Modelling and Simulation of Blood Circulation Towards Mitral Valve Replacement Risk Calculation
Vera Höfflin2,3, Amiliana M Soesanto1, Hananto Andriantoro1, Eko Supriyanto1,2, Jens Haueisen3
1Department of Cardiology and Vascular Medicine, Faculty of Medicine, Universitas Indonesia, / National Cardiovascular Center “Harapan Kita‘ Indonesia
2IJN-UTM Cardiovascular Engineering Centre, Faculty of Bioscience and Medical Engineering, Universiti Teknologi Malaysia
3Institute of Biomedical Engineering and Informatics, Technische Universität Ilmenau, Germany (Corresponding author: eko.supriyanto@gmail.com)
Abstract The operative mortality of a mitral valve replacement surgery in patients who suffer from a rheumatic heart disease and who have a concurrent pulmonary arterial hypertension accounts for 15 to 31 %. The complexity of the disease and of the cardiovascular system itself makes it difficult to find a specific reason for the high operative mortality. In a first approach to this problem we modelled and simulated the blood circulation under normal and diseased conditions. Herewith, the blood circulation can be analysed as a whole in spite of its complexity. For this purpose, we decided to model with the help of an electrical circuit analogy because it is not "spatially" dependant, fast and computational economic. The model includes the systemic and the pulmonary blood circulation with the main vessel-types. Hereby, we used the Multisim software to model and simulate. Simulation results show that our model matches the theoretical mitral flow patters from other resources. This model has positive signs to be used for a mitral valve replacement risk classification.
Keywords: Rheumatic Heart Disease, Mitral Valve Replacement, Blood Circulation, Lumped Parameter Model, Simulation.
I.INTRODUCTION
Acute rheumatic heart fever is an inflammatory disease that is caused by an abnormal autoimmune response to a group A streptococcal infection. It occurs mostly in children in the age between 5 and 14 and may affect connective tissues such as the heart, joints, the brain and the skin. This form of the disease can still be treated with antibiotics. However, in 60 to 90 % of cases it leads to a chronic form, the so called rheumatic heart disease (RHD). There are about 1.96 to 2.21 million cases of RHD in Asia and it causes up to 233,000 deaths per year worldwide [1]-[3]. The disease can often be found in patients who have previously been asymptomatic and who come to see the medical doctor in the age between 20 and 50 because of shortness of breath [4]. Even though RHD shows a high variety of manifestations, the function of the mitral valve is affected most commonly. In about 25 % of all cases, mitral stenosis will be developed as a recurrent inflammation of the valve. About 40 % suffer from a combined mitral stenosis and mitral regurgitation. A mitral stenosis is characterized by a narrowing
of the valve and a mitral regurgitation by an insufficiency or leakage of the valve [5][6]. In either instance, when the function of the mitral valve is not sufficient any more the repair or the replacement of the mitral valve can be indicated among other treatment possibilities. In a late stage of the disease only a replacement with an artificial valve might be helpful in order to prevent heart failure [7][8][9]. In a late stage, however, the disease often comes along with pulmonary arterial hypertension, which is considered a risk factor for the mitral valve replacement surgery [10][11].The operative mortality1of mitral valve replacement surgeries in patients that suffer from RHD and that show pulmonary arterial hypertension ranges from 15 to 31 % [10]-[12]. In these cases the condition of the patient is not becoming stable after the diseased mitral valve has been replaced with an artificial valve. This can mean, that the medical doctors can not relieve him from the ventilation machine or that he dies within a few hours or days. The surgery itself is considered as having been successful. This means that there have not been any difficulty regarding the bypass or another complication like the occurrence of an inflammation.
The complexity of the disease, and of the cardiovascular system itself, makes it difficult to find a specific reason for the high operative mortality and to find the cause of the described outcome of the surgery. The unpredicted blood circulation after the surgery might be a part of the problem. More precisely, the diseased physiological system that previously adapted to this non-healthy state of RHD (by e.g. the compensatory work of the left ventricle) is suddenly after the replacement confronted with a properly working valve.
