Keywords
history - heart valve surgery - cardiopulmonary bypass
Introduction
From the 17th century onwards, the Bernoullis were one of the most distinguished families
in the history of science.[1] Their scientific accomplishments, mainly in mathematics and physics at the University
Basel, Switzerland, are outstanding. In particular those of Daniel Bernoulli, who
applied mathematical physics to medicine to further his understanding and to explain
a variety of physiological mechanisms, had and have a significant impact in daily
clinical practice. This is true even in today's high-end medicine field of cardiac
surgery.
The Bernoulli Family
The scientific record of the Bernoulli family in Basel starts with Jakob Bernoulli (1654–1705).[2]
[3] He was appointed professor of mathematics in Basel in 1687 and was one of the first
mathematicians to work on the calculus of variations and mathematical probability.
Johann Bernoulli (1667–1748), the younger brother of Jakob, was required to study medicine, but he
was more interested in mathematics and physics.[2]
[3] He became a brilliant scientist and was one of the most famous mathematicians of
his time, after Newton and Leibniz. His three sons, Nicolaus (1695–1726), Daniel (1700–1782),
and Johann II (1710–1790), were also scientists.[2]
[3] Of the three, Daniel Bernoulli was the most famous with achievements not only in
mathematics and physics but also in a variety of natural sciences including medicine.
Daniel Bernoulli, Scientist
Daniel Bernoulli, Scientist
First Scientific Works
At a young age, Daniel Bernoulli already showed interest in mathematics and physics.[1]
[2]
[3] However, his father tried to force him into a career as a merchant. Daniel strongly
opposed this, but then agreed to study medicine at the University Basel, then Heidelberg,
Germany, and Strasbourg, France. Meanwhile, Johann continued teaching Daniel mathematics
and physics. Among the many topics they discussed, one was to have a substantial influence
on Daniel's future discoveries. It was the “Law of Vis Viva Conservation,” known today
as the “Law of Conservation of Energy.”[1] What Daniel learned about the conservation of energy he applied in medicine to solve
physiological questions. Thus, in his doctoral dissertation, he explained part of
the mechanics of breathing with the help of geometric constructions and appropriate
calculations. In 1721, aged 21, he published his results in a book entitled “De Respiratione”[1]
[4]:
… as the shape formed by the ribs corresponds to an ellipse and the ribs, when stacked
one above the other, form an elliptical cylinder, one can use the basic elements of
geometry to demonstrate that the cylinder formed when the ribs have risen has a greater
volume than that which is formed by the ribs in their lowered position. The diaphragm,
after expiration, is convex to the lobes of the lungs and has the function of making
the chest space smaller or larger, sinking with inspiration and rising with expiration
aided by the diaphragmatic muscles…
Daniel Bernoulli also showed a strong interest in fluid dynamics, and in this context
he was especially attracted to the scientific work of the English physician William
Harvey.[1] In the 17th century, in his book “De Motu Cordis” Harvey was the first to describe
the human blood circulation and the heart as a pump forcing the blood to flow like
a fluid within the body.[5]
Daniel wanted to follow an academic career after his dissertation, like his father,
and applied for two chairs at the University Basel. Those were then decided by drawing
lots, but Daniel had no luck. As a consequence, he left Basel for Venice to study
practical medicine with Michelotti, one of the most respected physicians of the time,
who also studied the blood flow within the body.[1] Additionally, Daniel worked on mathematics and physics and published his results
in the book “Exercitationes Quaedam Mathematicae”[6] in 1724. This was a work of four parts, of which the second concerns the flow of
water from a hole into a container including a discussion of Newton's theories and
corresponding to his medical work on the flow of blood and blood pressure.[1] This publication launched Daniel's scientific career and made him famous across
Europe, more in mathematics than in medicine.
Master Work: Hydrodynamics
In 1725, Daniel Bernoulli, together with his older brother Nicolaus, was offered a
chair of mathematics at the newly founded “Imperial Academy” at St. Petersburg, Russia.[1]
[2]
[3] Less than a year after they arrived there, Nicolaus died, aged 31, as a result of
an intestinal infection. Daniel thought of returning to Basel, but his father arranged
for another promising student from Basel, Leonard Euler, to go to St. Petersburg to
work with Daniel. Euler arrived in 1727 and the subsequent period in St. Petersburg
was a productive time for both.
Daniel continued his studies on the flow of fluids in tubes producing, for the first
time, the correct analysis of water flowing from a hole into a container.[1] In addition, he revealed the relationship between the velocity at which blood flows
and its pressure. Daniel experimented by puncturing the wall of a pipe with a small
open-ended straw and found that the height to which the fluid rose was related to
the fluid pressure at this point. The explanation for this effect was based on the
principle of conservation of energy, which he had studied with his father long before.
Bernoulli, with Euler's help, proved mathematically what he had observed physically.
