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                    Master Thesis

A Variable Gain
Physiological Controller
for Rotary Ventricular
Assist Devices
by Luís Felipe Vieira Silva
advised by

Prof. Ph.D. Thiago Damasceno Cordeiro

Federal University of Alagoas
Computing Institute
Maceió, Alagoas
February 11, 2022

ii

Catalogação na Fonte
Universidade Federal de Alagoas
Biblioteca Central
Divisão de Tratamento Técnico
Bibliotecário: Marcelino de Carvalho Freitas Neto – CRB-4 - 1767
S586v

Silva, Luís Felipe Vieira.
A variable gain physiological controller for rotary ventricular
assistent devices / Luís Felipe Vieira Silva. – 2022.
44 f. : il.
Orientador: Thiago Damasceno Cordeiro.
Dissertação (mestrado em Informática) - Universidade Federal de
Alagoas. Instituto de Computação. Maceió, 2022.
Bibliografia: f. 42-44.
1. Coração auxiliar. 2. Controladores fisiológicos. 3. Sistema
cardiovascular. I. Título.
CDU: 004.358:612.1

​UNIVERSIDADE FEDERAL DE ALAGOAS/UFAL
Programa de Pós-Graduação em Informática – PPGI
Instituto de Computação/UFAL
Campus A. C. Simões BR 104-Norte Km 14 BL 12 Tabuleiro do Martins
Maceió/AL - Brasil CEP: 57.072-970 | Telefone: (082) 3214-1401

Folha de Aprovação

LUIS FELIPE VIEIRA SILVA

UM CONTROLADOR FISIOLÓGICO DE GANHO VARIÁVEL PARA DISPOSITIVOS DE
ASSISTÊNCIA VENTRICULAR

Dissertação submetida ao corpo docente do Programa
de Pós-Graduação em Informática da Universidade
Federal de Alagoas e aprovada em 11 de fevereiro de
2022.

Banca Examinadora:

________________________________________
Prof. Dr. THIAGO DAMASCENO CORDEIRO
UFAL – Instituto de Computação
Orientador

__________________________________________
Prof. Dr. ICARO BEZERRA QUEIROZ DE ARAUJO
UFAL – Instituto de Computação
Examinador Interno

________________________________________
Prof. Dr. ANTONIO MARCUS NOGUEIRA LIMA
UFCG – Universidade Federal de Campina Grande
Examinador Externo

FEDERAL UNIVERSITY OF ALAGOAS
Computing Institute

A VARIABLE GAIN PHYSIOLOGICAL CONTROLLER FOR
ROTARY VENTRICULAR ASSIST DEVICES

Master Thesis submited to the Computing Institute fom Federal University of Alagoas as a partial requirement to obtain the degree in Master
in Informatics.

Aproved in February 11, 2022:

Thiago Damasceno Cordeiro,
Prof. Ph.D., Advisor

Antonio Marcus Nogueira Lima,
Prof. Ph.D., UFCG

Ícaro Bezerra Queiroz de Araújo,
Prof. Ph.D., UFAL

Where there is a will, there is a way. After all, miracles happen every day, do they not?
Final Fantasy XIV

Resumo
Este trabalho envolve o projeto de uma lei de controle fisiológico adaptativo para um
dispositivo de assistência ventricular turbodinâmico (TVAD) usando um modelo variante
no tempo de parâmetros concentrados que descreve o sistema cardiovascular. O TVAD
é uma bomba de sangue rotativa acionada por um motor elétrico. A simulação do sistema também inclui o controlador de realimentação adaptativo, que fornece uma saída
cardíaca fisiologicamente correta sob diferentes condições de pré-carga e pós-carga. A
saída cardíaca é estimada a cada batimento cardíaco e o objetivo de controle é alcançado
alterando dinamicamente a referência do controlador de velocidade do motor com base
no erro da pressão sistólica. TVADs fornecem suporte para a circulação sanguínea em
pacientes com insuficiência cardíaca. Diversas estratégias de controle foram desenvolvidas ao longo dos anos, com destaque para as fisiológicas, que adaptam seus parâmetros
para melhorar a condição do paciente. Neste trabalho, uma nova estratégia é proposta
utilizando um controlador fisiológico de ganho variável para manter a saída cardíaca em
um valor de referência sob alterações tanto na pré-carga quanto na pós-carga. Um modelo computacional é usado para avaliar o desempenho desta técnica de controle, que tem
apresentado melhores resultados de adaptabilidade do que controladores de velocidade
constante e controladores de ganho constante.
Palavras-chave: Dispositivos de Assistência Ventricular; Controladores Fisiológicos; Sistema Cardiovascular.

3

Abstract
This work involves designing a physiological adaptive control law for a turbodynamic
ventricular assist device (TVAD) using a lumped parameter time-varying model that describes the cardiovascular system. The TVAD is a rotary blood pump driven by an electrical motor. The system simulation also includes the adaptive feedback controller, which
provides a physiologically correct cardiac output under different preload and afterload
conditions. The cardiac output is estimated at each heartbeat, and the control objective
is achieved by dynamically changing the motor speed controller’s reference based on the
systolic pressure error. TVADs provide support for blood circulation in patients with
heart failure. Several control strategies have been developed over the years, emphasizing
the physiological ones, which adapt their parameters to improve the patient’s condition.
In this work, a new strategy is proposed using a variable gain physiological controller to
keep the cardiac output in a reference value under changes in both preload and afterload.
Computational models are used to evaluate the performance of this control technique,
which has shown better adaptability results than constant speed controllers and constant
gain controllers.
Keywords: Ventricular Assist Devices; Physiological Controller; Cardiovascular System.

4

List of Figures
2.1

Representation of the heart with the course of blood flow through the heart
chambers and heart valves. [Hall and Hall, 2020] . . . . . . . . . . . . . . .
2.2 Cardiac cycle events for the left ventricle shows left atrial, left ventricular and aortic pressure, the ventricular volume, electrocardiogram, and
phonocardiogram. [Hall and Hall, 2020] . . . . . . . . . . . . . . . . . . . .
2.3 Pressure-Volume loop diagram. [Hall and Hall, 2020] . . . . . . . . . . . .
2.4 The representation of coupling between 0D, 1D and 3D model of the arterial tree [Malatos, 2016]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 The two-element Windkessel model. . . . . . . . . . . . . . . . . . . . . . .
2.6 The three-element Windkessel model. . . . . . . . . . . . . . . . . . . . . .
2.7 The four-element Windkessel model, parallel configuration. . . . . . . . . .
2.8 The four-element Windkessel model, series configuration. . . . . . . . . . .
2.9 A variable capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Elastance function, E(t), of the left ventricle. . . . . . . . . . . . . . . . .
2.11 The 0D model of a cardiac valve. . . . . . . . . . . . . . . . . . . . . . . .
2.12 Coupling of the left side of the heart in the form of a 0D circuit. . . . . . .
2.13 0D model of the left side of the heart. . . . . . . . . . . . . . . . . . . . . .

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12
13

3.1
3.2
3.3
3.4

A PVAD from [Timms, 2011]. . . . . . . . . . . . . . . . . . . . . . . . . .
A RVAD from [Timms, 2011]. . . . . . . . . . . . . . . . . . . . . . . . . .
0D model of the left side of the heart with a coupled LVAD. . . . . . . . .
Closed loop block diagram for the speed controller. . . . . . . . . . . . . .

15
15
16
18

4.1
4.2

Average block diagram for LVAD control. . . . . . . . . . . . . . . . . . . . 20
Block diagram describing the speed update closed-loop (area within the
gray dashed line); the kSP updating (area within the black dashed line);
and the control law of the SP controller (area with gray background). . . . 23

5.1
5.2
5.3

PV loops representing preload by changing the mitral valve resistance, Rm . 26
PV loops representing afterload by changing the systemic resistance, Rs . . 26
Step response of the brushless DC motor. . . . . . . . . . . . . . . . . . . . 27

5

5

5
7

5.4

Pump Flow signal from human cardiovascular model with constant pump
speed of 9000 rpm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Pump Flow signal from human cardiovascular model with pump speed
increasing linearly from 12000 rpm to 18000 rpm. Suction phenomenon
occurs at 15500 rpm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Simulation result of the Pump Inlet Pressure (PIP) state variable. . . . . .
5.7 Preload increase effect on PIP. . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Preload decrease effect on PIP. . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Afterload increase effect on PIP. . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Afterload decrease effect on PIP. . . . . . . . . . . . . . . . . . . . . . . .
5.11 Preload and Afterload variation through Rm and Rs . . . . . . . . . . . . .
5.12 Preload and Afterload simultaneously on PIP. . . . . . . . . . . . . . . . .
5.13 Systolic Pressure (SP) value extracted from PIP using the peak detection
algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.14 Cardiac Output of the SP Controller with preload variation. . . . . . . . .
5.15 Cardiac Output of the SP Controller with afterload. . . . . . . . . . . . . .
5.16 Physiologic CO compared with the Variable and Fixed Gain SP Controllers. Increased preload from 10s to 30s of simulation. . . . . . . . . . .
5.17 Physiologic CO compared with the Variable and Fixed Gain SP Controllers. Increased preload from 50s to 80s of simulation. . . . . . . . . . .
5.18 Physiologic CO compared with the Variable and Fixed Gain SP Controllers. Decreased afterload from 80s to 110s of simulation. . . . . . . . .
5.19 Physiologic CO compared with the Variable and Fixed Gain SP Controllers. Preload and afterload variation from 0s to 120s of simulation. . . .
5.20 Physiologic CO compared with different values of KSP (t). . . . . . . . . . .

