Download mathematical model for optimizing ecmo settings in case of

MATHEMATICAL MODEL FOR
OPTIMIZING ECMO SETTINGS IN CASE OF
RESPIRATORY FAILURE
GROUP 874, 3rd SEMESTER MASTER, BIOMEDICAL ENGINEERING, 2013
Department of Health Science and Technology
Aalborg University
Frederik Bajers Vej 7D2
9220 Aalborg Øst
http://st.hst.aau.dk
Synopsis:
Title:
Mathematical model for optimizing ECMO settings in case of respiratory failure
Subject:
Applied Biomedical Engineering
and Informatics
Project period:
3rd Semester Master
Spring 2013
Project group:
874
Members:
Anna Doležalová
Supervisor:
Stephen Edward Rees
Copies: 3
Number of pages: 44
Completion date: May 29th 2013
Venovenous extracorporeal membrane oxygenation (VV ECMO) is a modified form of
cardiopulmonary bypass in which blood is
drained from the venous system of the patient,
pumped through the oxygenator where the
oxygenation takes place and re-infused to the
venous bloodstream [Betit, 2009]. VV ECMO
is used when the cardiac output is sufficient
and complete or partial lung support is needed
[Lindstrom et al., 2009]. The aim of this project
was to create a mathematical model of blood
circulation, lungs with insufficient function,
tissues and VV ECMO device in order to investigate optimal ECMO settings in case of
respiratory failure. The model was created in
the programming language called Modelica.
Set of experiments was designed in order to
perform the simulations, test the model and
find optimal ECMO flow and partial pressures
settings. The results have shown that optimal
ECMO flow, which is the major determinant
of blood oxygenation, is between 60 and 70%
of systemic blood flow. The optimal partial
pressure settings are a carbon dioxide partial
pressure (pCO 2 ) of 4.5kPa and an oxygen partial
pressure (pO 2 ) of 45kPa. From work on the
current project it has become apparent that
mathematical modeling could be considered as
a useful tool for expressing basic physiological
processes and functions of medical devices.
The content of this report is freely available but publication (with source reference) must only be published after agreement with the author.
P REFACE AND READING INSTRUCTION
P REFACE
This report is written by group 874 at the 3rd semester of the master degree in Biomedical Engeering at Aalborg University. The overall theme for the project period is ’Applied Biomedical
Engineering and Informatics’ and the title is ’Mathematical model for optimizing ECMO settings
in case of respiratory failure’.
The intended target group is primarily researchers with interest in extracorporeal membrane
oxygenation and mathematical modeling. The secondary target group is supervisors, examiner
and fellow students. It can be an advantage to have a basic knowledge about particularly physiology and general Biomedical Engineering techniques aswell.
R EADING INSTRUCTION
Some selected terms in the report are abbreviated the first time they are mentioned with the
abbreviation in a parenthesis after the word. For example: extracorporeal membrane oxygenation (ECMO).
The references are stated after the Harvard-method with the authors last name first comma the
year for the publication. In the bibliography the references are listed in alfabetic order according the last name of the author.
By signing here the author of the current project give her consent that the report is made by:
Anna Doležalová
TABLE OF CONTENTS
Preface and reading instruction
i
Table of contents
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Results
3.1 Experiment 1 - sO 2 in relation to ECMO flow . . . . . . . . . . . . . . . . . . . . .
3.2 Experiment 2 - pO 2 in relation to ECMO flow . . . . . . . . . . . . . . . . . . . . .
3.3 Experiment 3 - pCO 2 in relation to ECMO flow . . . . . . . . . . . . . . . . . . . .
3.4 Experiment 4 - sO 2 in relation to ECMO flow and pulmonary shunt . . . . . . . .
3.5 Experiment 5 - sO 2 in relation to ECMO flow, pulmonary shunt and Vt . . . . . .
3.6 Experiment 6 - sO 2 in relation to ECMO flow and V CO 2 and V O 2 . . . . . . . . .
3.7 Results summary of experiments 1 - 6 . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Experiment 7 - arterial blood properties in relation to pCO 2 and pO 2 in ECMO .
3.9 Experiment 8 - model behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Discussion
4.1 Recapitulation . . . . . . . .
4.2 Results summary . . . . . .
4.3 Limitations . . . . . . . . . .
4.3.1 Acid-base chemistry
4.3.2 ECMO . . . . . . . .
4.3.3 Shunt . . . . . . . . .
4.4 Conclusion . . . . . . . . . .
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2
Introduction
1.1 History . . . . . . . . . . . . . . . .
1.2 Problem statement . . . . . . . . .
1.2.1 Aim . . . . . . . . . . . . . .
1.3 ECMO models . . . . . . . . . . . .
1.4 Experimental protocol . . . . . . .
1.4.1 Experiments . . . . . . . . .
1.5 Methods - programming language
Methods
2.1 General description of the model
2.1.1 Blood . . . . . . . . . . . .
2.1.2 Shunt . . . . . . . . . . . .
2.2 Segment description . . . . . . .
2.2.1 Circulation . . . . . . . .
2.2.2 Lungs . . . . . . . . . . . .
2.2.3 Tissues . . . . . . . . . . .
2.2.4 ECMO . . . . . . . . . . .
2.2.5 Entire model . . . . . . .
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43
7
CHAPTER
1
I NTRODUCTION
Extracorporeal membrane oxygenation (ECMO) is a modified form of cardiopulmonary bypass in which blood is drained from the venous system of the patient, pumped through the
oxygenator where the oxygenation takes place and is re-infused to the patient [Betit, 2009].
There are some differences between ECMO and cardiopulmonary bypass. ECMO is often introduced by cervical cannulation, which can be provided under local anesthesia. Conversely,
the transthoracic cannulation, which has to be instituted under general anesthesia, is usually
used for cardiopulmonary bypass. ECMO is frequently used for long-term support measured
in days (3-10), but cardiopulmonary bypass can be used just for a short period measured in
hours. There is also a difference in the purpose of use. Cardiopulmonary bypass is usually used
for the heart and lung support during various types of cardio surgical procedures. ECMO can
provide complete or partial support of the heart and/or lung function and give the time for
recovery after disease, surgery, injury or respiratory or cardiac failure [Rodriguez-Cruz, 2011,
Lindstrom et al., 2009]. ECMO is a final option for the patients whose condition is refractory to
other management, for instance mechanical ventilation [Lindstrom et al., 2009, Oshima et al.,
2010]. The most important parts of the basic ECMO circuit are vascular cannulae to drain and
return blood, circuit tubing, a pump, artificial organ - oxygenator and a heater or heater-cooler
that sustain the blood temperature (figure 1.1) [Lindstrom et al., 2009].
Figure 1.1: Description of ECMO system [Rodriguez-Cruz, 2011]
9
CHAPTER 1. INTRODUCTION
There are two possible methods or modalities of ECMO (figure 1.2) - venovenous (VV) and
venoarterial (VA). VV is used to support mainly pulmonary function and VA supports both cardiac and pulmonary function [Betit, 2009].
Figure 1.2: Examples of cannulation for venovenous (VV) and venoarterial (VA) extracorporeal membrane oxygenation (ECMO); blue cannulae represents deoxygenated blood and red
cannulae represents oxygenated blood [Lindstrom et al., 2009]
In case of VV ECMO, the blood is drained from venous bloodstream and re-infused back to
the venous system. This modality is used when the cardiac output is sufficient and complete
or partial lung support is needed [Lindstrom et al., 2009]. The exact placement of the cannulation depends on the size of the patient. In older children and adults, the deoxygenated blood
is drained from one or both femoral vessels and the oxygenated blood is returned through the
right-internal jugular vein. After the return to the venous system, oxygenated blood passes
through the pulmonary system and therefore the gas exchange is sufficient. The optimal ECMO
flow is assured by optimization of venous drainage [Betit, 2009]. VV ECMO can also be introduced by a single double-lumen cannula, which allows drainage and return through the right
internal jugular vein [Betit, 2009, Lindstrom et al., 2009]. The cannula consists of the two channels; one is used for draining the deoxygenated blood from the vena cava and the second is
placed into the right atrium where the oxygenated blood is sent. When the placement is proper,
the suitable gas exchange is achieved and just one vessel is affected. The insertion of doublelumen cannula is easier and the complications associated with femoral cannulation are eliminated. However, the flow limitations may occur [Betit, 2009].
In VA ECMO deoxygenated blood is drawn from venous system and oxygenated blood is reinfused into the arterial circulation, which provides gas exchange and cardiac support [Betit,
2009, Lindstrom et al., 2009]. There are two possible ways of cannulation - peripheral and central. With central cannulation, blood is drained from the right atrium and re-infused to the
proximal thoracic aorta. With peripheral cannulation, blood is drawn from femoral or jugular
vein and re-infused to the aorta via carotid, axillary or femoral artery. The femoral cannulation
is useful, because the compressible site can help in haemorrhage control, but these vessels are
usually too small in neonates, so it is most commonly used in adults. When high ECMO flow
is required, it is possible to use twin drainage cannulae - jugular and femoral [Lindstrom et al.,
10 of 44
CHAPTER 1. INTRODUCTION
2009]. VA ECMO, in general, is more commonly used in neonates, especially when both heart
and lung function is threatened. In neonates collateral circulation develops, so cannulation
and ligation of the carotid artery is tolerated and therefore cardiac and also pulmonary support
is achieved [Betit, 2009].
