School of Industrial and Information Engineering M.Sc. in Aeronautical Engineering
3D and reduced-order modelling of blood
flow in idiopathic pulmonary hypertension
for design of a Potts shunt
Supervisor:
Prof. Luca Formaggia Co-Supervisor:
Dr. Irène Vignon-Clementel
Author: Alberto Noferi 841396
2 rue Simone Iff 75589 Paris Cedex 12
I would like to thank my Italian advisor Prof. Luca Formaggia for giving me the opportunity to work abroad, for his suggestions and availability.
This thesis work was developed mainly at Inria Paris within the REO team. For this reason, my gratitude goes to Dr. Irène Vignon-Clementel for having selected me as her intern during my time in France.
I would like to thank also all the people from the third floor of Inria’s bâtiment A for those amazing six months spent together and all the people form tender which made a little bit more mindless this last months. A sincere thanks to the least common multiple of both French and Italian souls of this work, Matteo. In a short time, he was able to help and advise me on the academic side and at the same time to become a very good friend.
Evident, but still true, the gratitude towards my lifelong friends, friends from uni-versity and friends known in Milan, wherever all of them are.
I would like to thank my parents, for their love and continuous support even in the darkest moment of this long journey, and my sister without which this wonderful adventure called Politecnico would have never begun.
Idiopathic pulmonary arterial hypertension (IPAH) is a lung disorder, occurring without an identifiable underlying cause, in which blood pressure in the pulmonary artery rises above normal levels. As palliation, the Potts shunt has been proposed for children where medical and pharmacological therapies are inefficient and other surgical procedures not attainable. The Potts shunt connects the pulmonary artery with the descending aorta resulting in a right-to-left shunt.
Mathematical modelling can help in interpreting the blood flow in IPAH and the effects of the presence of the Potts shunt. Geometries are retrieved from computed tomography angiography imaging. Multiscale simulations, solving the 3D Navier-Stokes equations around the Potts shunt and describing the rest of the circulatory system by means of an electric analogy (i.e., lumped parameter models - 0D), are used in lieu of in vivo experiments to devise a 0D shunt formulation to be included in a 0D closed-loop. This 0D model represents the post-operative stage with the shunt model. Multiple shunt parameters such as diameter, length and location can be varied to evaluate the decrease of the pulmonary arterial pressure and other physiological quantities of the cardiovascular system. The results obtained with the 3D multiscale simulations confirm medical expectations, while the comparison with the fully 0D post-operative simulations points out a good agreement.
Keywords: Idiopathic pulmonary hypertension, Potts shunt, Multiscale simulation, SimVascular, CFD, Hemodynamics, Lumped parameter, Fluid dynamics.
L’ipertensione polmonare idiopatica (IPAH, idiopathic pulmonary arterial hyper-tension) è una condizione patologica, che compare senza apparenti cause, in cui la pressione arteriosa polmonare media supera determinate soglie di normalità. In alcuni casi, il Potts shunt viene proposto in pediatria come palliativo laddove trat-tamenti medici e farmacologici risultino inefficaci e altre procedure chirurgiche non siano di facile attuazione. Il Potts shunt è un collegamento che unisce l’arteria pol-monare con l’aorta discendente permettendo a una parte di sangue desaturato di unirsi al flusso sistemico ossigenato.
Modellare matematicamente il flusso sanguigno nei casi di IPAH e con la presenza del Potts shunt, permette di comprenderne meglio i meccanismi ed i risvolti. Le geometrie dell’area di interesse sono ricavate con tecniche di segmentazione effet-tuate su immagini ottenute per mezzo di tomografia angiografica computerizzata. L’obbiettivo di costruire un modello ad anello chiuso 0D (a parametri concentrati) è perseguito mediante l’uso di simulazioni multiscala che si sostituiscono ad es-perimenti in vivo. In quest’ultime, le equazioni 3D di Navier-Stokes sono risolte nell’intorno del Potts shunt, le condizioni al contorno ed il resto del sistema circo-latorio sono descritti per mezzo di analogie elettriche (modelli zero dimensionali). Questa procedura permette di derivare una formulazione a parametri concentrati che descrive il comportamento dello shunt da includere nel modello 0D a circuito chiuso. Il modello ridotto ottenuto permette quindi, al variare di parametri dello shunt quali diametro, lunghezza e collocazione, di valutare la caduta di pressione nell’arteria polmonare ed altre condizioni fisiologiche del sistema cardiovascolare. I risultati ottenuti con le simulazioni 3D multiscala sono in accordo con le aspettative mediche ed evidenziano una buona similarità con i corrispondenti risultati ottenuti dal modello 0D.
Parole chiave: Ipertensione polmonare idiopatica, Potts shunt, Simulazione multi-scala, SimVascular, CFD, Emodinamics, Parametri concentrati, Fluidodinamica.
Introduction 1
1 Clinical overview 5
1.1 Circulatory system . . . 7
1.2 Pulmonary hypertension pathophysiology . . . 8
2 A reduced model of the cardiovascular system: a 0D closed-loop model 13 2.1 Formulation of 1D governing equations . . . 15
2.2 Zero-dimensional models (lumped models) . . . 18
2.3 Lumped parameter model for IPAH. . . 29
3 Multiscale simulation: 3D-0D coupling 35 3.1 Why a geometrical multiscale simulation . . . 37
3.2 3D solver methodology . . . 38
3.3 3D-0D coupling . . . 43
4 An application to IPAH: numerical results 51 4.1 Geometry . . . 53
4.2 SimVascular: a tool for cardiovascular simulation . . . 55
4.3 Steady simulations . . . 63
4.4 Unsteady multiscale simulations. . . 68
4.5 Shunt law modelling . . . 78
4.6 0D closed-loop . . . 88
5 Conclusions 95 5.1 Comparison with clinical data . . . 97
5.2 Future developments . . . 98
A Appendix 103
1.1 A sketch of the cardiovascular system (Wikipedia) . . . 8
1.2 A heart section viewed from the front (Wikipedia) . . . 9
1.3 A representation of the Potts shunt procedure [BVB04]. . . 11
1.4 Transcatheter Potts shunt placement procedure [SdBKO14] . . . 12
1.5 Potts shunt position [SdBKO14]. . . 12
2.1 A sketch of a simple compliant tube [PV09] . . . 15
2.2 RC Windkessel and RCR Westkessel models . . . 22
2.3 RCRL1, RCRL2 and RLCRCLR model schemes. . . 22
2.4 Single fiber model: functions f (`) and g(ta) [PCB+16] . . . 25
2.5 A diode which represents the electric analog of a heart valve . . . 27
2.6 Closed-loop circuit of the whole body (pre-operative case). . . 31
2.7 Closed-loop circuit of the whole body (post-operative case). . . 32
3.1 A scheme of a 3D domain [MVCF+13] . . . 39
3.2 An example of backflow instability [BC14] . . . 42
3.3 A domain where the variational form is not exact (mean pressure) . 45 3.4 A scheme of the 3D-0D coupling [QVV16] . . . 48
3.5 Scheme of time advancing in both 3D and 0D domain [MVCF+13] . 50 4.1 Aorta angio TC . . . 53
4.2 Pulmonary artery angio TC . . . 54
4.3 Shunt 3D rendering from angio TC . . . 54
4.4 Area of interest . . . 56
4.5 3D pre and post-operative models. . . 57
4.6 Different shunt diameter size 3D models . . . 58
4.7 SimVascular pipeline . . . 59
4.8 Mesh sphere refinement . . . 60
4.9 Two views of the mesh . . . 61
4.10 Mesh refined by the sphere refinement . . . 62
4.11 A detail of the volumetric mesh . . . 62
4.12 Coupling closed-loop . . . 64
4.13 Pre and post-operative pressure field . . . 65
4.14 Range of values that the shunt can handle . . . 66
4.15 High pulmonary artery pressure: jet area formation . . . 66
4.16 Steady simulation: velocity profiles (a) . . . 67
4.17 Steady simulation: velocity profiles (b) . . . 68
4.19 A Pre and post aorta slice: velocity magnitude . . . 71
4.20 Different diameters aorta slice: velocity magnitude . . . 72
4.21 Pre and post slices focused on the shunt: velocity magnitude . . . . 73
4.22 Different diameter slices focused on the shunt: velocity magnitude. . 74
4.22 Different diameter slices focused on the shunt: velocity magnitude. . 75
4.23 Wall shear stress for the different diameters . . . 76
4.24 DAE state variables from the multiscale simulation (pre and post) . 77 4.25 Pressure changes depending on the shunt diameter . . . 79
4.26 Pressure drop across the different shunt diameters. . . 80
4.27 Pressure slices for shunt modelling . . . 81
4.28 Pressure drop across the shunt (3D) . . . 82