6

27

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32
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34
35
36
37
39

List of Tables
2.1

Possible configurations of the valves in the cardiac phases. . . . . . . . . . 13

3.1

CVS-LVAD model parameters . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.1

MSE and RMSE compared with Physiologic CO. Increased preload from
10s to 30s of simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MSE and RMSE compared with Physiologic CO. Increased preload from
50s to 80s of simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MSE and RMSE compared with Physiologic CO. Decreased afterload from
80s to 110s of simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . .
MSE and RMSE compared with Physiologic CO. Preload and afterload
variation from 0s to 120s of simulation. . . . . . . . . . . . . . . . . . . . .

5.2
5.3
5.4

7

35
36
37
38

List of Acronyms
CO
CVS
EDV
EDP
ESPVR
ESV
HCS
HR
LVAD
MSE
ODE
P
PI
PID
PIP
PV
PVAD
RMSE
RVAD
SP
SV
TVAD
VG

Cardiac output
Cardiovascular System
End-Diastolic volume
End-Diastolic pressure
End-systolic pressure-volume relationship
End-Systolic volume
Human cardiovascular system
Heart rate
Left ventricular assist device
mean squared error
Ordinary differential equation
Proportional
Proportional integral
Proportional integral derivative
Pump inlet pressure
Pressure-volume
Pulsatile ventricular assist device
root mean squared error
Rotary ventricular assist device
Systolic pressure
Stroke volume
Turbodynamic ventricular assist device
Variable Gain

8

Contents
1 Introduction
1.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2
3
3

2 Modeling the Human Cardiovascular System
2.1 The heart structure and the cardiac cycle . . . . . . . . . . . . . . . . . .
2.1.1 Pressure-Volume loop diagram . . . . . . . . . . . . . . . . . . . .
2.1.2 Cardiac output . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Preload and Afterload . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 0D Model of the Human Cardiovascular System . . . . . . . . . . . . . . .
2.2.1 The Windskessel model . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Ventricle Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Cardiac Valves Modeling . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 0D Modeling of the Left Side of the Heart . . . . . . . . . . . . . .
2.3 Chapter considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
4
6
6
7
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8
10
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12
13

3 Modeling of the Ventricular Assist Devices
3.1 Venticular Assist Devices (VAD) . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Pulsatile Ventricular Assist Devices (PVAD) . . . . . . . . . . . . .
3.1.2 Rotary Ventricular Assist Devices (RVAD) . . . . . . . . . . . . . .
3.2 0D Model of Rotary Ventricular Assist Devices . . . . . . . . . . . . . . .
3.2.1 Dynamic model of the LVAD’s rotary pump . . . . . . . . . . . . .
3.2.2 Tunning a PI controller for the LVAD’s rotary pump . . . . . . . .
3.3 Cardiac Output Calculation with LVAD . . . . . . . . . . . . . . . . . . .
3.4 Chapter considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Variable Gain SP controller
4.1 Control Systems applied to rotary VADs . . . . . . . . . . . . . . . . . . .
4.2 Systolic Pressure Controller . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Systolic Pressure estimation by peak detection . . . . . . . . . . . .

20
20
21
21

9

4.3
4.4

The Variable Gain SP Controller . . . . . . . . . . . . . . . . . . . . . . . 22
Chapter considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Results
5.1 Preload and Afterload Simulation . . . . . . . . . . . . . . . . . . . . . . .
5.2 Simulation results of the LVAD modeling . . . . . . . . . . . . . . . . . . .
5.3 SP Controller Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Cardiac Output simulation with SP Controller . . . . . . . . . . . . . . . .
5.5 Simulations results with Variable Gain SP Controller . . . . . . . . . . . .
5.5.1 Variable Gain KSP (t) Analysis . . . . . . . . . . . . . . . . . . . . .

25
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26
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32
34
38

6 Conclusion
40
6.1 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Bibliography

41

10

Chapter 1
Introduction
Cardiovascular problems such as heart failure are growing at an epidemic rate
[Mancini and Colombo, 2015] and heart transplantation remains the primary treatment
for terminally ill patients. However, as the number of patients needing heart transplants
increases, the number of available organs decreases [Ochsner et al., 2014]. Thus, the use of
alternative treatments such as ventricular assist devices (VADs) has become increasingly
present [Mancini and Colombo, 2015].
The technology of these devices has gained many improvements over the last 20
years [Daners et al., 2017]. However, these devices need to be safer and more robust
for the patient, considering that they can cause thrombosis and bleeding in the user.
These events are usually associated with the operating speed of rotary type devices
[von Platen et al., 2019]. Therefore, the authors say that the adaptation of devices with a
physiological control system is one aspect that needs to be improved in the use of VADs.
The human body can be characterized by a set of variables, some of which are the
individual’s physical structure, age, and weight, making the adaptability of the functioning of these devices essential. Constant modes of operation provide a good quality of
life for a given patient. However, they can present, for example, the phenomenon called
suction in other patients, sucking the wall of the ventricles and causing them to collapse. For this reason, physiological controllers applied to VADs, in addition to providing
adequate pumping for each patient, have the function of preventing undesirable events
[Ochsner et al., 2014].
Physiological control of pumps in LVADs is considered the most efficient way to deal
with pumping problems. However, automatic controllers with feedback are not being used
in practice [Petukhov et al., 2019][Petrou et al., 2017]. Instead, many algorithms are being used to monitor events such as suction detection and arrhythmia events, among others.
It is believed that these adverse events can be detected and prevented by controllers with
feedback [Petukhov et al., 2019].
In a review by [AlOmari et al., 2012], a large number of control strategies proposed
for VADs were listed, from more classic controllers like P, I, PI, and PID to more modern
1

Problem

2

ones like adaptive controllers. [Koh et al., 2019], for example, used a predictive control
capable of preventing suction while regulating the flow rate of a VAD pump applied
to a left ventricle (LVAD). [Wang et al., 2012] developed a feedback controller based on
the control current of an LVAD. The simulations in this work show that the controller
guarantees the physiological demands of a patient at different levels of activity while
avoiding the phenomenon of suction.
Many numerical models representing the entire human cardiovascular system have
been used for computational simulation to test such control techniques and algorithms.
Models that use electronic components to simulate the circulatory system in a simplified
way are called 0D models [Simaan et al., 2008]. The results obtained in the computational
simulations need to be tested in vitro, for this, hydraulic simulators are set up in the
laboratory so that the controller’s performance can be tested with real components.
One of the biggest challenges in developing physiological controllers is the selection of feedback signals for the control algorithm. The literature shows results obtained with controllers using estimated flow, blood pressure, or end-diastolic pressure.
[Petrou et al., 2016] proposed a controller that uses systolic pressure (SP), called an SP
controller, as the main parameter for updating the speed of a rotary VAD. The advantage
of this work is to obtain the systolic pressure signal, with the VAD’s sensor, without
invasive means.
The systolic pressure controller has already been evaluated in the literature
([Malchesky, 2017], [Petrou et al., 2018], [Daners et al., 2017]). However, by updating the
pump speed with constant gain, the controller has some limitations. One of the limitations
is the inability to adapt to physiological variations such as preload and afterload.

1.1

Problem

A rotary LVAD with the SP controller has the advantage of not requiring invasive means
to measure the primary hemodynamic variable that updates pump velocity. However,
this controller has limitations; its equation must be calibrated for a specific preload or
afterload value; if these values change over time, the controller will need a new calibration
due to the constant parameters of the main equation.
In view of this limitation, it’s interesting to develop a new algorithm for the SP
controller that allows the system to adapt to adverse conditions. As it’s known, a patient
may suffer physiological changes in clinical environments, e.g., when in a stressful situation
or due to cardiac arrhythmia.
The new algorithm for the SP controller needs to undergo the same tests performed
on the original controller and new tests with preload and afterload variation to show a
possible improvement in the performance when using variable parameters.
A control algorithm for LVADs needs a series of tests and validation steps before being

Proposal

3

actually implemented, the first step will be developed in this work with the design and
testing in silico of the algorithm. The other steps, that can be carried out in future works,
are: the implementation in vitro of the algorithm in hydraulic simulators; deployment and
testing in vivo of the algorithm in animals and the last step of validation is made by testing
in a small group of humans with LVAD. The last two steps are made with the approval of
an ethics committee of internationally recognized research institutions. Each step in the
development process of the algorithm depends on the previous step being validated and
approved.

1.2

Proposal

The purpose of this work is to present a new adaptable physiological controller for ventricular assist devices that face hemodynamic changes in both preload and afterload.
For this purpose, the SP controller is implemented and simulated using the computational model of the human cardiovascular system developed by Simaan et al.
[Simaan et al., 2008], with a set of parameters that represents an adult male. A new
strategy for this controller is designed to improve its adaptability.