1.1 History
Since 1953 when Gibbon used first extracorporeal oxygentation during open heart surgery,
ECMO and other extracorporeal technology have continued to develop [Lindstrom et al., 2009].
Recently, ECMO is most commonly used for patient with acute respiratory distress syndrome
(ARDS). Patients with the most severe forms of this disease usually have a bad prognosis and
mortality can be 60%. In these situations, ECMO may minimize the trauma caused by mechanical ventilation and might be a therapeutic option for the lungs to be recovered. Nevertheless,
at the beggining of ECMO use, most of the trials have shown no advantage of ECMO compared
to mechanical ventilation. The complications were mostly caused by severe bleeding due to
the use of anticoagulation and weak biocompatibility of the circuits [Schmidt et al., 2012]. In
1979 Zapol et al. made a randomized trial where nine medical centers collaborated in order
to find out if ECMO has a benefit as therapy for patients with acute respiratory failure. It has
been shown that ECMO did not increase the likelihood of long term survival [Zapol et al., 1979].
In 1994 Morris et al. have shown in a randomized study with ARDS patients that there was no
significant difference in survival with use of ECMO compared to mechanical ventilation [Morris et al., 1994]. The early attempts to use ECMO for adult patients were unsuccessful, but the
neonatal experience has led to better understanding and improvement in ECMO technology
[Betit, 2009]. In 1996 Green et al. made a study for investigating the impact of ECMO in pediatric patients with acute respiratory failure. They have found that ECMO had a positive effect
on survival; the mortality in patients treated with ECMO has been reduced [Green et al., 1996].
UK collaborative ECMO trial group has shown the effectiveness of ECMO in neonatal patients.
They demonstrated that ECMO should be considered as an option for neonates with severe
but potentially reversible respiratory failure [Group, 1996]. As early as 1985, it was shown that
ECMO improves survival in neonatal patients with severe respiratory failure compared to conventional ventilation [Bartlett et al., 1985]. However, the patient outcomes in children with respiratory failure associated with hematopoietic stem-cell transplantation have not been found
encouraging. The survival in these patients was reported to be poor [Gow et al., 2006]. In the
last ten years, outcomes in adult patients have improved. Linden et al. have shown a high
survival in patients with severe ARDS when treated with ECMO and pressure-supported ventilation [Linden et al., 2000]. In 2004 Hemmila et al. also proved the effectiveness of ECMO
in adult patients with ARDS. They have found ECMO as a good option for treatment in cases
when patients are not responsive to conventional approach [Hemmila et al., 2004]. Recently,
the encouraging results continue to appear mostly due to higher biocompatibility and performance as well as increased durability [Schmidt et al., 2012]. In 2009 after their randomized
trial, Mugford et al. have recommended ECMO as a good option for adult patients with severe
but potentially reversible respiratory failure [Mugford et al., 2009]. Good outcomes were also
reported in patients who received ECMO as a therapy during A(H1N1) influenza pandemic in
2009. Several studies have reported lower mortality in patients who received ECMO than in
patients who have not been treated with ECMO [Patroniti et al., 2011, Noah et al., 2011, Australia and Investigators, 2009]. These new encouraging results started an increased interest in
ECMO as a treatment for ARDS. In the past, the unsuccessful tries could be caused by a lack
of adequately educated and qualified medical, nursing and perfusion staff. The availability of
11 of 44
CHAPTER 1. INTRODUCTION
assistant devices such as echocardiography was also limited and the concerns about safety and
efficacy were presented [Lindstrom et al., 2009]. A lot of these concerns and deficiencies have
been improved in last ten years. The knowledge about ECMO has been developed as well as
the experience of the staff. The technology has been improved; devices are miniaturized and
therefore more available. All these changes and improved patient’s outcomes in last few years
signals a strong potential of ECMO application.
1.2 Problem statement
The purpose of using VV ECMO is to take the place of lung function providing time for the
lungs to recover from disease, injury or surgery [Rodriguez-Cruz, 2011, Lindstrom et al., 2009].
The most important functions of the lungs are oxygenation and carbon dioxide removal. The
gas exchange takes place in alveoli where the oxygen passes to the blood and the carbon dioxide passes to the air by diffusion. The air is afterwards breathe out. Adequate gas exchange
maintains balance of important parameters providing acid-base homeostasis of the blood. The
normal ranges for parameters relevant to acid-base homeostasis, as measured in arterial and
venous blood are listed in the tables 1.1 and 1.2 [Silbernagl and Despopoulos, 2009]. If the
lungs do not work sufficiently, ECMO has the responsibility to manage the gas exchange and
therefore maintain the acid-base homeostasis.
The main problems in introducing and leading ECMO are both anatomical and physiological and include issues such as bleeding usually caused by anticoagulation, insufficient perfusion of some parts of the patient’s body and the introduction of drainage. Furthermore, there
are some technical aspects which are necessary to adjust in order to achieve successful use of
ECMO. These are mainly ECMO flow (blood flow), gas flow and oxygen level, which are dependent on the level of patient’s lung and heart function. The values have to be modified according
to the function present in the patient’s body [Schmidt et al., 2012]. A model of VV ECMO, blood
circulation, lungs with insufficient function and tissues may help with adjusting the values of
ECMO settings in order to achieve proper oxygenation and carbon dioxide removal. Since a
computer model is not a threat for the patient, it might be helpful for educational purposes to
implement a model with explanatory or predictive functions. It might be especially useful in
the training of new medical staff and for teaching students of medicine and biomedical engineering. Trainees might learn the basic functions and features of the device and understand
the way it works. More experience and better understanding of the technology can possibly
improve the outcomes of using this device and result in the saving of lives.
1.2.1 Aim
The aim of this project is to design computer model of blood circulation, lungs, tissues and VV
ECMO in order to find out which ECMO settings are optimal for achieving sufficient oxygenation and carbon dioxide removal in different states of lung function presented.
12 of 44
CHAPTER 1. INTRODUCTION
Variable
Normal value
Unit
Description
ph p
ph e
pO 2
mmHg
kPa
mmHg
kPa
mmol/l
Potential of hydrogen in plasma
Potential of hydrogen in erythrocytes
Partial pressure of oxygen
HCO 3−
7.35 - 7.45
7.19
74.3 - 108
10 - 13
35 - 45
4.67 - 6
22 - 26
CO 2,p
1.23
mmol/l
N B B (H )
17.2
mmol/l
BB
H N B B (H A)
41.7
6.3
mmol/l
mmol/l
t N B B (t A)
23.5
mmol/l
pK HCO 3
6.1
-
pK N B B
6.96
-
DPG
5
mmol/l
Hb
135 - 180
7.4 - 10.9
0.00023
g/l
mmol/l
mmol/l.Pa
0.00001
21
95 - 99
7.1 - 9.9
200
25.7
mmol/l.Pa
%
%
mmol/l
ml/l
mmol/l
pCO 2
αCO 2
αO 2
F iO 2
sO 2
tO 2
tCO 2
Partial pressure of carbon dioxide
Concentration of bicarbonate buffer
base in plasma
Concentration of physically dissolved
carbon dioxide in plasma
Concentration of non-bicarbonate buffer
base in plasma
Concentration of buffer base in plasma
Concentration of non-bicarbonate buffer
acid in plasma
Concentration of non-bicarbonate buffer
in plasma
Dissociation constant for bicarbonate
in plasma
Dissociation constant for non-bicarbonate
buffers in plasma
Concentration of 2,3-diphosphoglycerate
in blood
Amount of hemoglobin in the blood
Solubility coefficient of carbon dioxide
in plasma
Solubility coefficient of oxygen in plasma
Fraction of oxygen in the air
Percent of blood cells filled with oxygen
Total concentration of oxygen
in blood
Total concentration of carbon dioxide
in plasma
Table 1.1: Normal range of blood variables in arterial blood [Rees and Andreassen, 2005, Martin,
1999, council UK, 2012, Siggaard-Andersen et al., 1987]
13 of 44
CHAPTER 1. INTRODUCTION
Variable
Normal value
Unit
Description
ph p
ph e
HCO 3,p −
7.371
7.169
26.2
mmol/l
CO 2,p
1.403
mmol/l
N B B (H )
16.9
mmol/l
B Bp
B Be
BB
H N B B (H A)
43.1
70.7
55.3
6.6
mmol/l
mmol/l
mmol/l
mmol/l
t N B B (t A)
23.5
mmol/l
pK HCO 3
6.1
-
pK N B B
6.96
-
DPG
5
mmol/l
Hb
135 - 180
7.4 - 10.9
27.6
g/l
mmol/l
mmol/l
Potential of hydrogen in plasma
Potential of hydrogen in erythrocytes
Concentration of bicarbonate buffer
base in plasma
Concentration of physically dissolved
carbon dioxide in plasma
Concentration of non-bicarbonate buffer
base in plasma
Concentration of buffer base in plasma
Concentration of buffer base in erythrocytes
Concentration of buffer base in blood
Concentration of non-bicarbonate buffer
acid in plasma
Concentration of non-bicarbonate buffer
in plasma
Dissociation constant for bicarbonate
in plasma
Dissociation constant for non-bicarbonate
buffers in plasma
Concentration of 2,3-diphosphoglycerate
in blood
Amount of hemoglobin in the blood
tCO 2,p
Total concentration of carbon dioxide
in plasma
Table 1.2: Normal range of blood variables in venous blood [Rees and Andreassen, 2005]
1.3 ECMO models
1.4 Experimental protocol
The project consists of several goals which are necessary to achieve. First, it is required to build
a model which has particular functions. It should represent human blood circulation through
the lungs, tissues and additionally VV ECMO device.