4.29 Zoom of the pressure-flow rate relationship in diastole . . . 82
4.30 Shunt behavior: 3D vs 0D . . . 85
4.31 Results of the regression over the entire dataset available . . . 86
4.32 Shunt behavior: 3D vs 0D (excluded diameter regression) . . . 88
4.33 Shunt behavior: 3D vs 0D (alternative shunt law) . . . 89
4.34 Comparison of shunt flow rate: 3D vs 0D . . . 90
4.35 Comparison of shunt flow rate: 3D vs 0D (single diameter). . . 90
4.36 Comparison DAE state variables: multiscale vs 0D closed-loop (pre) 91 4.37 Comparison DAE state variables: multiscale vs 0D closed-loop (post) 92 4.38 The greatest errors: Qaov and Qpov . . . 92
4.39 DAE state variables from the 0D closed-loop simulation (pre and post) 93 5.1 Geometry recostruction of a different shunt shape . . . 99
5.2 Mesh and backflow problems (flow) . . . 100
1.1 Properties of the human systemic vessels [Car12] . . . 7
1.2 Properties of the human pulmonary circulation [SHH+73] . . . 7
2.1 Analogies between electric and hydraulic networks . . . 19
2.2 Analogy between electric and hydraulic network . . . 20
2.3 Legend of the elements in the circuits of Figure 2.6-2.7 . . . 29
4.1 Sarcomere material behavior: reference values.. . . 63
4.2 Reynolds numbers and velocities . . . 70
4.3 Notation for the DAE state variables represented in Figure 4.24-4.39 78 4.4 Parameter estimated by the linear regression analysis individually. . 84
4.5 Parameters for complete and alternative shunt law . . . 87
4.6 Cross-validation shunt law coefficients . . . 87
4.7 L2-norm error multiscale and 0D state variables . . . 91
5.1 Pressure in the aorta and pulmonary artery comparison (pre) . . . . 97
5.2 Pressure in the aorta and pulmonary artery comparison (post). . . . 98
A.1 Parameters adopted in the lumped parameter model . . . 104
Idiopathic pulmonary arterial hypertension (IPAH) is a rare disease character-ized by elevated pulmonary arterial pressure without apparent cause. This disease eventually leads to right heart failure with a high mortality rate. Medical procedures and pharmacological treatments are available for the less severe cases. However, in some patients, surgical procedures are required. Among those, this work focuses on the Potts shunt, an approach proposed as a palliation for children affected by IPAH. The Potts shunt is an anastomosis that connects the pulmonary artery with the de-scending aorta. Thanks to this artificial connection, the pressure in the pulmonary artery is considerably reduced and the right ventricle decompressed.
Nowadays this and similar practices are common techniques to treat pathological conditions in human cardiology. These connections have been used and developed for a while, new ones have been devised for paediatric applications during the years, but their assessment in terms of effectiveness and overall impact on the circulation is still an ongoing study.
The mathematical models help to establish predictive tools to guide clinicians in future decision-making regarding individual patients. Indeed, medical imaging techniques allow to reconstruct patient-specific geometries and with pressure and blood flow measurements, it is possible to calibrate the numerical model in order to match the patient condition.
Since the systemic circulation has a large number of vessels, 3D simulations are not attainable because of the high computational costs. Zero-dimensional (0D) models are simplified representations of the cardiovascular network and by adding models for the heart and the heart valves, it is possible to reproduce in a good qualitative way the real behaviour of the circulatory system.
In this work, we have developed a reduced model for the blood flow inside the Potts shunt based on a lumped parameters (0D) representation. To design and calibrate the formulation, we have used 3D-0D multiscale simulations. A 0D closed-loop model has been used to describe the whole systemic circulation while a more detailed 3D Navier-Stokes model has been used to zoom on the area around the Potts shunt. In fact, the 3D is able to better capture the complexity of the flow in the region of interest. The final 0D closed-loop model, that includes the 0D shunt formulation, can be used as a tool to assess the pulmonary arterial pressure drop and the overall system oxygenation by varying parameters such as shunt diameter, length and location.
In particular, in this study we focused on a 12-years old patient with suprasys-temic, therapy-resistant, IPAH treated at L’Hôpital Neker - Enfants malades in Paris. Most of this research took place at Inria Paris (Institut national de recherche en informatique et en automatique) within the REO team. This team works on
the numerical simulations of biological flows and cardiac electrophysiology. The main objectives of this team are the reduction of direct and inverse models and computational methods for the investigation of fluid-structure interaction problems, cardiac electrophysiology and electromechanical coupling and aerosol deposition in the respiratory tract.
In the first phase of this work, we generated a total number of six 3D geome-tries from computed tomography angiography. The geometry contains part of the aorta, the pulmonary artery and their branches. The Potts shunt is present in five of these geometries, which differ for the diameter of the shunt, the remaining one represents the pre-operative state. We performed the 3D-0D multiscale sim-ulations by using Simvascular, an open source software for blood flow simsim-ulations (http://simvascular.github.io/). We solved the blood flow around the Potts shunt by solving the 3D Navier-Stokes equations, while the rest of the circulatory system has been described via lumped parameter models (0D). We implemented a specific code for the exchange of the boundary conditions between the 3D and the 0D compartments. In particular, the 3D provides blood flow rates as an input for the 0D, while the 0D computes mean pressure values. As expected, the presence of the Potts shunt affects the blood circulation. The pressure in the pulmonary artery seems to decrease consistently and a decompression of the right ventricle afterload is obtained.
In the second phase of this work, we used the results of the 3D-0D multiscale simulations to calibrate and design a 0D lumped parameter model describing the shunt. In this way, we were able to replace the 3D compartments and to have a fully 0D closed-loop model for the entire circulation. The results showed that the fully 0D simulation is in good agreement with the corresponding multiscale simulation.
Contents and manuscript organization
This manuscript is organized as follows. Chapter 1 provides a brief description of the human cardiovascular system, introduces the main characteristics of the id-iopathic pulmonary arterial hypertension and presents the Potts shunt. Chapter 2 presents the lumped parameter models of the different compartments of the cardio-vascular system. These tools are then combined to build a 0D closed-loop model for the study of patients with suprasystemic therapy-resistant idiopathic pulmonary hy-pertension. We use heart and valves models to better simulate the whole circulation paying particularly attention on the parametrization of the vessels more involved in the Potts shunt region.
In Chapter 3, we describe the models used for the 3D solver. We introduce the weak form of the Navier-Stokes equations and the techniques employed for the coupling of the 3D formulation with the 0D models introduced in the previous chapter. The 0D compartment provides realistic boundary conditions and, since it contains the heart, it is also the forcing term of the system.
how the images are processed to reconstruct the 3D geometries and the subsequent meshing procedure. Then, the parameters setting of the 3D solver is discussed. Nu-merical results from both 3D-steady and 3D-0D-unsteady simulations are presented. Finally, the results of the 3D-0D simulations are used to build the 0D model of the shunt and its parameters are estimated via a least squares regression. The results obtained with the 0D model are compared with those of the 3D-0D simulations.
In the last chapter we summarize and discuss the results presented in this thesis and we also discuss the limitations of the models and the future perspectives.
Clinical overview
In this first chapter, we provide a quick description of how the cardiovascular system works. An introduction to idiopathic pulmonary arterial hypertension follows, what this disease comports and how it can be palliated.
