Hemodynamic Model of the Cardiovascular System During Valsalva Maneuver and Orthostatic Changes

Hemodynamic Model of the Cardiovascular System during Valsalva Maneuver and Orthostatic Changes

Niklas Moberg

November 23, 2011 Master’s Thesis in Engineering Physics, 30 credits Supervisors at Med-Uni Wien: Univ.-Prof. Dr. techn. H. Schima, Dipl.-Ing. Dr. techn F. Moscato Supervisors at TU Wien: Univ.-Prof. Dr. techn. A. Kugi, Dr.-Ing. W. Kemmetmüller, K. Speicher Examiner: Urban Wiklund

Umeå University Department of Engineering Physics SE-901 87 UMEÅ SWEDEN

Abstract

The goal of the Master’s Thesis was to extend an existing cardiovascular model to include the mechanics of the lungs, thus allowing to simulate breathing maneuvers such as the Valsalva maneuver and the Forced Vital Capacity maneuver. This included a remodeling of the pulmonary capillaries and of the existing interactions of the model with the intrathoracic pressure. The existing description of the vascular compartments was found to be insuﬃcient to describe the hemodynamic response to orthostatic changes and was extended to include a compartment representing the upper body. Stress relaxation was included into all the larger vascular compartments. The results showed an improved accuracy of the extended model when subjected to large intrathoracic pressure changes and during orthostatic stress. The internal responses of the newly modeled pulmonary capillaries were studied and veriﬁed against literature with satisfying results.

Sammanfattning

Målet med detta examensarbete var att utvidga en existerande kardiovaskulär modell till att inkludera lungmekanik, för att därigenom möjliggöra simulering av andningsmanövrar som Valsalva-manövern och Forced Vital Capacity-manövern. Detta inkluderade en omformning av de pulmonära kapillärerna samt hur modellen som helhet påverkades av det intrathorakala trycket. Den existerande vaskulära modellen ansågs vara otillräcklig för beskriva de hemodynamiska responsen under ortostatisk stress och utökades därför till att inkludera ett fack som representerade överkroppen. Stress relaxation inkluderas in i all större vaskulära kärlrum. Resultatet visade på en förbättrad noggrannhet hos den utökade modellen vid större intrathorakala tryckändringar och även vid ortostatisk stress. De interna svaren hos den utökade pulmonära kapillära modellen studerades och veriﬁerades gentemot litteratur med positiva resultat. Contents

1 Problem Description 2 1.1 ProblemStatement...... 2 1.2 Goals ...... 2 1.3 Purpose ...... 2 1.4 Methods...... 2

2 The Human Cardiovascular System 4 2.1 Basic Structure of the Cardiovascular System ...... 4 2.2 PhysiologyoftheHeart ...... 4 2.3 TheCirculatorySystem ...... 8 2.3.1 Arteries ...... 8 2.3.2 Arterioles ...... 9 2.3.3 Capillaries...... 9 2.3.4 VenulesandVeins ...... 9 2.3.5 SystemicCirculation ...... 10 2.3.6 Pulmonary Circulation ...... 10 2.4 RegulatoryMechanisms ...... 13 2.4.1 Arterial Baroreﬂex Controller ...... 13 2.5 PathologyoftheHeart...... 15 2.5.1 Hypertension ...... 15 2.5.2 SystolicDysfunction ...... 16 2.5.3 Cardiac Arrhythmia ...... 16 2.6 Mechanical Circulatory Assist Devices ...... 16 2.6.1 PulsatileDevices ...... 17 2.6.2 ContinuousBloodPumps ...... 17

3 Model Theory 19 3.1 Conservation of Mass/Volume ...... 19 3.2 ModelingofBloodVessels ...... 19 3.3 HeartModel...... 21 3.3.1 Active and Passive Pressure Functions ...... 22

i CONTENTS ii

3.3.2 Contractility Function ...... 24 3.3.3 Atria...... 25 3.3.4 Ventricles ...... 25 3.3.5 IntraventricularSeptum ...... 26 3.3.6 HeartValves ...... 28 3.4 BaroreﬂexController...... 28 3.4.1 Systemic Arterial Resistance Controller ...... 29 3.4.2 HeartRateController ...... 30 3.4.3 Unstressed Volume Controller ...... 31 3.5 SystemicCirculation ...... 33 3.5.1 Arterial Circulation ...... 33 3.5.2 VenousCirculation...... 34 3.6 LungsandAirwaysModel ...... 41 3.7 PulmonaryCirculation...... 46 3.7.1 PulmonaryArteries ...... 46 3.7.2 Pulmonary Capillary Model ...... 46 3.7.3 PulmonaryVeins ...... 49 3.8 LVADModeling...... 50 3.9 ValsalvaManeuver ...... 51 3.10FVCManeuver ...... 52

4 Matlab Simulink Implementation 55 4.1 Simulink...... 55 4.2 Matlab-code...... 55 4.3 SystemRequirements ...... 56

5 Results and Conclusions 57 5.1 Normal Physiological Conditions ...... 57 5.1.1 Breathing ...... 59 5.2 LeftHeartFailure ...... 61 5.3 OrthostaticStress ...... 62 5.4 ValsavaManeuver ...... 64 5.4.1 Pathological Conditions ...... 66 5.5 FVC-Maneuver ...... 68

6 Review 73 6.1 AchievedResults ...... 73 6.2 Limitations ...... 74 6.3 Futurework...... 74

7 Acknowledgements 76 CONTENTS iii

A Parameters and Variables 80

B Full Model Schematic 81 Introduction

This Master’s Thesis is divided into six main parts: a problem description, a basic introduction to the human cardiovascular system, the theory behind the model development, the model implementation, the results achieved and a discussion reviewing the results. The first chapter describes the aims of this Master’s Thesis and the methods used to accomplish them. The second chapter describes the working physiology of the human heart, the vessels, the body’s own control mechanisms and gives some basic information on the pathology of the heart. The third chapter describes the equations ruling the physical system, how they were derived and their physiological meaning. The fourth chapter explains how the model was implemented in Matlab-Simulink and the basic function calls within the code. The simulated results of the model are then presented and compared to experimental results from articles and books. The thesis then reviews the results including limitations, future work and includes a summary of the whole project. The main goal of this work is to build a comprehensive mathematical model that gives an accurate representation of the flows, pressures and blood volumes in the human cardiovascular system and its interaction with a Left Ventricular Assistant Device (LVAD). This model shall include the cardiopulmonary interaction with the lung mechanics and the response to orthostatic position changes. It is important that the hemodynamic responses of orthostatic position changes and breathing maneuvers, such as the Valsalva maneuver, are modeled correctly. This since the hemodynamic responses can be used to diagnose left heart failure in patients and also test reflex controller reactions. The Master’s Thesis is based on the Master’s Thesis done by Michael Baumann in cooperation with the Research Group in Cardiovascular Dynamics and Artificial Organs at the Medical University of Vienna situated at the General Hospital of Vienna and the Automation and Control Institute (ACIN) at Vienna University of Technology [1].

1 Chapter 1

Problem Description

1.1 Problem Statement

As mentioned in the introduction, this is a continuation from another Master’s Thesis done by Michael Baumann [1]. He modiﬁed and extended an existing mathematical model of the human cardiovascular system and its interactions with a cardiac support system developed by the Medical University of Vienna [30]. The goal of this project is to extend the model making it a more accurate representation of the human cardiovascular system.

1.2 Goals

I. Extend the existing mathematical model, found in [1], by including the interactions of the lungs and pulmonary capillaries.

II. Evaluate the new interactions by simulating changes in lung volume and pressure, and verifying the results against literature.

III. Extend the existing systemic venous model to better reproduce the hemodynamic responses associated with orthostatic stress.

1.3 Purpose

The purpose of including the cardiopulmonary interactions of the lungs is to be able to accurately simulate the inﬂuence of breathing maneuvers such as the Valsalva maneuver. Breathing maneuvers are non-invasive to the patients, the expected cardiovascular response is known and can therefore be used for diagnosis.

1.4 Methods

The Master’s Thesis starts with an introductory study of the human cardiovascular system and heart mechanics. This is needed to get a basic understanding of the hemodynamics of the body. The study is followed by a review of the equations describing the mathematical cardiovascular model and their physiological meaning as described in Baumann [1].

2 1.4. Methods 3

In order to extend the existing model to include the interactions of the lungs, a mathematical description needs to be created. This is done by searching for already existing models describing the cardiopulmonary interactions of the lungs and choosing a suitable one to implement. When a model is found, an in-depth study needs to be made to de- termine the important parts of the model, which simpliﬁcations can be made and how to include the new cardiopulmonary model into the entire cardiovascular system model. The Master’s Thesis then continues with extending the existing systemic venous circulation making it a more physiologically meaningful model. Again a literature review and an in-depth study needs to be made for existing models to ﬁnd a suitable model for implementation. Chapter 2

The Human Cardiovascular System

This chapter describes the basic structure of the cardiovascular system, the mechanics of the heart and common pathologies involving left heart failure.

2.1 Basic Structure of the Cardiovascular System

The human cardiovascular system is basically a complex array of valves and tubes representing the blood vessels. The heart is the pump that keeps the blood flowing through the system and the nervous system is the computer controlling the system. The heart can be found almost in the middle of the chest. There are blood vessels directly connected to the heart in which blood flows from the heart out into the rest of the body and through which blood returns back into the heart. Figure 2.1 shows a schematic of the human cardiovascular system. The figure shows how oxygenated blood flows from the left heart through arteries, capillaries, venules, veins, through the upper and lower vena cava into the right heart. This systemic circulation sup- plies the organs in the body with minerals and oxygen as is represented by the transition from oxygenated blood to oxygen-deprived blood in the systemic capillaries. Oxygen-deprived blood then flows from the right heart through the pulmonary circulation, also containing arteries, capillaries, venules and veins, and returns to the left heart. It is in the pulmonary circulation, more specifically in the capillaries, that blood is supplied with oxygen before returning to the left heart [13].

2.2 Physiology of the Heart

The heart can be divided into four parts: left and right ventricle and left and right atrium. The left and right ventricle are the two larger pumps, mentioned in the last section, that keep the blood flowing through the body and the lungs. The left and right atrium can be seen as two primer pumps that fill the ventricles with blood. The right and the left ventricle work basically in the same way, just under different pressures. The left heart pumps the blood out into the high pressure systemic arterial system and therefore needs to be able to create pressures higher than 100 mmHg. The right heart pumps the blood out into the low pressure pulmonary arterial system and just needs to create pressures up to 20 mmHg [13].

4 2.2. Physiology of the Heart 5

Pulmonary circulation–9%

Aorta Superior vena cava

Heart–7%

Inferior Systemic vena cava vessels Arteries–13%

Arterioles and capillaries–7%

Veins, venules, and venous sinuses–64%

Figure 2.1: Schematic of the human cardiovascular system with the direction of blood ﬂow going from the left heart (red) to the right heart (blue). Also marked is the percentage of blood volume stored in each part of the circulation [13].

HEAD AND UPPER EXTREMITY

Aorta

Pulmonary artery Superior vena cava Lungs

Right atrium Pulmonary Pulmonary vein valve Left atrium

Tricuspid Mitral valve valve Right ventricle Aortic valve Inferior Left vena cava ventricle

TRUNK AND LOWER EXTREMITY

Figure 2.2: Cross section of the heart and their respective inlets and outlets [13]. 2.2. Physiology of the Heart 6

Each ventricle can be thought of as an elastic chamber that allows for filling and emptying of the chamber through two one-way valves, one for input and one for output. Figure 2.2 shows a cross section of the heart detailing the different chambers, the heart valves and the blood vessels directly connected to the heart [13]. The heart’s pumping of the blood can be divided into two phases, a filling phase called diastole where blood from the atrium fills the ventricle and a contraction phase called systole where the ventricle contracts and pumps the blood out into the vessels. The amount of blood being pumped out into the body during a time unit is called the cardiac output qco. The cardiac output is determined by multiplying the heart rate fhr with the stroke volume Vsv. The heart rate is the number of beats the heart makes in one minute. The stroke volume is the amount of blood ejected by the ventricle during one contraction. An average person has normally a heart rate of 70 min−1 and a stroke volume of 70 ml which results in a cardiac output (qco = fhrVsv) of 4.9 l/min [17].

Pressure-Volume (PV) Diagram A important tool for studying the interactions of the heart and the circulatory system is the so called pressure-volume (PV) diagram. To better understand what the PV diagram is, first take a look at the time dependent pressures for one heart cycle of the left ventricle, which are depicted in figure 2.3. The PV diagram is created by plotting the pressure as a function of the volume, seen in figure 2.4.

Isovolumic relaxation Rapid inflow Ejection Atrial systole Isovolumic Diastasis contraction

120 Aortic Aortic valve valve closes 100 opens Aortic pressure 80

60 A-V valve A-V valve 40 closes opens

Pressure Hg) (mm Pressure 20 Atrial pressure a c v 0 Ventricular pressure 130 Ventricular volume 90 R

Volume (ml) 50 P

Figure 2.3: Time-dependent pressure and volume of the left ventricle [13].

The following explanation of the phases describes the working of the left ventricle corresponding to the illustrations shown in figures 2.3 and 2.4. However, it is important to remember that the right ventricle and the atria work in the same manner, the differences being that the pressures are lower and the names of the inlet/outlet valves of the heart chambers change. The points A, B, C and D represent important phases of the heart cycle. Point A in figure 2.4 indicates the so called end-diastolic-volume (EDV) and the end-diastolic-pressure (EDP). A is the point in time when the diastole ends and the systole begins. During the following period of the heart cycle, represented by moving from point A to B, the mitral valve (i.e. the left ventricle inlet valve) closes and the ventricle rapidly builds up pressure almost without changing its volume. This is called an isovolumic contraction (II). 2.2. Physiology of the Heart 7

300 SystolicESPVR pressure

250

200 Isovolumic relaxation Period of ejection 150 C III Isovolumic 100 B contraction EW 50 IV II A DiastolicEDPVR D pressure

Intraventricular pressure (mm Hg) Intraventricular pressure I 0 0 50 100 150 200 250 Period of filling Left ventricular volume (ml)

Figure 2.4: PV diagram of the left ventricle [13].

The point B is reached when the ventricular pressure exceeds the aortic blood pressure causing the opening of the aortic valve (i.e. the left ventricle outlet valve) and the ventricle ejects the blood into the blood stream. This phase is the so called ejection phase (III) and is represented by the segment B − C in the PV diagram. When the point C is reached, the aortic valve closes and the ejection phase is over. At point C the diastole begins with an isovolumic relaxation (IV) of the ventricle represented by the segment C − D. As the ventricular pressure falls below the pressure in the atrium the mitral valve opens and the ventricle starts to fill (I) along the curve segment D − A. When filling is complete, the PV diagram is back to its starting point A making a PV-loop. The stroke volume Vsv can easily be obtained from figure 2.4 by measuring the vertical distance B − C. Other things to note from the PV diagram in figure 2.4 are the two limiting functions for the PV-loop, which are the End-Systolic-Pressure-Volume-Relationship (ESPVR) and the End-Diastolic-Pressure-Volume-Relationship (EDPVR). The ESPVR is the top curve in the PV diagram and always coincides with the upper left shoulders of the PV-loop, point C. The ESPVR represents the contractility1 of the heart. The EDPVR is the bottom curve and always connects to the end-diastolic point A of the PV-loop. The EDPVR represents the passive stiffness of the heart and is a measure of the pressure-volume relation in a fully relaxed chamber. 1The maximum pressure that can be developed for a certain volume during an isovolumic contraction. 2.3. The Circulatory System 8

2.3 The Circulatory System

The human circulatory system is, as previously mentioned, divided into a systemic and a pulmonary circulation. When looking at the distribution of blood in the body it is said that about 84 % of the entire blood volume can be found in the systemic circulation and 16 % in the heart and lungs. Of the 84 % contained in the systemic circulation, 64 % can be found in the veins, 13 % in the arteries, and 7 % in the systemic arterioles and capillaries, see ﬁgure 2.1. Of the 16 % contained in the heart and lungs, the heart contains about 7 % and the pulmonary vessels 9 % of the blood [13].

120

100

60 Capillaries Venules veins Small veins Large cavae Venae arteries Pulmonary Arterioles Capillaries Venules veins Pulmonary

40 Pressure (mm Hg) (mm Pressure

20 Aorta arteries Large arteries Small Arterioles 0 0 Systemic Pulmonary

Figure 2.5: Typical pressure proﬁles in the diﬀerent vessels [13].

A schematic of the typical pressure profiles in the different vessels can be seen in figure 2.5. The reason for the oscillations observable in the circulation is the pulsatile nature of the heart which causes the systemic arterial pressure to fluctuates between a systolic (120 mmHg) and a diastolic (80 mmHg) pressure with an average pressure of 100 mmHg. The reason for the small increase in the amplitude of the pressure oscillations in the large arteries is due to wave reflection in the arterial tree2, for more information see Frans [8] and Milnor [29]. The pressure in the pulmonary circulation fluctuate for the same reason but at much lower values, between 25 and 8 mmHg with a mean of 16 mmHg [30]. The central venous pressure (CVP), measured in the inferior vena cava, has a mean value of about 4 mmHg. An illustration of the branching of the vessels of the systemic circulation can be seen in figure 2.6. It is illustrated how from the aorta, the arteries branch out into the arterioles, which further branches out into the smallest vessels, the capillaries. The capillaries then converge into larger vessels, the venules, which finally converge into the veins and the venae cavae.

