The Cardiovascular System: Mathematical Modeling .

conditions and of (novel) numerical strategies for a stable, reliable, and computationallyeffective solution of the global problem.Clinical data play a decisive role for CS models and, at the same time, they representa formidable challenge. Clinical radiological images (such as Computer Tomography andMagnetic Resonance Imaging) are necessary to construct the computational domains. Theprocedure of geometric reconstruction is difficult and, especially for the heart, requires advanced mathematical and numerical tools. Standard radiological images can sometimesbe useless: some cardiovascular components may have a size smaller than the spatial resolution featured by the imaging device (this is e.g. the case of the Purkinje network); inother cases the elevated brightness gap between fluid and vessel wall, makes the detectionof the latter very hard. Boundary data are also difficult to obtain. When the computational domain results from an artificial truncation, specific physical quantities (e.g. fluidvelocity or pressure) should be provided at those locations of the arterial tree corresponding to the artificial boundaries. However, this would require invasive measurements thatcannot be easily carried out. Finally, the huge inter- and intra-patient data variability anduncertainty represent further sources of concern toward model calibration and validation.In spite of all these difficulties, a wealth of models has already been successfully appliedto address both physiological and pathological instances. The aim is from one side abetter understanding of the physical and quantitative processes governing the CS, andon the other side the opening of new frontiers in therapeutic planning and the design ofimplantable devices (such as e.g. medical stents and cardiac defibrillators).The literature about the mathematical and numerical modeling of CS is huge (asthe reader would realize by browsing this paper’s references, a tiny subset of the overallexisting ones). In the forthcoming sections we will try to provide an outlook to the maincontributions in this field. Here, among the several books, monographes, and reviewpapers published so far, we mention [188, 540, 482] for the circulatory system and [448,521, 118, 468, 470] for the heart.This review paper consists of three main parts, i.e. i) modeling the arterial circulation(Sects. 2, 3 and 4), ii) modeling the heart function (Sects. 5, 6 and 7), and iii) solvinginverse problems and including uncertainty (Sects. 8, 9, 10 and 11). Both parts 1 and 2 arecomposed by an introductory section on physiology (Sects. 2 and 5), a section describingthe available data and their use (Sects. 3 and 6), and a final section on the mathematicaland numerical modeling (Sects. 4 and 7). Regarding the third part, in an introductorysection we underline the need of going beyond a single (forward) simulation in someapplications (Sect. 8). This represents the common denominator of three topics recentlyapplied to cardiovascular mathematics: control and optimization (Sect. 9), parameterestimation (Sect. 10), and uncertainty quantification (Sect. 11).When appropriate (in particular in Sects. 4, 7, 9, 10 and 11), we report some numericalresults to highlight the effectiveness of the numerical strategies here presented. Unlessotherwise specified, all our numerical results have been obtained using the Finite Elementlibrary LifeV, see www.lifev.org for more details.5