Contents
1.1 Circulatory system . . . 7
1.2 Pulmonary hypertension pathophysiology. . . 8
1.2.1 Overview . . . 9
1.2.2 Types of pulmonary hypertension. . . 9
1.2.3 Symptoms and signs of IPAH . . . 10
1.1
Circulatory system
The main components of the human cardiovascular system are the heart, the blood and the blood vessels (Figure 1.1). This is made up of two essential loops: the pulmonary circulation, where blood is oxygenated passing through the lungs; and the systemic circulation, which provides oxygenated blood to the rest of the body. The digestive system supplies the needed nutrients to keep the heart pumping to the circulatory system. Both systemic and pulmonary blood circuits are composed of three main compartments: arteries, capillaries and veins (Table1.1 and Table1.2).
Vessel Diameter of Wall Number of Blood Mean
lumen thickness vessels volume pressure
[mm] [mm] [%] [kPa] Aorta 25 2 1 2 12.5 Large Arteries 1 − 10 1 50 5 12 Small arteries 0.5 − 1 1 103 5 12 Arteriole 0.01 − 0.5 0.03 104 5 7 Capillary 0.006 − 0.01 0.001 106 5 3 Venule 0.01 − 0.5 0.003 104 25 1.5 Vein 0.5 − 15 0.5 103 50 1 Vena cava 30 1.5 2 3 0.5
Table 1.1: Properties of the human systemic vessels. Adapted from [Car12].
Diameter range Number of Volume Mean
[mm] vessels [mL] [mm/s] 30 (main PA) 1 64 110 8 − 30 10 21 155 1 − 8 103 37 104 0.1 − 1 0.25 · 106 19 44 0.02 − 0.1 20 · 106 5 23 0.1 (capillaries) 300 · 106 5 2
Table 1.2: Properties of the human pulmonary circulation. Adapted from [SHH+73].
The cardiovascular system of humans is a closed-loop, meaning that the blood never leaves the network of blood vessels. In contrast, oxygen and nutrients diffuse across the blood vessel layers and enter interstitial fluid, which carries oxygen and nutrients to the target cells, and carbon dioxide and wastes in the opposite direction. The other component of the circulatory system, the lymphatic system, is open.
Blood circulation conventionally starts when the heart (Figure 1.2) relaxes be-tween two heartbeats: blood flows from both atria (the upper two chambers of the heart) into the ventricles (the lower two chambers) which then expand. The
follow-Figure 1.1: A sketch of the cardiovascular system: the systemic and pulmonary circulation.
LA left atrium, LV left ventricle, RA right atrium, RV right ventricle. SourceWikipedia.
ing phase is called ejection period, which is when both ventricles pump the blood into the large arteries.
In the systemic circulation, the left ventricle (LV) pumps oxygenated blood into the aorta. The blood travels from the main artery to larger and smaller arteries into the capillary network. There the blood releases oxygen and nutrients and takes on carbon dioxide and waste substances. The blood, which has now a low concentration of oxygen, is collected in veins and travels to the right atrium (RA) and into the right ventricle (RV).
Now blood enters the pulmonary circulation: the right ventricle (RV) pumps blood that carries little oxygen into the pulmonary artery (PA), which branches off into smaller arteries and capillaries. The capillaries form a fine network around the pulmonary vesicles, grape-like air sacs at the end of the airways. This is where carbon dioxide is released from the blood into the air contained in the pulmonary vesicles and fresh oxygen enters the bloodstream. When we breathe out, carbon dioxide leaves our body. Saturated blood travels through the pulmonary vein and the left atrium (LA) into the left ventricle (LV). The next heart beat starts a new cycle of systemic circulation.
1.2
Pulmonary hypertension pathophysiology
Pulmonary hypertension (PH) is a condition of increased pressure in the pul-monary arteries. The pulpul-monary arteries are those vessels which carry blood from the heart to the lungs to pick up the oxygen [GHR,NHL,OSU].
Figure 1.2: A heart section viewed from the front. It is possible to distinguish: chambers (left atrium LA, left ventricle LV, right atrium RA, right ventricle RV); vessels (aorta Ao, pulmonary artery PA, superior vena cava SVC, inferior vena cava IVC, pulmonary veins PV); valves (tricuspid valve TcV, aortic valve AoV, pulmonary valve PoV, mitral valve
MvV). SourceWikipedia.
1.2.1 Overview
Pulmonary hypertension occurs because of the progressive thickening and con-striction of the small pulmonary arteries throughout the lungs. This phenomenon leads to an increase of the vascular resistance. To overcome the increased resistance, the blood pressure in the pulmonary artery and in the right ventricle of the heart increases. Since the heart is working harder than normal, the right ventricle may become strained and weak. After a while, the right ventricle (that is the chamber demanded to pump blood into the pulmonary artery) may fail in its function result-ing in a decreased cardiac output, which shows several symptoms and eventually death.
This disease is considered present when the mean pulmonary artery pressure is greater than 25 mmHg at rest or 30 mmHg during exercises [GHH+09].
1.2.2 Types of pulmonary hypertension
Pulmonary hypertension can be a disease without an identifiable reason (idio-pathic or primary pulmonary artery hypertension - IPAH) or can be associated to a variety of conditions such as cirrhosis, AIDS, connective tissue diseases, drugs etc. (APAH). Among the causes from which a secondary pulmonary hypertension can result, we can cite lung or heart disease, low oxygen level in the blood (hypoxia) and obstruction of the pulmonary vessels (clots, foreign body, etc.) Depending on
its causes, five groups of PH are distinguished [SRB+09]. Idiopathic pulmonary artery hypertension is rare condition (estimated incidence 1 in every 106 patients per year approximately) and most often occurs in young adults and is more common in women.
1.2.3 Symptoms and signs of IPAH
Idiopathic pulmonary arterial hypertension may present some of the following signs and symptoms: shortness of breath (dyspnea) during routine activities, tired-ness, chest pain, rapid heart rate, pain on the upper right side of the abdomen and decreased appetite. As IPAH worsens, signs and symptoms may include: dizzi-ness, swelling of the ankles or legs, faints and difficulty in carrying out the simplest everyday physical activities.
The definition of paediatric idiopathic pulmonary artery hypertension is the same defined in adults (1.2.1).
1.2.4 Treatment of IPAH
IPAH is a disease where no cure is possbile. Its course is often one of steady deterioration which brings to a reduced life expectancy. The patients who do not receive any treatment have a 68% of survival probability at one year, 48% at three years and 34% at five years [DBA91] (depending on the institute registry, the per-centages may change [PMD11]). APAH affected patients are treated with the same procedures of the ones affected by IPAH.
The cures vary considering the stage of the disease and evaluating every single specific case in order to plan the best therapy for an individual patient.
• Pharmacological treatment: a therapy consisting of treatment with medicines may be pursued in order to relax the vessels in the lungs and reduce the wall thickening of the small pulmonary arteries. By taking these kinds of drugs it is possible to increase blood flow through the blood vessels. In the presence of blood clots in the lungs or in general blood clotting disorders, clinicians can prescribe blood-thinning medicines which prevent the formation or the growing of such clots.
• Medical procedures: an oxygen therapy can be prescribed to the patients with a low oxygen level in the blood. This treatment can be carry out by prongs that fit into the nose.
• Surgical procedures: Unfortunately, some IPAH patients become unrespon-sive to the therapy mentioned above and a surgical procedures is required. The main surgical procedures are: the atrial septosomy that creates an opening be-tween the two atria, lung/heart-lung transplant and right-to-left shunt (Potts shunt).
Figure 1.3: A representation of the Potts shunt procedure. The left pulmonary artery is connected to the descending aorta by means of this anastomosis. Desaturated blood is allowed to reach the lower part of the body (path marked by the arrow) mixing with the oxygenated blood coming from the ascending aorta. In this sketch the right pulmonary artery passes in front of the ascending aorta because of an arterial-switch procedure. Source
[BVB04].