2.3.1 Arteries Arteries are thick-walled vessels and have a large amount of elastin and collagen ﬁbers. These ﬁbers allow for the expansion of the arteries and can thus provide an additional storage of blood. This expansion property is especially needed during systole when blood is rapidly pumped out into the arterial system. During diastole the diameter of the arteries is

2The pulsatile blood flow propagates as a wave through the arterial branches. Wave reflections are caused by the elasticity of the vessels and changes in the arterial geometry, stiffness and bifurcations [8]. 2.3. The Circulatory System 9

Aorta Vena Cava Small Artery Large Artery Capillaries Vein

Arteriole Venule

Figure 2.6: Schematic of the branching of the vessels in the systemic circulation [21].

in turn reduced, which contributes to the stabilization of the arterial pressure. The largest artery in the human body is the aorta with a diameter of about 25mm [17].

2.3.2 Arterioles As described in Heller [17], "the arterioles are narrower and structured diﬀerently than the arteries. In relation to their diameter, the arterioles have a greater wall thickness and a greater proportion of smooth muscle than the arteries". This allows the blood ﬂow to be regulated throughout the organs. The arterioles are referred to as resistance vessels because of their high and tunable resistance [17].

2.3.3 Capillaries The capillaries are the most narrow vessels in the entire cardiovascular system. The capillaries have a diameter of only 3 µm. In order to get a better understanding of this dimension remember that the blood cells in the human body have a diameter of 7 µm. Thus, the blood cells have to deform in order to travel through the capillaries. The capillaries, in contrast to the arteries and arterioles, have no smooth muscles and can therefore not change their diameter [17].

2.3.4 Venules and Veins The venules and veins have very thin walls when compared to their diameter. This makes them very sensitive to small changes in transmural pressure, i.e. the diﬀerence between internal and external pressure of the vessel. As the arteries, the venules and veins also have smooth muscles in the walls which allow a change of their diameter. Compared to the arteries and arterioles, the venules and veins have a much higher compliance, which means that they can expand a lot more. This in turn, allows to store much blood in these vessels, which is the reason why so much of the blood in the body (64 %) can be found in the veins. Most of the larger venous vessels also have one-way valves that prevent reverse ﬂow. This is of great importance to the workings of the cardiovascular system when standing up or lying down [17]. A summary of the number, the cross-sectional area and some physical characteristics of all vessels can be found in table 2.1. 2.3. The Circulatory System 10

Table 2.1: Summary of the number, the cross-sectional area and some physical characteristics of the circulatory vessels [17]. Arteries Arterioles Capillaries Venules Veins Internal diam. 2.5 cm 0.4 cm 30 µm 5 µm 70 µm 0.5 cm 3 cm Wall thick. 2 mm 1 mm 20 µm 1 µm 7 µm 0.5 mm 1.5 mm Number 1 160 5 · 107 1010 108 200 2 Cross-area 4.9 cm2 20 cm2 400 cm2 4500 cm2 4000 cm2 40 cm2 14 cm2

2.3.5 Systemic Circulation The systemic circulation starts at the left ventricle where oxygenated blood is ejected into the aorta. The blood is then distributed over the systemic arteries and arterioles into the systemic capillaries supplying blood to the organs. The oxygen-deprived blood is then transported through the systemic venules and veins into the right heart.

Orthostatic Stress The pressures in the vessels are highly dependent on the body’s position. Orthostatic stress is something a body is subjected to when changing body position, e.g. from lying down to standing up. In this case a redistribution of blood volume will occur due to hydrostatic pressure changes. This pressure comes from the weight of the blood in the vessels. When lying down, all the organs and limbs are on the same level when compared to the ground. When standing up this will no longer be the case. In the lower part of the body an increase in pressure will occur, as opposed to the upper body where a decrease will occur. The hydrostatic reference point that does not experience any effects lies in the throat and neck. The systemic vessels supplying blood to the lower body experience a pressure rise of up to 90 mmHg as can be seen in figure 2.7. This pressure rise will cause the vessels to distend, especially the more compliant veins, which will cause an accumulation of blood. Since the opposite is true for the upper body, a displacement of blood will occur throughout the systemic vessels [13]. This is the reason why sometimes people get light headed and even faint when standing up too quickly. The one-way valves in the larger venules and veins help to redistribute the blood after standing up by preventing back flow when the blood is expelled [17].

2.3.6 Pulmonary Circulation The pulmonary circulation carries the blood from the right heart via the pulmonary arteries, through the capillaries in the lungs and through the pulmonary veins into the left side of the heart. When the blood passes the lungs it is supplied with oxygen and delivered to the organs through the systemic circulation. The pressures in the pulmonary circulation are much smaller than the ones in the systemic circulation because of the fact that the structure of the pulmonary circulation is diﬀerent from the systemic circulation. The pulmonary circulation only feeds blood to one single organ, the lungs. This, in combination with the fact that the dimensions of the vessels in the pulmonary circulation are larger than those in the systemic circulation, causes the overall resistance to be much less thereby lowering the pressures [17]. 2.3. The Circulatory System 11

Sagittal sinus -10 mm

0 mm 0 mm

+ 6 mm

+ 8 mm

+ 22 mm

+ 35 mm

+ 40 mm

+ 90 mm

Figure 2.7: Schematic of orthostatic eﬀects present in a fully upright person, where mm stands for [mmHg] [13]. 2.3. The Circulatory System 12

Cardiac Output Dependence of Pulmonary Arterial Pressure When the cardiac output increases, pressures in the blood vessels rise due to an increase in the blood volume flowing through the vessels. The rise of pulmonary arterial pressure pap is, however, limited because of the fact that a rise in cardiac output qco causes a decrease in the pulmonary capillary resistance. This is possible since, during normal conditions, some capillary vessels are collapsed and not yet "open" to blood flow. When the cardiac output increases more blood is forced through the pulmonary capillaries opening more capillaries and expanding them. This is the cause of a decrease in pulmonary resistance which counteracts the increase of the pulmonary arterial pressure [13]. The dependence of pap on qco can be seen in figure 2.8.

Normal value 20

10 pressure (mm Hg) (mm pressure Pulmonary Pulmonary arterial

0 0 4 8 12 16 20 24 Cardiac output (L/min)

Figure 2.8: The relation between pulmonary arterial pressure and cardiac output [13].

Lungs Breathing happens because the human body can control the pressure surrounding the lungs. The intra-thoracic pressure (pith) or also called the pleural pressure is the pressure difference between the pressure in the thoracic cage and the pressure in the lungs [25]. When no strain is put on the lung, it is collapsed and is nearby empty of air. A person can never fully empty the lungs even when one forcibly exhales to the maximum. The lungs will still retain some volume and this volume is called the residual volume. When a person wants to inhale, the lung needs to expand. As a result, in order to inflate the lung, a negative pressure in the thorax relative to the pressure in the lungs is needed. During inspiration the pressure in the thorax decreases and the intra-thoracic pressure becomes more negative and pulls on the lung from the outside, thus expanding it and drawing in air. During expiration the intra-thoracic pressure increases and the lung contracts, thus emptying it of air [39]. During inspiration the intra-thoracic pressure pith is at rest at about −3 mmHg and during inspiration at about −6 mmHg [39]. Figure 2.9 shows the working volume of the lungs. The tidal volume is the normal working volume during normal breathing. The tidal volume is normally around 500 ml [13]. As can be seen from the figure, the total lung capacity is the maximum lung volume that can enter the lungs, the residual volume is the volume that the lungs retain even after fully exhaling and the vital capacity is the total capacity minus the residiual volume. The residual 2.4. Regulatory Mechanisms 13 volume of an average person is about 1200 ml [13]. Figure 2.10 shows the interactions of

6000 Inspiration

5000 Inspiratory Inspiratory Vital Total lung reserve capacity capacity capacity 4000 volume

Tidal 3000 volume

Lung volume (ml) 2000 Expiratory Functional reserve volume residual capacity 1000 Residual Expiration volume

Time

Figure 2.9: Illustration of the working area of the lungs [13].

the lungs with the pulmonary circulation and how the blood is supplied with oxygen (O2). It also illustrates how the oxygenated blood is transported through the heart and carries oxygen to the rest of the body.

2.4 Regulatory Mechanisms

The human circulatory system has several regulatory mechanism that respond to and counteract physiological changes. These changes include the maintenance of a sufficient cardiac output and perfusion pressure for the organs [39]. The regulatory mechanisms can be sepa- rated into three different kinds of mechanisms: short-, medium- and longterm mechanisms. As the name implies, the shortterm mechanisms respond to rapid changes in local blood flow and react within seconds. An example of a shortterm regulatory mechanism is the pressure sensitive baroreflex mechanism. As for the medium- and longterm mechanisms, they respond to the local blood flow changes very slowly. The setting of the optimum operating point can take, for the mediumterm mechanism, from minutes up to several hours and for the longterm mechanisms from days, weeks up to several months [13]. An example of a slower reacting mechanism is the stress relaxation of the systemic veins. Figure 2.11 shows how the regulatory mechanisms are divided according to their response time.

2.4.1 Arterial Baroreflex Controller As is stated in Heller [17], "appropriate systemic arterial pressure is perhaps the single most important requirement for proper operation of the cardiovascular system." The aortic pressure is regulated through many different coupled systems, the most important one being the baroreflex controller [17]. The baroreflex is a short-term pressure sensitive controller that adjusts the aortic pressure through changes in the systemic arterial resistance, the unstressed volume of the systemic venous compartments, the heart rate and the heart contractility 2.4. Regulatory Mechanisms 14

expired air inspired air O2 O2

CO 2 CO 2

alveolar air

O2 CO 2

CO 2

O2-poor blood O 2-rich blood

O2 O2 CO 2 CO 2

tissue fluid

CO 2 O2 CO 2 internal respiration gas transport external respiration

respiring tissues

Figure 2.10: The process of gas exchange between the lungs and the cardiovascular system [27]. 2.5. Pathology of the Heart 15

Renin-angiotensin-vasoconstriction • • !!

e r CN d u 11 S o s s is o c l l h e 10 b r o e r m – p t l ic n 9 a e r o e n s m c p e 8 o u l Baroreceptors n R se o 7 v 6 5 Chemorec 4 eptor erone s ldost 3 A on xati 2 rela ess 1 Str Acute change in pressure at this time pressure in Acute change 0 CapillaryFluid shift • Maximum feedback gain feedback Maximum at optimal pressure 015301248163212 4 816 124 816

Seconds Minutes Hours Days Time after sudden change in pressure

Figure 2.11: The diﬀerent regulatory mechanisms in the human body and their individual response times [13].

accordingly. The sensors, so called baroreflex receptors, are located in the walls of the aortic arch, the carotid sinus and in the major thoracic arteries [39]. In order to better understand how the regulation of arterial pressure works with the baroreflex controller, the regulatory responses to a sudden drop in arterial pressure is explained. When the arterial pressure drops, the baroreflex receptors in the arterial wall tell the controller to increase sympathetic nerve activity. This in turn increases the heart rate, increases the contractility of the heart, increases venous tone and constricts the arterial vessels. These adjustments increase the arterial pressure towards the optimal value [17].

2.5 Pathology of the Heart

When a patient is suﬀering from heart failure it simply means that there is a failure of the heart to pump blood as well as it normally does. There are a lot of causes for and types of heart failures and only a few will be presented in this section. The intention is mainly to give the reader an introduction to the most common causes of heart failure, for more information see Guyton [13] and Heller [17].

2.5.1 Hypertension Hypertension is deﬁned as a chronic increase of the arterial blood pressure above 140/90 mmHg (systole/diastole values). It is not actually a pathology of the heart but it greatly increases the risk of a coronary artery disease, heart failure, stroke and other cardiovascular 2.6. Mechanical Circulatory Assist Devices 16 diseases. Hypertension is one of the most common cardiovascular problems in the world, aﬀecting over 20 % of the adult population in the western world. In most cases (up to 90 %), the primary cause of hypertension is unknown and the only thing that can be treated is the symptom of high blood pressure [17].

2.5.2 Systolic Dysfunction Systolic heart failure is defined as a "lower than normal cardiac function curve" [17], which means that at any given filling pressure a reduction, compared to normal, in cardiac output can be observed. The reduction in cardiac output causes a lowered arterial pressure which in turn causes reflex activation of the sympathetic nerves (baroreflex). This increased reflex activation counteracts the heart failure and returns the cardiac output near to its normal working value. But in the long term, the cardiovascular operation cannot remain at this increased reflex activation but will cause an accumulation of cellular fluid in the body, an increase in the end-diastolic volume (EDV) and a dilation of the heart muscles. Excessive cardiac dilation will impair the functionality of the heart and thus put more strain on the cardiac muscles [17]. When the heart becomes even more severely damaged, the reflex mechanisms will not be able to reestablish a sufficient cardiac output. In this case the accumulation of fluid has reached a level where any further retention will not help in cardiac output recovery but instead cause increasingly severe edema3 [15]. In this situation the heart normally can no longer recover and a heart transplant is needed. When an immediate transplant is impos- sible, Ventricular Assist Devices (VADs) can be used to establish hemodynamic stability and the opportunity for rehabilitation before transplant. For some patients with incurable diseases as cancer or severe infections, who are therefore not viable for transplant, VAD support is the only chance for survival [30]. For more information on systolic dysfunction and cardiac failure see Guyton [13] and Heller [17].

2.5.3 Cardiac Arrhythmia Cardiac arrhythmia is defined as a disturbance in the heart rhythm [39]. Most cardiac arrhythmias are benign4, e.g. those where the heart sometimes skips a beat or has an extra beat. Cardiac arrhythmia come from the disturbance of the electrical signals that are sent to the heart, some might be delayed or even blocked completely. The disturbances have the effect that the heart does not contract rhythmically and therefore cause the heart to pump less efficiently [13]. An example of cardiac arrhythmia is ventricular tachycardia which occurs when the ventricles are driven at unusually high rates as a compensatory mechanism for heart failure.

2.6 Mechanical Circulatory Assist Devices

A mechanical circulatory assist device is a cardiac support system that is intended to help a failing heart to pump the blood out into the limbs and organs. This section will concentrate on the so-called left ventricular assist devices (LVADs). Cardiac support systems have as a main goal the improvement of the survival rate of transplant patients by bridging them to the transplantation. Other aspects of usage are for the improvement in "quality of life",

3Swelling in the hands, arms, legs and feet caused by fluid retention [17]. 4Of no danger to health; not recurrent or progressive; not malignant, [44] 2.6. Mechanical Circulatory Assist Devices 17 for ventricular recovery and maybe in the future can be used as a longterm or permanent support system for cardiac patients [30]. The research on mechanical circulatory assist devices started as late as in the 1960s and has evolved into a standard therapy supporting patients with end-stage heart disease. The first left ventricular assist devices were intended for short-term usage in weaning patients from heart-lung machines. The first LVAD used as a bridge to transplantation was a pneu- matic LVAD in 1986. The first electrical powered LVAD used was the HeartMate LVAD implanted in the U.S. in 1991. Since then, the research on electrically powered LVAD has continued to this day [9]. Based on their flow characteristics, LVADs are generally divided into two categories, pulsatile and continuous flow pumps [38]. The pulsatile flow pumps mimic the pulsatile nature of the heart by pumping out the blood in pulses while the continuous flow pumps supply a continuous blood flow.

2.6.1 Pulsatile Devices Pulsatile pump systems consist of a chamber that is ﬁlled during the cardiac cycle and emptied during the ejection phase. The pulsatile devices are built to follow the beating of the human heart. A lot of devices use a bag made of polyurethane or a shell/membrane combination. A schematic of an implanted pulsatile LVAD can be seen in ﬁgure 2.12.

A Pulsatile-Flow LVAD

Aorta External battery One-way outflow One-way inflow Left pack valve (closed) valve (open) ventricle Blood- Pump Skin pumping housing Flexible entry chamber diaphragm site

Blood flow External system controller Actuator Motor Pusher Pulsatile-flow Percutaneous bearing plate Percutaneous LVAD lead lead

Figure 2.12: Schematic of an implanted pulsatile LVAD and a cross section of the device [38].

The pulsatile LVAD works by alternately filling and emptying the bag, thereby pumping the blood out into the body. The biggest advantage of using a pulsatile LVAD in comparison to a continuous flow LVAD is the generation of pulsatile blood flow, which generates a flow profile more similar to the human heart blood flow. The biggest disadvantage lies in the large dimensions and in the complex mechanism involved [30]. Moreover, they are usually more expensive than the continuous flow LVADs [1].

2.6.2 Continuous Blood Pumps Continuous blood pumps or rotary blood pumps, as they are also called, work, as the name suggests, with an impeller that rotates at high speeds and creates a continuous blood flow. They are characterized by their compact design and simple functionality. They have only 2.6. Mechanical Circulatory Assist Devices 18 one moving part, the impeller, which makes them very reliable and it is the compact design that produces less bodily stress on implant patients than in the case of the larger pulsatile LVADs. Rotary LVADs are connected to the left ventricle and pump the blood via an inflow cannula through a rotor and out through an outflow cannula into the aorta. Figure 2.13 shows a schematic of an implanted rotary LVAD.

B Continuous-Flow LVAD

From left To aorta ventricle

Motor Pump housing Outlet stator and diffuser

Percutaneous Blood lead flow Continuous- flow LVAD Rotor Inlet stator and blood-flow straightener

Figure 2.13: Schematic of an implanted rotary LVAD and a cross section of the pump [38].

Rotary pumps can be divided into three categories: radial (centrifugal), diagonal and axial rotary pumps, see ﬁgure 2.14.