1.3

Structure

This work consists of 6 chapters. Chapter 2 shows the basics of the cardiovascular system,
like preload, afterload, cardiac cycle, heart rate, and also presents how the modeling of
the human cardiovascular system is made for simulation. Chapter 3 brings the modeling
of ventricular assist devices, adds the dynamic model to the pump to get more realistic
simulation results, and presents a way to estimate cardiac output when using this type of
device. Chapter 4 presents the SP Controller with fixed gain for LVAD and the algorithm
of the new proposed controller, the Variable Gain SP Controller. The performance of
both controllers is evaluated in experiments using the root mean squared error and the
mean squared error as performance indices. Chapter 5 shows all simulations results of
this work and Chapter 6 contains the conclusions and the future works of this work 1 .

1

All the algorithms used in this work can be found at https://github.com/lfelipev/vgsp-controller.

Chapter 2
Modeling the Human Cardiovascular
System
This chapter introduces the human cardiovascular system with basic concepts so that the
reader can acquire basic knowledge in the area. This chapter also describes the metrics
used when analyzing the cardiovascular system, the cardiac cycle is detailed, and the
concept of cardiac output, preload, and afterload is also shown.
The chapter also shows how the human cardiovascular system can be modeled for
computer simulations through 0D modeling that uses differential equations drawn from
electrical circuits.

2.1

The heart structure and the cardiac cycle

The heart can be thought of as two separate pumps, one on the right side that pumps
blood through the lungs and one on the left side that pumps blood through the peripheral
organs. The ventricles supply the main pumping force that propels blood circulation. The
heart has unique mechanisms that cause a continuous succession of contractions, called
heart rhythmicity, which characterize the heart’s rhythmical beat [Hall and Hall, 2020].
The events that happen from the beginning of one beat to the beginning of the next
are called the cardiac cycle. More specifically, the cycle consists of a period of relaxation
called diastole, where the heart fills with blood, followed by a period of contraction called
systole. The duration of the cardiac cycle is the heart rate (HR). For example, if HR is 70
beats/min (bpm), the cardiac cycle duration is 1/70 beats/min or 0.84 seconds per beat.
The events during the cardiac cycle for the left side of the heart can be visualized by
the curves shown in Figure 2.2. The three curves at the top show the pressure behavior in
the aorta, left ventricle, and left atrium, respectively. The fourth curve shows the changes
of volume in the left ventricle and the following two curves are the electrocardiogram and
phonocardiogram signal, respectively.

4

The heart structure and the cardiac cycle

5

Head and upper extremity
Aorta
Pulmonary artery
Superior
vena cava

Lungs

Pulmonary
valve

Pulmonary
veins
Left atrium
Mitral valve
Aortic valve
Left ventricle

Right ventricle

Trunk and lower extremity

Figure 2.1: Representation of the heart with the course of blood flow through the heart
chambers and heart valves. [Hall and Hall, 2020]

Pressure (mm Hg)

Isovolumic
contraction

Isovolumic
relaxation
Rapid inflow
Atrial systole
Ejection
Diastasis

Aortic
valve
opens

A-V valve
closes
Aortic pressure
A-V valve
opens

A-V valve
closes

Volume (ml)

a

c

v

Atrial pressure
Ventricular pressure
Ventricular volume

R
P
1st

2nd

3rd

Q

S

T

Electrocardiogram
Phonocardiogram

Systole

Diastole

Systole

Figure 2.2: Cardiac cycle events for the left ventricle shows left atrial, left ventricular
and aortic pressure, the ventricular volume, electrocardiogram, and phonocardiogram.
[Hall and Hall, 2020]

The heart structure and the cardiac cycle

6

The functioning of the ventricles as pumps can be explained using the four stages of
the cardiac cycle. The stages are the same for both ventricles. The first stage is the
filling during diastole, characterized by the large amount of blood accumulated in the
atria. At this stage, the growth of the volume curve can be seen in Figure 2.2.
The subsequent stage is the isovolumic contraction. At this moment, the volume
in the ventricle is kept constant while the heart undergoes an abrupt increase of pressure.
When the pressure rises to a certain level, the cycle enters the ejection stage, where
blood is expelled out of the ventricle.
The last stage before the cycle restarts is the isovolumic relaxation allowing the
intraventricular pressure to decrease rapidly while the volume is kept constant.
During the stages of the cardiac cycle, some measurements must be taken to obtain
relevant information about the cycle: (1) the End-Diastolic Volume (EDV), which is the
increase of blood volume in the ventricle during the diastole, (2) the End-Systolic Volume
(ESV), which is the remaining volume in the ventricle, and (3) the Stroke Volume (SV),
which is the decrease of volume during the systole. The values of these measurements
can be used as, for example, heart performance analysis for cardiologists or parameters
for controllers, as will be seen in section 4.

2.1.1

Pressure-Volume loop diagram

In the literature, a helpful tool for analyzing left ventricular pumping is the PressureVolume (PV) loop diagram, shown in Figure 2.3. The diagram is divided between the
four stages of the cardiac cycle. It is possible to obtain information such as the value
of EDV, ESV, and SV, or even information about the opening and closing of the heart
valves.
Cardiologists can use the PV loop to identify physiological conditions. However, it
can also be used as a performance comparator of ventricular assist devices (VAD), devices
that can assist cardiac pumping in cases of heart failure.
A large area in the diagram indicates that the heart pumps a large amount of blood,
so the ventricle fills with more blood during diastole, extending the diagram to the right.
With that, in the contraction stage, the pressure will increase. The extension of the
diagram to the left indicates the ejection of blood volume out of the ventricle, then decay
of pressure will follow in isovolumetric relaxation.

2.1.2

Cardiac output

The cardiac output (CO) can be defined by the amount of blood that the heart transports through the circulatory system in one minute [King and Lowery, 2017]. One way
to calculate CO is by measuring the stroke volume (SV). The CO relates the SV and the
heart rate (HR) by the following equation:

The heart structure and the cardiac cycle

7
Period of ejection

120
Aortic valve
closes

Left intraventricular pressure (mm Hg)

100

D
Aortic valve
opens

EW

80

C

Isovolumetric
relaxation

60

Isovolumetric
contraction

Stroke volume
40

20

0

Mitral valve
opens
0

50

End-systolic
volume
Period of
A
filling

End-diastolic
B
volume

70
90
110
Left ventricular volume (ml)

Mitral valve
closes
130

Figure 2.3: Pressure-Volume loop diagram. [Hall and Hall, 2020]

CO = SV × HR

(2.1)

To calculate the SV, it is necessary to know the end-diastolic volume (EDV) when
the ventricle is filled with blood and the end-systolic volume (ESV) when the ventricle
ejects it. Therefore the SV can be calculated by the difference in the volume of these two
moments as in the following equation:
SV = EDV − ESV.

2.1.3

(2.2)

Preload and Afterload

The degree of tension in the muscle when it starts to contract is called preload
and is usually considered to be the end-diastolic pressure when the ventricle is filled
[Hall and Hall, 2020]. The afterload indicates the difficulty that the heart will have to
eject the blood, which is directly related to the tension in the muscular wall of the ventricle
and uses systolic pressure as an indicator.
Variations in preload and afterload directly affect stroke volume, influencing the cardiac output and the overall heart function. Therefore, understanding preload and afterload are essential to understanding overall cardiac physiology. For example, the abnormal
values of these measurements are seen in several conditions, such as heart failure and mitral regurgitation [O’Keefe and Singh, 2020].

0D Model of the Human Cardiovascular System

2.2

8

0D Model of the Human Cardiovascular System

The cardiovascular system obeys the laws of mass and momentum conservation, and the
blood interaction with the arterial wall can be modeled in different levels of complexity.
The most low-level type of modeling, called 0D modeling, uses a coupled system of ordinary differential equations (ODEs) to represent pressure, flow, and blood volume. The
1D models are based on simplified fluid flow equations and can reveal the pressure and
flow changes along the whole length of a vessel.
The other higher dimensional models (2D and 3D) are developed with computational
fluid dynamics to detail the hemodynamic properties within vessels. These models consider the local vascular geometry derived from reconstructions of medical screening data.
Therefore the computational complexity of these models is demanding.

Figure 2.4: The representation of coupling between 0D, 1D and 3D model of the arterial
tree [Malatos, 2016].
The 0D model is usually used for control purposes because it demands less computational power than the other models. It is represented with differential equations, thus
being represented with mathematical models such as transfer functions and state-space
models.

2.2.1

The Windskessel model

Figure 2.5 shows the simplest 0D model of the HCS formulated by [Otto, 1899], the socalled two-element Windkessel model. It consists of a resistor (R) that represents the
resistance of the arterial system to the blood flow and a capacitor (C) representing the
elasticity of the arteries. With the Windkessel model, the voltage and current in the

0D Model of the Human Cardiovascular System

9

circuit are normally calculated using Kirchhoff’s laws, with the electric current behaving
like the blood flow (Q) in the artery and the voltage behaving like the pressure (P).

