .

The simulated pressure-time graph of left ventricle is shown in Fig9, where the waveform varies between 120–11 mmHg (volt). The results are in complete agreement with experimental observation of physiological text _{9} which is brought in the same figure. The waveform starts from 11 mmHg and the peak is in 120 mmHg.

The calculated and real pressure changes of ascending aorta artery are shown in Figure 10. This graph shows that calculated pressure for aorta varies between 120–68 mmHg (volt) (systole-diastole) and the results are in exact agreement with physiological article _{9} (real pressure pulse). Even, two peaks in pressure graph of ascending aorta are earned the same as real measurements.

Because of these agreements between real and calculated graphs for ascending aorta, right and left ventricles the citation of our circuit and modeling would be verified. So we can count on answers strongly.

### Results

Also, other vessels pressure graphs are so exact but showing all of them is out of scope of this paper.

Now, in different vessels, pressure wastes graphs because of resistances and bifurcations are shown.

It is obvious that in downer parts because of decrease in radius of vessels, wastes would be increased. For main arteries this waste is obtainable below. It should be reminded that each one Volt is equal to one mmHg.

1 volt =1 mmHg

The pressure waste changes between 0 mmHg and 400 mmHg. The average is so near to 0. It is vivid that the waste is so little because if aorta wastes lots of energy, other vessels wouldn't have sufficient pressure.

As vessels start to get smaller the resistance against the blood flow would becomes larger and because of this wastes increases.

After upper arteries descending arteries are considered. In these arteries the additional movement of vessels increases the amount of pressure which it is modeled with additional resources in their equivalent circuit.

From figures 16 to 20 it is understood that by moving down wastes are increased. Also after femoral this leakage would be larger until blood flow reaches to arteriols and capillary which would have the greatest amount of lack. The earned graphs are shown in figures 21 to 24.

After right ventricle flow would cross from pulmonary which from its figure little waste is obvious.

The blood should cross from capillary of pulmonary. The small radius of these parts would waste all the pressure that right ventricles produces.

This decrease would reach to the amount of 20 mmHg which is almost the right ventricle pressure.

{image:31}

{image:32}

{image:33}

The blood flow after this part would enter in to left ventricle and again needed pressure would be produced.

### Conclusion

It should be noted that using complex electronic circuit to model human cardiovascular system, is so useful for studying blood, different vessels and heart behaviors. As a result by knowing exact pressure graph of each vessel and its wastes, prevention from diseases would be simpler. Finally it should be said that the circuit has this tendency to be more accurate and useful by adding more compartments and details to it.