1. Hemorheology and Hemodynamics

Perhaps the most emblematic case is the miracle of the “liquefaction” of Saint Januarius’ blood. It is less known that the blood of many other saints periodically exhibits the same phenomenon. Saint Januarius (San Gennaro), the patron of Naples, died as a martyr around 305 ad, beheaded at the Solfatara Crater. According to the legend, his blood was collected by a pious woman (Eusebia) and preserved till our days (after many vicissitudes) in a sealed transparent container. A thousand of years later the saint’s relics ended in Naples and in 1389 chronicles started reporting the miracle. In our days the blood is exposed three times a year and upon mild agitation (most of the times) the reddish dust turns into liquid. Many explanations have been attempted for this intriguing phenomenon (including some thixotropic mixture prepared in the middle ages [93]). A professor of molecular biology of the University Federico II of Naples (prof. G. Geraci) has performed experiments with a sample of old blood saved in a sealed vial and with his own blood, suitably aged, reproducing the same “miracle”, thus concluding that “liquefaction” may well be just a blood property [94].

For a history of ancient Egyptian medicine see e.g. [59].

Though all kinds of body fluids were thought to flow through the heart. For instance semen was believed to be provided to testicles by two dedicated vessels (Ebers Papyrus).

The translation [21] with splendid reproductions of the Smith Papyrus is available on line.

The English translation [22] (parallel to [21]) is available on line.

Dates largely uncertain.

One of the oldest surviving text is Charaka Samhita (Fig. 1.5), written in Sanskrit (fifth to third century bc).

Two classes of medical approaches can be adopted to recuperate the broken equilibrium: Samana (for light alterations), and Shodana aimed at expelling the corrupted doshas. Panchakarma belongs to the second class.

Physical manifestations of yin and yang are natural opposites like dark and light, male and female, life and death, moist and dry, sun and moon, etc.

One more proof of how humans have tried to interpret life and the physical world on the basic of recurrent principles. Striking similarities can be found also among the many myths explaining creation.

A legendary, semi-divine figure between myth and history, he was credited with the invention of almost anything which started Chinese civilization.

In this paper it is reported how extremely heavy bloodletting accompanied the last days of Charles II and of George Washington.

Erasistratus (304–250 bc), a renowned Greek physician active in Syria, came to the conclusion that heart is a pump. Though he made some remarkable progress in understanding the role of brain and nerves, he believed (as many others) that arteries carried the “spiritual substance” (pneuma). In other words, arteries were full of air and if by chance they were contaminated by blood it would have caused illness. It is amazing that at that time there was a controversy about the path of drunk liquids: somebody claimed that they went to the trachea (a name probably due to Erisistratus himself) eventually reaching the lungs. He stated instead that anything which is ingested goes through the esophagus to the stomach. He opposed the practice of bloodletting. Herophilus (335–280 bc) established that the brain, and not the heart, was in control of motion.

The practice was legal in Alexandria.

He was particularly adversed by his former teacher, Franciscus de la Boe (1478–1555), better known as Jacobus Sylvius or Jacques Duboi, who was an irreducible worshiper of Galen’s works, to the point that if he had to admit that something in the human body was different from what Galen had described, then the mistake was in the body, in the sense that it had modified since Galen’s time. Sylvius was nevertheless the author of important discoveries in brain anatomy.

The list of scientists condemned by the Catholic Church for heresy is impressive. The case of Galileo Galilei is emblematic (1632). Despite their reputation, non-catholic Christian churches were not more tolerant about heresy, which most of the times consisted just in interpretations of the Scriptures (or of real world) different from the ones officially adopted by this or that Confession. An emblematic case was the one of the Spanish born Michael Servetus (1509–1553), Vesalius contemporary, and also a physiologist, who dared oppose Galen’s authority, providing his own description of pulmonary circulation [218]. Serveto was burnt at the stake (alive and with sulfur on his head) in Geneva, victim of the fanatic hate of John Calvin because of his refusal of the concept of Trinity.