R

P
Q
C

Figure 2.5: The two-element Windkessel model.
In systole, it was shown [Wetterer, 1940] that the two-element Windkessel model
poorly predicts the relation between pressure and flow. With Fourier analysis, it was
possible to notice the necessity to add an impedance whose modulus, in high frequencies,
is equal to the proximal aorta impedance —leading to the creation of the three-element
Windkessel model (Figure 2.6).

Z

P

R
Q

C

Figure 2.6: The three-element Windkessel model.
To reduce the errors in low frequencies, a fourth element was added to the Windkessel
model proposed by [Burattini and Gnudi, 1982]. In this model shown in Figure 2.7, the
fourth element consists of an inertance that represents the total inertance of the arterial
system.

Z

R

P
L

Q
C

Figure 2.7: The four-element Windkessel model, parallel configuration.
The simplified version of the four-element Windkessel model is shown in Figure 2.8.

0D Model of the Human Cardiovascular System

Z

L

P

10

R
Q

C

Figure 2.8: The four-element Windkessel model, series configuration.

2.2.2

Ventricle Modeling

As the ventricle works as a reservoir that charges with volume in the filling phase and
discharges this volume in the form of blood flow in the ejection phase, in the 0D model,
it can be represented by a variable capacitor C(t) (Figure 2.9) where voltage represents
volume, and electrical current represents blood flow.

C(t)
Figure 2.9: A variable capacitor
Capacitance variation needs to be modeled according to a function that represents the
behavior of the ventricle. A possible function for this purpose is the elastance function,
proposed by [Suga and Sagawa, 1974], which relates the pressure change to a given volume
change as
1
Pv (t)
E(t) =
=
(2.3)
C(t)
Vv (t) − Vo
where E(t) is the time varying elastance (mmHg/ml), Pv (t) is the ventricular pressure
(mmHg), Vv (t) is the ventricular volume (ml) and Vo is the initial ventricular volume (ml)
or the theoretical volume in the ventricle at zero pressure.
To implement the elastance function, the following mathematical representation can
be used
E(t) = (Emax − Emin )En (tn ) + Emin

(2.4)

where Emin and Emax are constants related to the end-systolic pressure volume relationship (ESPVR) and end-diastolic pressure volume relationship (EDPVR), respectively, and
En (tn ) is an analytic function normalized between zero and one (in tn = 1), the so-called
”double hill” function defined as
[

tn 1.9
( 0.7
)
En (tn ) = 1.55
tn 1.9
1 + ( 0.7 )

][

1
tn 21.9
1 + ( 1.17
)

]
(2.5)

0D Model of the Human Cardiovascular System

11

t
where the normalization is calculated by tn = Tmax
, Tmax = 0.2 + 0.15Tc and Tc is the
cardiac cycle interval, e.g., Tc = 60/HR. The Figure 2.10 shows the elastance function
for Emax = 2.0, Emin = 0.06, and HR = 75 bpm.

2

1.5

1

0.5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 2.10: Elastance function, E(t), of the left ventricle.

2.2.3

Cardiac Valves Modeling

Heart valves allow the blood to flow in only one direction, so in 0D modeling, they can be
represented by a diode in series with a resistance (Figure 2.11). The following equation
can calculate the value of blood flow Q that passes through the valve
D(P1 − P2 )
(2.6)
R
this equation means that if the voltage at point P1 of the valve is greater than at point
P2 , the valve is open, so D = 1. Otherwise, if the voltage at point P1 is less than at point
P2 , the valve is closed, so D = 0.
Q=

P1

R

D

P2

Q

Figure 2.11: The 0D model of a cardiac valve.
The regurgitation phenomenon is a reverse blood flow that occurs when the valves are
closed. This phenomenon cannot be represented in the model of Figure 2.11 due to the
physical characteristics of the ideal diode. In this work, the regurgitation phenomenon
was not considered to simplify the simulations.

0D Model of the Human Cardiovascular System

2.2.4

12

0D Modeling of the Left Side of the Heart

Putting all the elements of the left ventricle together as in the Figure 2.12, there is the left
atrium (I) as a blood reservoir, then represented by a capacitor; The atrium is directly
connected to the mitral valve (II), represented by the resistor and the diode; the left
ventricle (III) represented by the variable capacitor; the ventricle is also connected to
the aortic valve (IV) and finally the aorta (V) which also behaves as a blood reservoir,
therefore represented by a capacitor. Finally, the circuit is connected to the systemic
circulation (VI) represented by the Windkessel RLC assembly.

II

IV
IV

I

III

V

V -Aorta
VI - Systemic
Flow

I - Left atrium
II - Mitral valve
IV - Aortic valve
III - Left ventricle

Figure 2.12: Coupling of the left side of the heart in the form of a 0D circuit.
The left ventricular model with the variables and parameters used for calculations
is shown in figure 2.13. The original formulation of this model uses the left ventricular
pressure, Plv (t), as a state variable. However, this work adapted this model to use the
left ventricular volume, Vlv (t). This modification was made to avoid using the derivative
of time-varying capacitor C(t) to avoid possible numerical instabilities. Using equation
(2.3), Plv (t) might be calculated as follows:
Plv (t) = E(t)(Vlv (t) − Vo ).

(2.7)

Thus, the five state variables of this system without LVAD are: Pao (t), the aortic
pressure; QT (t), the total flow; Vlv (t), the left ventricular volume; Ps (t), the systemic
pressure and; Pla (t), that is the left atrial pressure. The left atrium is represented by
the capacitor Cla ; the mitral valve is represented by the resistor Rm and diode Dm ; and
the aortic valve is represented by the resistor Ra and diode Da and the aortic flow is

Chapter considerations

13

Table 2.1: Possible configurations of the valves in the cardiac phases.
Valve
State
Phases
Mitral Aortic
1
Closed Closed Isovolumic Relaxation
2
Open
Closed
Filling
1
Closed Closed Isovolomic Contraction
3
Closed
Open
Ejection
Open
Open
Not feasible
represented by Qa (t). The behavior of these valves is modeled using ideal diodes taking
values of either 1, if valve is open, or 0, if valve is close (see table 2.2.4). The aortic
compliance is represented by Cao and the systemic arterial system is modeled using a
four-element Windkessel model comprising Rc , L, Cs and Rs .
Rs

Pla(t)

Rm

Dm

Plv(t)

Ra

Da

HR

Qa(t)

Vlv(t)
Cla

C(t)

Pao(t)
Cao

Rc

L
QT (t)

Ps(t)
Cs

Figure 2.13: 0D model of the left side of the heart.

2.3

Chapter considerations

This chapter introduces the cardiac physiology necessary for understanding the concepts
used in this work. The metrics used when analyzing the cardiovascular system were
displayed, such as the pressure and volume curves and the pressure-volume loop diagram.
In addition, the cardiac cycle was detailed, and the concepts of preload and afterload were
described. The chapter also shows how the cardiovascular system is modeled for use in
computational calculations.
Then the 0D modeling that uses electrical circuits is detailed. The next chapter will
describe ventricular assist devices, such as their function is in the cardiovascular system
and how they can be modeled.

Chapter 3
Modeling of the Ventricular Assist
Devices
This chapter opens the discussion of Ventricular Assist Devices (VAD). It presents two
types of devices, the pulsatile and the rotary. Then, the rotary LVAD 0D modeling is
done with an approach similar to the one in chapter 2. The dynamic model of the DC
motor controlled by a PI controller is included in the modeling.

3.1

Venticular Assist Devices (VAD)

Cardiovascular diseases are growing at an epidemic rate, and heart transplantation remains the primary treatment for terminally ill patients [Mancini and Colombo, 2015].
However, as the number of patients needing heart transplantation increases, the number of available organs decreases [Ochsner et al., 2014]. Thus, the use of alternative
treatments such as Ventricular Assist Devices (VADs) becomes increasingly present
[Mancini and Colombo, 2015].
VADs are mechanical devices used to support the blood circulation of patients with
heart failure. These devices support the right ventricle (RVAD) or the left ventricle
(LVAD) and are used for myocardial recovery, bridge to heart transplantation, or for
long-term cardiac treatments [Timms, 2011].
Mechanical circulatory assist devices can be classified according to their outflow as
pulsatile or continuous [Timms, 2011].

3.1.1

Pulsatile Ventricular Assist Devices (PVAD)

Pulsatile Ventricular Assist Devices (PVAD) pump blood into the circulatory system
through pulses that are pneumatically or electrically generated by diaphragms, bags, or
push plates. These devices have a flow characteristic similar to the human heart and are
commonly called volume displacement devices [Timms, 2011].
14

Venticular Assist Devices (VAD)

15

Due to their pulsatile mode of operation, these devices are more likely to present
problems with durability and reliability, leading to an increase in the development of
rotary ventricular assist devices.

Figure 3.1: A PVAD from [Timms, 2011].