The model of the lungs should contain mechanical ventilation which supports the insufficient pulmonary function together with ECMO. The lung insufficiency should be represent by
a pulmonary shunt. Two important processes of gas exchange should be described in the lungs
and ECMO - carbon dioxide removal and transfer of oxygen into the bloodstream. The model
of the tissues should describe opposite processes than lungs and ECMO - oxygen uptake and
carbon dioxide production. In order to demonstrate changes of oxygen saturation, oxygen and
carbon dioxide concentrations and partial pressures during the passage throughout the circulation, the model should describe the basic acid-base chemistry of the blood.
Since the aim is to find the optimal settings of ECMO in order to keep the acid-base balance,
it is necessary to design a set of experiments which will provide the answer. The experiments
should simulate the dependency of blood properties on ECMO settings and the level of pulmonary shunt. My focus is on ECMO flow and partial pressures of oxygen and carbon dioxide
14 of 44
CHAPTER 1. INTRODUCTION
in the ECMO compartment and how these variables should be set to give normal properties of
arterial blood.
When the experiments are designed, the last step of the project is to perform the simulations
and test the model to find the optimal settings under different conditions.
1.4.1 Experiments
The following experiments have one common feature, the purpose of performing them is to
find out what is the optimal ECMO flow and partial pressures settings in ECMO.
In total eight experiments will be performed. Experiments 1 to 6 will be focused on optimization of ECMO flow. Experiment number 7 will be oriented on finding ideal settings of partial
pressures in the ECMO compartment. Experiment number 8 will be concentrated on investigating if the model has satisfactory physiological behavior.
Schmidt et al. have investigated the relationship between ECMO flow/cardiac output and
oxygen saturation (sO 2 ) in different parts of circulation. They have also reported the changes in
oxygen partial pressure (pO 2 ) and carbon dioxide partial pressure (pCO 2 ) in relation to ECMO
flow/cardiac output. In order to compare results from the former study and my project, first
three experiments represent the relationship between sO 2 , pO 2 , pCO 2 and ECMO flow.
Experiment 1 will be performed in order to find out what ECMO flow is needed to achieve
good value of sO 2 in the peripheral and pulmonary artery.
Experiment 2 will be related to the first experiment and the question is what ECMO flow is
necessary in order to achieve a good value of pO 2 in the peripheral and pulmonary artery.
Experiment 3 is analogous to experiment 2, but in this case, I am interested in finding optimal
ECMO flow in order to have a good value of pCO 2 in the peripheral and pulmonary artery.
Experiment 4 will be performed in order to find out if or how the ECMO flow should be
changed compared to results from experiment 1 in cases of different pulmonary shunt, specifically 20, 30, 40 and 50%. In other words, the task will be to search for the ideal value of ECMO
flow for achieving well saturated arterial blood under different conditions of lung function.
Experiment 5 will be carried out with a view to find what the optimal ECMO flow is for different settings of mechanical ventilation under various states of pulmonary shunt. During this
experiment, pulmonary shunt will be 20, 30, 40 and 50%. For each of these pathological states,
I would like to investigate if there is different optimal settings of ECMO flow when tidal volume
(Vt ) settings in mechanical ventilation vary.
During experiment 6 the production of carbon dioxide, and therefore the oxygen uptake in
the tissues, will be changed. I would like to find out what ECMO flow is optimal in order to keep
the oxygen supply and carbon dioxide concentration in arterial blood at a sufficient level. The
goal is to find out the ideal ECMO flow for different states of patient’s metabolism.
Experiment 7 will be performed in order to find the optimal settings of pO 2 and pCO 2 in
ECMO subcompartment. I assume that partial pressures are analogous to the fraction of gases
in the real ECMO device. For instance, if the oxygen fraction in real ECMO is 50%, I assume that
it is equal to pO 2 of 50kPa in my mathematical model. pO 2 and pCO 2 in the ECMO subcompartment will be set on different values and the ideal settings for achieving good arterial blood
properties will be investigated. The blood properties which will be observed are pCO 2 , pO 2 ,
sO 2 , total concentration of carbon dioxide (tCO 2 ) and total concentration of oxygen (tO 2 ).
Under normal conditions, pO 2 and pCO 2 in the ECMO subcompartment are set to be 50 and
3kPa, respectively.
Experiment 8 will be carried out in order to observe behavior of the model. The settings of the
model will be chosen due to the results from experiments 1 to 7. Specific blood properties will
be investigated during the passage throughout the body and compared with the physiological
values. The blood properties which will be observed are pCO 2 , pO 2 , sO 2 , tCO 2 and tO 2 .
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CHAPTER 1. INTRODUCTION
1.5 Methods - programming language
The model will be created in the programming language called Modelica. The effort to design
an object oriented modeling language was initiated by Hilding Elmqvist in 1978 and started
in 1996 within an action of the ESPRIT project "Simulation in Europe Basic Research Working
Group" [Otter and Elmqvist, 2001]. The language has been created by the developers of objectoriented modeling languages such as Allan, Dymola, NMF, ObjectMath, Omola, SIDOPS+, Smile
as well as practitioners in different domains. The first version used in real application, version
1.3. was finished in December 1999. One year later in December 2000, an update version 1.4.,
was made [Otter and Elmqvist, 2001].
Modelica is an object-oriented, equation-based programming language. This language specializes in high complex computational applications which require high performance. There
are four important features of this programming language which make it powerful and easy to
use. Firstly, Modelica is primarily based on equations instead of assignment statements and it
allows acausal modeling. Equations do not specify certain data flow direction so it provides better reuse of classes. Modelica class is able to adapt to more than one data flow context [Fritzon,
2004]. Secondly, Modelica is a multidomain programming language, meaning that it is possible to describe and connect model parts from a varying domains such as electrical, mechanical,
thermodynamic, hydraulic, biological, and control applications. Thirdly, Modelica has a general class concept that unifies classes, generics and general subtyping into a single language
construct. This feature makes it easier to reuse components and aids in development of the
models. Finally, Modelica has a strong software for designing and connecting components,
meaning that this language is suitable for an architectural description of complex physical and
software systems [Fritzon, 2004].
16 of 44
CHAPTER
2
M ETHODS
In this chapter, the way of building the model will be described. Firstly, the explanation about
the general overview of the model will be given. Secondly, a description about each part of the
model will be provided.
2.1 General description of the model
The entire model consists of several major parts as shown in the figure 2.1. The first major part
is the model of human circulation which includes six segments - pulmonary veins, pulmonary
arteries, system veins, system arteries, left and right heart. The second main part is the model
of the lungs which is composed of four segments - alveolar space, arterial blood, resistor and
shunt. In the alveolar space, mechanical ventilation helps to fulfill the processes of oxygenation
and carbon dioxide removal. In the arterial blood compartment, venous and capillary blood
are mixed. The compartments shunt and resistor represent the pulmonary shunt. The third
major part of the system is the compartment tissues where two important processes take place
- oxygen uptake and carbon dioxide production. The last main segment of the model is the
ECMO compartment where the same processes as in the lungs take place - oxygenation and
carbon dioxide removal. ECMO contains four segments - ECMO, blood mixing, resistor and
shunt. These four parts have analogous functions as the subcompartments in the lungs. As the
the figure 2.1 shows, each compartment or subcompartment has its own code which describes
the particular processes happening inside.
The entire model demonstrates a human whose lungs are not working properly and have to
be supported with slight mechanical ventilation and VV ECMO.
17
CHAPTER 2. METHODS
Figure 2.1: Hierarchy of the model with labeled parts, each of the compartments has its own
parts and each part has its own code, the particular parts are manually connected as you can
see on the picture
Before focusing on the particular parts of the model it is necessary to describe two essential
components. First, the way the blood is represented in the system is explained. Second, the
way the shunt is expressed in the model is described.
2.1.1 Blood
For the purpose of modeling the acid-base balance of blood, there are two, nearly separate,
parts of the blood description - carbon dioxide and oxygen part. Carbon dioxide content is
described by two chemical equations (2.1, 2.2) which take place in plasma and six mathematical equations (2.3, 2.4, 2.5, 2.6, 2.7, 2.8) which explain these processes in plasma. There are
some assumptions which have been made in order to make very complex system manageable.
Concentrations of buffer base (B B ) and non-bicarbonate buffer (t A or t N B B ) in plasma are
set to be parameters with values 42.3 and 23.5 mmol/l, respectively. Values of dissociation constant for bicarbonate in plasma (pK HCO 3 ), dissociation constant for non-bicarbonate buffers
in plasma (pK A − or pK N B B ) and solubility coefficient of carbon dioxide (αCO 2 ) are listed in
the introduction of the report in the table 1.1.