CENTRIFUGAL DIAGONAL AXIAL

Figure 2.14: A schematic of the direction of ﬂows through the three types of rotary LVADs, radial, diagonal and axial LVAD [30].

As can be seen in the schematic shown in figure 2.14, radial rotary pumps have an inflow path that is parallel to the rotation axis while the outflow path is orthogonal to it, e.g. the HeartWare R Ventricular Assist System, Heartware International Inc., Framingham, Mas- sachusetts, USA. Diagonal rotary pumps have an inflow that is parallel to the rotational axis while the outflow path is diagonal to it, e.g. the Deltastream, Medos Medizintechnik, Stol- berg, Germany. Axial rotary pumps have both inflow and outflow parallel to the rotational axis, e.g. the DeBakey, Micromed Cardiovascular, Houston, Texas, USA. Chapter 3

Model Theory

This chapter presents the mathematical model of the cardiovascular system. The model described in this chapter is based on an existing model [1]. Therefore, parts of the theory in this chapter have been taken from the work of Baumann [1]. The major contributions of this Master’s Thesis are the pulmonary circulation model and the systemic venous model. A schematic of the model of the full cardiovascular system can be seen in the Appendix in ﬁgure B.1. This can be compared with the old schematic created by Baumann [1] in ﬁgure B.2.

3.1 Conservation of Mass/Volume

For all chambers and vessels the law of mass conservation can be applied. When assuming that the fluid flowing within the chambers and vessels, in this case blood, is an incompressible fluid, volume conservation also applies. This means that, for a given point in the cardiovascular circulation, the difference between the volume flow of blood coming in qin and going out qout has to account for the change in volume V . This can be expressed as dV = q − q . (3.1) dt in out 3.2 Modeling of Blood Vessels

A segment of any blood vessel can be represented as a tube with blood flowing mostly in a laminar way, according to figure 3.1 [17]. Blood flow through a vessel segment can only occur when there is a pressure difference between p1 and p2. This difference is the driving force that moves the blood through the segment. When the blood moves through the segment, friction will develop between the blood and the stationary walls, thereby resisting the movement. This is expressed as a so-called vascular resistance R. The stationary flow q through a tube can be calculated as the pressure drop ∆p over the tube divided by R [17]. ∆p p − p q = = 1 2 . (3.2) R R According to the equation of Poiseuille the resistance R can, for a fluid of viscosity η, be expressed as 8ηL R = . (3.3) πr4

19 3.2. Modeling of Blood Vessels 20

pa A

p1 r p2

pi qout qin

Figure 3.1: Schematic of a cardiovascular vessel with length S, cross-section area A, radius r, ﬂow q and pressure diﬀerence p1 − p2 > 0. The pressure inside the vessel is described as pi and the external pressure as pa [17].

This means that a change in the radius r of the vessel greatly aﬀects the resistance R. This is an important principle e.g. in baroreﬂex control [17]. It is important to note that blood does not have constant viscosity but the viscosity of blood is dependent on the shear rate and the blood composition [1]. About 40 % of blood is made up of blood cells. These cells are suspended in a plasma1 that accounts for the remaining 60 % [17]. Blood vessels are elastic and there exists a relationship between the distending pressure2 and the blood volume contained within the vessel. The distending pressure is assumed to be linearly correlated with the active volume, as described in [20]

V − V0 = C(pi − pa), (3.4) where V is the volume in the vessel, V0 is the dead volume at pi − pa = 0 and C is the so-called compliance of the vessel. In order to better understand and describe the modeling of the cardiovascular system, the vessels can be expressed using an electrical equivalent. The electrical description, equivalent of figure 3.1, can be seen in figure 3.2. In figure 3.2 the flow q through the vessel is equivalent

qin R1 L qout R2 pi p1 p2

0 mmHg Figure 3.2: Electrical schematic of a cardiovascular vessel segment where R1 + R2 = R.

1The plasma has approximately the same viscosity as water [17]. 2 Pressure difference between internal pressure pi and external pressure pa of the vessel. 3.3. Heart Model 21 to the current I, the pressures p to the voltage V , the resistance to blood flow R to electrical resistance R and the inductance L caused by blood acceleration to electrical inductance L. The resistance R of the tube has, in figure 3.2, been divided into two parts R1 and R2, where R1 +R2 = R. The values of R1 and R2 can be set as to model the flow characteristics the vessel should demonstrate. The internal pressure of the vessel pi can be calculated from equation (3.4). V − V p = 0 + p . (3.5) i C a

The outﬂow qout of the vessel can be calculated from equation (3.2)

pi − p2 qout = . (3.6) R2 To calculate the inflow into the vessel one must consider the effect of inertia for blood acceleration. This is represented as the inductance L in the figure 3.2, which is important to consider at points where large accelerations in blood flow takes place, such as during opening of the heart valves. Assuming a flat velocity profile the inductance L can, according to [34], be expressed as ∆p L = , (3.7) q˙ where ∆p is the pressure drop over the inductance and q˙ is the acceleration in blood flow. (3.7) can be used to express the change in inflow to the vessel as a differential equation

p − p − q R q˙ = 1 i in 1 . (3.8) in L Equation (3.4) can be used to set up the differential equation describing the pressure change within the vessel due to blood flow. By taking the time derivative and assuming that the dp change in external pressure is negligible a = 0 and C is constant, the differential equation dt can be expressed as V˙ = Cp˙i. (3.9) Using equation (3.1), the differential equation for pressure change within a vessel becomes q − q p˙ = in out . (3.10) i C 3.3 Heart Model

The equations for the modeling of the heart have been taken from Baumann [1]. The important eﬀects that need to be considered when modeling the human heart are:

I. The pressure and volume changes in the atria and ventricles and the contractility of the heart.

II. The intraventricular septum which is an elastic wall that separates the left and right ventricle.

III. The inﬂow and outﬂow valves of the ventricles and atria.

In the following subsections, these points will be considered in detail. 3.3. Heart Model 22

3.3.1 Active and Passive Pressure Functions According to Campbell [3] the ventricles and atria can be modeled using a non-linear time- varying elasticity and an internal resistance. The relation between the pressure and the volume inside one of the chambers can be described by

p(t)= pe (V (t) ,t) − R · q, (3.11) where t represents the time, V (t) the blood volume inside the chamber, p (t) the instanta- neous blood pressure inside the chamber, R the internal resistance and q the flow out of the chamber. The internal resistance R represents the drop in pressure in the ventricle during the ejection phase and comes from friction effects. The resistances for the left and right ventricle have been taken from Ursino [40]. The friction effect for the atria is negligible and thus the resistance term is set to zero. The pressure function pe (V (t) ,t) depends on the non-linear time-varying elastance of the chambers. The mathematical description of pe (V (t) ,t) can be given in the form

pe (V (t) ,t)= pp (V (t)) + pa (V (t)) Fiso (t) , (3.12) where pp (V (t)) and pa (V (t)) represent the passive and the active pressure-volume relationship, respectively, and Fiso (t) represents the normalized ventricular contraction function. pp is also the end-diastolic pressure-volume relation (EDPVR) and pp + pa the end-systolic pressure-volume relation (ESPVR). For the left and right atria, pp is described as a linear function of V pp,a (Va)=(Va − V0,a) Emin,a + p0,a, (3.13) where Va is the volume in the atrium, V0,a is the dead volume, p0,a an offset pressure and Emin,a the elasticity of the atrium. For the left and right ventricle respectively, the non-linear passive pressure-volume relationship pp is approximated by a hyperbolic function λ pp,v (Vv)= + kVv + pλ, (3.14) Vsat − Vv where λ is a weighting factor, Vsat the maximum blood volume in the ventricle, Vv the actual blood volume and pλ is a pressure offset. The linear coefficient k is determined through the least squares method as to minimize the square error between the pressure in the ventricle pp,v(Vv) and the line Emin,v(V − V0)+ p0 in the low volume region. Emin,v is the linear elastance of the ventricles in the low volume region, see figure 3.3. The additional hyperbolic term is added to describe the limited volume of the ventricle. When blood is filling the ventricle, Vv increases and the ventricle expands. But the ventricle can not expand to an infinite size. As filling continues and Vv → Vsat the pressure pp,v will rise drastically eventually preventing any further expansion of the ventricle. The active pressure-volume relation pa is, for all chambers, modeled by using a parabolic relation as described in [2].

∗ 2 V − V (t) ∗ p (V (t)) = 1 − p . (3.15) a V ∗ − V " 0 # This relation is characterized as a downward concavity with a maximum value at (V ∗, p∗) and passing through the point (V0, 0), where V0 is the so called ventricular dead volume, see ﬁgure 3.4. The point (V ∗, p∗) describes the maximum contractility3 of the respective ventricle or atrium. Figure 3.4 shows the pressure-volume relationship for the active pressure part 3.3. Heart Model 23

10 Pressure [mmHg]

EDPVR 5

(V0,p0) Emin,v(V − V0)+ p0 0 0 50 100 150 200 250 Volume [ml]

Figure 3.3: An example of the passive pressure (pp)volume relationship (EDPVR) of the ventricle [23, 30].

300 (V ∗,p∗)

250

200

150 (V ∗,p∗)

100 Pressure [mmHg]

0 (V0,0)

−50 0 100 200 300 400 500 600 Volume [ml]

Figure 3.4: An example of the active pressure (pa) volume relation of the left ventricle, both in the physiological case (solid line) and in the pathological case (dashed line) [1].

of the left ventricle. The scope of the curve shown in ﬁgure 3.4 is in the physiological case from V0 to 150 ml and in the pathological case from V0 to 250 ml [1]. The parameter values describing the ventricles and atria can be found in Colacino [5] and [30] for the physiological case and in [23] for the pathological case simulating left heart failure.

3The point at which the hearts generates its maximum pressure in an isovolumic contraction. 3.3. Heart Model 24

For more information on the derivation of the passive and active pressure volume relations of the ventricles and atria see Baumann [1].

3.3.2 Contractility Function

The normalized contractility function Fiso describes the contraction of the ventricles and the atria. Fiso is basically a time-varying function and has the same shape for the ventricles as for the atria. The only diﬀerence is that the contractility function for the atria has a time displacement due to the fact that the atria contract before the ventricles. In Baumann [1] and Moscato [30] Fiso has been determined from data given in Senzaki [36] and approximated by polynomial interpolation. For detailed information on the derivation of the contractility function Fiso see Baumann et al. [1, 30, 36] and the works cited therein.

(t2, 1) 1

0.8

[1] 0.6 ) t ( iso

F 0.4 (t1,Fiso(t1))

0.2 T sys , 0 T 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t/T [1]

Figure 3.5: The approximated contractility function Fiso [30].

Figure 3.5 shows the approximated contraction function Fiso according to [30]. Three polynomials are used in the normalized time intervals [0, t1], [t1, t2] and [t2, Tsys/T ]. The heart period T is set as the sum of Tsys and Tdia [1]

T = Tsys + Tdia, (3.16) where Tsys is the time of the systolic phase of the heart and Tdia is the time of the diastolic phase of the heart. Both Tsys and Tdia depend on the heart rate fhr. The relationship between the heart rate and the diastolic part Tdia was calculated as in Vollkron [41]

1 − − − − T = e 0.01207(fhr 40) + e 0.038(fhr 40) . (3.17) dia 2 3.3. Heart Model 25

The parameters, t1, t2 and Fiso (t1) are set to the values given in Moscato [30] The time t2 2 corresponds to the value T /T . The point (t ,F (t )) is set to the value (0.32t , 0.42). 3 sys 1 iso 1 2 At the point (t1,Fiso (t1)), the slope is set to fs = 1/t2. Using piecewise deﬁned cubic polynomials, Fiso can then be expressed as

3 2 a13τ + a12τ 0 <τ

3.3.3 Atria It is known from the electrocardiogram4 (ECG) analysis of the P-wave5 that the contraction of the atria occurs before the contraction of the ventricles. For more information on ECG and on the P-wave see [17]. The time lag of the atrial contraction to the contraction of the ventricles is about 20 % of the cardiac cycle. Equation (3.1) can be used to set up the diﬀerential equation for the atria. This gives for the left atrium:

V˙la = qvp − qin,lv with Vla(0) = Vla,0, (3.19) where qvp is the flow coming from the pulmonary system into the left atrium, qin,lv is the flow out of the atrium into the left ventricle and Vla,0 is the dead volume in the atrium. The differential equation for the right atrium is described in the same manner.

V˙ra = qvs − qin,rv with Vra(0) = Vra,0, (3.20) where qvs is the ﬂow coming from the systemic venous system into the right atrium. By using (3.12), (3.13), (3.15) and, since the heart is within the thoracic cage, adding the intrathoracic pressure pith, the expression for the pressure-volume relationship of the left and right atrium takes the form

∗ 2 V − V (t) ∗ p =(V − V ) E + p + 1 − ia ia p F (t)+ p , (3.21) ia ia 0,ia min,ia 0,ia V ∗ − V ia iso,a ith " ia 0,ia # where i = {l, r} represent the left and right atrium [30].

3.3.4 Ventricles Equation (3.1) is used to set up the diﬀerential equation for the ventricles. This gives for the left ventricle V˙lv = qin,lv − qout,lv with Vlv(0) = Vlv,0, (3.22)

4ECG is a diagnostic tool that measures and records the electrical activity of the heart [17]. 5The P-wave is a pressure-wave that can travel through the blood [17]. 3.3. Heart Model 26

where qin,lv is the ﬂow coming from the left atrium into the left ventricle, qout,lv is the ﬂow out of the left ventricle into the aortic system and Vlv,0 is the dead volume in the ventricle. The equation for the right ventricle can again be described in the same manner

V˙rv = qin,rv − qout,rv with Vrv(0) = Vrv,0. (3.23)

By using (3.12), (3.14), (3.15), adding the intrathoracic pressure pith and remembering to subtract the friction force in (3.12), the pressure-volume relationship of the left and right ventricle takes the form

∗ 2 λ V − V (t) ∗ p = + k V + p + 1 − iv iv p F (t)+ iv V − V iv iv λ,iv V ∗ − V iv iso,v (3.24) sat,iv iv " iv 0,iv # pith − Rivqout,iv, where Riv is the internal resistance of the ventricles, i ∈ {l, r} [30].

3.3.5 Intraventricular Septum The following sections discuss the interaction of the ventricles with the intraventricular septum. The intraventricular septum is an elastic wall separating the left and right ventricle. Since the pressures in the left and right ventricle differ, the intraventricular septum will be displaced depending on this difference and this, in turn, influences the volumes in the ventricles [1]. When not taking the effect of the intraventricular septum into account the pressures in the ventricles can be expressed according to equation (3.11) and (3.24)

piv (Viv)= pp,iv (Viv)+ pa,iv (Viv) Fiso,v (t) − Rivqout,iv + pith, i = {l, r}. (3.25)

When including the intraventricular septum, the volumes Viv in the ventricles need to be adjusted in the form Vlv = Vlvf + Vspt (3.26a)

Vrv = Vrvf − Vspt, (3.26b) where Vspt is the volume displacement in the ventricles caused by the displaced intraventricular septum and Vivf the non-adjusted ventricular volumes, see figure 3.6. As can be seen from figure 3.6, Vspt is actually a fictitious volume with the initial value Vspt,0 = 0. In reality there are just the left ventricular Vlv and right ventricular Vrv volumes. The intraventricular septum is displaced due to the pressure difference in the ventricles

pspt = plv − prv. (3.27)

The dynamics of the intraventricular system is described in a similar way as for the ventricles. The pressure pspt is divided into an active and a passive part, where the active part is aﬀected by the contractility function for the ventricles Fiso,v

pspt = pp,spt + pa,sptFiso,v. (3.28)

Assuming that the intraventricular septum has a linear pressure-volume relationship for both the active and the passive part, pp,spt and pa,spt can be described by

pp,spt = Espt,minVspt (3.29a)

pa,spt = Espt,maxVspt, (3.29b) 3.3. Heart Model 27

Vspt

Vrvf Vlvf prv plv

Figure 3.6: Schematic cross-section of the ventricles illustrating the displacement of the intraventricular septum [1].

6 [ml]

spt 4 V

0 0 20 40 60 80 100 120 pspt [mmHg]

Figure 3.7: Simulated VP loop of the intraventricular septum [1].

where Espt,min is the minimum elastance of the intraventricular septum and Espt,max the maximum elastance. The parameters Espt,min and Espt,max have been optimized to ﬁt the Volume-Pressure loop of the intraventricular septum observed in literature [1, 4] and seen in ﬁgure 3.7. Adjusting the new volumes in the ventricles Vivf and using (3.11), (3.29a), (3.29b), equation (3.27) can be expressed as

Espt,minVspt + Fiso,vEspt,maxVspt = plv (Vlvf ,Fiso,v) − prv (Vrvf ,Fiso,v) (3.30) 3.4. Baroreﬂex Controller 28

Since plv and plv are both non-linear functions of Vspt, (3.30) is a non-linear equation. The calculation of the septum volume Vspt is therefore done using the Newton-Raphson method. For more information on the derivation of the equations for the intraventricular septum, see Baumann [1].