3.1.2

Rotary Ventricular Assist Devices (RVAD)

Rotary Ventricular Assist Devices (RVAD) are operated with continuous flow via a rotating impeller housed within a small pump chamber. The direction to which the blood
enters and leaves the impeller determines the type of the RVAD, which can be axial, radial
(centrifugal), or mixed flow.
The speed at which the impeller spins depends on the fabricant, but it can vary
between 2000 and 15000 rpm to deliver blood flow to the circulatory system. These
devices are capable of providing full circulatory support.

Figure 3.2: A RVAD from [Timms, 2011].

0D Model of Rotary Ventricular Assist Devices

3.2

16

0D Model of Rotary Ventricular Assist Devices

The characteristics of LVADs depend on their manufacture, but all have the purpose of
helping the blood flow of patients with cardiac comorbidities. To provide this blood flow,
LVADs use certain types of pumps. The most common types of pumps are the centrifugal
pump and the axial flow pump.
Rs
Rk

PIP(t)
LVAD

Ri

Pla(t)

Rm

Li
Dm

QLVAD (t)

Lo
Plv(t)

Ra

Da

HR

Qa(t)

Vlv(t)
Cla

Ro

C(t)

Pao(t)
Cao

Rc

L
QT (t)

Ps(t)
Cs

Figure 3.3: 0D model of the left side of the heart with a coupled LVAD.
An LVAD can be coupled to the 0D model as in Figure 3.3. This coupling represents
a rotary blood pump described in [Choi et al., 1997]. The coupling of this device to
the circuit adds the state, QLVAD (t), that represents the blood flow through the LVAD.
Resistors Ri and Ro and inductors Li and Lo represents the inlet and outlet resistances
and inertances, respectively. The parameter, Rk , is a time-varying, nonlinear, pressuredependent resistor that simulates the phenomenon of suction and is described as
{
Rk (t) =

α(Plv (t) − Plv−suc ), Plv (t) ≤ Plv−suc
0,
otherwise

(3.1)

where α is an LVAD-dependent weight parameter and Plv−suc is a threshold pressure.
All parameter values of the CVS-LVAD model and their descriptions are listed in Table
3.1. The pressure difference (inlet-outlet) across the pump, H, is defined by the following
equation:
dx6
+ β2 ω 2
(3.2)
H = β0 x6 + β1
dt
where ω is the pump speed, and β0 = −0.17070, β1 = −0.02177 and β2 = −9.3 × 10−5 ,
which are LVAD-dependent parameters.

0D Model of Rotary Ventricular Assist Devices

17

Table 3.1: CVS-LVAD model parameters
Resistances (mmHg s/ml)
Rs
1.0000
Systemic vascular resistance
Rc
0.0398
Characteristic resistance
Rm
0.0050
Mitral valve resistance
Ra
0.0010
Aortic valve resistance
Ri
0.0677
Inlet Resistance of Cannulae
Ro
0.0677
Outlet Resistance of Cannulae
Rk
Eq. (4)
Suction Resistance with parameters
α = −3.5s/ml and x̄1 mmHg
Compliances (ml/mmHg)
C(t) Time
Left ventricular Compliance
varying
Cae 4.4000
Left atrial compliance
Cs
1.3300
Systemic compliance
Cao 0.0800
Aortic compliance
Inertances (mmHg s2 /ml)
L
0.0005
Inertance of blood in aorta
Li
0.0127
Inlet Inertance of LVAD Cannulae
Lo
0.0127
Outlet Inertance of LVAD Cannulae
LVAD-dependent parameters
β0
-0.17070
β1
-0.02177
β2
-9.3×10−5

3.2.1

Dynamic model of the LVAD’s rotary pump

The simulations results shown in [Simaan et al., 2008] do not take into account the mechanical dynamics of the pump described in [Choi et al., 1997]. It is driven by a brushless
DC motor described as:
dω
= Te − Bω − Tp
(3.3)
J
dt
where ω is the rotor speed, J = 0.916 × 10−6 is the inertia of the rotor, B = 0.660 × 10−6
is the damping coefficient, Te is the motor torque. Tp is the load torque on the pump
that is dependent of the pump rotor speed and the LVAD flow. This load torque can be
defined as:
Tp = f (ω, QLVAD ) = a0 ω 3 + a1 ω 2 QLVAD
where a0 = 0.738 × 10−12 and a1 = 0.198 × 10−10 are motor model parameters.

(3.4)

Cardiac Output Calculation with LVAD

3.2.2

18

Tunning a PI controller for the LVAD’s rotary pump

A PI controller can be tuned to obtain a stable step response and a zero steady-state
error; for this consider its closed-loop transfer function to be given by
(
PI(s) = Kp

1
1+
Ti s

)
(3.5)

One way of tuning the Kp and Ti parameters is considering the system of the Figure
3.4, if Tp is not taken into account for simplification purposes and the system is in openloop, then
)
(
ω(s)
1
1
= Kp 1 +
×
ωdes (s)
Ti s
Js + B

(3.6)

after some calculations it can be seen that
T (s) =

T i Kp s + Kp
Ti s(Js + B)

(3.7)

QLVAD
f(ω,QLVAD)
Eω

ωdes +

PI

Te +

Tp
-

-

1
Js+B

ω

Figure 3.4: Closed loop block diagram for the speed controller.
So to cancel the pole (Js + B) and to obtain a zero steady-state error in the open-loop
system, the assumptions are made:
Kp = B
J
J
Ti =
= .
Kp
B

3.3

(3.8)
(3.9)

Cardiac Output Calculation with LVAD

In this work, CO is calculated as the integral of the sum of QLVAD (t) and Qa (t) and this
value is kept constant during one cardiac cycle. Let the k-th cardiac cycle beginning at
t = Tck and ending at t = Tck+1 , for k = 1, 2, .... The value of CO is only calculated at
t = Tck+1 , called COk , using the values of QLVAD (t) and Qa (t) of the previous cardiac
cycle, i.e., from t = Tck−1 to t = Tck . Hereafter, COk is kept constant during the k-th

Chapter considerations

19

cardiac cycle, i.e., until t = Tck+1 . Discrete and continuous values of CO, COk and CO(t)
are defined as:
[∫
COk =

Tck
Tck−1

]
(QLVAD (t) + Qa (t)) dt × HR

CO(t) = COk , for Tck ≤ t < Tck+1

(3.10)
(3.11)

In real clinical situations, CO(t) can be obtained by estimation strategies
[Petrou et al., 2020] or flow sensors.

3.4

Chapter considerations

This chapter introduced ventricular assist devices (VAD), showing pulsatile and rotating
devices. Then, the 0D modeling of a rotating LVAD was made, including the dynamic
model of the pump, so that the simulation is closer to reality.
The next chapter will present a control strategy for rotary LVAD velocity using systolic
pressure and cardiac output as the main parameters.

Chapter 4
Variable Gain SP controller
This chapter describe the SP controller with constant gain and the proposed strategy with
the Variable Gain SP controller, emphasizing the limitation without using the controller
with gain variation.

4.1

Control Systems applied to rotary VADs

Figure 4.1 shows how VAD control is usually done. The cardiovascular system model
(CVS-model) provides one or more hemodynamic variables that are compared with a
previously defined reference signal. The error signal e(t) generated by the difference
between these signals will serve as input to the controller to determine the device’s speed
w(t). The device’s speed determines the flow of the pump to move blood through the
system. The CVS model can also provide hemodynamic variables to modify the VAD
velocity as the VAD is coupled to the system.
What differentiates VAD controllers in the literature is the strategy used to modify the speed w(t). Some controllers use classical or modern control theory, and others
use computational algorithms or state machines. The objective of the VAD will usually
influence which control strategy will be used.
Reference
Signal

e(t)

+

Controller

ω(t)

Pump Speed

Hemodinamic
Variable

VAD

Pump Flow

CVS-MODEL

Figure 4.1: Average block diagram for LVAD control.
20

Systolic Pressure Controller

4.2

21

Systolic Pressure Controller

Petrou et al. developed a control strategy called SP controller that uses the pump inlet
pressure (PIP) to calculate the maximum systolic pressure (SP), which is detected within
a fixed time interval of 2 seconds to ensure that SP value is detected even for low heart
rates [Petrou et al., 2016]. Using Kirchoff’s Law, the PIP(t) value is given the equation
4.1.
P IP (t) = Plv (t) − Li Q̇LV AD (t) − Ri QLV AD (t)

(4.1)

where Plv (t) is the left ventricular pressure, Li is the Inlet Inertance of LVAD Cannulae,
QLV AD (t) is LVAD flow and Ri is the Inlet Resistance of Cannulae. The SP value is used
to update the pump speed (ωdes ) according to
ωdes = kSP (SP − SPref ) + ωref

(4.2)

where kSP (rpm/mm Hg) is a proportional gain. The values of SPref (mm Hg) and ωref
(rpm) are reference values obtained during a calibration process, which is done in a simulation environment to identify the pump speed that keeps the desired CO at rest. The
value of kSP and the pump speed are adjusted until the desired CO value is obtained. The
rotation speed and the systolic pressure value that satisfy the target CO are the values
of SPref and ωref , respectively, of the equation (4.2).
The performance of the SP controller was evaluated considering changes in preload,
afterload, and ventricular contractility (VC). The variables end-diastolic pressure (EDP),
CO, and SP were observed and compared with reference values previously defined by a
physiological response. The results have shown that the behavior of these variables using
the SP controller is better than using constant speed. However, the values were not the
same as the physiological reference values. The hypothesis proposed by this work’s author
is that this fact occurs because of the constant gain kSP . Moreover, the changes in preload,
afterload, and VC were tested separately, but these changes can co-occur simultaneously.