In order to get more insight into the conditions and circumstances before and after a mitral valve replacement, a model of the blood circulation has been built. Herewith, haemodynamic changes can be examined and risk factors of the surgery may possibly be classified with the help of a
1"Operative mortality has been defined as any death, regardless of cause, occurring (1) within 30 days after surgery in or out of the hospital, and (2) after 30 days during the same hospitalization subsequent to the operation." [33]
simulation with the provided model. In a first approach to this method the focus lies on modelling the mitral valve under normal and diseased conditions. Hereby, the mitral valve as one component can be changed within the blood circulation model.
Firstly, it has been decided to model the mitral valve with the help of a lumped parameter model using an electrical analogy approach. Secondly, in order to investigate the circumstances of the mitral valve replacement surgery, the scope of the blood circulation model includes the model of the mitral valve under five conditions: Normal, mild mitral stenosis, severe mitral stenosis, mild mitral regurgitation, severe mitral regurgitation.
II.METHODS
A. Overview
The blood circulation has been modelled before by other research teams. Some of them make use of the lumped parameter approach and work with an equivalent electrical circuit [13]-[23]. Therefore, this work could start with referring to existing models in the literature. Consequently, the first step included the implementation of a normal blood circulation that was already available. For this purpose the work of GHASEMALIZADEH et al., 2014 [13] has been chosen. After- wards, it needed to be adjusted to the requirements of the question at hand. First, the complexity of the systemic arterial system was simplified compared to other models. Discussions with medical doctors and a literature review resulted in the assumption, that the outcome of the mitral valve replacement surgery is more dependent of the pulmonary, than of the systemic circuit [10]. Second, the modifications of the model also included the addition of the atrial contraction that has not been implemented in the existing model. Following these adjustments, simulations of normal pressure and flow graphs during the cardiac cycle at different points of the circulation were possible. The pathological mitral valve was realized by changing the electrical parameters of the diode that represents the mitral valve, so that diseased conditions could be simulated as well.
B. Electrical Analogy
The main idea of the research approach is to model the components of the blood circulation by an equivalent electronic circuit. The first simplification lies in using a lumped parameter system that only considers one spatial point for one functional component of the system. The second assumption is to regard the haemodynamic system of the blood circulation with an electrical analogy. This might be possible in some applications when the Hagen-Poiseuille's and the Ohm's law are regarded equivalently. The Ohm's law states for the electrical current I, the electrical voltage V and the electrical resistance R [24]:
R
I = V
(1)The law of Hagen-Poiseuille is given with the volumetric flow rate Q• that is proportional to the pressure-difference P
between the beginning and the end of the tube with the flow resistance R [25]:
R Q D P
& =
(2)The particular parameters need to be converted in order to build a model based on this assumption [26]. The flow resistance depends on the length l, the inner radius r of the tube and the viscosity !"of the liquid [26]:
4
8 r R l
×
×
= × p
h
(3)The blood inertia L can be calculated with the knowledge of the blood vessel's length l, the blood density # and the cross sectional area A of the vessel [26]:
A L l
×
×
= × 4 9 r
(4)
In order to calculate the value for the electrical capacitance that is equivalent to the haemodynamic compliance the elasticity E and the thickness h of the vessel's wall are needed additionally [26]:
h E
r C l
×
×
×
×
= × 2
3
p
3(5)
For the implementation at hand these parameters are taken from literature [13][19][27], the values were calculated and after a conversion used for the model. If available, the electrical parameters as well as each a constant value for the blood’s"density and the" blood’s" viscosity"(# is 1050 kg/m³;
$ is 0.0035 kg/m.s) were obtained directly from literature. In order to realize the circulation electrically the circuit design software Multisim of National Instruments was used. It provides a SPICE simulation environment that allows the user to design and test electrical circuits.
C. Model of valves
One of the purposes of the heart valves is to prevent a back- flow. In doing so, a one directional blood flow is guaranteed.