Potts shunt As briefly introduced in Section 1.2.4, despite the recent improve-ments in pharmacological therapy, the probability of right heart failure and sudden death because of idiopathic pulmonary arterial hypertension is still high. Patients who have an elevated pulmonary vascular resistance do not tolerate atrial septosomy (i.e. creation of a small hole between the upper two chambers of the heart) because this may result in a non-sufficient pulmonary blood flow and hypoxemia (i.e. oxygen deficiency in arterial blood) due to a massive right-to-left shunting. Furthermore lung transplantation is not always achievable or it is considered only a last attempt. For patients who are refractory to medical procedures, performing a Potts shunt (Figure 1.3) is a valid alternative for IPAH palliation.
This kind of shunt was reported in 1946 for the first time working as a left-to-right shunt meaning a systemic-to-pulmonary shunt [PSG46], while the Potts shunt we are treating is a pulmonary-to-systemic shunt. This intervention seek to reduce right ventricular afterload and to improve right ventricular function creating a direct side-by-side anastomosis from the left pulmonary artery to descending aorta.
The surgical creation of a Potts shunt compared to the atrial septosomy results in an immediate decrease in the right ventricular afterload. The surgery can be performed through a left thoracotomy, with or without cardiopulmonary bypass [BSL+12,GE16,BVB04], a sternotomy, which are the practices with most significat risks, or through a transcatheter shunt implantation in which a stent is placed
between the vessels [BBB+14,ESC+13,Ven13].
(a) (b) (c)
Figure 1.4: Snapshots from animations showing anatomical aspects of the transcatheter Potts shunt placement procedure using a transcatheter delivery. The most optimal place where to implant the anastomosis is between the distal part of the central left pulmonary artery, just before its branching, and the opposite descending aorta. RPA/LPA: right/left pulmonary artery; DAo: descending; PV: pulmonary vein; AAr: aortic arch, (animal
surgery [SdBKO14]).
In Figure 1.4is shown a 3D reconstruction of a Potts shunt placement through a transcatheter procedure while Figure1.5shows how this connection links the two vessels.
Since the using of a Potts shunt procedure in case of IPAH dates to the last decades, there are not sufficient data in order to express a precise verdict about the probability of survival. Patient with an Eisenmenger’s syndrome (i.e. the condition which a Potts shunt procedure leads to) have a survival probability of 87% in 5 years, 77% in 15 years [SBJ+94] and a median life expectancy of 53 years [CHM+99].
Figure 1.5: Potts shunt position between the distal part of the central left pulmonary artery, before its branching, and the opposite descending aorta. RPA/LPA: right/left pul-monary artery; DAo: descending; PA: pulpul-monary artery; AAo: ascending aorta. Source
A reduced model of the cardiovascular
system: a 0D closed-loop model
The objective of this chapter is to develop a reduced model of the cardiovascular system in the presence of idiopathic pulmonary arterial hypertension in paediatrics. In particu-lar, the so-called zero-dimensional (0D) models permit to represent the human circulatory system quickly and eventually to study diseases or problems that may affect this system. The 0D models are most commonly expressed through hydraulic-electrical analogies where resistor (R), inductor (L) and capacitor (C) of the circuit theory are the flow resistance, the fluid inertia and the vessel compliance respectively. The overall hemodynamic can be de-scribed approximately by means of resistance consideration in a Poiseuille flow, inertia and capacitance parameters are fundamental in the understanding of the blood flow pulsatile nature. Models for the heart and the heart valves are elaborated and an initial model for
Contents
2.1 Formulation of 1D governing equations . . . 15
2.1.1 One-dimensional models . . . 15
2.2 Zero-dimensional models (lumped models) . . . 18
2.2.1 Systemic vasculature models . . . 21
2.2.2 Lumped parameter models for the heart . . . 23
2.2.3 Models for the heart valves . . . 27
2.3 Lumped parameter model for IPAH . . . 29
2.1
Formulation of 1D governing equations
In this section, governing equations for a 0D model are derived following the steps developed in [PV09]. The formulation reduction starts describing the one-dimensional (1D) form of the Navier-Stokes equations where the spacial dependence is reduced only to axial coordinate.
2.1.1 One-dimensional models
Several approaches for the derivation of such equations are available. In [BHTV66], performing an asymptotic analysis of the Navier-Stokes equations, an assumption about the radius of the vessel R0 is made: R0 is small compared to the
length of the vessel ` so that R0
L 1. In this way, it is possible to discard the
higher order terms R0
L and simplify the governing equations. As another option, it
is possible to integrate the Navier-Stokes equations on a generic section considering cylindrical symmetry [OOL04].
The most general approach, which does not require any simplifying assump-tions concerning the geometry of the vessel section, is the one pursued in [HL73] and [Hug74]. A way to represent a human vessel (artery in particular) is a simple compliant tube as the one shown in Figure 2.1. The axis of the vessel is assumed rectilinear and coinciding with the x−axis. The derivation of the 1D governing equations starts from the Reynolds’ transport theorem for an arbitrary control vol-ume Vt with boundary ∂Vt and outer normal n [QF04]. For a continuous function
f = f (t, x), we have d dt Z Vt f dV = Z Vt ∂f ∂tdV + Z ∂Vt f ub· n dσ,
Figure 2.1: A sketch of a simple compliant tube. The tube goes from x = x1 to x = x2,
section varies from S = S1to S = S2, Vtis the volume of the tube and ∂Vt,w is the surface
where x stands for (x,y,z) and ub is the velocity of the boundary of the volume Vt. The volume we are considering is composed by the arterial wall ∂Vt,w and the
sections S1, S2 perpendicular to the axis. On S1 and S2 the normal component of
ub is equal to zero, while on ∂Vt,w velocity ub ≡ uw, where uw is considered to
be different from fluid velocity u = (u1, u2, u3) in order to consider the eventual
presence of a permeable lumen. It is possible to consider w = uw − u as the
relative velocity between the arterial wall and the fluid at the lumen and accounts for possible perfusion across the wall. We extend w to zero on S1 and S2.
Area-averaged values for the main variables of the equations are considered to obtain the one-dimensional formulation of the governing equations. Such value is denoted as ¯f and is computed as
¯ f = 1 A Z S f dσ,
where A = A(x, t) = RSdσ is the area of the cross section S. Using this notation, after several steps it is possible to write the Reynolds’ transport theorem in the 1D form ∂ ∂t A ¯f + ∂ ∂xA(f u1) = Z S ∂f ∂t + ∇ · (f u) dσ + Z ∂S f w · n dγ. (2.1) Mass conservation
The continuum equation in a flexible tube is derived considering f = 1 in (2.1), then if we consider the incompressibility of the fluid ∇ · u = 0, we obtain
∂A ∂t + ∂ ∂x(A¯u1) = Z ∂S w · n dγ, (2.2)
the term on the right side of the equation can be treated as a volumetric outflow per unit length and unit time.
Momentum equation
To derive the balance of momentum equation, f is considered to be f = u1 in
(2.1). Assuming again an incompressible fluid, we get
∂ ∂t(A¯u1) + ∂ ∂x A¯u 2 1 = Z S ∂u1 ∂t + u · ∇u1 dσ + Z ∂S u1w · n dγ, (2.3)
making use of the material derivative (DtD = ∂t∂ + u · ∇), we can rewrite the (2.3) as ∂ ∂t(A¯u1) + ∂ ∂x A¯u 2 1 = Z S Du1 Dt dσ + Z ∂S u1w · n dγ. (2.4)
It is possible to recover another formulation for the momentum balance for a control volume Vt Z Vt D Dt(ρu)dV = Z Vt ρ fbdV + Z ∂Vt T n dσ,
where fb represents the body force per unit volume and T is the Cauchy stress tensor. Making use of the divergence theorem, considering ρ = const and the constitutive equation for the fluid T = −pI + D (p pressure, I identity tensor and D deviatoric stresses due to the viscosity of the fluid), with several substitution on the (2.4), we arrive at the final formulation
∂ ∂t(A¯u1) + ∂ ∂x(α¯u 2 1) = Af1b− A ρ ∂ ¯p ∂x − KRu¯1+ Z ∂S u1w · n dσ, (2.5)
where α is a momentum-flux correction coefficient (also called the Coriolis coef-ficient) which assumes value α = 1 for a flat velocity profile and α = 4/3 for a parabolic one and KR is a strictly positive quantity which represents the viscous resistance of the flow per unit length of tube.