4.2.1

Systolic Pressure estimation by peak detection

As the peak pressure values of PIP give the systolic pressure (SP), then a peak detection
algorithm is required to obtain the SP value used in the equation 4.2.
The pressure graph is composed of curves with positive and negative peaks. One way
of detecting positive peaks is to analyze the moments when the curve reaches its maximum
positive values. In this work, the moment in which the second derivative of PIP changes
its sign from positive to negative was used, indicating that the curve went from increasing
to decreasing so that a local maximum was detected. This peak value is maintained until
the subsequent detection. This procedure is summarized in Algorithm 1.

The Variable Gain SP Controller

22

Algorithm 1 SP calculation by peak detection algorithm
k
Input: PIPk−1 , dPIPk−1 , Pve
, Li , Q̇kLVAD , Ri , QkLVAD
k
k
1: PIP = Pve
− Li Q̇kLVAD − Ri QkLVAD
k
k
k−1
2: dPIP = (PIP − PIP
)/h
k−1
3: if (dPIP
≥ 0 and dPIPk < 0) then
4:
SPk = PIPk
5: else
6:
SPk = SPk
7: end if
k
8: return SP

4.3

The Variable Gain SP Controller

The SP controller works well when calibrated for a certain afterload or preload; however,
the controller has steady-state error for situations other than calibration. The Equation
4.2 has three constant values, i.e, SPref , ωref and kSP . Of these three values, the gain
kSP can be changed as a fine adjustment after the calibration process. Therefore, this
work suggests the Variable Gain SP Controller, whose equation is given by
ωdes = kSP (t)(SP − SPref ) + ωref .

(4.3)

Thus kSP (t) is a time-varying gain. To elaborate the strategy that will define how
kSP (t) will vary, the physiological cardiac output (COphy ) and the controlled (CO) behavior was used so that kSP (t) vary as a function of the error (ECO ) between these hemodynamic variables.
When ECO is high, kSP (t) needs to undergo significant changes so that CO approaches
COphy , therefore the growth or decrease of kSP (t) must be high if ECO is high.
To accelerate the minimization process of this error signal, ECO , positive and negative
+
−
+
thresholds, Eth
and Eth
were empirically defined. If ECO is greater than Eth
, it means the
kSP value must decrease faster. Thus, its value is updated with a factor that is called of
+
high-threshold Delta (∆high ). However, if ECO is still positive, but smaller than Eth
, the
kSP value is updated with a factor that is called low-threshold Delta (∆low ). For negative
values of ECO the idea is similar, but the kSP should be increased. This logic is described
+
−
in Algorithm 1. The values of the Eth
and Eth
and the values of ∆high and ∆low are also
empirically defined.
Figure 4.2 shows the block diagram of the variable gain SP controller. The area within
the dashed gray line is the dynamic modeling of the rotary pump with a PI controller in a
closed-loop. The calculation of CO for updating kSP is done in the area delimited by the
black dashed line. The area with gray background is where the SP controller actuates.

Chapter considerations

23

Algorithm 2 kSP updating
+
−
Input: COkphy , COk , kSP , ∆high , ∆low , Eth
, Eth
k
k
1: ECO = COphy − CO
+
2: if ECO > Eth then
3:
kSP = kSP − ∆high
+
4: else if 0 < ECO < Eth
then
5:
kSP = kSP − ∆low
−
< ECO < 0 then
6: else if Eth
7:
kSP = kSP + ∆low
−
then
8: else if ECO < Eth
9:
kSP = kSP + ∆high
10: else if ECO = 0 then
11:
kSP = kSP
12: end if
13: return kSP

SP
+

SPref -

COphy

SP
Calculation

PIP
Qa

COk
ksp
CO
Updating
Calculation

ksp

ωref

+

CSV-LVAD
Model

ω

HR
QLVAD

f(ω,QLVAD)
ωdes

Eω

+

PI

Te +

-

Tp
-

1
Js+B

ω

Figure 4.2: Block diagram describing the speed update closed-loop (area within the gray
dashed line); the kSP updating (area within the black dashed line); and the control law of
the SP controller (area with gray background).

4.4

Chapter considerations

The chapter shows the SP controller among its advantages and the facility to obtain
the primary hemodynamic variable to change the LVAD’s pump speed when extracting
the pressure value at the Pump Inlet Pressure (PIP). However, the SP controller has a
constant gain adjusted for a specific preload and afterload value. The controller can not
supply the expected cardiac output demand when these hemodynamic values change. For

Chapter considerations

24

this reason, the Variable Gain SP controller was proposed, and an algorithm to update
the gain was developed and tested.

Chapter 5
Results
This chapter shows the computational simulations for both preload and afterload changes.
Simulations are also performed to show how the rotary VAD model implementation works
and to show that the closed-loop for the speed controller works as expected with constant
or variable rotation speed. Lastly, results shows that the Variable Gain SP controller does
have good adaptability under variations in preload and afterload.

5.1

Preload and Afterload Simulation

Performing simulations with varying preload and afterload means changing the patient’s
physiological condition to test the robustness of some control system applied to LVADs.
One way to perform these variations was done in Simaan et al. and consists in change
the value of the resistance of the Mitral valve Rm for the preload; and change the value
of the Systemic Vascular Resistance (Rs ) for the afterload [Simaan et al., 2008].
Pressure-Volume (PV) loops, as shown in Figure 5.1, can be used to analyze the effect
of changes in both preload and afterload. Changing the value of Rm while keeping constant
left ventricular parameters such as Emax , Emin , and V0 , it is possible to observe the effect
in the PV loop and the linear end-systolic pressure-volume relationship (ESPVR). In
Figure 5.2 it is illustrated the effect on the PV loop with changes in afterload by varying
the value of Rs .

25

Simulation results of the LVAD modeling

26

120
100
80
60
40
20
0

0

20

40

60

80

100

120

Figure 5.1: PV loops representing preload by changing the mitral valve resistance, Rm .

120
100
80
60
40
20
0

0

20

40

60

80

100

120

Figure 5.2: PV loops representing afterload by changing the systemic resistance, Rs .

5.2

Simulation results of the LVAD modeling

Figure 5.3 shows the result of a rotary pump simulation with the projected PI in section
3.2.2. As the objective speed was 1000 rpm, the result shows that the pump has zero
steady-state error. The controller design using this method has been tested and satisfies
the values obtained in [Choi et al., 1997].

Simulation results of the LVAD modeling

27

1200
1000

(rpm)

800
600
400
200
0

0

2

4

6

8

10

Time (s)

Figure 5.3: Step response of the brushless DC motor.
Figure 5.4 shows the result of a 60-second simulation of the human cardiovascular
system model with LVAD at a constant velocity of 9000 rotations per minute. The graph
makes it possible to see the pump flow alternating between 6 ml/s and 175 ml/s.

200

150

100

50

0

10

20

30

40

50

60

Figure 5.4: Pump Flow signal from human cardiovascular model with constant pump
speed of 9000 rpm.
Figure 5.5 shows the result of a 60-second simulation of the CVS-LVAD model with
the pump speed increasing linearly from 12000 rpm to 18000 rpm. In the figure, it is
possible to see the pump flow as a function of the pump speed. Around 15500 rpm, the
phenomenon of suction occurs. The suction is a life‐threatening event that damages the
walls of the cardiovascular system and should be avoided.

SP Controller Simulation

28

Figure 5.5: Pump Flow signal from human cardiovascular model with pump speed increasing linearly from 12000 rpm to 18000 rpm. Suction phenomenon occurs at 15500
rpm.

5.3

SP Controller Simulation

Figure 5.6 shows the simulation of the pump inlet pressure (PIP), a state variable needed
to extract the systolic pressure.

100

Pump Inlet Pressure (PIP)

50

0

20

30

Figure 5.6: Simulation result of the Pump Inlet Pressure (PIP) state variable.
The SP controller responds to changes of preload and afterload in the cardiovascular
system; these phenomena can also be visualized through the graph of pressure over time.

SP Controller Simulation

29

Figure 5.7 shows a pressure increase when decreasing the preload through Rm value from
0.1 to 0.05.
100

Pump Inlet Pressure (PIP)

50

0

20

30

Figure 5.7: Preload increase effect on PIP.
The effect of increasing Rm from 0.1 to 0.5 can be seen in Figure 5.8, where there is
a decrease in pressure.

100

Pump Inlet Pressure (PIP)

50

0

20

30

Figure 5.8: Preload decrease effect on PIP.
Similarly, the afterload can be observed in the PIP graph when changing the value of
Rs from 1 to 1.5. The result is seen in the Figure 5.9.

SP Controller Simulation

120

30

Pump Inlet Pressure (PIP)

100

50

0

20

30

Figure 5.9: Afterload increase effect on PIP.
Decreasing the value of Rs from 1 to 0.5 causes the afterload decrease, as shown in
the simulated result of the figure 5.10.