The same principle applies to electrical diodes. This is why diodes can be used in order to model heart valves in the haemodynamic-electrical analogy. Figure 1 illustrates three basic arrangement of diodes and resistors in order to represent a specific valve condition. Figure 1A shows an ideal diode that can be used for modelling a normal or even ideally working valve. Figure 1B and C of the same figure illustrate possibilities to model diseased valves. A resistor in series to the diode allows us to increase the valve's resistance (see figure 1B). This is necessary to model a stenotic valve. In addition to that, the flow behaviour of a mitral valve regurgitation is achieved by adding a second diode anti-parallel to the first one and in changing the resistance in series (see
figure 1C). By changing the particular parameters of the resistors it is possible to model the severity of the disease.
Fig. 1: Models of heart valves. A: Ideal electrical diode as a model for a normal working valve. B: An ideal diode and an electrical resistor in series as a model for a valve stenosis. C: Two ideal electrical diodes and each an electrical resistor in series arranged parallel as a model for a valve regurgitation.
The described conditions of the valve (normal working valve, valve stenosis, and valve regurgitation) can theoretically simulated with the help of the model for all four heart valves.
However, as the focus of the work at hand lies on the mitral valve disease, this form is described in the following. As RHD is considered a complex disease, more manifestations of the illness are practically possible. For instance, some patients suffer from a mixed form of the mitral valve disease where the mitral valve is stenotic as well as regurgitated. For reasons of simplification only the three basic forms are considered. The typical flow patterns for each case can be seen in figure 2.
Fig. 2: Qualitative mitral flow pattern over one cardiac cycle for a normal working mitral valve (solid black), regurgitated mitral valve (red dash-dotted), and for a severe stenotic mitral valve with an effective orifice area of 0.8 cm² (green dotted) [28]-[30]. While the mitral valve is supposed to be closed during time period A (systole), it is supposed to be open during period B (diastole).
Under normal conditions the flow through the mitral valve is zero while it is closed. This is mainly the case during the systole. During the diastole the mitral valve opens and the flow
value raises as a result of the hypotension of the left ventricle.
The increased flow curve is known as E-wave. Following this, the flow decreases again for a short period of time until it increases again as a result of the pressure originating in the left atrium. This characteristic is typically lower than the E-wave and is called A-wave. Afterwards, the mitral valve closes again and the next cardiac cycle starts. The main difference of the flow behaviour under presence of a mitral valve stenosis is, that the flow does not increase in the same amount. This is because the effective orifice area is significantly smaller than in the normal case. The valve is not able to open entirely. As it is not in the same way flexible as under normal conditions, the flow value does not decrease to zero between the E- and the A- wave. For a mitral valve regurgitation one difference in the flow behaviour can already be observed during systole. In this phase, when the mitral valve is supposed to be closed, back- flow can be measured. In sum, the desired flow behaviour as a simulation result is orientated on the qualitative pattern that is shown in figure 2. [22][28]-[30]
D. Summary of Assumptions
As the model is a simplification of the complex physiological system, a few assumptions are necessary. The main model technique is the analogue use of an electrical circuit as a haemodynamic system. Basically, the requirements for the Hagen-Poiseuille's law need to be fulfilled. A laminar flow is not guaranteed in the entire blood circulation system.
The blood may include turbulent characteristics while flowing through the heart valves and chambers as well as at some point of valve bifurcations. The vessels are more complex than the considered tubes for the physical law of Hagen-Poiseuille.
Moreover, the blood density and viscosity is assumed to be constant over the entire time in the circuit. Furthermore, the vessels in the pulmonary and the systemic circuit are condensed to each five compartments. This means for instance, that the entire amount of capillaries are represented by one compartment in the electrical system. In addition to that, they are assumed to be in one plane. Considering the Bernoulli principle the pressure value of two points also depends on their height difference. Apart from the physical assumptions, there are also some limitations regarding the simulation of the disease. For instance, during the simulation of the mitral valve stenosis, the right heart chambers remain like normal as sufficient data has not been found in literature. It is also notable that the data that is in-putted in the example simulation is not consistent. It is not taken from one patient and one measurement device. It is taken from different literature resources.
E. Simulation Principle
Basically, the input values of the simulation are the electrical parameters for the resistors, the capacitors, the inductors and the diodes as well as the voltage graphs of the four heart chambers that have previously been converted from the blood vessel’s characteristics. The vascular resistance of the pulmonary circuit can possibly be raised and the typical pressure graphs of the mitral condition can be in-putted to the
model. After this, the simulation can be started and the flow behaviour can be measured at any point of the electrical model.