H + + HCO 3− ←→ CO 2 + H2O
(2.1)
H + + A − ←→ H A −
(2.2)
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CHAPTER 2. METHODS
p H = pK HCO 3 + l og 10
p H = pK A − + l og 10
HCO 3−
CO 2
A−
H A−
(2.3)
(2.4)
B B = HCO 3− + A −
(2.5)
tCO 2 = HCO 3− +CO 2
(2.6)
t A = A− + H A−
(2.7)
CO 2 = αCO 2 · pCO 2
(2.8)
In the case of oxygen, a dissociation curve, which represents the relationship between oxygen
saturation and oxygen partial pressure, has been made. Several assumptions have been made
in order to make the computations less complex. For instance, concentration of hemoglobin
(H b) was set to be fixed as a parameter with the value 9.3mmol/l and the fraction of methemoglobin (F Met H b) and fraction of carboxyhemoglobin (F COH b) were set to be 0. F Met H b
is the ratio between the concentration of methemoglobin (Met H b) and total concentration of
H b. F COH b is the ratio between the concentration of carboxyhemoglobin (COH b) and total
concentration of H b. The value of solubility coefficient of oxygen (αO 2 ) can be found in the
table 1.1 in the introduction. The most important relations which describe oxygen status of the
blood are equations about free dissolved oxygen (2.9) and total concentration of oxygen (2.10).
O 2 = αO 2 · pO 2
(2.9)
tO 2 = O 2 + sO 2 · (H b − (F Met H b · H b) − (F COH b · H b))
(2.10)
The acid-base chemistry of the blood described in the model is simplified. The focus is centered on oxygenation and carbon dioxide removal. Since the strong base is left out from the
description, B B and t A are assumed to be constants. Carbon dioxide content is described just
in the plasma, the content in erythrocytes is ignored. Oxygen content is defined in the blood in
general.
2.1.2 Shunt
In order to represent lung failure in the system, it was necessary to model the insufficiency of
gas exchange. This was provided by creating pulmonary shunt in the compartment lungs. Pulmonary shunt is the pathological state of the lungs when deoxygenated blood is mixed with
oxygenated blood and thereby the overall efficiency of gas exchange in the lungs is reduced
[Lovering and Goodman, 2012]. The shunt was modeled as two parallel resistances, the first resistance was incorporated to the alveolar space compartment and the second was simply made
as a resistor (figure 2.2). The compartment shunt is a control panel which sets the values of resistances in the alveolar space and the resistor according to pulmonary shunt. The function of
the compartment shunt is described by two equations (2.11, 2.12).
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CHAPTER 2. METHODS
Figure 2.2: Segments of the lung compartment where the representation of the shunt is seen
1
R t ot al
=
1
R al veol ar
+
1
R r esi st or
shunt · R r esi st or = (1 − shunt ) · R al veol ar
(2.11)
(2.12)
The same system was used when the ECMO compartment was created. The shunt was used
as a tool for setting how much blood would go through the ECMO, so in other words for setting
ECMO flow.
2.2 Segment description
In this section each part of the model will be described separately.
2.2.1 Circulation
For the purpose of the model of circulation, a simple model from my home university in Prague
was used [Tribula et al., 2013]. This model was adjusted and upgraded for my own use.
The circulation consists of six parts. All four blood-vessel compartments (pulmonary veins,
pulmonary arteries, system veins, system arteries) have the same basis, but they have different
initial values and different values of the parameters; all of the them are shown in the table 2.1. In
these compartments, change of the total volume (Vt ot al ) is described as sum of inflow (Q i n ) and
outflow (Q out ) (equation 2.13). It is necessary to note that the blood-vessels have unstressed
volume (VU ), the volume when the transmural pressure is equal to zero. The stressed volume
(VS ) means the remainder of the volume when the transmural pressure is above zero. VU is a
parameter defined in each of the four blood-vessel compartments, Vt ot al is a variable which
is initialized and then calculated as previously mentioned and VS is a variable which is computed by substracting VU from Vt ot al (equation 2.14). The next variable which is defined in
the blood-vessel compartments is pressure (p) which is calculated by dividing VS and compliance (C ) (equation 2.15). There is one assumption what has been made in the context of the
blood-vessel compartments, that the concentration of the blood properties which are transported through the entire system are not changed by passing through the veins and arteries.
There are just different initial values of the mass (M ) of the substances. In each of the bloodvessel compartments, the change of the mass is calculated as a multiple of Q i n and inflowing
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CHAPTER 2. METHODS
concentration (c i n ) plus multiple of Q out and outflowing concentration (c out ) (equation 2.16).
The last important equation to be mentioned is the calculation of concentration (c) which is
described as a division of M by VS (equation 2.17).
d (Vt ot al )
= Q i n +Q out
dt
(2.13)
VS = Vt ot al − VU
(2.14)
p=
VS
C
(2.15)
d (M )
= Q i n · c i n +Q out · c out
dt
c=
(2.16)
M
VS
(2.17)
Variable
Unit
Pulmonary
veins
System
arteries
System
veins
Pulmonary
arteries
Compliance
Initial volume
Unstressed volume
Initial mass CO 2
Initial mass O 2
l/Pa
l
l
mmol
mmol
0.000225
0.7
0.4
5
2
0.000012
0.67
0.5
3.8
1.5
0.0015
3.9
2.7
25
6
0.0000225
0.37
0.3
1.8
1
Table 2.1: Parameters and their values which have been used for modeling blood-vessels
[Tribula et al., 2013]
The left and right heart are similar and very simple segments of the circulation. They use the
pressure (p i n ) produced by veins to build the flow (Q) in the whole system. The equations in
the left and right heart have one common parameter - starling slope (SS). SS is approximated
by the linear relationship between p i n and cardiac output (Q) (equation 2.18) [Tribula et al.,
2013]. The values of SS for both left and right heart are listed in table 2.2.
Q = p i n · SS
Side of the heart
Starling Slope Value [l/Pa.s]
Right
Left
0.00012751
0.00007501
(2.18)
Table 2.2: Values of the starling slope in both sides of the heart [Tribula et al., 2013]
2.2.2 Lungs
Gas exchange in the lungs in the present model is insufficient and therefore their function has
to be supported by mechanical ventilation with low Vt .
The lung compartment consists of four segments - alveolar space, resistor, shunt and arterial
blood (figure 2.2). In the alveolar space the oxygenation and carbon dioxide removal take place.
In this part of the lungs, all of the equations describing the carbon dioxide (2.3, 2.4, 2.5, 2.6, 2.7,
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CHAPTER 2. METHODS
2.8) and oxygen (2.9, 2.10) content are used. The alveolar space also contains the mechanical
ventilation output which is the source for calculating the partial pressures of oxygen and carbon dioxide. Mechanical ventilation is described by nine equations. There are few parameters
which are necessary to know in order to be able to compute those nine equations and they are
listed in table 2.3.
Parameter
Unit
Value
Description
F iO 2
F iCO 2
Vt
f
Vl ung s
%
%
l
breaths/s
l
21
0
0.1
0.2
2
Fraction of inspired oxygen
Fraction of inspired carbon dioxide
Tidal volume
Frequency of breathing
Volume of the lungs
Table 2.3: Values of the mechanical ventilation parameters
Few assumptions have been made when the partial pressures have been calculated. The
fraction of alveolar oxygen (F AO 2 ) and carbon dioxide (F ACO 2 ) are considered to be equal with
partial pressures, so if pCO 2 or pO 2 is 5kPa, the fraction is 5%. Minute volume (Vm ) is assumed
to have different units by approximating l/s into mmol/s by multiplying by 40, so that 1l/s is
equal to 40mmol/s. The same approximation is used when the mass of oxygen and carbon
dioxide are calculated.
The first two equations of the mechanical ventilation set describe the equality between fraction and partial pressures (2.19, 2.20). They are adjusted in order to have the value of the partial
pressures in Pa. The next equation describes the calculation of Vm . The fourth and fifth calculation are used to define the mass of carbon dioxide (MCO 2 ) and oxygen (MO 2 ) in the alveolar
space (2.22, 2.23). The next equations are describing the addition of oxygen (mV O 2 ) and removal of carbon dioxide (mV CO 2 ) in the lungs, so the difference between the concentrations
inflowing (tO 2,i n and tCO 2,i n ) multiplied with Q i n and outflowing concentrations (tO 2,out and
tCO 2,out ) multiplied with Q out (2.24, 2.25). The last couple of equations are the final change
of mass dependent on inspired fraction, alveolar fraction, minute volume and also addition or
removal of the substance (2.26, 2.27).
pCO 2 = 100000 · F ACO 2
(2.19)
pO 2 = 100000 · F AO 2
(2.20)
Vm = Vt · f · 40
(2.21)
MCO 2 = F ACO 2 · Vl ung s · 40
(2.22)
MO 2 = F AO 2 · Vl ung s · 40
(2.23)
mV CO 2 = Q i n · tCO 2,i n −Q out · tCO 2,out
(2.24)
mV O 2 = Q i n · tO 2,i n −Q out · tO 2,out
(2.25)
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CHAPTER 2. METHODS
d (MCO 2 )
dt
= Vm · F iCO 2 − Vm · F ACO 2 + mV CO 2
(2.26)
d (MO 2 )
= Vm · F iO 2 − Vm · F AO 2 + mV O 2
(2.27)
dt
The resistor is one of the necessary parts for building pulmonary shunt. The function of this
segment is expressed by two equations. The first equation (2.28) is definition of the pressure
drop (pD), which is the difference between inflow pressure (p i n ) and outflow pressure (p out ).