3.3.6 Heart Valves Each heart valve is modeled using an inductance and two different resistances. One small value resistance in the normal direction of blood flow and one inverse resistance, which has a much higher value, in the other. A schematic of the electrical equivalent of the heart valves can be seen in figure 3.8, where L is the inductance of the valve, Rd is the direct flow resistance and Ri is the inverse flow resistance, D1 and D2 are diodes only letting one direction of flow through. The index y = {m,a,t,p} represent the different heart valves,

Rd,yv D1 Lyv p1 p2 Ri,yv D2 qin

Figure 3.8: Schematic of the heart valves [30]. mv = mitral valve, av = aortic valve, tv = tricuspid valve, pv = pulmonary valve. The values for the inductance Lyv were taken from Baumann [1] and Moscato [30]. For more details on the derivation of the inductance values see Baumann [1]. The diﬀerential equations for the heart valves can be expressed by using (3.8)

p1 − p2 − qinRd,yv for qin > 0 Lyv q˙in =  , (3.31) p1 − p2 − qinRi,yv  for qin < 0  Lyv  where qin is the ﬂow through the heart valve. By using (3.31), the diﬀerential equation for the mitral valve can be expressed in the form

pla − plv − qin,lvRd,mv qin,lv > 0, mitral valve open Lmv q˙in,lv = . (3.32)  pla − plv − qin,lvRi,mv  qin,lv < 0, mitral valve closed  Lmv The diﬀerential equations for the ﬂows through the other heart valves can be expressed in the same manner [30].

3.4 Baroreﬂex Controller

As mentioned in chapter 2, the baroreflex controller is a pressure controller sensitive to changes in aortic pressure pao. In order to correctly model the regulation of the aortic pressure, three different controllers are implemented in the model: a controller that adjusts the total systemic arterial resistance, a controller that changes the heart rate and a controller that controls the unstressed volume in the systemic veins. 3.4. Baroreflex Controller 29

3.4.1 Systemic Arterial Resistance Controller The structure of the systemic arterial resistance controller has been taken from Ursino [40]. The controller reacts to changes in the mean transmural aortic pressure from the reference value and counteracts these changes by altering the systemic arterial resistance Ras. The transmural pressure is the pressure diﬀerence between pressure within and outside the vessel. This is important since vessels within the thoracic cage, as the aorta, are subject to an outer pressure of pith. Therefore the mean transmural aortic pressure is expressed as

p¯ao,tm =p ¯ao − pith. (3.33) The model of the control loop for the systemic arterial resistance controller can be seen in ﬁgure 3.9.

e xRas σRas ∆Ras

+ Delay 1 + pao,tm GRas DRas 1+ τ s + Ras - Ras

pao,ref

Ras,const

Figure 3.9: Schematic of systemic arterial resistance controller [40].

The error e in the transmural aortic pressure is multiplied by a factor GRas

xRas = GRas (¯pao,tm − pao,ref ) , (3.34) where pao,ref is the reference value for the transmural aortic pressure. The limiting function σRas described by the saturation block in figure 3.9 is expressed as xRas − R + R exp kRas R − R σ = as,min as,max with k = as,max as,min , (3.35) Ras xRas Ras − 4SRas 1+ exp kRas where Ras,max/Ras,min is the max/min change in the systemic arterial resistance caused by the controller, xRas is the weighted error, SRas is the slope of the limiting function at e = 0 [40]. The limiting function σRas with Ras,min = −0.6, Ras,max = 0.6 and SRas = 1 can be seen in figure 3.10. The differential equation for the change in systemic arterial resistance ∆Ras can finally be expressed as 1 ∆R˙ as = (σRas (t − DRas) − ∆Ras) , (3.36) τRas where τRas is the time constant for the transfer function in figure 3.9 and DRas is the time delay. The total systemic arterial resistance can be expressed as a constant mean value plus the change caused by the controller

Ras = ∆Ras + Ras,const, (3.37) where Ras,const is the mean systemic arterial resistance as described by Guyton [13] and Moscato [30]. The parameter values Ras,min, Ras,max, SRas for the systemic arterial resistance controller have been taken from Ursino [40]. 3.4. Baroreﬂex Controller 30

1 Normal working point 0.5

Ras,max 0 Ras,min [mmHgs/ml]

Ras −0.5 σ

−1 −100 0 100 e [mmHg]

Figure 3.10: Saturation function σRas with pao,ref = 100 − pith,mean = 104.5 mmHg [1].

3.4.2 Heart Rate Controller The structure of the heart rate controller has been taken from Ursino [40]. The heart rate controller works in a similar way to the systemic arterial resistance controller, it is only a little more complex. The heart rate controller model the changes in heart rate fhr present in patients who are subjected to changes in the aortic pressure. The controller works by altering the heart rate when changes in the mean transmural aortic pressure p¯ao,tm from the reference value pao,ref are present. The controller is made out of two parts, a sympathetic and a vagal one. The vagal part is fast reacting and the sympathetic part needs a couple of seconds to react [13, 40]. A schematic of the model of the heart rate controller can be seen in ﬁgure 3.11.

vT xTv xT

+ b Delay 1 pao,tm GaTv T DTv 1+ τT vs + - + pao,ref sT

Delay 1 GaTs DTs 1+ τTss xTs

Figure 3.11: Schematic of heart rate controller [40].

Each part xT i, i = {v,s} is calculated in the same way as for the baroreflex controller. The weighted error of the mean transmural aortic pressure from the reference value is expressed as iT = GaT i (¯pao,tm − pao,ref ) , i = {v,s}, (3.38) where GaT i is the weighting factor for the vagal and sympathetic parts. The differential 3.4. Baroreflex Controller 31 equation for the two parts is then 1 x˙ T i = (iT (t − DT i) − xT i) , i = {v,s}. (3.39) τT i

The total heart period T can be expressed with the variable xT , where

xT = xTs + xTv, (3.40) and with the limiting function

xT − T + T exp kT T − T T = min max , with k = max min . (3.41) xT T − 4ST 0 1+ exp kT The parameters for the heart controller have been taken from Ursino [40].

3.4.3 Unstressed Volume Controller The structure of the unstressed volume controller has been taken from Heldt [16]. The unstressed volume controller regulates the vascular tone in venous compartments thus mod- ifying the pressure-volume relationships in the compartment and consequently the amount of blood stored. A schematic of the unstressed volume controller can be seen in ﬁgure 3.12.

σusv khead ∆Vhead,usv xusv ∆Vusv,tot

kkid ∆Vkid,usv + Delay 1 p¯ao,tm Gusv Dusv 1+ τusvs k ∆V — sp sp,usv

pao,ref kleg ∆Vleg,usv

Figure 3.12: Schematic of unstressed volume controller [40].

The control loop has been derived from Ursino [40]. The control loop is made up by a weighting factor Gusv, a time delay Dusv, a linear saturation function σusv and a transfer function of the ﬁrst order with the time constant τusv. The weighted error xusv of the mean transmural aortic pressure from the reference value is expressed as

xusv = Gusv (¯pao,tm − pao,ref ) . (3.42)

The implemented limiting function σusv is a simple linear function with a minimum and a maximum value

Lusv xusv(t − Dusv) Uusv, where xusv(t − Dusv) is the weighted error with the included time delay Dusv, Uusv the maximum value and Lusv the minimum value for the limiting function σusv. The value for 3.4. Baroreﬂex Controller 32

Dusv has been taken from Ursino [40] and τusv, Uusv, Lusv were set as to minimize the overshoot and for the rapid decay of any oscillations. In comparison to Baumann [1] the structure of the controller stayed the same but all parameters related to the controller were recalculated since the systemic venous compartments have been changed and an additional compartment representing the head and upper body has been added to the previous model configuration. The factors khead, ksp, kkid, kleg are the volume percentiles that give the change in unstressed volume in each compartment [16]. The weighting constant Gusv has been recalculated, when multiplied with the factors khead, ksp, kkid, kleg, to correspond to the gain values for each compartment given in Heldt [16]. The final change in unstressed volume is given as ∆Vhead,usv for the head, ∆Vkid,usv for the kidneys, ∆Vsp,usv for the splanchnic and ∆Vleg,usv for the legs. The differential equation for the total change in unstressed volume in the venous compartments is 1 ∆V˙usv,tot = (σusv (t − Dusv) − ∆Vusv,tot) . (3.44) τusv Figure 3.13 shows the pressure-volume relation in the systemic veins according to Guyton [13] and what happens when the controller adjust the unstressed volume. pi is the pres-

pi,2

pi,1

Vusv0,i Vact,i ∆Vusv,i

Figure 3.13: Pressure-volume relation in the systemic veins. sure in a vascular compartment, ∆Vusv,i is the change in unstressed volume caused by the controller, Vusv0,i is the initial unstressed volume and Vact,i is the active volume in the venous compartment. The active volume Vact,i is the volume in the compartment that, due to streching of the compartment vessels according to (3.4), give rise to a linear increase in pressure pi. The change ∆Vusv,i in unstressed volume, following the control loop in ﬁgure 3.12, eventually increases the pressure from pi,1 to pi,2. 3.5. Systemic Circulation 33

3.5 Systemic Circulation 3.5.1 Arterial Circulation The model describing the systemic arterial circulation has been taken from Baumann [1]. The electrical schematic of the arterial circulation can be seen in ﬁgure 3.14. Rao represent

Ras qo,lv Lao pCas qRas pao pv qCao qLao qCas Rao C pCao as

Cao

pith pith

0 mmHg 0 mmHg Figure 3.14: Electrical schematic of the arterial systemic circulation model [1]. the resistance to flow of the aorta and the branched off arteries and Cao the respective compliance. The inductance Lao represents the inertia of the blood in the aorta. Ras represents the total resistance and Cas the total compliance of the systemic arterial vessels. The systemic arterial resistance Ras is expressed as a sum of a constant part Ras,const and an adjustment ∆Ras controlled by the baroreflex controller (3.37). As explained earlier, a change in Ras is equivalent to a change in the diameter of the vessels. The flow over the inductance Lao can be described by

pao − pCas q˙Lao = , (3.45) Lao where pao is the pressure in the aorta and pCas the pressure in the arteries. From ﬁgure 3.14 it can be seen that the pressure in the aorta can be described as

pao = pCao +(qo,lv − qLao) Rao, (3.46) where pCao is the transmural pressure over the aorta. The pressure pCao can be described by the diﬀerential equation

pao − pCao p˙Cao = , with pCao(0) = pCao,0, (3.47) CaoRao where pCao,0 is the initial pressure over the aorta. The pressure pCas in the arterial system can be described through

qLao − qRas p˙Cas = , with pCas(0) = pCas,0, (3.48) Cas 3.5. Systemic Circulation 34

where Cas is the total compliance of the arterial vessels and pCas,0 is the initial pressure in the arterial system. qRas is the ﬂow over the systemic arterial resistance Ras and can be calculated from pCas − pv qRas = , (3.49) Ras where pv is the pressure in the veins. The component values of the equations above can be calculated from the physiological conditions of the human cardiovascular system at rest to

Ras,const = 1.066 mmHgs/ml

Cas = 1.6 ml/mmHg

Rao = 0.048 mmHgs/ml

Cao = 1.48 ml/mmHg 2 Lao = 0.0099 mmHg s /ml, where Ras,const is the estimated average value for the systemic arterial resistance, Cas the systemic arterial compliance, Rao the resistance and Lao the inductance of the aorta [1, 28, 30, 32]. For more information on the derivation of the parameters of the systemic arterial circulation see Baumann et al. [1, 30, 28, 32].

3.5.2 Venous Circulation The important eﬀects that have been considered in modeling the systemic venous circulation are:

I. The eﬀect of the orthostatic stress on the more compliant veins. Due to hydrostatic eﬀects blood displacement will occur and the veins will expand/contract.

II. The eﬀect of stress relaxation where according to Heusden [18] "stress relaxation refers to the ability of the veins to stretch slowly when the pressure rises and to contract slowly when the pressure falls".

III. The effect of changes in abdominal pressure pabd on the kidneys and intestines. In the following subsections, these points will be considered in detail. The basis for the model of the systemic venous circulation has been taken from Baumann [1] but has been extended to include an extra venous compartment representing the head and upper body, the effect of stress relaxation and the effect of external abdominal pressure pabd.

Simple Model To first get a general idea of the resistances and blood flows that make up the systemic venous system and to estimate some parameters, a simple model of the system is studied. The schematic of the simple model can be seen in figure 3.15. qRas is the flow coming from the system arterial circulation, qvsin is the eventual flow coming from an infusion, qvs is the flow out of the systemic venous circulation into the right atrium, Rvs is the total systemic venous resistance and Cvs the total systemic venous compliance. According to Coleman et al. [14, 30] Rvs can be calculated by 0.164 Rvs = , (3.50) pv 3.5. Systemic Circulation 35

qvsin Rvs qRas qvs pv pra

Cvs

pith

0 mmHg Figure 3.15: Simple model of the systemic venous circulation [14, 30].

where pv is the pressure in the vena cava. By using equation (3.50) and estimating pv to 4 mmHg as in Baumann [1], Rvs is calculated to 0.164 R = = 0.041 mmHgs/ml. (3.51) vs 4

The value for Cvs has been taken from earlier estimations in Guyton [13, 30].

Cvs = 50 ml/mmHg. (3.52)

When extending the model to include all the venous compartments the idea is to keep the total systemic venous compliance Cvs and resistance Rvs constant. This to keep the total inﬂow, outﬂow and blood volume stored in the veins the same as in the simple model.

Extended Model In order to correctly simulate orthostatic effects, such as hydrostatic pressure changes and blood volume displacement an extended model of the one seen in figure 3.15 needs to be derived. The model divides the systemic venous system into several compartments modeling different parts of the body. This makes it possible to simulate the different hydrostatic pressure changes and blood volume displacements in the different parts of the body. The extended model is based on the model presented in Heldt [16] with the venous system partitioned into four compartments: the head, the kidneys, the splanchnic and the legs. The structure of the extended model can be seen in figure 3.16. Blood from the systemic arterial system qRas and fluid from an infusion qvsin enter the systemic venous system and is distributed into the four major compartments of the human body. The blood then flows from the head into the superior vena cava and from the lower compartments into the inferior vena cava . Both the inferior and superior vena cava are vessels that lie within the thorax and are therefore subjected to the pressure pith due to breathing. The inferior and superior vena cava finally transport the blood into the right atrium. 3.5. Systemic Circulation 36

Rhead,1 Rhead,2 qhead,1 qhead,2

portho,head

phead pith Csvc Chead psvc qsvc 0 mmHg 0 mmHg

Rsvc Rkid,1 Rkid,2 qkid,1 qkid,2

qvs pra portho,kid

pkid

Ckid Rivc

pith qivc pabd

pivc 0 mmHg Civc Rsp,1 Rsp,2 0 mmHg qvsin qsp,1 qsp,2 qv,low

portho,sp

psp

Csp

pabd

0 mmHg

Rleg,1 Rleg,2 qRas qleg,1 qleg,2 pv

portho,leg

Cv pleg

Cleg

0 mmHg 0 mmHg

Figure 3.16: Schematic of the model of the extended systemic venous system [16]. 3.5. Systemic Circulation 37

Stress Relaxation

The pressure pi in each compartment is calculated from the sum of an artificial pressure pstr,i, simulating the effect of stress relaxation and the pressure change caused by a change in unstressed volume ∆Vusv,i. Stress relaxation of the systemic veins has mainly been added in order to simulate the hemodynamics during orthostatic stress more accurately. As mentioned in section 3.4.3, a pressure rise can also occur when the controller changes the unstressed volume as seen in 3.13. Because a change in unstressed volume is caused by the unstressed volume controller and not due to a displacement of blood volume, it is assumed that the only volume under the effect of stress relaxation is the active volume Vact,i. According to the model described in Heldt [16] only the compartments representing the kidneys and splanchnic are thought to be within the abdomen and are therefore subjected to an external pressure pabd. The pressure in each vascular compartment can therefore be expressed as 1 pi = pstr,i − Vusv,i, i = {leg, head} (3.53) Ci 1 pj = pstr,j − Vusv,j + pabd, j = {kid,sp} , (3.54) Cj where Ci is the compliance value for each of the compartments. The equation describing the stress relaxation variable pstr,i were taken from Heusden [18]. The equation describes the step response in pressure pstr,i(t) to a change in volume V0,i occuring at time t = 0 in a compartment

V 5 − p (t)= 0,i 1+ e t/τ i = {kid,sp,leg,head}, (3.55) str,i C 3 i where τ = 30 s is the time constant. (3.55) is a step response to a volume change which means the active time-dependent volume Vact,i(t) in a compartment can be expressed as

u(t) = 0 t ≤ 0 V (t)= V u(t) . (3.56) act,i 0,i u(t) = 1 t> 0

The related linear diﬀerential equation describing pstr,i can be determined by making a Laplace transformation L(pstr,i(t))(s) and inserting the Laplace transform of the step function Vact,i(s)= V0,i/s

Vact,i(s) (8/3s + 1/τ) pstr,i(s)= , i = {kid,sp,leg,head}. (3.57) Ci s + 1/τ

The linear diﬀerential equation of p˙str,i(t) for a time-dependent volume Vact,i(t) representing (3.57) is expressed as

pstr,i Vact,i 8 (qi,1 − qi,2) p˙str,i = − + + , i = {kid,sp,leg,head}, (3.58) τ Ciτ 3 Ci where Vact,i is the active volume in each systemic venous compartment. The inﬂow qi,1 and outﬂow qi,2 of the vascular compartments are calculated by

pv − (pi − portho,i) qi,1 = , i = {kid,sp,leg,head} (3.59a) Ri,1 3.5. Systemic Circulation 38

(pi − portho,i) − px qi,2 = , i = {kid,sp,leg,head}, x = {ivc,svc}, (3.59b) Ri,2

where px = psvc when i = {head} and px = pivc when i = {kid,sp,leg}. The diﬀerential equations describing changes in pivc and psvc are calculated using (3.10)

qhead,2 − qsvc p˙svc = (3.60a) Csvc

qv,low − qivc p˙ivc = , (3.60b) Civc where Csvc is the compliance of the superior vena cava, Civc is the compliance of the inferior vena cava, qsvc the ﬂow out of the superior vena cava and qivc the ﬂow out of the inferior vena cava. qivc and qsvc are calculated using (3.2)

pivc − pra qivc = (3.61a) Rivc

psvc − pra qsvc = , (3.61b) Rsvc where Rivc is the resistance of the inferior vena cava and Rsvc is the resistance of the superior vena cava. qv,low is the summarized flow of the lower compartments that flows into the inferior vena cava qv,low = qkid,2 + qsp,2 + qleg,2. (3.62) The total inflow into the right atrium is expressed as

qvs = qivc + qsvc. (3.63)

Orthostatic Pressure

The pressure portho,i represents the hydrostatic pressure change in a venous compartment due to a change in body position. A change in body position could be, e.g. standing up. A way to measure the eﬀect of orthostatic stress is often done through the so-called Head- Up-Tilt (HUT) test. During the test a patient lies on a tilt table that gradually increases its angle of tilt compared to the ground, slowly putting the patient in an upright position and then returning them back to the initial lying position. The hydrostatic pressure change portho,i for each compartment is calculated by

portho,i = pbias,i sin(α), (3.64) where portho,i is dependent on the angle of tilt α of one’s body compared to the ground, see figure 3.17. pbias,i represents the hydrostatic effects that are present when a person is fully o upright, i.e. α = 90 . The values for pbias,i have been taken from Guyton [13], see figure 2.7 in section 2.3. For pbias,leg an estimated value, lying between the hydrostatic pressure around the hips and the pressure in the feet in figure 2.7, was used

pbias,kid = 15.0 mmHg (3.65a)

pbias,sp = 22.0 mmHg (3.65b)

pbias,leg = 70.0 mmHg (3.65c)

pbias,head = −10.0 mmHg. (3.65d) 3.5. Systemic Circulation 39

Tilt Plate

Ground α

Figure 3.17: Schematic of a tilt table, tilted at angle α compared to ground.