Assuming these model's input-parameters are measured in the diseased patient, various situations can be simulated in order to get more insight of the current behaviour at any location of the model - and hereby of the flow behaviour of the blood circulation. By changing the four voltage sources into current sources the simulation principle can be turned around. In this case, the pressure values are measured and considered as the output information of the simulation. This decision depends on the underlying question of interest.
III.RESULTS
One result of the provided work is the adjustment of the normal model of the blood circulation system. Hereby, a tool has been made available in order to simulate the situation before and after the mitral valve replacement surgery. This means, the haemodynamic behaviour before and after the surgery can be simulated with the help of the provided model.
The four main parts of it are described in the following; the systemic and the pulmonary circuit as well as the right and the left heart. The values for the electrical components refer to the model of GHASEMALIZADEH [13].
An electrical model of the systemic circulation is illustrated in figure 3. It includes the equivalent components of the systemic arteries, arterioles, capillaries, and veins. All of these components are electrical RLC- or RC- circuits. The arteries can be seen as a general electrical low pass. While the arterioles as resistance-vessels are characterized by a high resistance of 72 k•, the veins as capacitance-vessels are built with high capacitors with a capacitance of 210 F and 450 F.
The capillaries are modelled with an inertia of 1 H.
Fig. 3: Blood circulation model of the systemic circulation, values mainly [13].
The pulmonary circuit is constructed in a similar way as the systemic circuit. While the arteries have a high resistance, the veins are built with relatively high capacitors. Figure 4 shows the RLC- and RC-elements of the modelled pulmonary circulation in series. Here, the pulmonary arteries are divided into three blocks that are supposed to represent the smaller, the medium and the bigger kind of pulmonary arteries. While the resistance value is increasing from 1 k•, over 4 k• to 8 k•, the capacitance is enlarged from 1 F over 3 F to 27 F. Only the first kind of pulmonary arteries is modelled with the help of an electrical inductor with the inertance of 0.1 H. The pulmonary veins are made of two blocks with the same capacitance value of 10 F. The resistance of the second
compartment is lower and it has an additional inductance of 0.1 H.
Fig. 4: Blood circulation model of the pulmonary circulation, values [13].
In the model, the right heart is made up of two valves, the tricuspid and the pulmonary valve as well as of two heart chambers, the right atrium and the right ventricle. The valves are modelled with the help of electrical diodes. The heart chambers are constructed like any other vessel in the circuit model but with an additional voltage source. The pumping of the chambers is realized by using configurable voltage sources.
The voltage signal of the right ventricle is, like the pressure in the physiological context, higher than the voltage signal in the right atrium. The arrangement of components can be seen in figure 5.
Fig. 5: Blood circulation model of the right heart.
The structure of the left heart is the same as the one of the right heart. It consists of the mitral and the aortic valve as well as of the left atrium and the left ventricle. The difference can be modelled by changing the parameters of the electrical components. In this context, especially the parameters of the diode that represents the mitral valve are changed, as well as the input signals of the voltage sources that model the pumping of the chambers. Figure 6 shows the equivalent electrical circuit of the left heart. The input voltage-signal, that represents the pressure values originating in the chambers, was taken from literature [28][31]. There are three sets of input signals; the graph of the normal patient, the graph of a patient with mitral valve stenosis, and the graph of a patient with mitral valve regurgitation.
Fig. 6: Blood circulation model of the left heart.
In order to assess the model's accordance to the physiological system a graph of different pressure values over
the entire circuit under normal conditions has been drawn (see figure 7). Herewith, an overview of the pressure behaviour of the model is given. Like in the physiological context, it can be observed that the pressure value is still low in the left atrium.
However, it raises from the left ventricle on where the main pumping work originates from. It is attenuated after the aortic valve and the systemic arteries. Starting from the arterioles that are considered as resistance-vessels, the value significantly decreases. In the systemic capillaries and veins the value continues to decrease and is attenuated again, as it is typical for exchange- and capacitance-vessels. Compared with the physiological values that can be found in literature [32], the quality of the modelled pressure is in good accordance, assuming 1 mmHg being equivalent to 1 V.