Part ITHE ARTERIAL CIRCULATION2Basic facts on quantitative physiologyThe cardiovascular system is a close circuit that carries oxygenated blood to all the tissuesand organs of the body. Functionally, it can be regarded as made by three compartments:the heart, the systemic and pulmonary circulations, and the microvasculature. In thissection we will recall the most important features about the physiology of the systemiccirculation characterizing the mathematical models that will be introduced later on. Wewill also highlight the main peculiarities of the pulmonary circulation. Heart physiologywill be addressed in Section 5.The systemic circulation is composed by the arteries, that carry the oxygenated bloodejected by the left heart to the living tissues, and the veins that allow the non-oxygenatedblood to returning to the right heart. The exchange of oxygen between blood and the bodytissues occurs in the microvasculature, which in fact separates the systemic arterial treefrom the venous systems. In the pulmonary circulation, non-oxygenated blood ejectedby the right heart flows in the pulmonary arteries towards the lungs where it becomesoxygenated and goes back to the left heart through the pulmonary veins.Blood is composed by plasma (about 55% of its total volume) which consists of water(about 92% of plasma volume), proteins and ions. The remaining part of blood correspondsto the blood cells, whose 97% of volume is occupied by erythrocytes (red blood cells) thatcarry the oxygen in oxygenated blood. The other cells are leukocytes (white blood cells)and platelets. The diameter of blood cells is approximately 10 3 cm, whereas that of thesmallest arteries/veins is about 10 1 cm. This is the reason why blood in the systemicand pulmonary circulations is often considered as Newtonian, i.e. characterized by alinear relationship between internal forces and velocity gradients [443, 188] However, in thesmallest arteries, such as coronaries (the arteries perfusing the heart and the correspondingveins, see Figure 1, right), or in presence of a vessel narrowing (stenosis), a non-Newtonianblood rheology is more appropriately assumed, see, e.g., [93] and references therein.Thanks to the heart contraction, the blood flow is pulsatile and blood is pumped intothe two circulations by means of discrete pulses with a pressure usually varying during anheartbeat in the ranges 70 130 mmHg and 20 30 mmHg for the systemic and pulmonarynetworks, respectively (1 mmHg 133.3 P a 1333 g/(cm s2 )).In the systemic circulation, blood first enters the aorta (the largest artery with diameterequal to about 2.5 cm in adults, see Figure 1, left) and then flows through a network ofhundreds of branching arteries of decreasing size, reaching all the regions of the body.Dimensions and numbers of veins are comparable with those of arteries. The waveformof the flow rate as a function of time is characterized by different peak values whenmoving downstream towards the smallest arteries. In particular, the flow rate peak valueis about 200 cm3 /s in aorta, 80 cm3 /s in the abdominal aorta, 15 cm3 /s in the carotids(the arteries supplying blood to the brain, see Figure 1, middle), and 1 cm3 /s in coronaries6

(corresponding to a maximum blood velocity of about 150 cm/s in aorta, 100 cm/s in theabdominal aorta, 80 cm/s in the carotids, and 40 cm/s in coronaries). Also the shape ofthe waveforms changes while moving downstream, see Figure 2, left. In particular, in theascending aorta, after the systolic peak the flow rate decelerates assuming null or evennegative values, whereas in the abdominal aorta and in carotids is more spread out andalways positive. In any case, we can distinguish the systolic phase, i.e. the interval ofacceleration and deceleration of blood flow, and the diastolic phase, i.e. the interval ofalmost constant or negative flow 1 . A different situation occurs in coronaries, where thepeak flow rate is reached during diastole, see Figure 2, right. Coronaries are not directlyfed by the heart; indeed, blood in the proximal part of the aorta (the sinuses of Valsalvafrom which coronaries originate) during diastole is allowed to enter the coronaries thanksto the elastic response of the aorta (see below for more details).Figure 1: Visualization of the aorta (left), carotids (middle), and coronaries (right)In the pulmonary circulation blood first enters the pulmonary artery (diameter equalto about 3.0 cm in adults) and then flows into another network of branching arteries ofdecreasing size reaching the lungs. The waveforms and peak intensities are similar to thoseof systemic arteries.The different characteristics of blood flow in the arteries of the systemic circulationρ DUresult in different values of the Reynolds number Re f µ (ρf being the blood density,D and U characteristic vessel dimension and blood velocity, respectively, and µ the fluidviscosity), a dimensionless quantity which quantifies the importance of the inertial termsover the viscous ones. In particular, Re 4000 in the aorta and Re 400 in coronaries,1The previous definition of systole and diastole is formulated from the point of view of the arteries. Analmost equivalent definition could be given from the point of view of the heart, see Section 5.7

Figure 2: Typical flow rate waveforms in ascending aorta, abdominal aorta and carotids(left), and in coronaries (right)with intermediate values when moving downstream the aorta. Thus, blood covers a rangeof Reynolds numbers where both the inertial and the viscous components of the flow arerelevant. Although in the aorta Re is higher than the critical value of 2000 above which theflow would not be laminar any longer in a straight pipe, the pulsatile nature of blood flowdoes not allow fully transition to turbulence to develop. It is debated whether in aorta atleast transitional-to-turbulence effects may occur. In this respect, some authors speculatethat the helicoidal velocity pattern in aorta induced by the torsion of the heart contractioninhibits any transition to turbulence, thus supporting the thesis that in healthy conditionsturbulence is never observed in the cardiovascular system [383]. This is not necessary thecase for some pathological

Mathematical and numerical modeling of the cardiovascular system is a research topic that has attracted a remarkable interest from the mathematical community be-cause of the intrinsic mathematical difficulty and due to the increasing impact of cardiovascular diseases worldwide. In this review article, we will address the two prin-