Estimation of Parameters

In the estimation of the parameters Ri,1, Ri,2, Ci, Rivc, Rsvc, Civc and Csvc for the systemic venous circulation as seen in figure 3.16, the values given in Heldt [16] were used as a basis. There is one important difference between the model described in Heldt [16] and the model in this Master’s Thesis. The calculated parameter values presented in Heldt [16] have been calculated to represent both the systemic arterial and the systemic venous part of the circulation. The systemic venous model has divided the systemic arterial and systemic venous circulation into two separate parts. In order to obtain the same flow patterns and volume ratios as presented in Heldt [16], the same parameter ratio is used but the parameter values themselves need to be recalculated.

Rkid,tot,lit = Rkid,1,lit + Rkid,2,lit = 4.4 mmHgs/ml (3.66a)

Rsp,tot,lit = Rsp,1,lit + Rsp,2,lit = 3.18 mmHgs/ml (3.66b)

Rleg,tot,lit = Rleg,1,lit + Rleg,2,lit = 3.19 mmHgs/ml (3.66c)

Rup,tot,lit = Rhead,1,lit + Rhead,2,lit + Rsvc,lit = 4.19 mmHgs/ml, (3.66d) where Ri,tot,lit, i = {kid,sp,leg} is the total resistance of each lower body vascular compartment and Rj,1,lit is the inﬂow resistance and Rj,2,lit, j = {kid,sp,leg,head} the outﬂow resistance of each compartment. Rup,tot,lit is the sum of the resistances in the vascular compartment representing the head and the resistance of the superior vena cava Rsvc,lit. All resistances with index lit are given according to the values in Heldt [16]. The total resistance Rv,low,lit of the lower body compartments can be calculated using 1 1 1 1 = + + ⇒ Rv,low,lit = 1.253 mmHgs/ml (3.67) Rv,low,lit Rkid,tot,lit Rsp,tot,lit Rleg,tot,lit

The ﬁnal resistance values that need to be calculated from Heldt [16] are

Rlow,tot,lit = Rv,low,lit + Rivc,lit = 1.278 mmHgs/ml (3.68a)

Rvs,lit = 0.979 mmHgs/ml, (3.68b) where Rlow,tot,lit is the total resistance of the lower body venous compartments and the resistance of the inferior vena cava Rivc,lit. Rvs,lit is the total venous resistance according to Heldt [16]. It is, from the calculated values, clear that the model in this report and 3.5. Systemic Circulation 40

the one in [16] describe two diﬀerent systems since according to 3.51 Rvs was calculated to be 0.041 mmHgs/ml which clearly diﬀers from Rvs,lit = 0.979 mmHgs/ml. The important ratios between the resistances in the upper and lower body are

Rup,tot,lit/Rvs,lit = 4.2788 = r1 (3.69a)

Rlow,tot,lit/Rvs,lit = 1.30499 = r3 (3.69b)

Rkid,tot,lit/Rv,low,lit = 3.512 = r5 (3.69c)

Rsp,tot,lit/Rv,low,lit = 2.538 = r7 (3.69d)

Rleg,tot,lit/Rv,low,lit = 3.1128 = r9. (3.69e) The internal ratios of each compartment are

Rhead,2,lit/Rsvc,lit = 3.8333 = r2 (3.70a)

Rhead,1,lit/Rsvc,lit =65= r4 (3.70b)

Rkid,1,lit/Rkid,2,lit = 13.667 = r6 (3.70c)

Rsp,1,lit/Rsp,2,lit = 16.667 = r8 (3.70d)

Rleg,1,lit/Rleg,2,lit =12= r10 (3.70e)

Rv,low,lit/Rivc,lit = 50.12 = r12. (3.70f)

The ratios r1,...,r12 are important because in order to keep the ﬂow pattern the same in the implemented model as the one described in Heldt [16] these ratios need to stay the same. By using r1,...,r12 and the total systemic resistance Rvs = 0.041 mmHgs/ml the resistances for the extended model can be calculated. The calculation of the resistances in the upper body compartment and the superior vena cava is done by using the fact that the ratio between the total resistance of the upper body Rup,tot and Rvs should be as described in 3.69a (Rup,tot/Rvs = r1) where

Rup,tot = Rhead,1 + Rhead,2 + Rsvc. (3.71)

Next by using the calculated values for the ratios between Rhead,1/Rsvc = r4 and Rhead,2/Rsvc = r2, where Rhead,1 is the inﬂow resistance, Rhead,2 the outﬂow resistance of the upper body compartment and Rsvc the resistance of the superior vena cava, the individual resistance can be calculated. The ratios for the upper body can be rewritten as

Rup,tot = r1Rvs (3.72a)

Rhead,1 = r4Rsvc (3.72b)

Rhead,2 = r2Rsvc. (3.72c) By using the relations above and putting them into (3.71) the resistance values in the upper body are calculated to

Rsvc = 0.0025 mmHgs/ml R = R + r R + r R = r R =⇒ R = 0.1633 mmHgs/ml (3.73) up,tot svc 2 svc 4 svc 1 vs  head,1  Rhead,2 = 0.0096 mmHgs/ml.  3.6. Lungs and Airways Model 41

The resistances for the remaining compartments are calculated in the same manner by using the ratios from Heldt [16] and the fact that Rvs = 0.041 mmHgs/ml

R = 0.0525 mmHgs/ml R + r R = r R =⇒ ivc,1 (3.74) ivc,2 12 ivc,2 3 vs R = 0.0010 mmHgs/ml ivc,2 R = 0.1717 mmHgs/ml R + r R = r R =⇒ kid,1 (3.75) kid,2 6 kid,2 5 vs R = 0.0126 mmHgs/ml kid,2 R = 0.1256 mmHgs/ml R + r R = r R =⇒ sp,1 (3.76) sp,2 8 sp,2 7 vs R = 0.0075 mmHgs/ml sp,2 R = 0.1507 mmHgs/ml R + r R = r R =⇒ leg,1 . (3.77) leg,2 10 leg,2 9 vs R = 0.0126 mmHgs/ml leg,2 The calculations of the capacitances are done in the same way

Cvs,lit = Chead,lit + Csvc,lit + Cv + Ckid,lit + Csp,lit + Cleg,lit + Civc,lit = 141 ml/mmHg, where Cvs,lit is the total systemic venous capacitance, Ci,lit i = {kid,sp,leg,head} is the capacitance for each compartment, Csvc,lit is the capacitance of the superior vena cava and Civc,lit is the capacitance of the inferior vena cava. Cv = 2 ml/mmHg is a small artiﬁcial capacitance, taken from Baumann [1], representing the capacitance of the vena cava but is not present in Heldt [16]. The values indexed lit are all values taken from Heldt [16]. The new values for the capacitances are a bit simpler to calculate, just using the ratios and the fact that according to Moscato [30] Cvs = 50 ml/mmHg.

Chead,lit/Cvs,lit = 0.0567 = c1 ⇒ Chead = c1Cvs = 2.835 ml/mmHg (3.78a)

Csvc,lit/Cvs,lit = 0.1067 = c2 ⇒ Csvc = c2Cvs = 5.320 ml/mmHg (3.78b)

Cv,lit/Cvs,lit = 0.0142 = c3 ⇒ Cv = c3Cvs = 0.710 ml/mmHg (3.78c)

Ckid,lit/Cvs,lit = 0.1064 = c4 ⇒ Ckid = c4Cvs = 5.320 ml/mmHg (3.78d)

Csp,lit/Cvs,lit = 0.3901 = c5 ⇒ Csp = c5Cvs = 19.505 ml/mmHg (3.78e)

Cleg,lit/Cvs,lit = 0.1348 = c6 ⇒ Cleg = c6Cvs = 6.740 ml/mmHg (3.78f)

Civc,lit/Cvs,lit = 0.1915 = c7 ⇒ Civc = c7Cvs = 9.575 ml/mmHg (3.78g)

3.6 Lungs and Airways Model

When modeling the lungs and airways it is important to consider parameter dimensions. In contrast to the cardiovascular modeling, where dimensions such as [mmHg] and [ml] are standard, airway models as Liu et al. [25, 26] often use dimensions such as [cmH20] and [l]. The relation between [cmH20] and [mmHg] is deﬁned by

1 mmHg = 1.36 cmH20. (3.79)

In order to use equations given in literature [25] the intrathoracic pressure pith in this section only is expressed in [cmH20]. The goal of the lung model is to accurately depict the interactions between the lungs and the cardiopulmonary system. The lung model in this Master’s Thesis is based on an earlier work by Liu [25]. The model described by Liu 3.6. Lungs and Airways Model 42

[25] is a three part model: the first part describing the airway mechanics of the lungs and airways, the second part describing the intra-alveolar pulmonary capillaries and the third part describing the gas exchange taking place between the lungs and the surrounding tissue. Only the first two parts of the model are relevant for the purposes of this Master’s Thesis. The aim is to create a model that is able to reproduce the effect of normal respiration and the dynamics of cardiopulmonary interaction with particular focus on the hemodynamic response of the Valsalva maneuver which is a forced expiration against a closed glottis. A schematic of the airway mechanics model from Liu [25] can be seen on the right in figure 3.18 together with a schematic cross-section of the lungs on the left in figure 3.18. Figure

Mouth

Upper Airways qair,1 pref Collapsible RC (VC ,qair,1) Airways ptm Thorax pC

qair,2 pel ptm(VC ) pith Lungs pith R (V ,p ) palv alv alv ith

pel(Valv,pith) RLti 0 mmHg ptmb palv Intra-alveolar capillaries

Figure 3.18: Schematic cross-section of the lungs, redrawn and partially modiﬁed from Liu [25] (left) . Schematic of the model in Liu [25] describing the airway mechanics (right).

3.18 shows how air enters through the mouth at reference pressure pref , flows through the upper and collapsible airways at pressure pC and into the lungs at pressure palv. The lungs are characterized by a variable resistance Ralv(Valv,pith) and by a transmural pressure pel(Valv,pith). The collapsible and upper airways are characterized in terms of a volume and flow dependent resistance Rcoll(VC ,qair,1) and by the transmural pressure ptm(VC ) [25]. The equations describing the transmural pressures pel and ptm were taken from Liu [25]. pel is characterized by two parts, one inspiration function pel,I and one expiration function pel,E representing the outer boundries of the pel-loop depicted in figure 3.19.

V − V + 0.001 3 p = p alv RV (3.80) el,E el,max V max − V + 0.001 l RV p + n ln(n /n ) p = ξ el,max d a b , (3.81) el,I ξ + 0.001 max where Vl is the total lung capacity and VRV is the minimum lung volume, also called the residual volume. According to Liu [25] the pel,E and pel,I curves were generated by assuming that pith is held constant at pith,max and pith,min during the expiration and inspiration, respectively. The mean of these two curves refers to the equilibrium curve corresponding to pith = pith,mean = 4.5 mmHg = 4.5 · 1.36 cmH20. The dependence of pel on pith was 3.6. Lungs and Airways Model 43 calculated using linear interpolation through grading by eﬀort between the equilibrium curve and the pel,E and pel,I curves respectively. na, nb, nc and nd are then calculated according to max Vl − VRV + 0.1 na = max − 0.99 (3.82) Vl − VRV + 0.001 max Vl − VRV + 0.1 nb = − 0.99 (3.83) Valv − VRV + 0.001 V max − V + 0.1 n = l RV − 0.99 (3.84) c 0.001 pel,max + 25 nd = , (3.85) ln (nc/na) where pel,max is the maximum value of pel and has been taken from measurements in Liu [25]. pel is ﬁnally expressed as a sum of pel,E and pel,I

(0.5+ n )p + (0.5 − n )p for p (t) ≥ 0 p = e el,E e el,I ith (3.86) el (0.5 − n )p + (0.5+ n )p for p (t) < 0. e el,E e el,I ith ne is calculated by 0.5 (pith,mean − pith(t)) for pith(t) ≥ 0 pith,mean − pith,max ne =  0.5 (3.87)  (pith,mean − pith(t)) for pith(t) < 0.  pith,mean − pith,min  In order for thepel equations to create a full loop during maximum eﬀort, a further condition on pel needs to be set pel ≥ 0 ∀ Valv,pith (3.88)

In [25] inspiration is defined as pith ≤ 0 and expiration as pith > 0. During normal conditions, when pith = pith,mean, pel is the same during inspiration and expiration. This can be seen by setting pith = pith,mean in (3.87). Figure 3.19 shows pel as a function of the active lung volume Valv and graded by effort (different pith). The parameters in figure 3.19 max have been set according to [25] as Vl = 5.3 l, VRV = 1.24 l and pel,max = 35 cmH20. According to Liu [25] and Olender [33], the transmural pressure ptm is expressed as

VC,max VC 5.6 − lbptm ln − 0.999 for > 0.5 VC VC,max p =  2 , (3.89) tm V V  s − s C − 0.7 for C ≤ 0.5  aptm bptm V V C,max C,max  where VC,max is the maximum volume of the airways. The constant lbptm is calculated by p − 5.6 l = tm,max . (3.90) bptm 6.908

VC The constants saptm and sbptm are calculated using the conditions of continuity at = VC,max 0.5

2 ptm = 5.6 − lbptm ln(2 − 0.999) = saptm − sbptm (0.5 − 0.7) . (3.91) V C =0.5 VC,max

3.6. Lungs and Airways Model 44

max (pel,max,Vl )

pith = pith,max [l] RV - V

alv pith = pith,min V

pith = pith,mean (equilibrium curve)

pel [cmH20]

Figure 3.19: Transmural pressure pel plotted against lung volume Valv and for diﬀerent values of intrathoracic pressure [25].

dp 4l 0.4s tm = bptm = bptm . (3.92) dVC V 1.001VC,max VC,max C =0.5 VC,max

Combining (3.91) and (3.92), sbptm and saptm can be expressed by 4l s = bptm (3.93) bptm 0.4004 saptm = 5.6 − lbptm ln(1.001) + 0.04sbptm. (3.94)

The transmural pressure ptm as a function of the volume VC in the airways can be seen in ﬁgure 3.20. By assuming the air in the lungs and airways is incompressible and using (3.1) the diﬀerential equations for the lung volume Valv and the airway volume VC can be expressed by

V˙alv = qair,2 (3.95)

V˙C = qair,1 − qair,2. (3.96)

The ﬂows qair,1 and qair,2 are calculated using (3.2)

pref − pC qair,1 = (3.97) RC pC − palv qair,2 = , (3.98) Ralv where pC is the pressure in the airways and palv is the pressure in the lungs. The resistances RC and Ralv are calculated according to Liu [25]. The total airway resistance RC is expressed by V 2 K K 2 R = K C,max + 1 + 1 + | p − p |. (3.99) C 3 V 2 2 ref C C s 3.6. Lungs and Airways Model 45

20 0] 2 15

10 [mmH 5 tm

p 0

−5

−10

−15 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 VC [l]

Figure 3.20: Transmural pressure ptm plotted against the airway volume VC [25].

K1 and K3 are constants taken from [25]. The airway resistance Ralv is expressed by

 Valv − VRV  RSa max  Vl − VRV  Ralv = RSme + RSc, (3.100) where RSc is calculated by

RSc,max−0.02 p (pith − pith,max)+ RSc,max pith ≥ 0 RSc = ith,max (3.101) ( 0.02 pith < 0.