We must recall that about three centuries earlier Ala-al-Din Abu al-Hasan Ali Ibn Abi al-Hazm al-Qarshi al-Dimashqi (known as Ibn Al-Nafis, 1213–1288), among his many other discoveries, described pulmonary circulation [238].

This is a good place to mention a remarkable scientist, the German physiologist Adolf Eugen Fick (1829–1901), mainly known for two things: (1) the Fick’s principle for the determination of the cardiac output on the basis of the ratio between oxygen consumption and the arterio-venous oxygen difference; (2) the formulation of Fick’s law for diffusion, paralleling Fourier’s law for heat conduction.

Malpighi first described RBC’s as fat corpuscles (1663). Malpighi was also the discoverer of capillaries (1661) and of the filtrating units in kidneys, bearing his name. It is worth mentioning here an extraordinary character, Athanasius Kircher (1602–1680), a German Jesuit scholar, who wrote a great number of books in different areas. Kircher is mostly remembered for having built a machine for automatic music composition (the Arca Musurgia, 1650). An expert microbiologist, he was rightly convinced that the plague was caused by a microorganism (today known as the Yersinia pestis bacterium) that he thought to have found in blood with the help of the microscope in 1658. Most probably he had instead observed RBCs, the same year as Swammerdam.

It seems however that even Anthony Leeuwenhoek had observed them as early as 1678. Many others described cells of that kind in pus and other physiological fluids (see [220]). The French anatomist Joseph Lieutaud (1703–1780) called them “globuli albicans” (1749) [135]. The lymphatic system was then described by William Hewson (1739–1774) [109].

He formerly identified groups A, B, C, but “C” was later changed to “0” (zero). He was also the discoverer of the polio virus. The story of the ABO groups is actually more complicated. It is today recognized that the Czech serologist Jan Janský (1873–1921) had provided the complete 4-group classification (including group AB) before Landsteiner, who had nevertheless worked independently.

From the name of the monkey species (Rhesus) used in their tests.

The isovolumic contraction is a short transient phase preceding actual ventricles contraction, just before the sudden aortic pressure raise, during which the mitral valve closes (first heart sound). The isovolumetric relaxation is an equally short phase, accompanying the rapid aortic pressure drop, during which the mitral valve closes (second heart sound).

The values presented in this table are very crude estimates. It is quite difficult to find a complete and coherent table of this type when we compare all data provided by the different authors (see e.g. [166, 226, 227]). We also quote the paper [240]. Systolic values are of course much larger (e.g. peak velocity in aorta 65 cm/s, in venae cavae 38 cm/s, or more, though there are differences between superior and inferior v.c.). A simple calculation of the peak velocity in aorta can be done by dividing the blood volume ejected by the left ventricle (heart stroke = 50 ml) by the ejection time (300 ms) and by the aortic valve area (2.5 cm²), which gives 67 cm/s. Papers referring specifically to these main vessels are [153, 240].

The SI derived unit for pressure is the Pascal. It is equivalent to one Newton per square meter. The unit is named after Blaise Pascal (1623–1662), the eminent French mathematician, physicist and philosopher. 1 mmHg = 133. 32 Pa. The mmHg is also named Torr after Evangelista Torricelli (1608–1647), Italian physicist and mathematician.

A particular role in flow through veins is played by valves, representing a help in vessels with moderate pressure gradients. Some veins and venules are equipped with valves, whose importance will be discussed in several instances [15]. A study of the dynamics of vein valves has been performed in [141].

Knowing the velocity and pressure fields it is possible to obtain stresses, in particular the wall shear stress (WSS) which is the force per unit area exerted by the fluid tangentially to the vessel wall. WSS may cause alterations in the endothelium and has a great influence in many inflammatory diseases, including atherosclerosis, the development of aneurysms and clotting.

Studies on 1D models of blood flow were first introduced by the Swiss mathematician Leonhard Euler (1707–1783) in his seminal work [64].