120

Pump Inlet Pressure (PIP)

100

50

0

20

30

Figure 5.10: Afterload decrease effect on PIP.
A combination of preload and afterload is also possible by changing the value of Rm
and Rs (Figure 5.11), and the 120-second simulation result can be seen in Figure 5.12.

SP Controller Simulation

31

0.1
0.08

0.04
0

16 20

30

40

120

1
0.75
0.5
0

60

70

80

90

120

Time (s)

Figure 5.11: Preload and Afterload variation through Rm and Rs .

Pump Inlet Pressure (PIP)

120
100

50

0

20

30

40

60

70

80

90

120

Figure 5.12: Preload and Afterload simultaneously on PIP.
The detection result using the Algorithm 1, proposed in this dissertation, can be seen
in Figure 5.13.

Cardiac Output simulation with SP Controller

32

Pump Inlet Pressure (PIP)

120

SP
PIP

100

50

0

20

30

40

60

70

80

90

120

Figure 5.13: Systolic Pressure (SP) value extracted from PIP using the peak detection
algorithm.

5.4

Cardiac Output simulation with SP Controller

The SP controller is designed to adapt to preload or afterload exclusively in the simulation
from 10 to 25 seconds shown in Figure 5.14 shows an SP controller designed to have a
cardiac output next to 4L and preload with Rm = 0.08. Initially the simulation starts
with the preload of Rm = 0.1, then between 16 and 20 seconds the preload varies to
Rm = 0.08. In the cardiac output graph, the adaptation of the controlled cardiac output
(CO) is seen getting closer to the previously defined reference (COphy ). It can be seen
from the figure that initially, the controller has a steady-state error when the preload is
with Rm = 0.1, which is expected since the controller was configured for a preload of
Rm = 0.08.
The SP controller configured for a afterload with Rs = 0.5 has the simulation result
shown in Figure 5.15. In this case, as the controller was designed for afterload with
Rs = 0.5 the controlled CO is able to follow the reference well (COphy ), but When
changing the afterload value with Rs = 0.75 the controller presents steady state error.
The steady state errors presented in both simulations of Figures 5.14 and 5.15 happen
due to the constant parameters of the Equation 4.2 (SPref , ωref and kSP ), so a new control
technique with kSP variable is presented in the next section.
The same behavior of Figures 5.14 and 5.15 can be observed in vitro tests in the
literature [Petrou et al., 2016].

Cardiac Output simulation with SP Controller

5

33

Cardiac Output (CO)

4
CO
COphy

3
10

15

20

25

15

20

25

0.1
0.08
10

Figure 5.14: Cardiac Output of the SP Controller with preload variation.

6

5

4
10

Cardiac Output (CO)

CO
COphy

15

20

25

15

20

25

0.75

0.5
10

Figure 5.15: Cardiac Output of the SP Controller with afterload.

Simulations results with Variable Gain SP Controller

5.5

34

Simulations results with Variable Gain SP Controller

To test the performance of the SP controller with variable, calibration was done for a
default preload with Rm = 0.1. The exact calibration was done for the SP controller with
fixed gain, and then both controllers were compared.
In SP controller simulations [Daners et al., 2017], the curves of hemodynamic variables
are used as comparison parameters, so this section will show the simulation results for
the cardiac output versus time.
In the simulation shown in Figure 5.16 the preload starts with Rm = 0.1 and varies to
0.08 in the range of 16 to 20 seconds. As expected, the fixed-gain SP controller experiences
steady-state error when the system assumes a preload that it was not designed for.
On the other hand, the variable gain controller has its cardiac output closer to the
physiological reference. In Figure 5.16 it is possible to see Algorithm 2 working because
when the error between the physiological and the controlled cardiac output is greater, the
controller follows the reference more quickly, in the middle of the simulation, when the
error is smaller, the controller signal approaches the reference more slowly.
Table 5.5 shows that both the Mean Squared Error (MSE) and the Root Mean Squared
Error (RMSE) of the variable-gain controller are smaller than those of the fixed-gain
controller.

4.1
4.05
4
3.95
3.9
3.85
Physiologic
Variable Gain
Fixed Gain

3.8
3.75
10

15

20

25

30

Time (s)
Figure 5.16: Physiologic CO compared with the Variable and Fixed Gain SP Controllers.
Increased preload from 10s to 30s of simulation.

Simulations results with Variable Gain SP Controller

35

Table 5.1: MSE and RMSE compared with Physiologic CO. Increased preload from 10s
to 30s of simulation.
SP Controller
Measure

Variable Gain

Fixed Gain

MSE

0.0054

0.0266

RMSE

0.0732

0.1631

Figure 5.5 shows another moment of the simulation where the afterload varies from
Rs = 1 to Rs = 0.5. In this simulation, the controller with a variable gain was able
to approach the Physiologic CO more quickly and minimize steady-state error. The
afterload variation also influenced the controller’s behavior with fixed gain. However, this
one continues with greater error in steady-state, but as this controller was designed to
operate with a pre-set preload and afterload, this result was already expected. The table
5.5 shows the MSE and RMSE obtained in this simulation time.

7.5
7
6.5
6

Physiologic
Variable Gain
Fixed Gain

5.5
5
4.5
4
50

55

60

65

70

75

80

Time (s)
Figure 5.17: Physiologic CO compared with the Variable and Fixed Gain SP Controllers.
Increased preload from 50s to 80s of simulation.

Simulations results with Variable Gain SP Controller

36

Table 5.2: MSE and RMSE compared with Physiologic CO. Increased preload from 50s
to 80s of simulation.
SP Controller
Measure

Variable Gain

Fixed Gain

MSE

0.0219

0.4923

RMSE

0.1481

0.7016

In the simulation interval shown in Figure 5.18 the afterload value was changed from
Rs = 0.5 to Rs = 0.75 in the interval between 80 and 90 seconds. The controller with
a variable gain was able to adapt to this afterload variation in CO and maintained a
lower error in the steady-state than the controller with fixed gain. The table 5.5 shows
the MSE and RMSE obtained in this simulation time, where the controller values with
variable gain are smaller.

7.5

Physiologic
Variable Gain
Fixed Gain

7
6.5
6
5.5
5
4.5
80

85

90

95

100

105

110

Time (s)
Figure 5.18: Physiologic CO compared with the Variable and Fixed Gain SP Controllers.
Decreased afterload from 80s to 110s of simulation.

Simulations results with Variable Gain SP Controller

37

Table 5.3: MSE and RMSE compared with Physiologic CO. Decreased afterload from 80s
to 110s of simulation.
SP Controller
Measure

Variable Gain

Fixed Gain

MSE

0.0132

0.5061

RMSE

0.1150

0.7114

The complete simulation is shown in Figure 5.19. The variable gain controller was
able to reduce the steady-state error of CO both in the presence of preload and afterload
compared to the fixed gain controller.
This result shows that the SP controller, which can adapt the gain kSP according to
the expected CO, can obtain better results than the controller with fixed gain. These
results were published in the proceedings of the 43rd Annual International Conference of
the IEEE Engineering in Medicine and Biology Society [Silva et al., 2021].

7.5
7
Physiologic
Variable Gain
Fixed Gain

6.5
6
5.5
5
4.5
4
3.5

20

40

60

80

100

Time (s)
Figure 5.19: Physiologic CO compared with the Variable and Fixed Gain SP Controllers.
Preload and afterload variation from 0s to 120s of simulation.

Simulations results with Variable Gain SP Controller

38

Table 5.4: MSE and RMSE compared with Physiologic CO. Preload and afterload variation from 0s to 120s of simulation.
SP Controller

5.5.1

Measure

Variable Gain

Fixed Gain

MSE

0.0216

0.3674

RMSE

0.1469

0.6061

Variable Gain KSP (t) Analysis

In this subsection, an analysis was made with the gain KSP (t) of the controller. The
experiment consists of CVS-LVAD simulations with a maximum elastance of 1.2 and a
minimum of 0.06. The simulations aim to make the system reach a cardiac output of 4.7
L/min, for this the preload and afterload were defined at Rm = 0.1 and Rs = 1.
It was chosen the values of KSP (t) equal to 20, 30, 50, 100, 200 and 500. The result
can be seen in Figure 5.20, the value of KSP (t) = 500 caused instability in the system,
so it was omitted from the figure. The result shows that the smaller the value of KSP (t),
the faster the controller will reach the expected cardiac output and the larger the value
of KSP (t)), the slower the controller will be to achieve the same goal.
With KSP (t) = 10 and KSP (t) = 20 it is seen that the controller has reached the
cardiac output of 4.4 L/min in parity. However, with KSP (t) = 20 the rise is more linear
than with KSP (t) = 10. This happens, because with smaller KSP (t) more updates are
made in the loop of the equation that modifies the LVAD speed.
As a very high KSP (t) can cause instability in the system or slow down the controller,
Algorithm 2 proposes a limiter for the gain KSP (t). The lower bound of KSP (t) is set to
zero, meaning that the Equation 4.3 will only answer the reference velocity (ωref ).