RSm is the magnitude of Ralv − RSc at Valv = VRV , RSa is a parameter characterizing the curvature of Ralv and RSc,max is RSc at the instant of pith = pith,max. The parameter values for RSm, RSa and RSc,max have been taken from Liu [25]. The pressure in the airways pC is calculated by pC = ptm + pith. (3.102)

The pressure in the lungs palv can, according to [25], be expressed by

palv = pel + pith + RLtiqair,2, (3.103) where the term RLtiqair,2 represents the pressure drop over the resistance RLti. Solving (3.103) for palv yields p + RLti p + p el Ralv C ith palv = . (3.104) 1+ RLti/Ralv The pressures in the lungs and airways are all relative to the atmospheric pressure and the reference pressure pref has therefore been set to zero in all equations. 3.7. Pulmonary Circulation 46

3.7 Pulmonary Circulation 3.7.1 Pulmonary Arteries

The pulmonary arteries are represented by a single resistance Rpul and a single compliance Cpul as can be seen in ﬁgure 3.21. The model is based on the pulmonary artery model described in [1]. The diﬀerential equations describing the system are calculated using equation

qout,rv Rpul pap pcap,a qlp Cpul

pith

0 mmHg Figure 3.21: Model representing the pulmonary arteries and pre-capillaries [1].

(3.10) qout,rv − qlp p˙ap = with pap(0) = pap,0 (3.105) Cpul where qout,rv is the outﬂow of the right ventricle, qlp is the ﬂow into the pre-capillaries and pap,0 is the initial pressure in the pulmonary arteries.

3.7.2 Pulmonary Capillary Model The pulmonary capillaries can be divided into two parts: the first part describes the pre- capillaries representing the capillary vessels outside the alveolar region, the second part describes the intra-alveolar capillaries. The alveolar region is the region where the gas exchange between the air in the lungs and the blood takes place. The model describing the pulmonary capillaries has been completely modified from the one described in Baumann [1] to include the interactions of the lungs. The model is based on the pulmonary capillary model presented in Liu [25], which describes the pulmonary capillary vessels and the interactions of the lungs with the pulmonary capillary resistances. A schematic of the model can be seen in figure 3.22, where Vpc is the blood volume and ppc the pressure in the intra- alveolar capillaries, ptmb is the transmural blood pressure over the capillary wall, pcap,a is the pulmonary pre-intra-alveolar capillary pressures and pvp is the pulmonary venous pressure.

As can be seen in figure 3.22, the pulmonary capillary resistance is divided into three parts: a variable pre-intra-alveolar capillary resistance Rcap,a, a variable pulmonary capillary resistance Rpc and a variable post-intra-alveolar capillary venous resistance Rcap,v. The pulmonary capillary resistance is divided into two parts, an inflow resistance Rpc/2 and an outflow resistance Rpc/2. The capacitance of the pulmonary capillaries is divided into two parts, an intra-alveolar volume-dependent capacitance represented by a transmural pressure ptmb(Vpc) and an extra-alveolar capacitance Ccap. As the intra-alveolar capillaries lie within the alveolar region, the external pressure ef- fecting them is assumed to be the alveolar pressure palv that was calculated in (3.104). The 3.7. Pulmonary Circulation 47

Rpc(Vpc, q¯out,rv) Rpc(Vpc, q¯out,rv) 2 2 pcap,a qlp qin,pul ppc qout,pul pvp

Rcap,a(Valv,pith, q¯out,rv) Rcap,v(Valv,pith, q¯out,rv)

Ccap ptmb(Vpc)

pith palv

0 mmHg 0 mmHg Figure 3.22: Electrical schematic of the pulmonary capillary circulation [25]. idea is that due to the direct proximity of the capillary vessels to the lungs it is the pressure within the lungs palv, not the pressure within the thoracic cage pith, that interacts with the intra-alveolar capillaries. It is important to remember that the alveolar pressure palv is different from the intrathoracic pressure pith assumed to affect the other vessels within the thoracic cage. The intra-alveolar pulmonary capillaries are modeled as a single tube with a variable volume. Rcap,a and Rcap,v are assumed to be inversely proportional to the alveoli volume Valv but proportional to the intrathoracic pressure pith. Rpc is assumed to be only dependent on Vpc [25]. Another effect of significance is the effect of opening of pulmonary capillaries and the expansion of the vessels, described in section 2.3.6, [13]. This is modeled by setting the pulmonary capillary resistances dependent on cardiac output. Which means that when the cardiac output of the right ventricle rises, more capillary vessels will expand and open in parallel thereby reducing the total pulmonary resistance. The differential equation describing the pressure in the pre-capillaries pcap,a can be calculated using equation (3.10)

qlp − qin,pul p˙cap,a = with pcap,a(0) = pcap,a,0, (3.106) Ccap where qin,pul is the ﬂow into the capillaries and pcap,a,0 is the initial pressure in the pre- capillaries. Using (3.1), the state equation for the volume in the intra-alveolar pulmonary capillaries can be expressed as

V˙pc = qin,pul − qout,pul, (3.107) where qout,pul is the ﬂow out of the capillaries. qin,pul can be expressed in the form p − p q = cap,a pc . (3.108) in,pul R (V ) R (V ,p )+ pc pc cap,a alv pl 2 qout,pul can, in a similar manner, be expressed in the form p − p q = pc vp . (3.109) out,pul R (V ) R (V ,p )+ pc pc cap,v alv pl 2 3.7. Pulmonary Circulation 48

Rcap,a,lit, Rcap,v,lit and Rpc,lit are dictated by observations, showing that the extra-alveolar resistances decrease while the intra-alveolar resistances increase as Valv rises and are described by Liu [25] p R =(k (V − V max)+ R ) 1+ ith with i = {a,v} (3.110) cap,i,lit v,φ alv l 0,i k p,φ V 2 R = R pc,max , (3.111) pc,lit pc,max V pc where kv,φ is a parameter characterizing the volume dependence of Rcap,i,lit and kp,φ is a max parameter characterizing the pressure dependence of Rcap,i,lit. Vl is the maximum alveoli volume and R0,i is the estimated mean of Rcap,i,lit during normal breathing. Rpc,max is the maximum capillary resistance and Vpc,max is the maximum capillary volume. The calculated pulmonary capillary resistances above need to be adjusted to include the eﬀect of opening of pulmonary capillaries and the expansion of the vessels, as described in section 2.3.6. This eﬀect is modeled by changing the pulmonary capillary resistances according to the mean cardiac output of the right ventricle q¯out,rv. The capillary resistances are expressed by

Rcap,i,lit Rcap,i = i = {a,v} (3.112) α(¯qout,rv) Rpc,lit Rpc = , (3.113) α(¯qout,rv) where α(¯qout,rv) is a cardiac output dependent parameter. α(¯qout,rv) is described by

1 for q¯out,rv < 5 l/min q¯ − 5 α(¯q )= 1+ k out,rv for 5 ≤ q¯ < 16 , (3.114) out,rv  a 16 − 5 out,rv   1+ ka for q¯out,rv ≥ 16 where q¯out,rv is the mean cardiac output for one heart cycle of the right ventricle, ka is a constant characterizing the relationship of q¯out,rv with the pulmonary capillary resistances. In the case of left heart failure the pressure in the pulmonary circulation is already higher than normal implying that most capillary vessels are already open and somewhat distended. This means that the relation between the pulmonary capillary resistance and the cardiac output α(¯qout,rv) will change. It is assumed that this change can be described by changing ka. In the physiological case ka = 2 has been optimized, through simulation, as to fit the pap — q¯out,rv curve as close as possible to the one seen in figure 2.8 in section 2.3.6. In the pathological case ka = 0.5 has been optimized, through simulations, as to fit the pap — q¯out,rv curve to the observations made by Janicki [19]. The equation describing the transmural pressure ptmb(Vpc) V − 0.001 p = m − m ln pc,max − 0.999 , (3.115) tmb c b V − 0.001 pc was taken from Liu [25]. The constants ma, mb and mc are calculated according to

Vpc,max − 0.001 ma = (3.116) Vpc,max − 13.6Cpc − 0.001 13.6 mb = (3.117) 6.908+ln(ma − 0.999)

mc = 20.4 − 6.908mb, (3.118) 3.7. Pulmonary Circulation 49

15 Normal working point

5 [mmHg]

tmb 0 p

−5

−10

−15 0 10 20 30 40 50 60 70 Vpc [ml]

Figure 3.23: Transmural pressure ptmb from [25] used in modeling the pulmonary capillaries.

−4 where Cpc = 6.9 · 10 [l/mmHg] is the mean compliance of the intra-alveolar pulmonary capillary tube during normal respiration [25]. The transmural pressure ptmb as a function of the volume Vpc can be seen in ﬁgure 3.23.

3.7.3 Pulmonary Veins A schematic of the pulmonary veins can be seen in ﬁgure 3.24. The model for the pulmonary veins has been taken from Baumann [1].

qout,pul qvp Rvp pvp pla

Cvp

pith

0 mmHg Figure 3.24: Schematic of the pulmonary veins [1].

The diﬀerential equations describing the system are calculated in the same way as for the pulmonary arteries, using equation (3.10)

qout,pul − qvp p˙vp = , with pvp(0) = pvp,0, (3.119) Cvp where qout,pul is the outﬂow from the pulmonary capillaries, qvp is the ﬂow into the left atrium and Cvp is the compliance of the pulmonary veins. pvp,0 is the initial pressure in the pulmonary veins. 3.8. LVAD Modeling 50

3.8 LVAD Modeling

The LVAD modeled in this Master’s Thesis is a DeBakey axial ﬂow pump from MicroMed (MicroMed Cardiovascular Inc., Houston, TX, USA). The model for the LVAD has been taken from Baumann et al. [1, 31, 41]. The pump is connected to the left ventricle and the aorta. The blood is pumped from the ventricle through the LVAD and into the aorta. Figure 3.25 shows a schematic of the ﬂows and pressures in the LVAD.

R L p p L R plv ican ican qi i qvad Axial ﬂow qvad o qo ocan ocan pao pump qci qco

Cican Cocan i ω

PI-controller pith pith

ωset 0 mmHg 0 mmHg

Figure 3.25: Schematic of the internal ﬂow mechanics of the Debakey LVAD [1]

The pressures pi and po from figure 3.25 are the pressures at the inlet respectively the outlet of the rotational pump. Two cannulas carry the flow of blood to and from the LVAD. qi is the flow of blood through the inflow cannula that connects the LVAD to the left ventricle. qo is the flow of blood through the outflow cannula and into the aorta. The inflow cannula is very stiff and therefore has a very small compliance Cican. The outflow cannula is made up of a 23 centimeter flexible tube which has a larger compliance Cocan than the inflow cannula. The hydraulic resistances Rican and Rocan and the inductances Lican and Locan are determined from geometrical dimensions [30]. A PI-controller is used to set the rotational speed ω of the pump to maintain a constant value ωset using the current i as a control variable. The rotational speed of the LVAD causes in turn a flow qvad through the pump. The differential equations describing the model can be determined by looking at figure 3.25. For the inflow

qi − qvad p˙i = with pi(0) = pi,0 (3.120) Cican plv − pi − qiRican q˙i = with qi(0) = qi,0, (3.121) Lican where pi,0 is the initial inlet pressure and qi,0 is the initial inlet ﬂow. In the same way the equations for the outﬂow are

qvad − qo p˙o = with po(0) = po,0 (3.122) Cocan po − pao − qoRocan q˙o = with qo(0) = qo,0, (3.123) Locan 3.9. Valsalva Maneuver 51

Table 3.1: Working range for the DeBakey axial ﬂow pump [31]. Parameter Working Range

qvad 0 — 180 ml/s po — pi 0 — 120 mmHg ω 7500 — 12500 min−1

where po,0 is the initial inlet pressure and qo,0 is the initial outlet flow. According to Baumann [1] and Schima [31] the change in flow through the LVAD qvad can be expressed as 1 2 q˙vad = −b0qvad + b2ω + pi − po , (3.124) b1 where b0,...,b2 are constants taken from Schima [31]. For information on the internal mechanics of the axial pump and the derivations of the mechanical equations see Schima [31]. Table 3.1 show the working range of the axial flow pump.

3.9 Valsalva Maneuver

The Valsalva maneuver is a test which is used to evaluate the overall hemodynamic response of the cardiovascaular system to fast and large changes in intrathoracic pressure pith. This maneuver is used to test the model and its overall hemodynamic response. During the Valsalva maneuver, a person forcibly exhales against a closed glottis. This elevates the intrathoracic pressure pith to about 40 mmHg and is kept for about 15 seconds. The expected hemodynamic response of a healthy person performing the Valsalva maneuver can be seen in ﬁgure 3.26. The ﬁgure shows the changes of the systemic arterial pressure pap and the heart rate fhr. The oscillations visible in pap are from the pulsatile nature of the heart, as mentioned in chapter 2.

Figure 3.26: Arterial pressure response of a healthy person performing the Valsalva maneuver [26].

As can be seen in ﬁgure 3.26, the expected hemodynamic response to the Valsalva maneuver can be divided into four phases. Phase 1 starts at the beginning of the maneuver

3.10. FVC Maneuver 52

Table 3.2: Summary of the expected hemodynamic changes, for a healthy person, during the Valsalva maneuver [26]. Measured Data during Valsalva Maneuver Normal value Phase 1 Phase 2 Phase 3 Phase 4 Cardiac Output [l/min] 4.2 4.07 1.64 1.51 4.49 Heart Rate [min−1] 72 65 97 105 57 Pulmonary arterial pressure 23/12 53/46 50/44 22/11 28/15 Systole/Diastole [mmHg] with a rapid increase in intrathoracic pressure due to forced exhalation against a closed glottis. This is accompanied by a rapid rise in arterial pressure, due to the increase in intrathoracic pressure, followed by a drop in heart rate. Phase 2 is the maintaining of the raised intrathoracic pressure. This starts with an initial reduction in arterial pressure and an initial rise in heart rate. It continues with a small recovery of the arterial pressure and a continued rise in heart rate. Phase 3 is the opening of the glottis and the rapid fall of the intrathoracic pressure. This is followed by a rapid fall in arterial pressure and a further rise in heart rate. Phase 4 is the recovery phase after the termination of the maneuver. This starts with an initial overshoot in arterial pressure accompanied by a rapid drop in the heart rate. It ends with a recovery of the arterial pressure and the heart rate to normal working values. Further hemodynamic changes, that can be expected during the maneuver, are presented in table 3.2 The expected hemodynamic response of the Valsalva maneuver is for a healthy person, [7, 11, 24, 26, 37], and in the pathological case [43], well documented. In figure 3.27 the hemodynamic responses to the Valsalva maneuver for different sever- ities of left heart failure can be seen. Figure 3.27 clearly shows the differences between the hemodynamic response to the Valsalva maneuver of a healthy person and of a person suffering from left heart failure. For patients with severe left heart failure there is no arterial pressure decrease during phase 2 of the Valsalva maneuver as in the healthy situation. There is also no apparent overshoot in arterial pressure after the termination of the maneuver. This is the so-called square wave arterial pressure response and is indicative of left heart failure [43].

3.10 FVC Maneuver

The Forced Vital Capacity maneuver (FVC maneuver) is used to evaluate the cardiovascular response of the intra-alveolar pulmonary capillaries to rapid changes in intrathoracic pressure. This maneuver is also used to test the model and in particular the cardiopulmonary coupling. The maneuver is performed by first fully exhaling to the residual volume, then in- max haling to the total lung capacity Vl followed immediately by a full exhalation to residual volume once again and holding this for about 15 seconds [26]. The changes to intrathoracic pressure pith during the maneuver can be seen in figure 3.28. The expected internal responses of the intra-alveolar capillaries can be seen in figure 3.29. The responses from figure 3.29 are hereby explained. As one inhales i, starting from residual lung volume VRV , the pulmonary pre-intra-alveolar capillary resistance Rcap,a and post-intra-alveolar capillary resistance Rcap,v decrease due to inflation of the lungs which is followed by an increase in pulmonary capillary inflow qin,pul and outflow qout,pul. The difference between inflow and outflow cause a drop in capillary volume Vpc which in turn 3.10. FVC Maneuver 53

Figure 3.27: Arterial pressure responses during Valsalva maneuver. (A) Arterial pressure response, healthy patients; (B) absent overshoot arterial pressure response, light to mod- erate left ventricular dysfunction; (C) square wave arterial pressure response, severe left ventricular dysfunction [43]. 0] 2 [cmH ith p

Time [s]

Figure 3.28: Measurement of changes in intrathoracic pressure, taken from a healthy person, when performing the FVC maneuver. The dashed lines mark the start of the ﬁrst forced expiration e, the inhalation i and the ﬁnal full expiration e∗ [25]. 3.10. FVC Maneuver 54

Rpc Rcap,a

[mmHgs/l] Rcap,v R

qin,pul [l/min] q qout,pul [ml] pc V

Time [s]

Figure 3.29: Simulated pulmonary intra-capillary responses to the FVC-maneuver [25].

increases the pulmonary capillary resistance Rpc. When one starts the final forced expiration ∗ e , the pressure in the intra-alveolar pulmonary capillaries ppc is relatively low which cause a greater inflow than outflow and Vpc increases. The increase in pith causes an increase in Rcap,a and Rcap,v which in turn lowers inflow qin,pul and outflow qout,pul. The flow rates stabilize at around 50 % of normal value for the rest of the maneuver and recover when normal breathing resumes [25]. Chapter 4

Matlab Simulink Implementation

This chapter describes the implementation that is done in the computational program Mat- lab Simulink. The system requirements are also shortly explained.