System (1.4) was proposed for the first time by Claude-Louis Navier in 1822 and later by George Stokes in 1845.

A hyperelastic material is characterized by a nonlinear stress-strain response [18, 45].

Among the papers in which a shell model has been adopted we quote [32], containing a large introduction to the subject.

As for the fluid, it is not clear from physical arguments which should be the appropriate boundary conditions to impose on the artificial sections Γ _Σ, in ⁰ and Γ _Σ, out ⁰. It is important to refer that the geometrical multiscale approach has provided satisfactory answers [80]. Moreover, since the structure equations will be coupled with the fluid equations, the initial velocity for the fluid and the displacement should be compatible, i.e. \(\frac{\partial \eta _{0}} {\partial t} = \mathbf{u_{0}}\) on Γ _w ⁰. 

The ALE approach will be considered only at the discrete level. In fact, due to time discretization, the domain changes from one time step to another and the ALE method through the introduction of the ALE time derivative that we will define here, allows to follow the evolution of quantities associated to the mesh nodes.

The problem with slip boundary conditions would require the specification of the slip velocity in terms of the normal stress at the vessel wall. As far as we know this is still a debated question.

The term multiscale is often used whenever two or more time and/or scales are present, like in turbulence modeling or when molecular and cellular-level biological phenomena are combined with macro-scale fluid and tissue mechanics. The term geometrical has been added to identify the present multiscale strategy, avoiding ambiguities.

The theoretical basis of this approach developed by Naghdi and co-authors (see e.g. [97, 99]) is similar to that used for rods in solid mechanics [98].

We will see in various instances that flows in vessels whose length is much larger, that the radius satisfy these conditions up to corrections which are of the same order as the radius/length ratio.

More complex algebraic relations can be considered (e.g. [108]). Here we prefer to use definition (1.16) since it separates the effects of the wall mechanical properties (Young modulus) from the geometrical ones (A ₀).

In this Eq. (1.20) the term \(\frac{\partial A_{0}} {\partial z}\) may account for a variation of the area at the rest state, as in the case of tappering or stenosis, and the term \(\frac{\partial \beta } {\partial z}\) is related to the longitudinal variation of the mechanical properties of the vessel.

Under physiological hemodynamic conditions the flow regime is assumed to be subcritical and the typical values for c ₁ and Q∕A are such that c ₁ ≫ Q∕A. Indeed, characteristic values for c ₁ are of the order of 10³ m/s [236] whereas | Q∕A | is of the order of 10 [165].

This model was introduced by the German physiologist Otto Frank (1865–1944) in the late 1800s who described the heart and the systemic arterial system as a closed hydraulic circuit. He called it Windkessel in analogy with the device made by a water pump connected to a chamber, filled with water except for a pocket of air. As it is pumped, the water compresses the air, which in turn pushes the water out of the chamber.

It means the property of a system of being in a stable equilibrium as a result of interacting processes.

Proteins heavier that albumin, divided in four groups: α ₁, α ₂, β, and γ.

Hemoglobin was discovered in 1840 by Friedrich Ludwig Hünefeld (1799–1882), a German physician and chemist. Drying the blood of an earthworm between two glass slides he observed the formation of red crystalline structures. The role of hemoglobin, however, became clear much later. Felix Hoppe-Seyler (1825–1895), German physiologist and chemist, gave it its name (1864) and recognized that it combines with oxygen. Among the many scientists that gave fundamental contributions to the study of hemoglobin we mention the celebrated French physician Claude Bernard (1813–1878) and the Austrian born Nobel laureate molecular biologist Max Perutz (1914–2002), who established its structure consisting of four elements, each with an iron atom, able to bind with oxygen. CO ₂ and CO bind to other sites of the molecule. Hemoglobin is known to be synthesized by numerous cells other than erythrocytes for various purposes. See the review paper [207].