Simulations results with Variable Gain SP Controller

39

5
4.5
4
3.5
3
2.5
2
1.5
1

20

40

kSP = 10

kSP = 50

kSP = 20

kSP = 100

kSP = 30

kSP = 200

60

80

100

Time (s)
Figure 5.20: Physiologic CO compared with different values of KSP (t).
The results showed that the Variable Gain SP controller is able to better adapt to
preload and afterload variations.
As can be seen, increasing the controller gain leads to slower behavior, which is
counter-intuitive when it comes to control theory, e.g., proportional controller. However, this happens because the increase in gain increases the pump speed and the blood
flow through the pump. Also, the blood flow in the cardiovascular system is inversely
proportional to left-ventricular pressure. Hence, as the controller equation uses systolic
pressure, the increase in gain decreases the systolic pressure. This compensation is seen
in the sluggishness of the cardiac output response.

Chapter 6
Conclusion
This work presented the implementation of a new controller for left ventricular assist
devices (LVAD), the variable gain (VG) systolic pressure controller. The 0D cardiovascular
system model, based on the Windkessel model, was used. This model uses electrical
circuits to simulate hemodynamic variables through differential equations.
The new variable gain controller presented in this work is an improvement on the
fixed gain systolic pressure (SP) controller [Petrou et al., 2016], which was tested using
a mock circulatory cardiovascular system. It was used numeric models to reproduce the
behavior of the SP controller, representing changes in preload and afterload. As a result,
a wide range of different strategies and patient conditions can be tested more efficiently.
The mechanical dynamics of the LVAD were also implemented, along with a PI controller
designed to calculate the pump speed.
Although the reference value for the cardiac output, COphy , has been previously defined, it is not necessarily fixed, i.e., its value can be changed any time for experts. Besides,
it was proved that the proposed control strategy minimizes the steady-state error in the
presence of both preload and afterload changes. This fact did not appear using the SP
controller, which operates with a constant gain kSP and was tested either for preload or
afterload changes.

6.1

Future works

The SP controller with a constant gain is already well known in the literature with in
vitro tests. Although the VG controller has presented satisfactory results that indicate
the improvement in the performance of the SP controller, these results were made through
computational simulations, that is, in silico. Implementing the variable gain controller
strategy in a hydraulic simulator for research testing is one of the future works.
Another point to be analyzed is the update of the variable gain kSP that can be improved using classical techniques of adaptive control theory or even intelligent techniques.
Concerning the values of ∆low and ∆high , other approaches can be evaluated in the future
40

Bibliography

41

to improve the convergence of the CO, e.g., fuzzy techniques that are capable of changing
the contribution of these ∆’s. Furthermore, in real situations, the steady-state error can
be relaxed for something around ±5%. Ultimately, as future work, the detection of the
peaks to calculate SP can be improved, eliminating the use of derivatives.

Bibliography
[AlOmari et al., 2012] AlOmari, A.-H. H., Savkin, A. V., Stevens, M., Mason, D. G.,
Timms, D. L., Salamonsen, R. F., and Lovell, N. H. (2012). Developments in control
systems for rotary left ventricular assist devices for heart failure patients: a review.
Physiological measurement, 34(1):R1.
[Burattini and Gnudi, 1982] Burattini, R. and Gnudi, G. (1982). Computer identification of models for the arterial tree input impedance: comparison between two new
simple models and first experimental results. Medical and Biological Engineering and
Computing, 20(2):134–144.
[Choi et al., 1997] Choi, S., Boston, J., Thomas, D., and Antaki, J. F. (1997). Modeling
and identification of an axial flow blood pump. In Proceedings of the 1997 American
Control Conference (Cat. No. 97CH36041), volume 6, pages 3714–3715. IEEE.
[Daners et al., 2017] Daners, M. S., Kaufmann, F., Amacher, R., Ochsner, G., Wilhelm,
M. J., Ferrari, A., Mazza, E., Poulikakos, D., Meboldt, M., and Falk, V. (2017). Left
ventricular assist devices: challenges toward sustaining long-term patient care. Annals
of biomedical engineering, 45(8):1836–1851.
[Hall and Hall, 2020] Hall, J. E. and Hall, M. E. (2020). Guyton and Hall textbook of
medical physiology e-Book. Elsevier Health Sciences.
[King and Lowery, 2017] King, J. and Lowery, D. R. (2017). Physiology, cardiac output.
[Koh et al., 2019] Koh, V. C., Ho, Y. K., Stevens, M. C., Ng, B. C., Salamonsen, R. F.,
Lovell, N. H., and Lim, E. (2019). A centralized multi-objective model predictive control
for a biventricular assist device: An in silico evaluation. Biomedical Signal Processing
and Control, 49:137–148.
[Malatos, 2016] Malatos, S. (2016). Advances in low-dimensional mathematical modeling
of the human cardiovascular system.
[Malchesky, 2017] Malchesky, P. S. (2017). Artificial organs 2016: a year in review. Artificial Organs, 41(3):276–304.

42

Bibliography

43

[Mancini and Colombo, 2015] Mancini, D. and Colombo, P. C. (2015). Left ventricular
assist devices: a rapidly evolving alternative to transplant. Journal of the American
College of Cardiology, 65(23):2542–2555.
[Ochsner et al., 2014] Ochsner, G., Amacher, R., Wilhelm, M. J., Vandenberghe, S.,
Tevaearai, H., Plass, A., Amstutz, A., Falk, V., and Schmid Daners, M. (2014). A
physiological controller for turbodynamic ventricular assist devices based on a measurement of the left ventricular volume. Artificial organs, 38(7):527–538.
[O’Keefe and Singh, 2020] O’Keefe, E. and Singh, P. (2020). Physiology, cardiac preload.
StatPearls [Internet].
[Otto, 1899] Otto, F. (1899). Die grundform des arteriellen pulses. Zeitung fur Biologie,
37:483–586.
[Petrou et al., 2020] Petrou, A., Kanakis, M., Magkoutas, K., De Vries, B., Meboldt, M.,
and Schmiddaners, M. (2020). Cardiac output estimation: Online implementation for
left ventricular assist device support. IEEE Transactions on Biomedical Engineering.
[Petrou et al., 2018] Petrou, A., Lee, J., Dual, S., Ochsner, G., Meboldt, M., and
Schmid Daners, M. (2018). Standardized comparison of selected physiological controllers for rotary blood pumps: in vitro study. Artificial organs, 42(3):E29–E42.
[Petrou et al., 2017] Petrou, A., Monn, M., Meboldt, M., and Daners, M. S. (2017). A
novel multi-objective physiological control system for rotary left ventricular assist devices. Annals of biomedical engineering, 45(12):2899–2910.
[Petrou et al., 2016] Petrou, A., Ochsner, G., Amacher, R., Pergantis, P., Rebholz, M.,
Meboldt, M., and Schmid Daners, M. (2016). A physiological controller for turbodynamic ventricular assist devices based on left ventricular systolic pressure. Artificial
organs, 40(9):842–855.
[Petukhov et al., 2019] Petukhov, D., Korn, L., Walter, M., and Telyshev, D. (2019). A
novel control method for rotary blood pumps as left ventricular assist device utilizing
aortic valve state detection. BioMed Research International, 2019.
[Silva et al., 2021] Silva, L. F., Cordeiro, T. D., and Lima, A. M. (2021). A variable gain
physiological controller for a rotary left ventricular assist device. In 2021 43rd Annual International Conference of the IEEE Engineering in Medicine & Biology Society
(EMBC), pages 5606–5609. IEEE.
[Simaan et al., 2008] Simaan, M. A., Ferreira, A., Chen, S., Antaki, J. F., and Galati,
D. G. (2008). A dynamical state space representation and performance analysis of a

Bibliography

44

feedback-controlled rotary left ventricular assist device. IEEE Transactions on Control
Systems Technology, 17(1):15–28.
[Suga and Sagawa, 1974] Suga, H. and Sagawa, K. (1974). Instantaneous pressure-volume
relationships and their ratio in the excised, supported canine left ventricle. Circulation
research, 35(1):117–126.
[Timms, 2011] Timms, D. (2011). A review of clinical ventricular assist devices. Medical
engineering & physics, 33(9):1041–1047.
[von Platen et al., 2019] von Platen, P., Ruschen, D., Leonhardt, S., and Walter, M.
(2019). Assistance ratio: an approach to quantify the hydraulic load distribution in
lvad therapy. In World Congress on Medical Physics and Biomedical Engineering 2018,
pages 745–748. Springer.
[Wang et al., 2012] Wang, Y., Faragallah, G., Divo, E., and Simaan, M. A. (2012). Feedback control of a rotary left ventricular assist device supporting a failing cardiovascular
system. In 2012 American Control Conference (ACC), pages 1137–1142. IEEE.
[Wetterer, 1940] Wetterer, E. (1940). Quantitative Beziehungen zwischen Stromstärke
und Druck im natürlichen Kreislauf bei zeitlich variabler Elastizität des arteriellen
Windkessels; Wetterer, Erik, Dr. med. JF Lehmanns Verlag.