Soon physicians all over Europe were measuring patients' blood pressure by sticking
point-ended glass tubes directly into their arteries.[2]
[3] Taking his discoveries further, Bernoulli realized that just as a moving body exchanges
its kinetic energy for potential energy when it gains height, a moving fluid exchanges
its kinetic energy for pressure. This was the background of Bernoulli's law, now known
as Bernoulli fluid pressure equation[7]:
p + ½ ρv
2 + ρgh = constant
where p is the pressure, ρ(rho) is the density of the fluid, v is its velocity, g is the acceleration due to gravity, and h is the height. It follows from this law that the pressure falls if the velocity of
the fluid increases.
Daniel completed his works on the flow of fluids in the book “Hydrodynamica” while
he was still in St. Petersburg but published the book later in 1738 ([Fig. 1]).[8] This was his most important and famous publication, and even the term “hydrodynamic”
was first used by him.
Fig. 1 Title page of the book “Hydrodynamica.”
Calculation of the Heart's Work
Daniel Bernoulli left St. Petersburg in 1733 and returned to Basel where he took the
chair in anatomy and botany at the University Basel.[1]
[2] His lectures focused particularly on the anatomy and the mechanical power of the
heart. He used for the first time the term “work” done by the heart and presented
a method of calculating it. He defined the work as “weight times lifting height” and
used for his calculation a stroke volume of 62 g (2 ounces), a value based on the
work of Harvey, and blood pressure.[1] While Daniel assessed the stroke volume correctly, the blood pressure he used was
much too low leading to an overestimate of the work done by the heart. In a prize-winning
paper presented to the Paris Académie des Sciences in 1753, he stated[9]:
… only the heart's work can be determined with reasonable accuracy; for it is known
that it beats ∼115200 times in a day and that it pushes approximately two ounces of
blood with each systole, and observations and experiments seem to prove that the blood
is ejected from the heart with a velocity that would allow it to reach a height of
approximately eight feet; this is the performance of the left ventricle, and that
of the right ventricle will make about a quarter of this. Thus one can estimate the
heart's daily work equal to that of elevating 144000 pounds to the height of one foot
[around 24000 m-kg]…
Daniel's concept of calculating the work of the heart was correct, but it took over
100 years until physiology understood and accepted his analysis leading to the values
accepted today (19,200–20,000 m kg).[1]
Bernoulli in the Operating Room
Bernoulli in the Operating Room
The Centrifugal Pump
Coronary artery bypass surgery (CABG) is the most common procedure in cardiac surgery.
At present, CABG with the use of cardiopulmonary bypass through a full sternotomy
is the standard surgical technique in most centers. While off-pump coronary artery
bypass grafting has become a promising alternative to conventional bypass surgery,
some patients may still require extracorporeal support. Concepts of extracorporeal
circulation using closed circuits with low-priming volume and less inflammatory response
were developed to meet the requirements of a minimized extracorporeal circuit (MECC).[10] The pump to maintain the blood flow and pressure within the MECC is a centrifugal
pump ([Fig. 2]), and the basic principle of the pump is based on the Bernoulli law.[11]
Fig. 2 Principle of a centrifugal pump; arrows demonstrate the fluid flow driven by the
impeller.
A centrifugal pump is essentially a velocity machine.[11] It works by energy transfer of a pivoting impeller to a fluid within an external
housing. The fluid reaches its maximum velocity as it reaches the pump impeller's
outer diameter. At this point, all of the increased velocity energy available from
the rotating impeller has been imparted to the fluid. When the fluid passes from the
relatively confined smaller space inside the pump with a relatively high velocity
to an outer region of increased area, its velocity decreases. As the fluid exits the
pump impeller into the lower velocity zone, there is little or no change in its static
elevation difference. Therefore, based on the Bernoulli law, and to maintain conservation
of energy with the reduced velocity head on exiting the pump, and the potential static
head relatively unchanged, then the only other energy component is the pressure component.[11] At this point in the pumping system, the velocity head energy is transformed into
pressure head energy, which maintains the patient's blood pressure during CABG.
Transit Time Flow Measurement of the Bypass Graft
The quality of surgical anastomoses in CABG is crucial to avoid early and late graft
failure. For optimal operative results, good and reliable methods for assessment of
the technical quality of the created bypasses are needed.[12] At present, transit time flow measurement (TTFM) is the most common intraoperative
method for assessment of the function of the graft. TTFM is convenient and assessment
of the absolute graft flow, related to graft type, vessel size, degree of stenosis,
quality of anastomosis, and outflow area, can predict technically inadequate grafts,
mandating graft revision. The method used to measure the flow within the grafts is
based in part on the work of Bernoulli as the blood flow within a “tube” is assessed.
The TTFM probe has a defined size and is laid around the graft ([Fig. 3]), the blood flow velocity is measured using the Doppler principle, and, as a result,
the volume flow through the graft can be calculated throughout the cardiac cycle.[12]
Fig. 3 Transit time flow measurement probe laid around the coronary bypass graft.