For a non-Newtonian fluid, independently of the specific rheological model, one can define an apparent viscosity as the quantity measured by a viscometer (which is somehow an average measure of the fluid resistance to flow) for shear rates in the expected natural range. The term relative viscosity is also used meaning the ratio of the suspension viscosity (apparent viscosity) to the viscosity of the suspending fluid (plasma). Commonly used viscosity units are: poise (P)-named after the French physician Jean Louis Marie Poiseuille (1799–1869), centipoise (cP), which is 0.01 P and pascal-second (Pa s), the SI unit of viscosity equivalent to newton-second per square metre (Ns/m²). One poise is exactly 0.1 Pa s.

Robin S. Fårhaeus (1888–1968) and Torsten Lindqvist (1906–2007) were Swedish hematologists who described the effect in 1931. An earlier paper by a German team (P. Martini, A. Pierach, E. Scheryer, 1930) on the same issue unfortunately went unnoticed (see [231]).

Blood viscoelasticity was mentioned for the first time in this paper.

There is a large variety of published definitions for thixotropy, in the fields of industrial or biological applications. The following definition can be found in [14]: “When a reduction in magnitude of rheological properties of a system, such as elastic modulus, yield stress, and viscosity, for example, occurs reversibly and isothermally with a distinct time dependence on application of shear strain, the system is described as thixotropic”. Fluids whose behavior is opposite to thixotropic fluid (i.e. thickening under stress) are called rheopectic.

On top of that, experimenting with blood out of the body can find many obstacles. The simple process of extracting blood may apply high stresses, altering the original rheological properties. Then partial coagulation, particularly in the absence of flow, can severely influence the value of yield stress.

For isochoric motions the first invariant I _D = tr D(u) is identically zero.

The presence of ϕ ₂ is necessary to match experimental results on “real” fluids. The dependence on the value of II _D is often neglected [7].

When not differently stated the data we report refer to room temperature.

Fenestrations in renal glomeruli are impervious to molecules whose molecular weight exceeds ∼ 12 kDa. The presence of albumin in urine is linked to pathological conditions (e.g. diabetes nephropathy).

The regulation of microvessels perfusion is an immensely complicated topic. See for instance the massive review [52] with its 34 pages of references. Section 6.2.1 is devoted to the role of vasomotion in flow autoregulation.

The non-solid complement of the nucleus in an eukaryote cell is the cytoplasm, which consist of a fluid, the cytosol, and of the organelles, performing various functions, and usually possessing a membrane. Among the organelles of eukaryotic cells the endoplasmic reticula, characterized by the presence of cisternae (in the form of sacs and tubes) have different structure in different organs, and in myocytes (muscle cells) we find the specialized version of sarcoplasmic reticulum, storing and releasing calcium ions.

A messenger molecule diffusing in the sarcoplasmic reticulum and triggering the opening of Ca ⁺⁺ channels in the cell membrane.

A protein playing a major role in cells motility and in particular in skeletal muscle contraction.

Formula (1.57) produces the simple form (1.59) of the dimensionless Navier-Stokes equations. The actual maximal velocity provided by the Hagen-Poiseuille formula is 1/4 of that.

There is some controversy about the discovery of vein valves. According to A. Caggiati (a renown scientist in the field) the first to describe them was the Catalan Ludovicus Vassaeus in his De Anatomen Corporis Humani tabulae quator (Venice, 1542). See [26‐28]. There were more or less contemporary or even earlier observations. In the old review paper [85] reference is made to unpublished findings on the subject by Giambattista Canano (1515–1579), professor of anatomy in Ferrara, that he reported to Vesalius in 1545. In the same year veins valves were described by the French anatomist Charles Estienne (1504–1564), who however claimed to have discovered them in 1538.

Notation is different from the case of arterioles, since it is preferable to have the maximum radius as a reference length.

Arteriosclerosis is the hardening, thickening or loss of elasticity of the artery walls [76].

Cytokines and chemokines are small proteins made by cells of the immune system, both important in regulating responses to infection or injury by using chemical signals [193].

Molecules able to produce oxidative damage to cells.