The second equation is an analogy to the Ohm’s law where voltage is represented as pD, resistance is simply blood resistance (bl ood R) with units Pa.s/l and finally the current is defined as
the blood flow (Q).
pD = p i n − p out
(2.28)
pD = bl ood R · Q
(2.29)
Resistances in the alveolar space and in the resistor are controlled by the compartment shunt.
The system of the shunt was described in detail in the section 2.1.2. The total pulmonary resistance is known (9333 [(Pa.s)/l] [Tribula et al., 2013]) and the shunt computes the resistances for
the alveolar space and the resistor according to pulmonary shunt. Based on parallel resistors
rule, if the pulmonary shunt is 50% both resistances are 18666 (Pa.s)/l.
The last segment of the lung compartment is the arterial blood. This compartment could be
considered as a transition point where the blood from alveolar space and ECMO compartment
(venous blood) are mixed and the balance of the blood is recalculated. The process of mixing
is described by equations 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 and 2.10. The output from arterial blood
compartment is oxygenated blood with new properties which is distributed throughout the
body.
2.2.3 Tissues
Two very important processes take place in the tissues - oxygen uptake and carbon dioxide production. Oxygen and carbon dioxide content in the tissues is expressed by equations 2.3, 2.4,
2.5, 2.6, 2.7, 2.8, 2.9 and 2.10. Moreover, four equations define oxygen uptake, carbon dioxide
production and total outflowing concentrations of oxygen and carbon dioxide. The amount of
carbon dioxide which is added to the blood from tissues (ad dCO 2 ) is calculated by dividing
the production of carbon dioxide (V CO 2 ) with the cardiac output (Q) (equation 2.30). Therefore, the resulting outflowing concentration of carbon dioxide (tCO 2 ) is inflowing concentration (tCO 2,i n f l ow ) plus ad dCO 2 (equation 2.31).
ad dCO 2 =
V CO 2
Q
tCO 2 = tCO 2,i n f l ow + ad dCO 2
(2.30)
(2.31)
In contrast with carbon dioxide, oxygen is in the tissues removed from the blood. Two equations describe this process. First, the amount of oxygen which is removed from the blood
(r emO 2 ) is defined as division of oxygen uptake (V O 2 ) by cardiac output (Q) (equation 2.32).
Second, the final outflowing concentration of oxygen (tO 2 ) is calculated as the subtraction of
r emO 2 from the inflowing concentration (tO 2,i n f l ow ) (equation 2.33).
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CHAPTER 2. METHODS
r emO 2 =
V O2
Q
tO 2 = tO 2,i n f l ow − r emO 2
(2.32)
(2.33)
V CO 2 and V O 2 are assumed to be equal - 10mmol/min; it means that the respiratory quotient (RQ) is equal to 1. RQ is the ratio between V CO 2 and V O 2 (equation 2.34) and it varies
between 0.7 and 1. If RQ is equal to 1, patient’s body is burning carbohydrates (figure 2.3).
RQ = V CO 2 /V O 2 [Si l ber nag l and Despopoul os, 2009]
(2.34)
Figure 2.3: Relationship between respiratory quotient and caloric equvivalent [Silbernagl and
Despopoulos, 2009]
The blood coming out of the tissues is deoxygenated (venous) blood with new values of concentrations, pH, partial pressures and oxygen saturation. Concentration and partial pressure
of carbon dioxide are higher, oxygen concentration, saturation and partial pressure are lower
and pH is lower than in arterial blood. Venous blood needs to be oxygenated and deprived of
carbon dioxide in ECMO and later on in the lungs.
2.2.4 ECMO
The last part of the model is the ECMO compartment (figure 2.4). It has similar functions and
structure as the model of the lungs. This segment is composed of four subcompartments ECMO, resistor, shunt and blood mixing. In the ECMO subcompartment, the oxygenation and
carbon dioxide removal take place. Except for one difference, it is structured as the alveolar
space in the lungs. Partial pressures in the alveolar space are calculated from mechanical ventilation, however in the ECMO subcompartment they are set to be parameters. pCO 2 and pO 2
are set to be 3kPa and 50kPa, respectively. The process which takes place in the ECMO subcompartment is an equilibrium of the blood balance towards new partial pressures of oxygen and
carbon dioxide. For the adaptation on the new partial pressures, all the mathematical equations describing acid-base balance in the system are used (equations 2.3, 2.4, 2.5, 2.6, 2.7, 2.8,
2.9, 2.10).
The shunt and resistor are used in similar way as in the lungs. The resistor is in parallel with
the ECMO subcompartment and both have their own resistance. The shunt is a control panel
which sets the resistances in the resistor and ECMO subcompartment according to needed
ECMO flow. For instance, desired ECMO flow is 50% and the total resistance is 500Pa, therefore
resistance in both the resistor and ECMO subcompartment is 1000Pa.
Last segment of the ECMO compartment is blood mixing, which is analogous to the arterial blood in the lungs. Two important processes take place in this subcompartment. First,
24 of 44
CHAPTER 2. METHODS
deoxygenated blood from the tissues and oxygenated blood from the ECMO subcompartment
are mixed. Second, the balance of the blood is recalculated by equations 2.3, 2.4, 2.5, 2.6, 2.7,
2.8, 2.9 and 2.10. The output from the ECMO compartment is blood partly oxygenated and deprived of carbon dioxide which continues to the lungs where oxygenation and carbon dioxide
removal are fulfilled. The level of oxygenation and carbon dioxide removal depends on ECMO
flow.
Figure 2.4: ECMO compartment and its parts
2.2.5 Entire model
All segments of the model have been described in the former sections and it is appropriate to
describe how the system works together. The flow in the model is mediated by left and right
heart by the pressure which is provided by the blood-vessels. When the flow is secured, the
blood can circulate through the entire system and the processes of oxygenation, deoxygenation
and carbon dioxide production and removal may begin: oxygen uptake and carbon dioxide
removal take place in the tissues, in other words, particular amount of the oxygen is taken by the
tissues and carbon dioxide is added to the blood. Afterwards, usually some part of venous blood
goes through ECMO where the opposite processes take place. Oxygen is added and carbon
dioxide is removed from the blood. Then the blood from ECMO and the rest from the tissues
which did not go through ECMO are mixed. The processes of oxygenation and carbon dioxide
removal are completed in the lungs. Even though pulmonary shunt is presented, venous blood
is partially oxygenated and deprived of carbon dioxide by ECMO, so the properties of arterial
blood should be sufficient in order to provide enough oxygen and normal amount of carbon
dioxide to the tissues and the entire body.
25 of 44
CHAPTER
3
R ESULTS
In this chapter, the results from all eight experiments will be presented.
3.1 Experiment 1 - sO 2 in relation to ECMO flow
In experiment 1, optimal value of ECMO flow for achieving good sO 2 was investigated. Mechanical ventilation Vt and pulmonary shunt are set to be 100ml and 35%, respectively. As we
can see on figure 3.1 sO 2 is highly dependent on the ECMO flow. When 20% of the blood is
going through the ECMO device, it means that ECMO flow is 1.2 l/min (cardiac output is 5.8
l/min), sO 2 is low in both the peripheral and pulmonary artery. sO 2 in the pulmonary artery is
in this case just 74% and in the peripheral artery 85%. As the ECMO flow is increased, sO 2 also
increases. As soon as ECMO flow is more than 70%, the blood is well oxygenated (sO 2 of >97%)
in both the pulmonary and peripheral artery.
Schmidt et al. have performed similar trial with patients. It was shown that as soon as the
ECMO flow/cardiac output is above 60%, sO 2 is always above 90% (figure 3.3). sO 2 has the
same behavior in the right atrium, pulmonary artery and also the peripheral artery during increasing of the ECMO flow/cardiac output. sO 2 increases with enhancement of ECMO flow
(figure 3.2) [Schmidt et al., 2012]. The results from a study of Schmidt et al. are corresponding
with results from my mathematical model, but sO 2 of >90% in the model occurs earlier than
during ECMO flow of 60%. sO 2 >90% occurs in the pulmonary artery when ECMO flow is 50%
and in the peripheral artery when ECMO flow is just 30%. Nevertheless, sO 2 of >97% occurs in
the pulmonary as well as peripheral artery if ECMO flow is 70%.
