# Modeling The Human Cardiovascular System and Peristaltic Motion of Descending Arteries Using The Lumped Method

O Ghasemalizadeh, M Mirzaee, B Firoozabadi

###### Keywords

cardiovascular systems, descending arteries, electrical analogy lumped method, peristaltic motion of vessels walls, rlc circuit

###### Citation

O Ghasemalizadeh, M Mirzaee, B Firoozabadi. *Modeling The Human Cardiovascular System and Peristaltic Motion of Descending Arteries Using The Lumped Method*. The Internet Journal of Bioengineering. 2007 Volume 3 Number 1.

###### Abstract

Modeling of the human cardiovascular system is always an important issue because by doing such modeling, investigation of the cardiovascular system and its abnormities would be simpler. One of the most effective ways to do such modeling is using the lumped method (Electrical analogy). In this paper The Lumped method is used for simulating a complete model. A 36-vessel model was chosen to make equivalent circuit of a lumped method from it. 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 to improve this method. Some of the most important considerations to advance this modeling method are exact modeling of right and left ventricles pressures, division of ascending aorta into 10 segments, and adding peristaltic motions of vessels in descending arteries (thoracic to femoral) as a further pressure supplier respect to ventricles. These points were not noted in previous researches. Another advantage of these considerations is increase in the accuracy of our simulation and consequently our results. At the end it is shown that the pressure graphs from our complex circuit in descending arteries are so close to real graphs from biological tests which are earned by advanced instruments. So citation of our modeling would verify strongly.

### Introduction

Human cardiovascular system health and its abnormities are so important for clinical researches. Because by knowing the influences of diseases on cardiovascular system properties such as pressure graph the prevention from serious health problems would be possible. 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 which will lead to accurate answers as a result of exact modeling. 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 effect of drugs on the blood properties and its circulation was studied later in a 19-compartment model _{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 Computational model was developed to provide boundary conditions for simulations of the effects of endoleak on the AAA wall stress _{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). In this method we have taken a quite different way to model the cardiovascular system and we had intention to develop our electrical models by applying more details from main arteries which more accurate results would be one of its effects. The model consists of about 90 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 10 segments to cross flow completely and without wasting. So more exact results for ascending and descending arteries properties are reachable. Furthermore, adding suppliers to model waveform movement of descending vessels made our modeling more and more accurate and citable. Pressure graphs which are shown in results would confirm the above statements.

Logically increasing the number of segments for each artery will increase the accuracy of modeling answers greatly.

So investigating cardiovascular system faults and effects of diseases would accompany precise results. At last, we should add that our modeling (Lumped method) is capable of considering different elasticity modulus (E) for each segment, so investigating effects of E variation in human cardiovascular system would be possible.

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 Introduction and its Principles

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 formulas 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.

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.

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 © is considered using capacitors:

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

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.

Where ρ is 1050 (kg/m3), µ is 0.0035 (kg/m.s) and n is number of each artery segments. We can obviously understand the density of blood is very close to water density, so their properties would be very similar.

Also, (h) is artery 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 Fig3 (Physiological text9).

Also we should note that heart with its moving muscles is the power supplier of blood circulation in whole body vessels.

To have a very exact modeling of blood and vessels behaviors, exact modeling of power suppliers is an important factor. So in our simulation left and right ventricles are modeled quite exactly the same as biological graph sources9.

Fig4 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 is in complete accordance with physiological data of reference9.

The simulated pressure-time graph of left ventricle is shown in Fig5, 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.

Atriums are simulated as part of the venous system without any contraction. Atriums and ventricles can be modeled like vessels as a RLC segment. These two important parts of heart have two courses of action, one is resting position (diastole) which muscles would take their maximum volume and the other is acting position (systole) which muscles reach to their minimum size. 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 in Fig7.

Also, bifurcations are important cases to have accurate modeling. For simulating these parts of cardiovascular system a special method has been used _{8}

. In brief, it can be said that in bifurcations properties of jointed vessels will combine together in a complicated way to show the blood current division effects.

### Peristaltic Motion of vessels

When several simultaneous measurements are done at different points all along the aorta, it appears that the pressure wave changes shape as it travels down the aorta. Whereas, the systolic blood pressure actually increases with distance from the heart. Thus the amplitude of the pressure oscillation between systole and diastole, which is pulse pressure, nearly doubles. Thereafter, both PP (pulse pressure) and MAP (mean aorta pressure) decrease rapidly _{10} .

Logically, wastes because of frictions and bifurcations should be caused decreasement of systolic pressure of descending arteries but as said above from biological texts, it is obvious from the figure6 ^{12} that by moving forward through these arteries the maximum point of pressure graph would be increased. The reason of this phenomenon is that these arteries have an additional pressure which is the result of waveform movement of vessels walls. These walls have a peristaltic motion which it would help the easier movement of blood in descending vessels.

To have more exact results from our circuit, in addition to precise simulation of left and right ventricles we used appropriate pressure suppliers to consider these peristaltic motions and their effects. To model these pressure suppliers in our electronic circuit for each artery and respect to its motion, appropriate voltage sources were chosen. These sources are visible Fig9.

### Circuit Description

Human Cardiovascular system equivalent circuit which its elements quantities have been earned from modeling principles is shown Fig7, Fig8, and Fig9. This circuit includes three main pages that each one shows one part of body (heart, aorta, upper body and downer body).

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.

### Heart Equivalent circuit

Heart (ventricles, atriums, pulmonary and shutters).

The complex circuit is shown in Fig7.

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 Fig4 for right ventricle and Fig5 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.

### Arteries Circuits

Upper part of body (Ascending aorta, hands and carotids)

Downer part of body (thoracic aorta and feet)

Complex circuit of upper and downer body is shown in Fig8 and Fig9.

Ascending aorta would be subdivided into 10 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.

### Additional explanation

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.

It should be added that “Vcc”s, 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 pulmonaries 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.

### Verification of results

In human body Cardiac current output and aortic pressure should be in turn 100 (ml/s) and 120-68 (mmHg) ^{12} . The calculated current from our circuit is exactly the same as sources.

Also as we can see, the calculated pressure changes of ascending aorta artery with 10 compartments are shown in Fig7. This graph shows that aorta pressure varies between 120–68 mmHg (volt) (systole-diastole) and the results are in exact agreement with physiological article _{9} (Fig8). 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.

### Results

The most important purpose of our circuit was to extend lumped method to reach actual results simply.

As we showed in verification part the precision of aortic pressure is one of our exact modeling reasons below texts and graphs would confirm more and more our modeling power.

The great work is done in this paper is reaching to exact pressure graphs of descending arteries that are obtainable below.

By referring to figures 8, 9, 10, 11, 12, 13, and comparing them with figure 6, it is easily notable that the calculated pressure quantities from our modeling is in great accordance with the real one. So by these results precision of our modeling would be greatly confirmed.

### Conclusion

As it was shown in our results the citation of our circuit is quite acceptable. It should be noted that using this complex electronic circuit to model human cardiovascular system with its all details, is so useful for studying of blood, different vessels and heart behaviors respect to each other and in different conditions such as health, diseases and abnormities. These abnormities may be obstructions, heart problems, vessels diseases or lots of different other things which out of scope of this paper.

Finally it should be said that our circuit has this tendency to be more accurate and useful by adding more compartments and details to it.