Fig. 7: Average (dotted black curve), minimum and maximum (solid red curve) voltage values in [V] at different points of the blood circulation model. The minimum and maximum voltage values refer to the systolic and diastolic pressure differences. LA: Left atrium, MV: Mitral valve, LV: Left ventricle, AV: Aortic valve, SysA: Systemic arteries, Arterio: Arterioles, Cap:
Capillaries, SysV1: Systemic veins 1, SysV2: Systemic veins 2, RA: Right atrium, TV: Tricuspid valve, RV: Right ventricle, PV: Pulmonary valve, PulA1: Pulmonary arteries 1, PulA2: Pulmonary arteries 2, PulA3: Pulmonary arteries 3, PulV1: Pulmonary veins 1, PulV2: Pulmonary veins 2
The second main result of the presented work is given by the simulation of five different mitral valve conditions; normal, a milder and a more severe form of a regurgitation, and a milder and a more severe form of stenosis. In the first case, the flow through the normal working valve is simulated with the help of an ideal diode. Like explained earlier, in the case of a present regurgitation the mitral valve was modelled with the help of two antiparallel diodes. The severity has been adjusted by changing the resistance parameters of the particular diode arrangement (see figure 1). While the resistance of the forward diode in the milder form accounts to 0.5 k•, it is 0.9 k• for the more severe form of the regurgitation. The resistance of the backward diode is, 70 k• in the milder and 40 k• in the more severe form of the disease. The simulation result is given in figure 8 by measuring the current after the modelled mitral valve. Its value can be converted to blood flow with the assumption that 1 A electrical current corresponds to 70,000 mL/s blood flow.
Fig. 8: Electrical current I in [A] through the mitral valve as a result of the simulation under normal (solid black), a milder (red dashed) and a more severe form of regurgitation (red dotted), as well as a milder (green dashed) and a more severe form of stenosis (green dotted) conditions. 1 A corresponds to 70,000 mL/s blood flow Q•. The illustration shows the simulation result of one cardiac cycle.
The stenotic conditions of the diseased mitral valve are again modelled by only one diode in forward direction. The difference to the normal case is realized by increasing the resistance value of the diode arrangement. A value of 4 k•
models a milder and a value of 10 k• a more severe characteristic of mitral valve stenosis. The graph of the voltage-current characteristics of the mitral valve for the diseased conditions can be seen in figure 9.
Fig. 9: Voltage-Current characteristics of the diode arrangements for milder regurgitation (dotted red), more severe regurgitation (solid red), milder stenosis (dotted green), and more severe stenosis (solid green).
The result of the mitral valve stenosis, which is also illustrated in figure 8, shows the measured current after the mitral valve-diode on the model circuit. It should be noted that not only the diode parameters but also the voltage values of the left atrium and the left ventricle have a high influence on the resulting simulation and hereby in the current signal through the mitral valve. This is one reason why the accordance to theoretical flow values might not match to 100 %. However, the flow pattern for each disease has been reached qualitatively.
IV.CONCLUSIONS
An equivalent electrical model of the human blood circulation has been implemented and modified. Herewith, it is possible to simulate various haemodynamic situations at different locations of the modelled blood circulation. The voltage values for the normal case of the electrical circuit are in good accordance to the physiological pressure values over the
systemic and pulmonary circulation. A simulation of the electrical current through the mitral valve is also in qualitative accordance with blood flow patterns of other resources. The simulation of the blood flow through the mitral valve has been conducted for five different conditions that are of interest for the research topic at hand. As the underlying problem is concerned about the mitral valve replacement surgery in patients with rheumatic heart disease, models of the normal, a milder and a more severe form of regurgitation and stenosis are provided respectively. In addition to that, the model also considers the arterial contraction that is not included in the same way in many of the previous developed models. This was included, because the activity of the left, as well as of the right atrium, can increase the information about the underlying cardiac condition. Hence, a functional model as a basis and as a tool for further investigations of the blood circulation in the context of mitral valve replacement is provided. The study can continue to first validate the model and second to classify the risk of the mitral valve replacement surgery. In order to classify the risk an analysis about the particular cause of death each time the patient does not get stable after the surgery, could help to find the reasons for the high operative mortality of the described surgery. Together with a simulation before and after the surgery, there can possibly be found a correlation between the haemodynamic behaviour in the blood circulation and the condition of the patient after the surgery.