100
pulmonary artery
peripheral artery
95
sO2 [%]
90
85
80
75
70
100
90
80
70
60
50
40
30
20
ECMO flow [%]
Figure 3.1: Experiment 1 - Relationship between oxygen saturation (sO 2 ) in the peripheral and
pulmonary artery and ECMO flow; tidal volume (Vt ) is 100ml, pulmonary shunt is 35%
27
CHAPTER 3. RESULTS
Figure 3.2: Impact of ECMO flow reduction on oxygen saturation (sO 2 ) in the right atrium,
pulmonary artery and peripheral artery [Schmidt et al., 2012]
Figure 3.3: Relationship between ECMO flow/cardiac output and oxygen saturation (sO 2 ) in
the peripheral artery [Schmidt et al., 2012]
3.2 Experiment 2 - pO 2 in relation to ECMO flow
Experiment 2 is related to the first experiment, since both sO 2 and pO 2 describe oxygenation
of the blood. This experiment was performed in order to determine the optimal value of ECMO
flow for achieving a sufficient value of pO 2 in the peripheral and pulmonary artery. Vt and the
pulmonary shunt are set to be 100ml and 35%, respectively. Schmidt et al. have shown that with
an increase of ECMO flow/cardiac output pO 2 rises in all of the three parts of the circulation
(figure 3.5). Results from the model follow the same pattern (figure 3.4). pO 2 is higher with
an enhancement of ECMO flow. The normal value of pO 2 is 13kPa or 100mmHg. pO 2 in the
model reaches normal value when ECMO flow is 63% in the peripheral artery and 73% in the
pulmonary artery compared to 80% or max flow in the study of Schmidt et al..
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CHAPTER 3. RESULTS
20
18
pO2 [kPa]
16
14
12
pulmonary artery
peripheral artery
10
8
6
80
75
70
65
60
55
50
45
40
35
30
ECMO flow [%]
Figure 3.4: Experiment 2 - Relationship between oxygen partial pressure (pO 2 ) in the peripheral
and pulmonary artery and ECMO flow; tidal volume (Vt ) is 100ml, pulmonary shunt is 35%
Figure 3.5: Impact of ECMO flow reduction on oxygen partial pressure (pO 2 ) in the right atrium,
pulmonary artery and peripheral artery [Schmidt et al., 2012]
3.3 Experiment 3 - pCO 2 in relation to ECMO flow
In experiment 3, ideal ECMO flow for achieving a good value of pCO 2 in the pulmonary and
peripheral artery was investigated. Vt and pulmonary shunt are set to be 100ml and 35%, respectively. As Schmidt et al. have shown, pCO 2 is increased with increasing of ECMO flow/cardiac output in the right atrium and also the pulmonary and peripheral artery (figure 3.7).
My model shows the same behavior in the peripheral and pulmonary artery (figure 3.6). When
ECMO flow is only 30%, pCO 2 in the pulmonary artery is 5.8kPa and in the peripheral artery
5.4kPa. With the enhancement of ECMO flow, pCO 2 decreases towards 3kPa which is the set
value of pCO 2 in ECMO. The normal value of pCO 2 is 5kPa or 37mmHg. Since the default value
of pCO 2 in the ECMO subcompartment is set to be 3kPa, pCO 2 reaches normal value already
during low ECMO flow of 40%. The results from the study by Schmidt et al. confirm my findings. In their trial, pCO 2 reaches a normal value when ECMO flow/cardiac output is 40%. As
both figure 3.6 and 3.7 display, ECMO flow from 40 to 100% provides good values of pCO 2 .
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CHAPTER 3. RESULTS
6
pulmonary artery
peripheral artery
5.5
pCO2 [kPa]
5
4.5
4
3.5
3
2.5
100
90
80
70
60
50
40
30
ECMO flow [%]
Figure 3.6: Experiment 3 - Relationship between carbon dioxide partial pressure (pCO 2 ) in the
peripheral and pulmonary artery and ECMO flow; tidal volume (Vt ) is 100ml, pulmonary shunt
is 35%
Figure 3.7: Impact of ECMO flow reduction on carbon dioxide partial pressure (pCO 2 ) in the
right atrium, pulmonary artery and peripheral artery [Schmidt et al., 2012]
3.4 Experiment 4 - sO 2 in relation to ECMO flow and pulmonary
shunt
This experiment was performed in order to determine optimal ECMO flow for achieving well
saturated arterial blood under different values of pulmonary shunt. Vt is set to be 100ml. Figure 3.8 displays the relationship between pulmonary shunt and sO 2 in arterial blood during
changes of ECMO flow. As the figure 3.8 shows, sO 2 decreases with an increase of pulmonary
shunt. When pulmonary shunt is 20%, an ECMO flow of 50% provides good arterial sO 2 of
>97%. With an increase of pulmonary shunt, increase of ECMO flow is needed. When the
shunt is 50%, an ECMO flow of 60 - 70% is necessary for achieving well saturated arterial blood.
30 of 44
CHAPTER 3. RESULTS
100
98
96
sO2 [%]
94
92
90
88
pulmonary shunt = 20%
pulmonary shunt = 30%
pulmonary shunt = 40%
pulmonary shunt = 50%
86
84
82
100
90
80
70
60
50
40
30
20
ECMO flow [%]
Figure 3.8: Experiment 4 - Relationship between oxygen saturation (sO 2 ) in arterial blood and
level of pulmonary shunt during changes in ECMO flow; tidal volume (Vt ) is 100ml
3.5 Experiment 5 - sO 2 in relation to ECMO flow, pulmonary shunt
and Vt
Experiment 5 was carried out in order to investigate what ECMO flow is sufficient for achieving
well saturated arterial blood (sO 2 >97%) when the pulmonary shunt is changed and how it is
affected by different values of Vt . Figures 3.9, 3.10, 3.11 and 3.12 show the relation between
sO 2 in arterial blood and ECMO flow during changes in Vt and in cases of different stages of
pulmonary shunt. In all states of lung insufficiency the process has similar behavior. When
ECMO flow is low, sO 2 is poor and it obviously depends on Vt . For instance, if pulmonary
shunt is 20%, ECMO flow is 20% and Vt is 100ml, sO 2 is just 87%. As Vt is increased towards
500ml, sO 2 is already 97% (figure 3.9). During an increase of ECMO flow, the dependency of
sO 2 on Vt is vanishing. For example, if pulmonary shunt is still 20%, ECMO flow is 70%, the
difference between sO 2 under variation of Vt is insignificant (figure 3.9).
All stages of pulmonary shunt show similar characteristics, but there is a slight disparity between the values of sO 2 during changes of pulmonary shunt. If ECMO flow is 20 or 30% sO 2
is shifted towards lower values with an increase of pulmonary shunt in all cases of Vt , but the
difference almost disappears as soon as the ECMO flow is between 60 and 70%. It means that as
soon as the ECMO flow is 60 - 70%, Vt can easily be low and ECMO provides good oxygenation
and carbon dioxide removal even when pulmonary shunt is 50%.
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CHAPTER 3. RESULTS
100
98
sO2 [%]
96
94
92
Vt = 100ml
Vt = 200ml
Vt = 300ml
Vt = 400ml
Vt = 500ml
90
88
86
100
90
80
70
60
50
40
30
20
ECMO flow [%]
Figure 3.9: Experiment 5 - Relationship between oxygen saturation (sO 2 ) in arterial blood and
ECMO flow during changes in tidal volume (Vt ); pulmonary shunt is 20%
100
98
sO2 [%]
96
94
92
90
Vt = 100ml
Vt = 200ml
Vt = 300ml
Vt = 400ml
Vt = 500ml
88
86
85
100
90
80
70
60
50
40
30
20
ECMO flow [%]
Figure 3.10: Experiment 5 - Relationship between oxygen saturation (sO 2 ) in arterial blood and
ECMO flow during changes in tidal volume (Vt ); pulmonary shunt is 30%
100
98
sO2 [%]
96
94
92
90
Vt = 100ml
Vt = 200ml
Vt = 300ml
Vt = 400ml
Vt = 500ml
88
86
84
100
90
80
70
60
50
40
30
20
ECMO flow [%]
Figure 3.11: Experiment 5 - Relationship between oxygen saturation (sO 2 ) in arterial blood and
ECMO flow during changes in tidal volume (Vt ); pulmonary shunt is 40%
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CHAPTER 3. RESULTS
100
98
96
sO2 [%]
94
92
90
88
Vt = 100ml
Vt = 200ml
Vt = 300ml
Vt = 400ml
Vt = 500ml
86
84
82
100
90
80
70
60
50
40
30
20
ECMO flow [%]
Figure 3.12: Experiment 5 - Relationship between oxygen saturation (sO 2 ) in arterial blood and
ECMO flow during changes in tidal volume (Vt ); pulmonary shunt is 50%
3.6 Experiment 6 - sO 2 in relation to ECMO flow and V CO 2 and V O 2
This experiment was performed in order to find out the optimal value of ECMO flow under different activity of patient’s metabolism. In other words, the experiment was an investigation on
which value of ECMO flow is good enough to achieve sufficient sO 2 in arterial blood (>97%)
when oxygen uptake (V O 2 ) and carbon dioxide production (V CO 2 ) in the tissues are changed.