The unknowns of the system composed by (2.2) and (2.5) are the pressure p, the section area A and the velocity ¯u1. If we consider this system, the number of
unknowns is greater than the number of equations and a closure relation is required. Usually, the added equation is an algebraic relationship between the pressure p and the area A such as
p = Pext+ β √ A −pA0 , where β = √ πh0E (1 − ν2)A 0 , (2.6) or in general p = Pext+ Φ(A; A0, β),
with Φ function of the vessel section area A, the reference area A0 and a mechanical parameter β. The main properties of this function are
∂Φ
∂A > 0, and Φ(A0; A0, β) = 0.
Obviously this relation can be modified so that it could be possible to consider dif-ferent hypothesis (equation (2.6) assumes that the wall is instantaneously in equi-librium with the pressure forces acting on it).
It is possible to add other hypothesis in order to have simple system of equations. In particular, we can assume that the lumen is impermeable w · n = 0 and that the body forces are negligible ¯f1b = 0. We simplify the notation by indicating the velocity ¯u1 as u and the pressure ¯p as p. Equations (2.2) and (2.5) can be rewritten
∂A ∂t + ∂Au ∂x = 0, ∂u ∂t + (2α − 1)u ∂u ∂x + (α − 1)u ∂A ∂x + 1 ρ ∂p ∂x+ KR u A = 0. Defining the mass flux across a section as
Q = Au = Z
S
u1 dσ,
the rearranged system is ∂A ∂t + ∂Q ∂x = 0, (2.8a) ∂Q ∂t + ∂ ∂x αQ 2 A +A ρ ∂p ∂x + KR Q A = 0, (2.8b)
for t > 0 and 0 < x < L, A and Q are called conserved variables since they come from the conservation principles. System (2.8) has to be supplemented by proper initial and boundary conditions. For its analysis we refer to [PV09].
2.2
Zero-dimensional models (lumped models)
In Section 2.1.1 a derivation of one-dimensional governing equations has been introduced, averaging these it is possible to derive the corresponding lumped model. This method is close to the physics of the problem and it is useful to understand what are the parameters of the model, their role and their quantification.
Considering the vessel Ω sketched in Figure 2.1 with a length equal to ` = |x2− x1|, we can proceed with the definition of several mean quantities, that are
ˆ Q = ρ ` Z Ω u1dv = ρ ` Z x2 x1 Z S(x) uxdσ ! dx = ρ ` Z x2 x1
Q(x)dx → mean flow rate,
ˆ p = 1 ` Z x2 x1 P dx → mean pressure, ˆ A = 1 ` Z x2 x1
Adx → mean area.
Integrating the continuity equation (2.8a) along the axial direction (x1 ≤ x ≤ x2), we get
`d ˆA
dt + Q2− Q1 = 0, (2.9)
For how it concerns the momentum equation, assumptions which simplify the equation are hypothesized:
• ∂x∂ αQA2
may be neglected. This consideration may be true in the peripheral circulation where the velocity of the blood flow is low.
• A and β vary with respect to x much less than P and Q. This is reasonable when the axial average is made over short segment, this permits to assume A = A0, where A0 is the constant value of the area at rest.
With these assumptions, the average over x of the (2.8b) carries to ρ` A0 d ˆQ dt + ρKR` A2 0 ˆ Q + P2− P1= 0, (2.10) where P1(t) = P (t, x1) and P2(t) = P (t, x2).
As previously seen, there is a closure problem for the system of equations (2.9) and (2.10). This is solved again by adding a wall mechanics law like (2.6). This permits to rewrite (2.9) as
k1`
dˆp
dt + Q2− Q1 = 0. (2.11)
Equations (2.11) and (2.10) represent the lumped parameter model for a vessel. Usually these equations are associated with electric network analogy (Table2.1)
Hydraulic Electric
Pressure Voltage
Flow rate Current
Blood viscosity Resistance R Blood inertia Inductance L Wall compliance Capacitance C
Table 2.1: Analogies between electric and hydraulic networks
In order to better explicate the parallelism with the electrical circuits, we can readjust equation (2.11) and (2.10) as
Cdˆp
dt + Q2− Q1= 0, (2.12a)
Ld ˆQ
dt + R ˆQ + P2− P1 = 0. (2.12b)
The parameters appearing in (2.12a) and (2.12b) are associated to the elements of a circuit as shown in Table2.2, where there are recalled the pressure-flow relationship.
R L C R P Q L P Q C P Q P = R Q P = L ˙Q Q = C ˙P
Table 2.2: Correspondence of the terminology in the analogy between electric and hy-draulic network.
Flow resistance R The coefficient R = ρKR`
A2 0
present in (2.12b), represents the resistance that the flow encounters because of the blood viscosity. As expressed with the formula in Table 2.2, the mean pressure and the flow rate are related by means of the propor-tionality coefficient R through Darcy’s law [Lev13]. Assuming a constant viscosity (term included in KR) in the larger arteries, this parameter is proportional to the length of the vessel and inversely proportional to the squared section area. It is possible to affirm that the largest part of the pressure drop of the entire circulatory system occurs in the micro-circulation, where small vessels and capillaries offer a very high resistance to the flow.
Resistance R does not contribute directly to the shape of the pressure and the flow rate waveform [Ala06], the instantaneous pressure-flow relations depend on the simultaneous combination of compliance and inertia. We got the above formula for the resistance but changing the velocity profile or considering a non-Newtonian fluid, it is possible to obtain different expression [RD67,WBVN69,FV03].
Blood inertia L The coefficient L = Aρ`
0 represents the inertial term in equation (2.12b). During
every cardiac cycle, blood accelerates in the systolic phase and decelerates in the diastolic phase. The acting force induces an inertia in the blood [WSN10]. This inertia prevents sudden changes in the flow rate through a vessel.
Vessel compliance C
The coefficient C = k1` represents the compliance of a vessel volume and its
capacity to store blood in (2.12a). The stored blood has a potential energy that is released during diastole. Vessels with a high compliance expand more under the same pressure increments than stiffer ones. From the equation in Table 2.2for the
compliance, it is possible to infer that this term governs the instantaneous pressure change.
The system (2.12) involves the mean flow rate and pressure over the concerned vascular segment and the boundary values of flow rate and pressure Qi, Pi with
i = 1, 2. The term boundary is not exactly appropriate, since the continuous space dependence has been lost (averaging axially), so it simply represents input/output quantities exchanged with the near system sections. So, the concept of boundary condition is connected with the identification of input data for the district at hand. It is thus possible to provide input data and assume that the output of the district is given by the state variables. For instance, if we suppose to know Q1 and P2, we
approximate ˆp ≈ P1 and ˆQ ≈ Q2 (reasonable for a short pipe) and the resulting
system becomes
CdP1
dt + Q2 = Q1, LdQ2
dt + RQ2− P1 = P2, where the input data have been put on the right hand side.
2.2.1 Systemic vasculature models
The elements introduced above are combined to describe the vessel behaviour [SLH11, SSW+03]. There are several 0D systemic vasculature models which allow the modelling of intra-luminal pressure and flow changes. The simplest and first network description is the two-element Windkessel model, which was first proposed by Stephen Hales in 1733 and later formulated by Otto Frank in 1899 [Li00]. Lots of models were derived, from the so-called Windkessel model to the Guyton model. In the former, the veins are considered as a sink of zero pressure and the vasculature is modeled from the aorta to the capillaries with a single capacitance C connected in parallel to a single resistance R. The latter represents almost all the main circulatory vessels taking into account also autonomic and hormone regulation effects [GCG72]. Such models are born to describe the whole vessel network, but it is not unusual to have a different segmentation of the cardiovascular network [TZG+97,HSKM02,
PBF97,POPS02] with different parameters for the various vessels.