ACKNOLEDGMENT
Thanks to National Cardiovascular Centre Harapan Kita Indonesia for all clinical support, Universiti Teknologi Malaysia for financial support through grant Vot Nr. 03G12, and Ilmenau University of Technology for technical support and DAAD financial support.
REFERENCES
[1] J. R. Carapetis, A. C. Steer, E. K. Mulholland, and M. Weber, “The global burden of group A streptococcal diseases,” The Lancet Infectious Diseases, vol. 5, no. 11, pp. 685–694, Nov. 2005.
[2] J. R. Carapetis, “Rheumatic Heart Disease in Asia,” Circulation, vol. 118, no. 25, pp. 2748–2753, Dec. 2008.
[3] E. Marijon, M. Mirabel, D. S. Celermajer, and X. Jouven, “Rheumatic heart disease,” The Lancet, vol. 379, no. 9819, pp. 953–964, Mar. 2012.
[4] K. Sliwa, M. Carrington, B. M. Mayosi, E. Zigiriadis, R. Mvungi, and S.
Stewart, “Incidence and characteristics of newly diagnosed rheumatic heart disease in Urban African adults: insights from the Heart of Soweto Study,” European Heart Journal, vol. 31, no. 6, pp. 719–727, Mar. 2010.
[5] J. E. Hall and A. C. Guyton, Guyton and Hall textbook of medical physiology, 12th ed. Philadelphia, Pa: Saunders/Elsevier, 2011.
[6] Amiliana M Soesanto, Yoga Yuniadi, Muchtaruddin Mansyur, Dede Kusmana, “A Novel Echocardiography Formula for Estimating Pulmonary Vascular Resistance in Mitral Stenosis,” 2014.
[7] S. Ray, R. Beynon, and A. Borg, “Mitral valve disease,” Medicine, vol. 38, no. 10, pp. 537–540, Oct. 2010.
[8] Z. G. Turi, “Mitral Valve Disease,” Circulation, vol. 109, no. 6, p. 38e–41, Feb. 2004.
[9] N. A. Boon and P. Bloomfield, “The medical management of valvar heart disease,” Heart, vol. 87, no. 4, pp. 395–400, Apr. 2002.
[10] M. Mubeen, A. K. Singh, S. K. Agarwal, J. Pillai, S. Kapoor, and A. K.
Srivastava, “Mitral Valve Replacement in Severe Pulmonary Arterial Hypertension,” Asian Cardiovascular and Thoracic Annals, vol. 16, no. 1, pp. 37–42, Feb. 2008.
[11] C. Ward and B. W. Hancock, “Extreme pulmonary hypertension caused by mitral valve disease. Natural history and results of surgery.,” Heart, vol.
37, no. 1, pp. 74–78, Jan. 1975.
[12] R. S. Bonser, Ed., Mitral Valve Surgery. London: Springer, 2010.
[13] O. Ghasemalizadeh, M. R. Mirzaee, B. Firoozabadi, and K. Hassani,
“Exact Modeling of Cardiovascular System Using Lumped Method,”
arXiv:1411.5337 [physics, q-bio], Nov. 2014.
[14] L. Pater and J. Berg, “An electrical analogue of the entire human circulatory system,” Medical Electronics & Biological Engineering, vol. 2, no. 2, pp. 161–166, Apr. 1964.
[15]I. Kokalari, T. Karaja, and M. Guerrisi, “Review on lumped parameter method for modeling the blood flow in systemic arteries,” Journal of Biomedical Science and Engineering, vol. 06, no. 01, pp. 92–99, 2013.
[16] M. Korurek, M. Yildiz, and A. Yuksel, “Simulation of normal cardiovascular system and severe aortic valve stenosis using equivalent electronic model,” Anadolu Kardiyoloji Dergisi/The Anatolian Journal of Cardiology, vol. 10, no. 6, Dec. 2010.