Pulmonary shunt and Vt are set to be 35% and 100ml, respectively. Under normal conditions
of the model, V O 2 and V CO 2 are set to be equal 10mmol/min. Figure 3.13 displays the dependency of sO 2 on ECMO flow and on changes of V O 2 and V CO 2 . It is clear that sO 2 rises as
the ECMO flow increases. The plot also displays that as soon as V O 2 and V CO 2 are higher, an
increase of ECMO flow is necessary in order to keep the blood saturated enough. If V O 2 and
V CO 2 are 10mmol/min, an ECMO flow of 60% is sufficient for well saturated arterial blood. As
soon as V O 2 and V CO 2 increase, higher ECMO flow is needed. When ECMO flow is 80%, sO 2
in arterial blood is sufficient for all values of V O 2 and V CO 2 .
100
95
90
sO2 [%]
85
80
75
70
65
VCO2 = VO2 = 10 mmol/min
VCO2 = VO2 = 15 mmol/min
VCO2 = VO2 = 22 mmol/min
60
55
80
75
70
65
60
55
50
45
40
35
30
ECMO flow [%]
Figure 3.13: Experiment 6 - Relationship between oxygen saturation (sO 2 ) in arterial blood and
ECMO flow during changes of oxygen uptake (VO2) and carbon dioxide production (VCO2)
(considering equality between V O 2 and V CO 2 ); tidal volume (Vt ) is 100ml, pulmonary shunt
is 35%
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CHAPTER 3. RESULTS
3.7 Results summary of experiments 1 - 6
The goal of experiments 1 to 6 was to optimize ECMO flow. Most of the experiments have shown
that sufficient ECMO flow is between 60 and 70% (3.5 - 4.1l/min) in order to achieve good arterial blood properties. Nevertheless, from experiment 3 it is clear that the optimal value of
ECMO flow for reaching satisfactory pCO 2 is 40%. However, an ECMO flow of 40% would not
be sufficient in order to achieve well saturated arterial blood and a normal value of pO 2 . Experiment 6 has shown that an ECMO flow of 80% is necessary to reach good sO 2 in arterial
blood if V O 2 and V CO 2 are 22mmol/l. Since default values of V O 2 and V CO 2 are set to be
10mmol/min, an ECMO flow of 60% is sufficient. The rest of the experiments have shown that
an ECMO flow of 70% or between 60 and 70% provides good arterial blood properties under
different conditions. In summary, optimal value of ECMO flow is between 60 and 70%.
3.8 Experiment 7 - arterial blood properties in relation to pCO 2 and
pO 2 in ECMO
Experiment 7 was carried out in order to find out which settings of partial pressures in the
ECMO subcompartment are optimal for achieving good arterial blood properties. Pulmonary
shunt, ECMO flow and Vt were set to be 35%, 70% and 100ml, respectively. A former study has
suggested the normal value of oxygen fraction in ECMO around 50% [Schmidt et al., 2012]. The
typical value of pCO 2 in arterial blood is 5kPa [Martin, 1999]. Therefore, similar values were
chosen to be potential optimal settings. The results are summarized in table 3.1. In comparison with table 1.1, the results show that all of the values in the chosen range provide good properties of arterial blood. There are no significant differences between combinations of partial
pressures and reached values of blood properties. Although the differences are not significant,
the ideal combination seems to be pCO 2 of 4.5kPa and pO 2 of 45kPa.
Partial pressures [kPa]
tCO 2 [mmol/l]
tO 2 [mmol/l]
sO 2 [%]
pCO 2 [kPa]
pO 2 [kPa]
pCO 2 = 3
pO 2 = 40
pCO 2 = 3
pO 2 = 45
pCO 2 = 3
pO 2 = 50
24.08
9.29
98.40
3.30
13.46
24.08
9.30
98.54
3.30
13.99
24.08
9.32
98.67
3.30
14.56
pCO 2 = 3.5
pO 2 = 45
pCO 2 = 4
pO 2 = 45
pCO 2 = 4.5
pO 2 = 45
pCO 2 = 5
pO 2 = 45
24.62
9.30
98.46
3.77
14.28
25.13
9.30
98.39
4.24
14.54
25.61
9.29
98.32
4.70
14.77
26.06
9.29
98.26
5.16
14.98
Table 3.1: Experiment 7 - Dependency of arterial blood properties on partial pressures settings
in the ECMO subcompartment; pulmonary shunt is 35%, ECMO flow is 70%, Vt is 100ml
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CHAPTER 3. RESULTS
3.9 Experiment 8 - model behavior
Since the optimal settings seem to be ECMO flow of 60 to 70% and pCO 2 of 4.5kPa and pO 2
of 45kPa, an observing experiment was performed. The purpose of this experiment was to
observe how the blood properties change during the passage throughout the circulation. Pulmonary shunt was set to be 35% to simulate severe damage of the lungs. Vt was set to be 100ml
to show low support from mechanical ventilation. The results are summarized in table 3.2.
Compared to tables 1.1 and 1.2 it was shown that the modeled blood has similar properties as
the physiological values.
Part of the circulation
tCO 2 [mmol/l]
tO 2 [mmol/l]
sO 2 [%]
pCO 2 [kPa]
pO 2 [kPa]
Tissues
ECMO
After ECMO
Alveolar space
Arterial blood
27.30
25.40
25.97
25.41
25.61
7.59
9.74
9.09
9.40
9.29
80.96
99.84
96.55
99.06
98.32
6.52
4.50
5.07
4.51
4.70
6.45
45.00
11.43
18.56
14.77
Table 3.2: Experiment 8 - Summary of blood properties in different parts of circulation under
chosen optimal settings; pulmonary shunt is 35%, Vt is 100ml, ECMO flow is 70%, pCO 2,ecmo
is 4.5kPa and pO 2,ecmo is 45kPa
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CHAPTER
4
D ISCUSSION
In this chapter, the purpose, aim and solution strategy of the project will be recapitulated. Later
on, the results from the experiments will be summarized and compared with the literature. Afterwards, the critical view on some aspects of my model will be presented. Finally, the conclusion will be drawn.
4.1 Recapitulation
The aim of this project was to build a mathematical model of human circulation, lungs, tissues and VV ECMO device and investigate the optimal ECMO settings in order to obtain good
oxygenation and carbon dioxide removal. The purpose of the present project was to create a
mathematical model which can be useful for understanding and explaining the functions and
processes of the physiological system and medical device.
The model was created in programming language called Modelica. Four major parts of the
model were built separately - blood circulation, lungs with insufficient function, tissues and VV
ECMO device. Set of experiments was designed in order to test the model and find the optimal
ECMO settings.
4.2 Results summary
Two important findings have emerged from this project. First, for a patient with insufficient
lung function and slight support by mechanical ventilation (Vt = 100ml) the optimal ECMO
flow is between 60% and 70% (3.5 - 4.1l/min). Under these conditions, arterial blood is well
saturated and deprived from carbon dioxide. Second, partial pressures ECMO settings in chosen range do not have a strong influence on arterial blood properties compared to ECMO flow.
However, pCO 2 of 4.5kPa and pO 2 of 45kPa are chosen to be the optimal settings of partial
pressures in the ECMO subcompartment.
Former study has investigated that using femoro-jugular ECMO settings and ECMO flow of
>60% of systemic blood flow is associated with arterial sO 2 of >90% [Schmidt et al., 2012]. Results from the present project confirm this finding. However, sO 2 in the peripheral artery of
90% is achieved already when ECMO flow is 30% of cardiac output. sO 2 in the pulmonary artery
was found to be lower than in the peripheral artery under the same conditions of ECMO flow
[Schmidt et al., 2012]. The same finding has emerged from the present experiment. Specifically,
sO 2 in the pulmonary artery of >90% is achieved when ECMO flow is set to be 50%. Nevertheless, sO 2 of >97% in both the pulmonary and peripheral artery is achieved when ECMO flow is
70%.
37
CHAPTER 4. DISCUSSION
The previous study has observed that an ECMO flow of >60% of systemic blood flow is associated with arterial pO 2 of >60mmHg (8kPa). It was shown that pO 2 is lower in the pulmonary
than in the peripheral artery under the same conditions of ECMO flow [Schmidt et al., 2012].
Similar findings have emerged from the present project. However, pO 2 of >60mmHg (8kPa)
occurs in the pulmonary artery and in the peripheral artery when ECMO flow is 50% and 30%,
respectively. Optimal pO 2 of 100mmHg (13kPa) is achieved in pulmonary and peripheral artery
when ECMO flow is 73% and 63%, respectively.
Schmidt et al. have shown that during an ECMO flow reduction to 40%, pCO 2 is unaffected.
This finding is consistent with results from the present project. It has emerged that an ECMO
flow of 40% provides normal values of pCO 2 (37mmHg or 5kPa). The default value of pCO 2 in
ECMO compartment is set to be 3kPa. Therefore, when ECMO flow increases pCO 2 decreases
towards the default value. Nevertheless, ECMO flow does not have a strong influence on arterial
pCO 2 .
Pulmonary shunt is a medical condition when oxygenated blood is mixed with deoxygenated
blood and therefore efficiency of gas exchange is affected [Lovering and Goodman, 2012]. The
higher the shunt is, the worse the patient’s health conditions expected to be. The result from
experiment 4 has confirmed such a theory. It was shown that with an increase of pulmonary
shunt, higher ECMO flow is needed in order to achieve well saturated arterial blood. If pulmonary shunt is 20%, an ECMO flow of 50% is sufficient for arterial blood sO 2 of >97%. Nevertheless, an ECMO flow of 60 - 70% is associated with arterial sO 2 of >97% for all states of
pulmonary shunt.