The Windkessel model (Figure 2.2 on the top-left) consists of a capacitor C (which represents the storage properties of large arteries) in parallel with a resistance R (which describes the dissipative nature of small peripheral vessels). A modification of the Windkessel model has been made by Landes [Lan43], who introduced an extra resistance Rp. This RCR model (sketched in Figure 2.2 on the top-right)
has been used extensively by Westerhof and his co-workers [WES71] so that the model is called Westkessel alternatively. The added proximal resistance represents the impedance characteristic of the arterial network (defined as the ratio of the
R C Rp Rd C Rd Rp C
Figure 2.2: In the figure, the following schemes are sketched: RC Windkessel model, com-posed of two parallel elements, a resistance R and a capacitance C; RCR Westkessel model,
that is a resistance Rp added to the Windkessel model; RCR2 model, it is a modification
of the RCR Westkessel model where the Rp is placed in series with the capacitance C.
Rp L Rd C Rp Rd C L R L R L R C C
Figure 2.3: Schemes for RCRL1, RCRL2 and RLCRCLR models are represented. RCRL1 shows the addition of the inertial term, RCRL2 is a variation of the former and RLCRCLR
oscillatory pressure and the oscillatory flow rate in the absence of reflective wave). The sum of Rp + Rd equals the resistance of the vascular system present in the
previous RC model. Even if only a resistance has been added, the performances of the model greatly improve, but still the RCR model does not match exactly the in vivo studies (meaning the aortic peak and the mean arterial pressure [BAS88]). In [BN98], it has been developed a variation of the RCR Westkessel model in which the proximal resistance Rp is placed in series with the capacitance C in order to
describe the visco-elastic property of the vessel wall (Figure2.2 on the bottom). Landes in [Lan43] extends the RCR model adding an impedance term L incor-porating the inertial effect of the blood flow, he assembles a model configuration of RLCR1 (Figure 2.3 on the top-left). A variation of it (RLCR2) is adopted in [SWW99] and [JWN65] (Figure 2.3on the top-right). Lots of in vivo studies prove that RLCR1 model reproduces better than the other models the characteristic of
the vascular impedance data [DCG80][SPM+00].
The reason why RCR and RC models are widely used compared to the four-element models lies in the easier parameter estimation and identification.
In the need to consider the contribution of the venous side to the overall haemo-dynamic, the RLCRCLR model has been derived where additional R, C and L pa-rameters are included to better reproduce the dynamic characteristics of the veins (Figure 2.3on the bottom).
2.2.2 Lumped parameter models for the heart
Recalling Section 1.1, the heart is composed of four chambers: left atrium, left ventricle, right atrium and right ventricle. The right side of the heart is delegated to the pulmonary circulation while the left side pumps the saturated blood into the systemic tree. The atria and the ventricles are separated by an atrioventricular valve: the tricuspid valve in the right side and the mitral valve in the left side. The role of these chambers is to receive blood at low pressure and to transfer it to a higher pressure region.
A varying elastance model for the ventricle
There are several ways to model mathematically the ventricle, one of the most common and more used model is the one proposed in [SSS73]. In this model a varying elastance model for the ventricle is suggested, the ventricular pressure is considered a function of the ventricular elastance and of the ventricular volume change from its unstressed value. The determination of the ventricular volume change is carried out computing the blood flowing into and out of the chamber while the ventricular elastance is defined as a time-varying function based on ventricular activity over a cardiac cycle measured in vivo. This model has been widely used such as in [SBLC97,PBF97,UFB96,Urs99,PMD+97,MPD+01] and more recently in [LTHL09, SSU+10, SLT+13]. A brief explanation with equations taken from [SBLC97], formulation also used in [LL05] and [BF13], is following below.
Blood pressure in each cardiac cycle is given by Ph(t) = E(t)(V − V0) + S ˙V ,
where V is the chamber volume, ˙V its time variation, V0 is the unstressed volume
and S is the viscoelasticity coefficient of the cardiac wall. E(t) represents the time-varying elastance given by
E(t) = EAe(t) + EB,
with EAthat is the amplitude of elastance, EB the baseline value of elastance, while e(t) is a normalized time-varying function of elastance and it is modelled differently in the atrium and in the ventricle. Namely, in the ventricle we have e(t) = ev(t)
where ev(t) = 1 2 h 1 − cos πTt vc i 0 ≤ t ≤ Tvc, 1 2 h 1 + cosπ(t+T −tvc) Tvr i Tvc≤ t ≤ Tvc+ Tvr, 0 Tvc+ Tvr < t ≤ T,
while in the atrium e(t) = ea(t) with
ea(t) = 1 2 h 1 + cosπ(t+T −Tar) Tar i 0 ≤ t ≤ tar+ Tar− T, 0 tar+ Tar− T < t ≤ tac, 1 2 h 1 − cos π(t−tac) Tac i tac ≤ t ≤ tac+ Tac, 1 2 h 1 + cosπ(t−tar) Tar i tac+ Tac≤ t ≤ T,
v and a stand for ventricles and atria, T duration of a cardiac cycle, Tvc, Tac, Tvr
and Tar are the durations of the ventricular/atrial contraction/relaxation, and tac,
tar are the times when atria begin to contract and relax. The volume is then related
to the inflow and outflow flow rate as indicated dVch
dt = Qi− Qo.
A single fiber model for a heart chamber
Left ventricle can be considered as a cavity enclosed by a fibrous wall that can exchange mechanical power to the environment by only changing its volume depending on the cavity pressure. Most of the deformation energy is stored and exchanged by the fibrous structures formed by muscle filaments. Therefore there is a relation between the absorption or delivery of mechanical power by the cavity and the generation of mechanical power carried out by the fibers in the wall. The following model presents a general relation between the pressure, the volume of the cavity and the load stress of the fibers in the wall. [PCB+16,ABPR91, ABDP03,
BBAvDV06] used this model in their research works.
The relationship between pressure and stress tensor is dominated by the ratio of the cavity volume V to the wall volume Vw, according to
σf P = 1 +3V Vw , (2.14)
where again P is the chamber’s pressure and σf is the stress in the fiber. The fiber stress σf is made of two components: an active part σa and a passive part σp which
are linked by the relation
σf = σa+ σp. (2.15)
in-Figure 2.4: The figure shows the behavior of some chamber activaction functions related
to the sarcomere properties and the cardiac cycle time: (a) function f (`) (2.17) of the active
component of the stress σa (2.16) and the passive stress σp (2.19); (b) activaction curves
for the atrium and the ventricle, T is the time of a cardiac cycle, ta for the atrium, tv for
the ventricle and t1 that is the overlap between atrium and ventricle activation. Source
[PCB+16].
fluence of the active part of the heart cycle, i.e., heart contraction, on the chamber pressure (visible in eq.(2.14)). To model the heart contraction, the following equa-tion is used [BBAvDV06]
σa= c Ta0f (`) g(ta) h(vs), (2.16)
where the several components are described below. • c is the contractility of the heart chamber. • Ta0 is the maximum sarcomere stress.
• f (`) is a function which depends on the sarcomere length `. The sarcomere is the fiber which makes the heart contracts. f (`) may assume different values for different ` f (l) = 0 if ` < `a0, (` − `a0)/(`am− `a0) if `a0 < ` ≤ `am, 1.0 if `am < ` ≤ `ae, (`af − `)/(`af − `ae) if ` > `ae, (2.17)
`a0 is the minimum length when there is a contraction and can be computed
introducing the definition of the fiber stretch ratio λ
` `0 = λ = 1 + (3V /Vw) 1 + (3V0/Vw) 13 ,
`0 and V0 are the sarcomere length and the volume of the chamber at a zero
• h(vs) is a function of the form
h(vs) =
1 − (vs/v0)
1 + cv(vs/v0)
,
where vs is the shortening velocity of the sarcomere which is given by (2.18),
v0 is the unloaded sarcomere shortening velocity and cv is a shape parameter
of the stress-velocity relation.
vs= d` dt = − ` Vw 1 +3V Vw −1 dV dt . (2.18)
• g(ta) is a function which can vary as
g(ta) = ( [12 1 − cos(2π ta tmax)] Ea if t a< tmax, 0 otherwise,
where tais the time of activation of the chamber and tmax is the total time of activation in a cardiac cycle.
For how it concerns the passive part which represents the passive filling of the heart chamber, this is described as
σp =
(
0 if λ < 1,
Tp0(exp[cp(λ − 1)] − 1) if λ ≥ 1,
(2.19) where Tp0 and cp are sarcomere material constants.