[17] M. Rupnik, F. Runovc, and M. Kordas, “The use of equivalent electronic circuits in simulating physiological processes,” IEEE Transactions on Education, vol. 44, no. 4, pp. 384–389, Nov. 2001.
[18] M. Korda<s, S. Leonardis, and J. Trontelj, “An electrical model of blood circulation,” Medical & Biological Engineering, vol. 6, no. 4, pp. 449–
451, Aug. 1968.
[19] Hassani, Kamran, Mahdi Navidbakhsh, and Mostafa Rostami. "Simulation of the cardiovascular system using equivalent electronic system."
BIOMEDICAL PAPERS-PALACKY UNIVERSITY IN OLOMOUC 150.1 (2006): 105.
[20] M. Abdi and A. Karimi, “A Computational Electrical Analogy Model to Evaluate the Effect of Internal Carotid Artery Stenosis on Circle of Willis Efferent Arteries Pressure,” Journal of Biomaterials and Tissue Engineering, vol. 4, no. 9, pp. 749–754, Sep. 2014.
[21] S. Ribaric and M. Kordas, “Teaching cardiovascular physiology with equivalent electronic circuits in a practically oriented teaching module,”
AJP: Advances in Physiology Education, vol. 35, no. 2, pp. 149–160, Jun.
2011.
[22] J. Dolenšek, T. Podnar, F. Runovc, and M. Kordaš, “Analog simulation of aortic and of mitral regurgitation,” Computers in Biology and Medicine, vol. 39, no. 5, pp. 474–481, May 2009.
[23] M. Abdolrazaghi, M. Navidbakhsh, and K. Hassani, “Mathematical Modelling and Electrical Analog Equivalent of the Human Cardiovascular System,” Cardiovascular Engineering, vol. 10, no. 2, pp. 45–51, Jun.
2010.
[24] Millikan, Robert Andrews, and Edwin Sherwood Bishop. Elements of electricity: a practical discussion of the fundamental laws and phenomena of electricity and their practical applications in the business and industrial world. American Technical Society, 1917.
[25] G. Thews, E. Mutschler, and P. Vaupel, Anatomie, Physiologie, Pathophysiologie des Menschen: mit 99 Tabellen, 4., durchges. Aufl., rev.
Nachdr. Stuttgart: Wiss. Verl.-Ges, 1991.
[26] Rideout, Vincent C. Mathematical and computer modeling of physiological systems. Englewood Cliffs, NJ: Prentice Hall, 1991.
[27] A. P. Avolio, “Multi-branched model of the human arterial system,”
Medical & Biological Engineering & Computing, vol. 18, no. 6, pp. 709–
718, Nov. 1980.
[28] J. R. Mitchell and J.-J. Wang, “Expanding application of the Wiggers diagram to teach cardiovascular physiology,” AJP: Advances in Physiology Education, vol. 38, no. 2, pp. 170–175, Jun. 2014.
[29] D. Tanné, L. Kadem, R. Rieu, and P. Pibarot, “Hemodynamic impact of mitral prosthesis-patient mismatch on pulmonary hypertension: an in silico study,” Journal of Applied Physiology, vol. 105, no. 6, pp. 1916–1926, Dec. 2008.
[30] M. A. Syed and R. H. Mohiaddin, Eds., Magnetic resonance imaging of congenital heart disease. London!: New York: Springer, 2012.
[31] R. E. Klabunde, Cardiovascular physiology concepts, 2nd ed.
Philadelphia, PA: Lippincott Williams & Wilkins/Wolters Kluwer, 2012.
[32] Smith, James John. "Circulatory Physiology-the essentials." (1980).
[33] J. P. Jacobs, C. Mavroudis, M. L. Jacobs, B. Maruszewski, C. I.
Tchervenkov, F. G. Lacour-Gayet, D. R. Clarke, T. Yeh, H. L. Walters, H.
Kurosawa, G. Stellin, T. Ebels, and M. J. Elliott, “What is Operative Mortality? Defining Death in a Surgical Registry Database: A Report of the STS Congenital Database Taskforce and the Joint EACTS-STS Congenital Database Committee,” The Annals of Thoracic Surgery, vol.
81, no. 5, pp. 1937–1941, May 2006.