The previous study has suggested that optimal mechanical ventilation Vt is 110ml [Schmidt
et al., 2012]. Experiment 5 has investigated an influence of mechanical ventilation Vt on optimal ECMO flow in different stages of pulmonary shunt. The results have shown that under low
ECMO flow (20%) and pulmonary shunt of 20%, Vt of 500ml is necessary in order to achieve arterial sO 2 of >97%. However, with an increase of pulmonary shunt, Vt of 500ml is not sufficient
for reaching well saturated arterial blood. As soon as pulmonary shunt is 30%, arterial sO 2 is
under 95% although Vt is 500ml. In order to keep sO 2 on sufficient level, ECMO flow has to be
increased with an enhancement of pulmonary shunt. Experiment 5 has shown that as soon as
ECMO flow is 70%, sO 2 of >97% is achieved for all states of pulmonary shunt and Vt of 100ml is
sufficient support of mechanical ventilation. Therefore, experiment 5 has suggested the same
level of mechanical ventilation support as the study by Schmidt et al..
During strenuous exercise the human body requires up to 500 times more oxygen than when
at rest. At the same time, the body has to get rid of carbon dioxide and other metabolic products
whose production suddenly significantly increases [Silbernagl and Despopoulos, 2009]. The
present study has investigated that ECMO flow has to be increased if the activity of the patient’s
metabolism increases. Although patients receiving ECMO are usually bedridden, metabolism
activity may be at some point slightly higher. For example, if the patient is firstly sedated
and afterwards able to communicate with relatives and doctors. When the patient is at rest,
V CO 2 is 10mmol/min [Silbernagl and Despopoulos, 2009]. I assume that V CO 2 is equal to
V O 2 under standard conditions. Under normal conditions, an ECMO flow of 60% is needed
in order to achieve arterial sO 2 of >97%. As soon as V CO 2 and V O 2 are increased towards
15mmol/min, an ECMO flow of 70% is necessary to keep arterial sO 2 on >97%. Nevertheless,
an ECMO flow of 80% is sufficient in order to reach arterial sO 2 of >97% for all the levels of the
patient’s metabolism.
A previous study has suggested that the optimal value of oxygen fraction in ECMO is 50%
[Schmidt et al., 2012]. The most commonly used optimal value of pCO 2 in arterial blood is
5kPa [Martin, 1999]. I assume that partial pressures in the ECMO subcompartment are analogous to a fraction of the gases. Experiment 7 has investigated the optimal settings of partial
38 of 44
CHAPTER 4. DISCUSSION
pressures in the ECMO subcompartment. It was shown that the chosen range of the partial
pressures provides good arterial blood properties compared to the values suggested by Rees
and Andreassen and Martin. The interval of potential optimal values was from 3 to 5kPa for
pCO 2 and from 40 to 50kPa for pO 2 . The results have suggested no significant difference between combinations of partial pressures. Although no significant difference was shown, pCO 2
of 4.5kPa and pO 2 of 45kPa could be considered as the optimal settings. The chosen optimal
settings are consistent with the values suggested by the literature.
Experiment 8 has observed behavior of the model under optimal settings which have emerged
from previous experiments. The conditions of this experiment are as follows: ECMO flow of
70%, pulmonary shunt of 35%, Vt of 100ml, pCO 2,ecmo of 4.5kPa and pO 2,ecmo of 45kPa. The
results have shown values of arterial blood properties which are consistent with values suggested by Rees and Andreassen and Martin. The model behaves in relation with expected outcomes. Carbon dioxide is added and oxygen is removed to/from the blood in the tissues. The
blood is deprived of carbon dioxide in the ECMO and lungs, oxygen is added to the blood in
the ECMO and lungs. The resulting arterial blood properties are a mix of venous blood and the
blood going from alveolar space according to shunt level.
A previous study has suggested that ECMO flow is the main determinant of blood oxygenation. It was shown that an ECMO flow of >60% of systemic blood flow obtain adequate arterial
oxygenation. An ECMO flow of 4 to 7 l/min is usually required to achieve sO 2 of >88-90% depending on patient size, cardiac output, oxygen consumption and lung shunt. Safe ventilation
supports the pulmonary function while on ECMO [Schmidt et al., 2012]. The present experiments do not consider all of the conditions. The goal during most of the experiments is to
achieve sO 2 of >97% and cardiac output in the model is 5.8l/min. Under these circumstances,
ideal ECMO flow is between 60 and 70%. All of the present experiments have shown that ECMO
flow is the major determinant of blood oxygenation.
4.3 Limitations
Mathematical or computer model is never able to fully substitute or express the real system, but
it may be very useful in describing and explaining basic processes and functions of any system.
My intention in this project was to try to reproduce the behavior of human blood circulation
when the lungs are not working properly and it is necessary to support their function with
slight mechanical ventilation and an ECMO device. After designing the model which expresses
the behavior of human body, the goal was to find optimal ECMO settings which are sufficient
in providing oxygen and carbon dioxide in particular amounts within the patient’s body. The
model works satisfactorily, but there are few limitations which are necessary to mention.
4.3.1 Acid-base chemistry
The first limitation is an imperfect buffering system in the model. Since a strong base is not
considered as a blood component in the model, concentrations of tA (tNBB) and BB are assumed to be constants.
Next inaccuracy is the description of oxygen and carbon dioxide content in the blood. Each
of these two substances are described almost separately and in different ways. Carbon dioxide content is described only in plasma, its part in erythrocytes is ignored. Therefore, tCO 2 in
the model means total concentration in plasma. Oxygen content is defined in the entire blood
volume, since H b concentration was set to be 9.3mmol/l, which is oxygen carrying capacity of
hemoglobin per liter of blood. As you can see, the ways of describing oxygen and carbon dioxide are not completely consistent. In summary, acid-base chemistry in the model is simplified.
39 of 44
CHAPTER 4. DISCUSSION
However, it shows the most important behaviors, processes and dependencies of concentrations, partial pressures and saturation in the lungs, tissues and ECMO.
4.3.2 ECMO
The next limitation is the simplified structure of the ECMO compartment. In real ECMO device,
air supply is an independent unit with its own flow. The air unit sets the fraction of oxygen
and carbon dioxide. In the model, oxygen and carbon dioxide are incorporated to the ECMO
subcompartment without any additional flow settings. I assume that fractions of the gases are
analogous to partial pressures. Therefore the settings of oxygen and carbon dioxide fraction are
changed by changing partial pressures.
4.3.3 Shunt
The last limitation which I want to mention is the mixing of venous blood in pulmonary shunt
and the shunt in the ECMO compartment. At some point, the oxygenation in the lungs does not
make physiological sense. This occurs specifically if the entire blood volume is going through
ECMO. The blood is well oxygenated with the appropriate concentration of carbon dioxide after passing through ECMO. Since the model represents VV modality and blood from ECMO
is returned to the venous bloodstream, venous blood becomes well oxygenated and deprived
of carbon dioxide. After passing through ECMO, venous blood comes to the lungs and it is
supposed to be more oxygenated and deprived of carbon dioxide. However, it is not possible,
because it already is very well saturated with oxygen and carbon dioxide concentration is appropriate. The next problem arises with pulmonary shunt when the entire blood volume is going through ECMO. Pulmonary shunt is a pathological state when oxygenated blood is mixed
with deoxygenated blood. The higher pulmonary shunt is, the more of the venous blood is
mixed with the arterial blood and the worse the patient’s condition is expected to be. However,
since venous blood in the model is under mentioned conditions well saturated and deprived
of carbon dioxide, it is actually better to have higher shunt which does not make any physiological sense. Nevertheless, it is not that common even in real ECMO device, that the entire
blood volume is going through, since drainage is most often located in femoral vein or inferior
or superior vena cava.
The problem about comparison of the model with the real body is also the fact that each
human is different and it is not possible to generalize the behavior. However, it would be interesting to use the model as a preparation for a particular patient by fixing the parameters and
settings according to specific details of his or her body and design the optimal use of ECMO.
My intention is to expand and elaborate the model in my future master thesis and it may lead
to possible improvements.
4.4 Conclusion
The aim of the project was to investigate optimal extracorporeal membrane oxygenation (ECMO)
settings by creating a mathematical model and performing testing experiments.
The present project can conclude that mathematical modeling could be considered as a useful tool for expressing basic physiological processes and functions of medical devices. The
model demonstrates basic behavior of human circulation, processes in tissues, lungs and venovenous ECMO (VV ECMO) with satisfactory results. Results from the experiments have shown
that ECMO flow is the main determinant of blood oxygenation. Optimal ECMO flow in order to
achieve good arterial oxygen saturation (sO 2 ) is between 60% and 70% (3.5 - 4.1l/min). Optimal
40 of 44
CHAPTER 4. DISCUSSION
partial pressures settings in ECMO are a carbon dioxide partial pressure (pCO 2 ) of 4.5kPa and
an oxygen partial pressure (pO 2 ) of 45kPa.
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