Thus, the relationship between P , V and dVdt of a heart chamber is given by (2.14)-(2.19).
This single fiber model for a heart chamber is the one used in this work as heart model. The main reason why this model has been adopted is because of its accuracy in the description of the cardiac biomechanics, for both ventricle and atrium. This model is particularly indicated for lumped parameter modelling because it has been demonstrated that under rotational symmetry the shape of the chamber and other geometric parameters do not affect so much the relationship between the cavity pres-sure and the fiber stress [ABPR91]. More functions are involved in the description of the beating motion compared to the other models. Moreover, the main parame-ters involved, i.e., myofiber passive and active behaviour, have been experimentally measured and reported in the literature allowing to focus on the other parameters which may be characteristic of a specific patient. Thus, the versatility of this model is very interesting in the assessment of patient-specific cardiac function and in the study of heart diseases.
The models described above are all models for a single chamber. There are models where the interactions between the different chambers are described [KS06a][KS06b]. In particular, during the diastole, the unbalance of pressure in the
Figure 2.5: A diode which represents the electric analog of a heart valve. Behaviour of a
valve depending on the pressure and the flow rate. Source [PV09].
two ventricles brings to a motion of the septum. This movement causes an interac-tion in the filling of the two cavities. Moreover the septum contributes to the cardiac output during the systolic phase, so that models of variable elastance which take into account to the specific response of the ventricular septum have been derived. Although these interactions have been deeply studied, is still not well established their mathematical modelling.
2.2.3 Models for the heart valves
In the heart, there are four valves (Section 1.1): the mitral, tricuspid, aortic and pulmonary valves. The function of these elements is to prevent blood backflow, in particular, the mitral and the tricuspid valves from the ventricle to the atria during systole, the aortic and pulmonary valves from the aorta and pulmonary arteries into the ventricles during diastole. The opening and closure movement of these valves are regulated by the pressure gradient across them.
A heart valve is usually represented by means of a diode (Figure2.5). where the ideal behavior would be the one sketched in Figure 2.5
P = 0 if Q > 0, Q = 0 if P < 0,
This would introduce a discontinuity in the algebraic differential model, so an alter-native model of valve has been derived using the Shockley equation,
Q = QS eαP − 1 .
Another, very commonly used, model for a heart model valve is composed of a diode plus a linear or non linear resistance [PMD+97,SSU+10,MPD+01,BCY+11,
CCP+11], such as
Θ∆P = RQ + KQ |Q| , (2.20)
where the coefficient Θ rules the non binary states of the valve, so that the valve has a resistance to the flow when the pressure gradient across it is positive and the flow
is stopped when the pressure gradients are negative. An inertance L multiplied for the derivative in time of the flow rate dQdt can be added in (2.20) as it has been done in [BF13] or a bit differently in [SYZ04]. More complicated models are available, where a larger number of parameters is required [KS06c,PMC+11].
In [ŽK96] a change in the heart valve resistance is taken into account. During the valve motion a time-dependent drag coefficient is used as function of the valve open area. The volume of the reverse flow during the closure phase, i.e., dead space volume, is included as a function of the valve leaflet opening angle in [WBH02]. [KS06c] proposes to improve the valve dynamics modelling by means of an ordi-nary differential equation that considers the pressure difference across the valve, the frictional effect of the tissue nearby, the interaction blood-leaflet, the action of the vortex flow downstream of the valve and the shear stress on the leaflet.
A trade-off between the above models has been proposed in [MDPS12] and used in [PCB+16] whose description follows below. This is also the model it has been used for the four heart valves in the mathematical model of the idiopathic pulmonary arterial hypertension of this thesis work.
The pressure drop across a valve for a fluid with density ρ is described by the Bernoulli equation [SSA+95]
∆p = Bq|q| + Ldq dt with B = ρ 2A2eff and L = ρ`eff Aeff , (2.21)
where Aeff and `eff are the effective area and the effective length of valve opening. In (2.21) the viscous loss are neglected because they are small [SSA+95,MDPS12]. The valve state variable ξ(t) commands the opening or the closing of the valves. depending on the pressure difference across the valve. ξ(t) has the following form:
( ˙ξ = (1 − ξ)Kvo∆p if ∆p ≥ 0, ˙
ξ = ξKvc∆p otherwise,
(2.22)
where Kvo and Kvc are the opening and the closing rate of the valve. ξ(t) is always
bewteen 0 and 1 (when ξ(t) = 0, the valve is fully close and when ξ(t) = 1, the valve is fully open). ξ(t) influences the area in the following way
Aeff= (Amaxeff − Amineff ) ξ(t) + Amineff . (2.23)
In the case of a patient with valve regurgitation due to prolapse, additional varia-tion case would have been added to (2.22), which would have allowed ξ(t) to became negative (until −1). Other considerations there would have been for Aeff too. For numerical reasons, it has been chosen (2.23) as expression of Aeff, that is the
ex-pression of a regurgitant valve because of an incomplete leaflet closure, instead of Aeff= Amaxeff ξ(t), that is the expression in case of no valve regurgitation. This choice
zero.
Notation circuits (Figure 2.6-2.7)
Id Element Id Element Id Element
LV Left Ventricle LS Left Subclavian Art. LB Lower Body
RV Right Ventricle DA Descending Aorta UB Upper Body
LA Left Atrium PA Pulmonary Artery VS Veins
RA Right Atrium AO Aorta MVV Mitral Valve
LPA Left Pulm. Art. TCV Tricuspid Valve AOV Aortic Valve
RPA Right Pulm. Art. POV Pulmonary Valve RL Right Lung
IN Innominate Art. SVC Sup. Vena Cava LL Left Lung
LC Left Carotid Art. IVC Inf. Vena Cava
Table 2.3: Legend of the elements in the circuits of Figure 2.6-2.7.
2.3
Lumped parameter model for IPAH
Lumped parameter descriptions have been reported in Section 2.2.1, 2.2.2 and
2.2.3 for the various compartments of the human cardiovascular system. A way to derive a model of this system is to connect these compartments through appropriate matching conditions. Such conditions derive from the mass conservation law and the momentum balance at the interfaces. In the lumped parameter models, the state variables are the flow rate Q and the pressure P so that interface conditions are about the continuity of these variables at the interface. Since an analogy with the electric network has been introduced, these relations coincide to the application of the Kirchoff laws, such as the conservation of the flow rate at the nodes (P Qin=
P Qout) and the conservation of the pressure at the nets (P Pi = 0). This is why
lumped parameter models can be referred to Kirchoff models.
We proceeded to the derivation of our model. The following lumped parameter elements are included:
• Single fiber model for the left atrium (LA), left ventricle (LV), right atrium (RA) and right ventricle (RV), Section 2.2.2;
• Valve model proposed in [MDPS12, PCB+16] for the mitral (MVV), aortic (AOV), tricuspid (TCV) and pulmonary (POV) valves (diode, variable resis-tance B and inducresis-tance L);
• Single capacitance C for the aorta (AO) and the pulmonary artery (PA); • RCR model for the branches of the aorta (innominate IN, left carotid LC, left
subclavian LS, descending aorta DA) and the pulmonary artery (first dira-mation of left pulmonary artery LPA1, second diradira-mation of left pulmonary artery LPA2, right pulmonary artery RPA), Section2.2.1;
• RC model for the upper (UB) and lower (LB) body and for the left (LL) and right (RL) lung.
As first attempt the shunt has been modelled by a simple resistance Rshunt.
In the following chapter it will be possible to verify that this representation does not correspond to the real behaviour of the shunt (Section 4.5). In Table 2.3, the nomenclature for the main cardiovascular compartments represented in Figure 2.6
and Figure2.7 are reported.
2.3.1 Numerical simulation of the lumped parameter model
Under a mathematical point of view, lumped parameter models are represented as a system of differential-algebraic equations (DAE) of the form
dy
dt = b(y, z, t) t ∈ (0, T ], G(y, z) = 0,
y|t=t0 = y0,
(2.24)
where y is the state variables vector, z is the vector of the other variables of the network, y0 is the vector of the initial conditions and G represents the algebraic equations derived from the Kirchoff laws.