4.1 Simulink

The full cardiovascular model is implemented in Simulink. Most of the equations describing the model lie within a Simulink model block and are written in Matlab within S-functions. Some of the equations describing the control loops with inherent delays and complex function have instead been implemented with Simulink blocks. These control loops include:

– Unstressed volume controller

– PI-controller for axial ﬂow pump

4.2 Matlab-code

Most of the equations detailed in this report were implemented in Matlab and written in C as to increase the computational speed. The equations were implemented as ﬁrst order diﬀerential equations. The Matlab-program is made up by the following functions

– mdlInitializeSizes: Initialization of arrays.

– mdlInitializeSampleTimesInitialize: Initialization of the sample time arrays.

– mdlInitializeConditions: Setting of initial states.

– mdlOutputs: Calculation of outputs.

– mdlUpdate: Computation of states. – mdlDerivatives: Calculation of diﬀerential equations.

– mdlTerminateClean: Clean up memory after end of simulation.

The sequence of the function calls can be seen in ﬁgure 4.1.

55 4.3. System Requirements 56

START SIMULATION

Called when mdlProcessParameters parameters change

Major Time Step

mdlGetTimeOfNextVarHit

Called if sample time

of this S-function varies mdlInitalizeConditions

mdlUpdate

mdlOutputs

Simulink Engine Integration

mdlDerivatives

mdlCheckParameters

mdlOutputs

Minor Time Step Called if this S-function

has continuous states Called when mdlDerivatives

parameters change

Zero Crossing Detection

mdlOutputs

mdlZeroCrossings

Called if this S-function

detects zero crossings

END SIMULATION

Figure 4.1: Function call of the Matlab-program during simulation.

4.3 System Requirements

To run the model one will need to have Matlab with the Simulink package installed on ones computer. Chapter 5

Results and Conclusions

In this chapter the simulation results will be presented. The results will include simulations of:

1. Physiological and pathological hemodynamic responses with and without LVAD support.

2. Hemodynamic responses to orthostatic stress.

3. Hemodynamic responses to the Valsalva maneuver, both in physiological and pathological situations.

4. Hemodynamic responses of the intra-alveolar pulmonary capillaries to the FVC-maneuver.

The results will mostly focus on the new parts of the extended model.

5.1 Normal Physiological Conditions

This section displays the simulation results during physiological conditions which correspond to a healthy person. Figure 5.1 shows, on the left, the pressure of the left ventricle during one beat. The pressures seen in the figure are the left ventricular pressure, which rises and falls rapidly during contraction and relaxation of the ventricle respectively, the aortic pressure which rises when blood is ejected from the ventricle and decreases slowly during the filling phase, and the mean aortic pressure averaged over one heart beat. On the right in figure 5.1, the associated PV-loop of the left ventricular pressure is drawn. The points A, B, C and D correspond to the same points in the figures and are the phases described in section 2.2. The stroke volume, Vsv ≈ 90 ml, can be read from the picture as the horizontal distance between point B and C in the PV-loop which agrees with literature [17]. The simulated heart rate during physiological conditions is 62 min−1, which together with the stroke volume would indicate a cardiac output of 5.6 l/min. Figure 5.2 shows the same responses for the right ventricle. The results show that pao has a systolic pressure of 140 mmHg and diastolic of 80 mmHg with an average of 100 mmHg. pap has a systole/diastole pressures of 32/13 mmHg with an average of 20 mmHg. These values are slightly higher than what was described in section 2.3. This is probably because of the adjustments that have been made to the systemic venous model. To optimize the systolic and diastolic pressures of the left and right ventricle a review of the hemodynamic parameters governing the systemic

57 5.1. Normal Physiological Conditions 58

150 160 plv 140 pao,mean C PV-loop pao 120 ESPVR 100 C 100 B B 80 [mmHg] lv

p 60 50 Pressure [mmHg] 40

20 EDPVR D A AD 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 160 Time [s] Vlv [ml]

Figure 5.1: Pressure proﬁles of the left ventricle (left) and the associated PV-loop (right).

40 35 prv 35 pap,mean 30 PV-loop pap 30 25 ESPVR C

25 C 20 B 20 [mmHg] 15 rv

15 p Pressure [mmHg] 10 10 B 5 A D 5 EDPVR D A

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 160 180 200 Time [s] Vrv [ml]

Figure 5.2: Pressure proﬁles of the right ventricle (left) and the associated PV-loop (right). 5.1. Normal Physiological Conditions 59 circulation is recommended. Figure 5.3 shows, on the left, a Pressure-Volume (PV) loop of the intraventricular septum and the corresponding change in intraventricular septum volume during one second of simulation.

8 8 Systole 7 7

6 6 [ml] 5

5 spt V 4 [ml] 4 spt

V 3 3

2 2 Septum volume 1 1 Diastole 0 Systole Diastole 0 0 20 40 60 80 100 120 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 pspt [mmHg] Time [s]

Figure 5.3: PV-loop of the intraventricular septum (left). Changes in Vspt during simulation (right).

5.1.1 Breathing

To simulate normal breathing, pith was chosen as a sinusoidal function pith = −4.5 + 2π sin t , where τ is the time period for one breath and was set to 5.5 seconds. Figure τ b b 5.4 shows the change in alveolar volume Valv during normal breathing calculated from the model. The tidal volume is approximately 500 ml which agrees with the results found in literature [13]. Figure 5.5 shows the eﬀect of breathing on the PV-loop for the left and right ventricle. The observable change in the PV-loops originates from the intrathoracic pressure pith and the eﬀect it has on the vessels and chambers that lie within the thoracic cage. During normal breathing, pith changes and thereby the ventricle pressures change as according to equation (3.24). 5.1. Normal Physiological Conditions 60

2.6

2.5

2.4 [l]

alv 2.3 V

2.2

2.1 Lung volume

1.9

0 2 4 6 8 0 2 4 6 8 0 Time [s]

Figure 5.4: Changes in lung volume Valv during normal breathing.

160 35

140 30 PV-loop PV-loop 120 ESPVR 25 ESPVR C C 100 B 20 B 80 [mmHg] [mmHg] 15 lv rv p 60 p

10 40

20 5 EDPVR D A EDPVR D A

0 0 0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160 180 200 Vlv [ml] Vrv [ml]

Figure 5.5: Eﬀect of breathing on the PV-loop for the left (left) and right (right) ventricle. 5.2. Left Heart Failure 61

5.2 Left Heart Failure

This section describes the simulated hemodynamic responses of the cardiovascular system of a person suﬀering from left heart failure. The results also include the expected hemodynamic changes for a patient with LVAD support when the LVAD rotor is working at a turning speed of 9600 rpm.

120 Without LVAD With LVAD

100 ESPVR

60 [mmHg] lv p 40

20 EDPVR

0 0 50 100 150 200 250 300 Vlv [ml]

Figure 5.6: PV-loop of left ventricle under left heart failure with and without LVAD support.

Figure 5.6 shows the PV-loop for the left ventricle of a patient suﬀering from left heart failure with and without LVAD support. The results show how the systolic pressure in the ventricle plv ≈ 100 mmHg are lower than normal and, compared to ﬁgure 5.1, how the working volume of the left ventricle greatly increases 200 ml < Vlv < 250 ml which is indicative of left heart failure [17, 13]. With LVAD support the working volume of the left ventricle is reduced 170 ml

45 Without LVAD With LVAD 40

30 ESPVR

25 [mmHg] 20 rv p 15

5 EDPVR

0 0 20 40 60 80 100 120 140 160 180 200 Vrv [ml]

Figure 5.7: PV-loop of right ventricle under left heart failure with and without LVAD support. 5.3. Orthostatic Stress 62

Figure 5.7 shows the same responses but for the right ventricle. Notice here the increased stroke volume and size of PV-loop under LVAD support thereby increasing the cardiac output towards a more normal working range. The changes in cardiac output with and without LVAD support are summarized in table 5.1

Table 5.1: Changes in cardiac output for the left and right ventricle when under LVAD support. Left Ventricle [l/min] Right Ventricle [l/min] LVAD [l/min] Without LVAD 4.0 4.0 0.0 With LVAD 1.5 5.5 4.0 Healthy 5.6 5.6 -

Table 5.1 shows the simulated cardiac output for the left and right ventricle for a patient suffering of left heart failure without LVAD support and the changes with LVAD support. An increase, back to the normal working range of > 5 l/min, in cardiac output of the right ventricle can be observed in the table. On the other hand the cardiac output of the left ventricle drops drastically to about 1.5 l/min and instead the blood flows through the LVAD at 4 l/min. This results in a total cardiac output from the left ventricle of 5.5 l/min. The change in intraventricular septum volume Vspt with and without LVAD support can be seen in figure 5.8. The reason for this change can be attributed to the effect the LVAD has on the ventricle pressures as observed in figures 5.6 and 5.7.

20 20 Without LVAD Without LVAD 18 With LVAD 18 With LVAD

16 16

14 14

12 12 [ml] [ml] 10 10 spt spt V V 8 8

6 6

4 4

2 2

0 0 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 pspt [mmHg] Time [s]

Figure 5.8: VP-loop of the intraventricular septum with and without LVAD support (left). Changes in Vspt during simulation with and without LVAD support (right).

5.3 Orthostatic Stress

This section describes the hemodynamic response of a healthy person when under orthostatic stress. This is done by simulating a patient performing a Head Up Tilt (HUT) test. A HUT test is done by having a patient lying on a tilt table, as described in section 3.5, and gradually increasing the tilt angle α = from 0 to 75◦ and then back to zero again. The response of the systemic venous system is simulated and displayed in order to evaluate the displaced volumes in the vascular compartments. 5.3. Orthostatic Stress 63

160 pao,mean

] 140 1 α − fhr 120 [min hr f 100 ], ◦ [

α 80

60 [mmHg], 40

ao,mean 20 p Up Down 0 100 150 200 250 300 350 400 450 500 550 600 Time [s]

Figure 5.9: Mean aortic pressure p¯ao,mean response when subjected to orthostatic stress through Head Up Tilt (HUT) test. The tilt angle α and heart rate fhr response is also displayed.

Figure 5.9 shows the simulated mean aortic pressure p¯ao,mean response to the HUT test. The dashed black lines mark the start of tilt up to 75◦ and the start of tilt down back to level position. p¯ao,mean is taken as the averaged aortic pressure of one heart beat. Also plotted in the same ﬁgure is the heart rate fhr and the tilt angle α to give an easy overview of where the test starts and ends. This does not agree to that described in literature [16, 18]. The simulated results shows how p¯ao,mean initially drops and then recovers. However p¯ao,mean does not fully recover to normal working value of 100 mmHg, as described in literature [16, 18], but stagnates at around 80 mmHg. This is probably because the modeled reﬂex mechanisms is not enough to fully compensate for the orthostatic pressure changes and volume distributions occuring during orthostatic stress. This displacement of p¯ao,mean from the reference value causes the simulated heart rate to stay elevated during the test, even though it should drop to about 70 beats/min according to literature [16, 18].

8 140 qo,lv Vsv,l

7 qo,rv 120 Vsv,r

6 100

5 80

4 60

3 Stroke Volume [ml] 40 Cardiac output [l/min]

2 20

Up Down 1 0 100 150 200 250 300 350 400 450 500 550 600 100 150 200 250 300 350 400 450 500 550 600 Time [s] Time [s]

Figure 5.10: Cardiac output (left) and stroke volume (right), averaged over one heart beat, for the left Vsv,l and right ventricle Vsv,r when subjected to orthostatic stress (right). 5.4. Valsava Maneuver 64

In figure 5.10, on the left, the cardiac output of the left and right ventricle during the HUT test and on the right, the stroke volume of the left and right ventricle are displayed. When compared to the tabulated values in [18] it can be seen that the drop in both Vsv,i and cardiac output, during the HUT test, is more than what would be expected. In Heusden [18] cardiac output decrease with about 16 — 27 % of original values and stroke volume about 30 — 45 % during the maneuver. In figure 5.10 the drop in cardiac output can be measured to about 50 % and for stroke volume to about 70 %. This measurement is taken as the difference to the stabilized value that is reached during the maneuver and not the initial undershoot that can be observed in the figure. These values are much higher than what has been observed in literature [18] and is probably because the current control reflexes implemented are not enough to fully counteract the hemodynamical changes of orthostaic stress. This would suggest that an improvement in the control loops is needed or an addition of a heart contractility controller.

5500 Up Down V head 5000 2000 Vkid Vtot 4500 Vsp

Vleg 4000 1500 3500

3000 Vsys,v 2500 1000

Volume [ml] 2000

Blood Volume [ml] 1500 500 1000 Vsys,a

500 V pul Vheart Up Down 0 0 100 150 200 250 300 350 400 450 500 550 600 100 150 200 250 300 350 400 450 500 550 600 Time [s] Time [s]

Figure 5.11: Volume displacements in the venous compartments (left) and blood volumes in diﬀerent parts of the body (right) when subjected to orthostatic stress.

Figure 5.11 shows the simulated volume displacements across the vascular compartments during the HUT test. The largest volume displacement occurs in the legs, about +400 ml. The displaced volume in the head is about -150 ml, in the splanchnic about +150 ml and in the kidneys +50 ml. When adding these volumes together it is clear that the total volume change in the systemic veins do not add up to zero. To compensate blood is redestributed from other parts of the cardiovascular circulation, which can be seen on the right in the ﬁgure. Some blood from the systemic arteries Vsys,a, the pulmonary circulation Vpul and the heart Vheart goes into the systemic veins Vsys,v during orthostatic stress. The total blood volume Vtot stays constant during the simulation which conﬁrms the conservation of blood in the body.

5.4 Valsava Maneuver

This section shows the simulated hemodynamics of the cardiovascular system when performing the Valsalva maneuver. Figure 5.12 shows the driving force that performs the Valsalva maneuver, the intrathoracic pressure pith. When pith rises, the maneuver begins and when it drops the glottis is opened. The ﬁgure also shows the mean aortic pressure response 5.4. Valsava Maneuver 65

160 pith 140 pao,mean

120

100

Pressure [mmHg] 40

130 140 150 160 170 180 190 200 Time [s]

Figure 5.12: Mean aortic pressure p¯ao,mean response during Valsalva maneuver plotted together with the changes in intrathoracic pressure pith.

p¯ao,mean during the maneuver. As can be seen, all the four phases described in paragraph 3.9 are present and agree with literature [7, 11, 26, 37]. The heart rate response during the maneuver can be seen in figure 5.13. According to literature [24, 26] there should be an initial decrease in heart rate followed by a steady rise to a value of about 150 % of normal. The simulated results show a very similar behaviour only that the rise in heart rate is up to 175 % of normal value. The reason for this difference could be due to slight differences in the heart rate controllers between literature and this Master’s Thesis. At the end of the maneuver there is a drop followed by a stabilization of the heart rate at the normal working value also agreeing with literature [26, 24]. Examining the results of the simulations more closely, the heart dynamics of the simulated results seem to be more fast reacting than that of the experimental results shown in Liang et al. [24, 26]. This conclusion is drawn from the fact that the initial drop and following increase in heart rate at the start of the maneuver is much steeper in the simulated results.

120

110

100

] 90 1 −

80 [min

hr 70 f

40 130 140 150 160 170 180 190 200 Time [s]

Figure 5.13: Heart rate fhr response during Valsalva maneuver.

Figure 5.14 shows the changes of cardiac output of the ventricles and the changes of systemic arterial resistance Ras during the maneuver. The resulting cardiac output, 60 % of 5.4. Valsava Maneuver 66 normal value, is higher during the maneuver than reported in Lu [26] where cardiac output drop to about 40 % of normal value. The increase in Ras during phase 2 of the maneuver is probably because of the drop in transmural aortic pressure pao,tm and is also a reason for the slow recovery in aortic pressure during the later stages of phase 2.

12 1.5 qo,lv q 1.4 10 o,rv

1.3

8 1.2

6 1.1

[mmHgs/ml] 1

4 as R 0.9 Cardiac output [l/min] 2 0.8

0 0.7 130 140 150 160 170 180 190 200 130 140 150 160 170 180 190 200 Time [s] Time [s]

Figure 5.14: Cardiac output (left) for the left and right ventricle during the Valsalva maneuver; Baroreﬂex changes of systemic arterial resistance Ras during the maneuver (right).

5.4.1 Pathological Conditions This section describes the simulated results of the Valsalva maneuver of patients suffering from left heart failure. Figure 5.15 shows the changes in p¯ao,mean during the maneuver. Notice the large difference between the pathological case here and the healthy case in figure

160 pith 140 pao,mean

120

100

60 [mmHg] ao Pressure [mmHg] 40 p

130 140 150 160 170 180 190 200 Time [s]

Figure 5.15: Mean aortic pressure p¯ao,mean response during Valsalva maneuver and intrathoracic pressure pith (left). Expected pathological aortic pressure response during the Valsalva maneuver [43] (right).

5.12. As described in literature [43] there is no visible reduction in p¯ao,mean during phase 5.4. Valsava Maneuver 67

2 of the maneuver and no overshoot after the release of the maneuver all indicative of left heart failure. Figure 5.16 shows the simulated heart rate during the Valsalva maneuver in the pathological case.

160

140

120 ] 1 −

100 [min hr f 80

40 130 140 150 160 170 180 190 200 Time [s]

Figure 5.16: Heart rate fhr response during Valsalva maneuver when suﬀering from left heart failure.