It is possible to solve the problem (2.24) numerically by means of suitable meth-ods, such as the Runge-Kutta methods. These schemes are one step multistage schemes, and in particular, we have considered explicit RK4 [QSS10].
Usually system (2.24) is rearranged in order to achieve a Cauchy problem of the form
dw
dt = a(w, t) t ∈ (0, T ], w(t0) = w0.
(2.25)
This is performed by differentiating the algebraic equations with respect to time dG(y, z) dt = Jy dy dt + Jz dz dt = 0,
where Jz and Jy are the Jacobian matrices with respect to y and z. Jz is assumed non singular and therefore the DAE system is said to be of index 1. An initial vector z0 is assumed available. Now in (2.25) there are the vectors w = [y, z]T
and a = [b, −Jz−1Jyb]T, and a fourth-order Runge-Kutta method can be employed
R ub vs qsv c Cub R in d R in p R ls d R ls p R lc d R lc p R da d R da p Cao Lao v Bao v L V Bm vv Lm vv LA R lb vs qiv c Clb Cpa Lp o v Bp o v R V Btcv Ltcv RA R lpa1 d R lpa1 p R lpa2 d R lpa2 p R rpa d R rpa p R ll vs qll Cll R rl vs qrl Crl Figure 2.6: Closed-lo op circuit of the whole b o dy (pre-op erativ e case).
R ub vs q sv c C ub R in d R in p R ls d R ls p R lc d R lc p R da d R da p C ao L ao v B ao v L V B m vv L m vv LA R lb vs q iv c C lb C pa L p o v B p o v R V B tcv L tcv RA R lpa1 d R lpa1 p R lpa2 d R lpa2 p R rpa d R rpa p R ll vs q ll C ll R rl vs q rl C rl R sh un t Figure 2.7: Closed-lo op circuit of the whole b o dy (p ost-op erativ e case).
k1= a(wn, tn), k2= a(wn+ 1 2k1∆t, t n+ 1 2∆t), k3= a(wn+ 1 2k2∆t, t n+ 1 2∆t), k2= a(wn+ k3∆t, tn+ ∆t), wn+1= wn+1 6∆t(k1+ 2k2+ 2k3+ k4).
We wrote the code for the solution of the lumped parameter model developed for the idiopathic pulmonary arterial hypertension using Fortran 90. We performed the Runga-Kutta method differently from how it has been described above. In par-ticular, we did not reduce the sistem of DAE (2.24) as (2.25). We built a subroutine which contains all the equations (both differential and algebraic). We allocated a vector of the state variables k and a vector of the state variable derivatives f in such subroutine. Recalling this subroutine at each stage of the numerical method, we advanced in time as
yn, k1= b(yn, zn, tn), where zns.t. G(yn, zn) = 0
yn+14 = yn+∆t 2 k1, k2= b(y n+1 4, zn+ 1 4, tn+∆t 4 ), where z n+1 4 s.t. G(yn+ 1 4, zn+ 1 4) = 0 yn+12 = yn+∆t 2 k2, k3= b(y n+1 2, zn+ 1 2tn+∆t 2 ), where z n+1 2 s.t. G(yn+ 1 2, zn+ 1 2) = 0 yn+34 = yn+ ∆tk3, k4= b(yn+ 3 4, zn+ 3 4, tn+3∆t 4 ), where z n+3 4 s.t. G(yn+ 3 4, zn+ 3 4) = 0 yn+1= yn+∆t 6 (k1+ 2k2+ 2k3+ k4),
with this sequence allocated in a for loop advancing with the time steps. At the end of every time step the new computed yn+1 becomes yn+1= yn.
Inside the subroutine for the numerical solution of the network, we needed a solver for non-linear equations because of the models adopted for the heart and its valves. This problem appears when the loop is closed. In fact, entering in the left atrium (LA), a non-linear relationship among quantities such as atrium pressure and blood flows coming from the left and the right pulmonary veins (LPV/RPV) rises. For this reason a 3 × 3 system of non-linear equations needed to be solved (also at the entrance of the right atrium). To this aim, we used an external library. This library is called MINPACK. MINPACK is a Fortran 90 library which solves systems of nonlinear equations or carries out the least squares minimization of the residual of a set of linear or nonlinear equations [MSHG84,MGH80]. Hybrid meth-ods are performed to solve the non-linear system. In particular, Powell’s method is used to seek a solution (Powell’s method is an algorithm for finding a local minimum of a function).
We just needed to supply a subroutine for the evaluation of the nonlinear function (the Jacobian is approximated directly by means of a forward-difference
Multiscale simulation: 3D-0D coupling
In this chapter, we give an introduction on multiscale simulations, their practical issues and their advantages in terms of cardiovascular dynamics accuracy. The equations which compose a mathematical solver for 3D Navier-Stokes equation are presented. We illustrate some problems that arise when a coupled simulation is attempted. Especially, in coupling with a lumped parameter model, defective boundaries must be treated properly. Backflow phenomenon, which is often happening in physiological flows in some vessels, may cause numerical instability and a suitable solution should be found. Finally, a time marching scheme for both 3D and 0D domain is presented following the software framework used in this thesis.
Contents
3.1 Why a geometrical multiscale simulation . . . 37
3.2 3D solver methodology . . . 38
3.2.1 Navier-Stokes equations . . . 39
3.2.2 Weak form - finite element discretization. . . 39
3.2.3 Backflow stabilization . . . 41
3.3 3D-0D coupling. . . 43
3.3.1 3D defective boundary problems . . . 43
3.1
Why a geometrical multiscale simulation
In the last years, mathematical simulations have been largely used for hemo-dynamics evaluation, clinical research and risks estimation. Such simulations are useful in the patient treatment planning, pre-surgery analysis and design optimiza-tion. Numerical modelling of the cardiovascular system has been adopted more often not only because of the general improvements of computer hardware but also because more refined and complete mathematical models have been developed.
Despite this, the treatment of the boundary conditions remains one of the most debated questions, in particular for two main reasons.
• Lack of available data, it may happen that a simulation is launched over a chopped vessel (for a computational reason), so that a boundary condition is needed and very often the mathematical boundary condition does not repre-sent physically the real one.
• Local and systemic dynamics reciprocal influence, when we use a simulation to evaluate a disease or a surgical operation in a specific part of the cardiovascular system, this usually has a global impact on the rest of the closed circulation and in the setting of the simulation this mutual influence has to be taken into account.
For the reasons mentioned above, we considered multiscale approach in this work. It is important also to specify how the word multiscale is used in our context. In this context multiscale is combined with geometrical because we are trying to couple differential problems on domains with different dimensions. In other many fields of mathematical and numerical modelling multiscale refers to the presence of two or more time/spatial scales.
Geometrical multiscale is an approach which allows modelling the cardiovascular system pointing out to specific regions of interest [FNQV99]. Indeed, very often we are interested only in blood flow behaviour in a particular part of the circulation. To reduce computational costs, it is therefore possible to have just a rough description of the whole system while shifting the focus on the area we want to study (bottom-up approach). Otherwise this multiscale can be used for computing correct boundary conditions at the boundaries of the district of interest (top-down approach).
For our aims, a coupling between 3D Navier-Stokes equations and 0D lumped parameter is pursued. The coupling consists, for instance, in the lumped parameter model providing a flow rate at the inlet of the area of interest simulated by means of 3D equations. Navier-Stokes equations require the whole velocity field at the boundary, while the 0D model provides just a single value for the mass flow rate. A practical approach is to choose a priori the velocity profile (flat, parabolic, etc.) to be fitted with the available flow rate. Similar consideration for the pressure in the case of a Neumann boundary conditions where the traction or the pressure has to be prescribed.
![Table 1.2: Properties of the human pulmonary circulation. Adapted from [SHH + 73].](https://thumb-eu.123doks.com/thumbv2/123dokorg/7495608.104115/21.892.276.641.702.879/table-properties-human-pulmonary-circulation-adapted-shh.webp)