The cardiac output and the changes to systemic arterial resistance Ras in the pathological case can be seen in ﬁgure 5.17. Since the patient is suﬀering from left heart failure the cardiac output is lower than normal, only 4.2 l/min. Notice the relatively constant cardiac

12 q o,lv 1.4 q 10 o,rv

1.3 8

1.2 6 [mmHgs/ml] 1.1

4 as R Cardiac output [l/min]

2 1

0 0.9 130 140 150 160 170 180 190 200 130 140 150 160 170 180 190 200 Time [s] Time [s]

Figure 5.17: Both figures represent the case of left heart failure. Cardiac output for the left and right ventricle (left) during the Valsalva maneuver. Baroreflex changes to systemic arterial resistance Ras during the maneuver (right). output during the maneuver. In comparison to the physiological case, where the cardiac output drops to about 60 % of normal value, the cardiac output has an initial drop but then returns to the normal pathological value. The reason for this is not clear but is probably connected to the left ventricle dynamics and how they differ from the pathological case and the physiological. This result together with the square wave form of the arterial pressure 5.5. FVC-Maneuver 68 response during the maneuver would indicate that the modelled pathological heart is less sensitive or less adaptable to changes in intrathoracic pressure. This could be connected with the already heightened reflex activity, Ras ≈ 1.1 mmHgs/ml, which is accompanied with left heart failure.

5.5 FVC-Maneuver

This section details the hemodynamic responses of the cardiovascular system when performing the FVC maneuver. As previously mentioned extra attention will be given to the response of the pulmonary capillary system.

80 palv pith

0 Pressure [mmHg]

−20

e i e∗ −40 0 2 4 6 8 10 12 14 16 18 20 Time [s]

Figure 5.18: Changes in alveoli pressure palv during FVC maneuver plotted together with the driving intrathoracic pressure pith.

Figure 5.18 shows the changes in the alveoli palv and intrathoracic pressure pith during the maneuver. Notice that they only differ when pith is negative. This is connected to the dynamics of the transmural pressure pel which is zero in the lower lung volumes that occur when intrathoracic pressure rises and air is pressed out of the lungs. During inspiration the lungs inflate due to a negative pith and pel increases causing a pressure difference between pith and palv as can be seen in the figure. Figure 5.19 shows, on the left, the changes in pulmonary capillary volume Vpc. The results concur well with the ones presented in Liu [25], which can be seen on the right in the figure. At the beginning of the maneuver, Vpc drops due to the inflation of the lungs, a drop in pre-/post capillary resistances Rcap,a/Rcap,v and the difference between inflow and outflow of the capillaries. The changes in pulmonary resistance can be seen in the top of figure 5.20 together with the expected results from Liu [25] on the bottom. The drop in Vpc is followed by an increase in pulmonary capillary resistance Rpc. As expiration starts the pulmonary outflow is larger than the inflow, as can be seen on the top in figure 5.21, which causes Vpc to recover and Rpc to drop once again. Rcap,a and Rcap,v increases due to increasing pith. Rcap,a and Rcap,v remains elevated until the end of the maneuver when pith and lung volume return to normal. The two sudden drops at 8 and 12 seconds indicate a numerical error somewhere in the solution. The continuity of the equations modeling the pulmonary capillaries needs to be reexamined. The shape of the curves in figure 5.20 corresponds well to each other, with the excep- tion that the simulated results from this Master Thesis has a slight numerical error 8 and 5.5. FVC-Maneuver 69

50 [ml] [ml] pc pc 40 V V

20 ∗ e i e 10 0 2 4 6 8 10 12 14 16 18 20 Time [s] Time [s]

Figure 5.19: Changes in pulmonary capillary volume Vpc during FVC maneuver (left). Expected changes in Vpc according to Liu [25] (right).

1200 Rpc

Rpv 1000 Rpa

800

Rpc 600

400

R 200 cap,a R Pulmonary resistances [mmHgs/l] cap,v

∗ 0 e i e

0 2 4 6 8 10 12 14 16 18 20 Time [s]

Rpc

Rcap,a

[mmHgs/l] Rcap,v R

Time [s]

Figure 5.20: Changes in pulmonary capillary resistances, Rpc, Rcap,a and Rcap,v during the FVC-maneuver Vpc (top). Expected changes to pulmonary resistances according to Liu [25] (bottom).

12 seconds into the simulation. The reason why the simulated results for the pulmonary resistances are much higher than that reported in Liu [25] is probably due to the slight differences in the shape of the pith curve used to simulate the maneuver. Even though Rpc is three times larger in figure 5.20 than reported in Liu [25] this should not be seen as an error 5.5. FVC-Maneuver 70 in the simulation. The reason for this relatively large difference is most likely because of the strong dependence of Rpc on Vpc which has the result that, in the low Vpc range, even a very small change in Vpc causes very large changes to Rpc. Another reason the model does not exactly replicate the results presented in Liu [25] is because the model in Liu [25] is for an isolated model of the pulmonary capillaries and in this Master Thesis the local pulmonary capillary model has been implemented in a larger model of the entire cardiovascular system which could have an effect on the absolute values of the resistances.

20 qin,pul qout,pul

5 Pulmonary ﬂow [l/min]

0 ∗ e i e

0 2 4 6 8 10 12 14 16 18 20 Time [s]

qin,pul q [l/min] qout,pul

Time [s]

Figure 5.21: Changes in pulmonary capillary ﬂow qin,pul and qout,pul (top). Expected changes to pulmonary capillary ﬂow according to Liu [25] (bottom).

The changes in pulmonary capillary flow can be seen on the top of figure 5.21 together with the simulated results from Liu [25] on the bottom of the figure. These results are somewhat harder to compare because in Liu [25] the transmural pulmonary arterial and venous pressures were assumed to be constant, which is not the case in the model presented in this Master Thesis. Because of the pulsatile nature of the heart, the pulmonary flows fluctuate between heart beats which makes an exact comparison between the figures incon- clusive. What can be extrapolated from the figures is that, during the fast inhalation, there is indeed a difference between inflow and outflow of the pulmonary capillaries which in turn causes the drop in Vpc as was seen in figure 5.19. This pulmonary flow difference shifts during the forced expiration increasing Vpc. The difference in pulmonary capillary inflow and outflow disappears as Vpc → Vpc,max and is kept zero until the end of the maneuver. That is why the curves for qin,pul and qout,pul overlap the last ten seconds of the maneuver. The two square peaks 8 and 12 seconds into the simulation are due to the same numerical error affecting Rcap,a and Rcap,v. This is clear since a sudden change in resistance directly affect the flow through a system. Figure 5.22 shows the change in lung volume Valv during the maneuver. Comparing the 5.5. FVC-Maneuver 71

5.5

4.5 [l]

alv 4 V [l]

3.5 RV

3 - V alv V 2.5 Lung volume

1.5 ∗ e i e 1 0 2 4 6 8 10 12 14 16 18 20 Time [s] Time [s]

Figure 5.22: Changes in lung volume Valv during the FVC-maneuver (left). Expected changes to lung volume according to Liu [25] (right)

simulated results with the ones from Liu [25] in ﬁgures 5.21 and 5.22 it can be seen that the results agree well with literature.

1200

1000

800

600 Expiration [mmHgs/l]

pc 400 [mmHgs/l]

R ¡ R 200 Inspiration

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Active lung volume Valv − VRV [l] Valv - VRV [l]

Figure 5.23: Changes in pulmonary capillary resistance Rpc plotted against changes in alveoli volume Valv (left). Expected results according to Liu [25] (right).

The left side of the figures 5.23 and 5.24 shows the changes to Rpc and Rcap,a/Rcap,v plotted against Valv − VRV . It can be seen that the overall behavior and shape of the resistance loops are as describe in Liu [25] on the right side of the figures but, as mentioned earlier, the shape of the loops are more square shaped and the values of the resistances are higher than described in Liu [25]. The flow rate qair,2 changes, due to changes in Valv − VRV as can be seen in figure 5.25. The small peaks that can be observed in the figure comes from numerical approximations in the equations describing the transmural pressure pel. The peaks are only one time step long and happen just as pith = 0 but seem to have no effect on the overall solution. This could be due to some error in the continuity of the equations describing the airways at pith = 0. 5.5. FVC-Maneuver 72

500

400

300 [mmHgs/l] [mmHgs/l] 200 Expiration cap,v cap,v , R /R 100 cap,a cap,a R R 0 Inspiration

−100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Active lung volume Valv − VRV [l] Valv-VRV [l]

Figure 5.24: Changes in pulmonary pre-/post-capillary resistances Rcap,a/Rcap,v plotted against changes in active lung volume Valv − VRV (left). Expected results according to Liu [25] (right).

6 Expiration

2 [l/s] 0 [l/s] 2 air, air,2

q −2 q

−4

−6

−8 Inspiration −10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Active lung volume Valv − VRV [l] VA - VRV [l]

Figure 5.25: Airway ﬂow response qair,2 plotted against changes in alveoli volume Valv (left). Expected results according to Liu [25] (right). Chapter 6

Review

This chapter contains a review over the results presented, over any limitations of the system and future work that could or needs to be done.

6.1 Achieved Results

The largest problem encountered when trying to compare simulated results to that of literature and experiments is that every person is an individual with small diﬀerences in their own hemodynamics, i.e. lung size, cardiac output, normal heart rate can all vary from person to person. This makes it hard to say that a result is false or correct. Therefore a more qualitative approach is needed with a quantitative comparison done by always considering inter-subject variability. When looking at the overall simulation results of the system one can observe a very good comparison with ones reported in literature. Both during normal conditions and pathological conditions with and without LVAD support the results are very similar to those reported in Baumann [1], which was the goal. There are three points in which a deeper discussion is needed because they are part of the new model and therefore important to understand:

– Cardiovascular response due to orthostatic stress.

– Cardiovascular response during the Valsalva maneuver.

– The response of the intra-alveolar pulmonary capillaries during the FVC maneuver.

The first results that will be discussed are the hemodynamic responses of the model to orthostatic stress. The most important thing that the human body controls is the mean transmural aortic pressure [17]. When looking in literature [16, 18], the aortic pressure is, when standing up or performing the HUT test, mostly constant. What is different in the simulated results from literature is the initial drop in aortic pressure at the beginning of the HUT test. By comparing the simulated results to [16, 18] one can observe a deviation from what has been reported. A possible reason for this would be accountable to the systemic arterial resistance controller, the heart rate controller and stress relaxation have been implemented as regulatory mechanisms. The truth is that there are several more mechanisms that help regulate the cardiovascular system, e.g. a changeable heart contractility due to baroreflex control. It could be that the existing regulatory mechanisms are not enough

73 6.2. Limitations 74 to immediately compensate for the large hemodynamic changes that take place during the HUT test, which makes this an area of improvement. The next results to be discussed into more detail are about the hemodynamic responses of the Valsalva maneuver. The results of the Valsalva maneuver both in the physiological and pathological cases are very promising. In the physiological case all four phases described in theory could be detected and coincided with earlier results [7, 11, 24, 26, 37]. In the pathological case it was also very positive to see the change in aortic pressure response to the square wave form indicative of left heart failure as according to literature [43]. The last results to be discussed relate to the internal hemodynamic responses of the intra- alveolar pulmonary capillaries. Even though the responses for the pulmonary capillaries were overall very good and in accordance to literature [25, 26] one small problem lay in the lack of comparable literature. Since the model was based on the same source as the comparable results, even though the results were similar it is hard to draw any conclusions. The parameter values describing the airway mechanics have been taken and veriﬁed from [25] for just one simulated person. This is something that should probably be varied and implemented for a number of people in order to more solidly verify the model. Due to the lack of time this was not possible in this Master’s Thesis. Nevertheless, even though the internal responses of the pulmonary capillary model were only veriﬁed against a single source this was not the only test performed. All the global tests done, during normal conditions, pathological conditions, Valsalva maneuver show very realistic hemodynamic responses. This can only be true when the equations describing the overall dynamics of the pulmonary capillaries are correct.

6.2 Limitations

When discussing the limitations of the model it is important to know that the model is only supposed to be a close approximation of the human cardiovascular system. The model is supposed to display as much as possible of the complex dynamics that exist in the cardiovascular system keeping the model as simple as possible. There are however some important limitations to the current model. One of the larger limitations of the model could be seen in the hemodynamic responses to orthostatic stress. There is also, to date, no way, to simulate physical stress with the present model, i.e. exercise and an increasing cardiac output. The only way to increase cardiac output is through an infusion of blood but there is a limit to how much blood can be infused before the model crashes. With infusion the model response of an increase in cardiac output of about 100 % (10 l/min) is measurable before it crashes. This is probably related to the equations governing the heart mechanics because with an increase in total blood volume the ventricles will ﬁll with more blood. When too much blood is forced into the ventricles the equations describing the ESPVR and the EDPVR are no longer valid and the simulation crashes. It also a realistic response since there is a limit to how much blood can be infused into a person.

6.3 Future work

There is still much that could be done in regards to future work. As mentioned earlier, a model of the human cardiovascular system is almost never finished. Even though model verifications have been done against literature a future Master’s Thesis could still be the verification of this model. This should be done by someone with a medical background since 6.3. Future work 75 there is much more to verification than just comparing the results to literary references. It requires a lot of foreknowledge and sound judgement. Something that could be done relatively quickly, but was left out due to a shortage of time, is the reimplementation of the unstressed volume controller in the Matlab-code. This only requires an implementation of the differential equation describing the control loop in the Matlab-code. Due to the complexity of the inner mechanics of the axial flow pump the PI-controller could be left as implemented in Simulink. Another field of further study could be the reduction of the model and the future development of automatic control strategies for LVADs. As the results in this Master’s Thesis have shown, the LVAD, when in use, almost completely takes over the workings of the left ventricle. Instead of supporting the heart, the LVAD takes over and in some cases reduces the chances of a normal recovery. With new control strategies, the LVAD could be adjusted to work with the heart and further recovery making them suitable as therapy oriented devices and not only for medium to long term support until transplantation. Chapter 7

Acknowledgements

This Master Thesis Project was done in cooperation with the Automation and Control Institute (ACIN) at Vienna University of Technology, the Research Group in Cardivascular Dynamics and Artiﬁcial organs at the Medical University of Vienna (MUV) situated at the General Hospital of Vienna and Umeå University. First and foremost I would like to thank Univ.-Prof. Dr. techn. A. Kugi (ACIN) and Univ.-Prof. Dr. techn. H. Schima (MUV) for their support and for the provition of a work area. A great thanks go out to my supervisors Dipl.-Ing. Dr. techn. Francesco Moscato (MUV), Dr.-Ing. Wolfgang Kemmetmüller (ACIN) and Katrin Speicher (ACIN) for supporting me, coming with suggestions and correcting my work. A special thanks also goes out to DI Michael Baumann.

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[44] The American Heritage Dictionary of the English Language, Fourth Edition, Houghton Miﬄin Company. (2009). Appendix A

Parameters and Variables

A list of all variables with explanatory comments can be found in the source code cvmodel.c. A list of all parameters and their values, also including comments, can be found in the parameter ﬁle data_CV_model_BAUMANN_Michael.m.

80 Appendix B

Full Model Schematic

81 82

Rhead,1 Rhead,2

portho,head qair,1 pref R pith coll C Csvc p head Airway C qair,2 Mechanics ptm pith 0 mmHg R Rkid,1 Rkid,2 0 mmHg S R p svc el RLti 0 mmHg portho,kid palv Ckid

p pao,ref ao pabd pao pao,ref Rivc 0 mmHg pith pao Control fhr Rsp,1 Rsp,2 pao Cannula and Pump p Control ortho,sp Civc 0 mmHg Csp pla plv pao pv pabd

Rd,mv D1 Rd,av D3 0 mmHg Ras Rleg,1 Rleg,2 Lmv Lav Lao Ri,mv D2 Ri,av D4 portho,leg Rlv Rao

Cla Cas

Cv Clv Cao Cleg

pith pith pith

0 mmHg 0 mmHg 0 mmHg 0 mmHg 0 mmHg 0 mmHg

pvp ppc pcap,a pap Rd,pv D7 prv Rd,tv D5 pra Rcap,v Rpc/2 Rpc/2 Rcap,a Rvp Rpul Lpv Ltv

Ri,pv D8 Ri,tv D6

Rrv

Cvp ptmb Ccap Cpul Cra

Crv

pith palv pith pith pith pith

0 mmHg 0 mmHg 0 mmHg 0 mmHg 0 mmHg 0 mmHg Figure B.1: Schematic of full cardivascular model. 83

Rkid,1 Rkid,2

portho,kid

Ckid pao pao,ref pao pao,ref 0 mmHg

pao Control fhr Rsp,1 Rsp,2 pao Cannula and Pump Control portho,sp

pla plv pao pv Csp

0 mmHg Rd,mv Lmv D1 Rd,av Lav D3 Ras Rleg,1 Rleg,2 Lao

Ri,mv Lmv D2 Ri,av Lav D4 portho,leg Rlv Rao

Cla Cas

Cv Clv Cao Cleg

pith pith pith pith

0 mmHg 0 mmHg 0 mmHg 0 mmHg 0 mmHg 0 mmHg

Rd,pv Lpv D7 Rd,tv L D5 pla pap pvp pap prv tv pra Rcap Rvp Lpul Rpul

Ri,pv Lpv D8 Ri,tv Ltv D6 pap

Control Rrv

Cvp Ccap Cpul Cra

Crv

pith pith pith pith pith

0 mmHg 0 mmHg 0 mmHg 0 mmHg 0 mmHg Figure B.2: Schematic representing the human cardiovascular system according to Baumann [1].