Heart Valve Surgery
Aortic valve stenosis is the most common valve disease, and aortic valve replacement
is the second most common procedure in cardiac surgery. Patients with severe aortic
stenosis should be promptly referred for surgical valve replacement, as survival is
poor unless outflow obstruction is relieved.[13] The severity of a stenosis is mainly defined through the increased pressure gradient
over the diseased valve, and to measure this gradient the “Bernoulli fluid pressure
equation” as mentioned above is used. This equation can be transformed and, through
insertion of the density of blood, be rewritten as what is now called the “modified
Bernoulli equation”[13]
[14]:
Δp ≈ 4v
2 (mm Hg)
where Δp represents the pressure gradient over the stenotic aortic valve, and this is only
a function of the blood velocity (v) at the aortic valve. The velocity in this area can easily be measured using transthoracic
echocardiography and the Doppler effect ([Fig. 4]).
Fig. 4 Transthoracic echocardiography with Doppler examination showing the blood flow velocity
at the aortic valve during the cardiac cycle: the white arrow marks the velocity curve;
the box on the left shows the measured and calculated values. AV Vmax, maximum flow
velocity at the aortic valve; AV Vmean, mean flow velocity at the aortic valve; AV
maxPG, maximum peak gradient at the aortic valve; AV meanPG, mean peak gradient at
the aortic valve.
Other parameters help in assessing the severity of the valve defect, for example,
the aortic valve orifice area, the maximal velocity across the valve, and others,
but the mean and peak gradients over the valve calculated using the “modified Bernoulli
equation” remain important parameters in evaluating the degree of aortic valve stenosis.[13]
[15]
Surgical Technique of Valve Implantation
Surgical Technique of Valve Implantation
The Bernoulli equation is important in determining the surgical technique used to
implant a valve prosthesis. The aim is to implant the prosthesis with a minimal valve
gradient.[15] A low gradient over the prosthesis maximizes the relief of pressure on the myocardium
and subsequently results in fast recovery and remodeling of the heart which subsequently
leads to improved mid- and long-term survival.[15] The prosthesis gradient is measured with echocardiography, again using the “modified
Bernoulli equation.”[14] A minimal prosthesis gradient is achieved through complete resection of the calcified
aortic leaflets and supra-annular implantation of a large prosthesis with a maximum
valve opening.[16] Of course, the gradient over the implanted valve depends not only on the surgical
technique but also on the type of implanted prosthesis.[17]
In addition to those described above, the Bernoulli law explains many other phenomena
both inside and outside medicine. It largely explains why an airplane wing gives lift
to the plane, why the atmospheric pressure inside a hurricane is low, and why the
roofs of houses are often simply lifted off during a hurricane; it describes the flight
path of a ball with spin (e.g., soccer, tennis, and golf), the mechanism of a jet
ski, and many other phenomena.[1]
[2]
[3]
[7]
[18]
Limitations of the Bernoulli Equation
Limitations of the Bernoulli Equation
The principle of conservation of energy used by the Bernoulli equation is a basic
concept in physics that describes hydrodynamic problems. It is important to realize
that in the derivation of the Bernoulli equation, several assumptions were made that
simplify reality. It also assumes steady, laminar flow as well as an incompressible
fluid. The situation for the heart and the circulatory system is more complex with
its pulsatile flow conditions, turbulence, and other frictional effects. Furthermore,
blood is a so-called non-Newtonian fluid, that is, it has viscoelastic properties
that give it a type of memory. Accordingly, the situation in rotary blood pumps is
more complex in reality and cannot be adequately described only by the principle of
conservation of energy of a fluid. In modern medicine, computational fluid dynamics
models based on the full equations and taking into account all the above-mentioned
effects are employed to study this complex system. Nevertheless, the Bernoulli equation
is a physical concept that sets the basis for understanding the mechanisms of blood
flow and basic hydrodynamic problems and it has been of great historical significance
to cardiac medicine.
Daniel Bernoulli's Later Works
Daniel Bernoulli's Later Works
In 1750, Daniel Bernoulli was appointed Chair of Physics at the University Basel and
taught physics until 1776.[1]
[2]
[3] He maintained his Chair in the medical faculty and became their Dean seven times.
Daniel produced further excellent scientific work during his years in Basel, not just
in physics and mathematics or medicine. Most impressive is his active and imaginative
mind, which got to grips with the most varied scientific areas. In 1760, for example,
Bernoulli presented a paper, using probability, to the Paris Académie on the revolutionary
concept of inoculating a population against smallpox.[2] Bernoulli calculated that at that time, roughly three-quarters of the population
of Europe had been infected with smallpox, and one-tenth of all deaths were due to
smallpox. He reported that anyone who had survived smallpox had an immunity to the
disease, and based on his statistics he recommended a process called variolation,
deliberately inducing what was then called “artificial smallpox” in people so that
they suffered a milder infection that would later protect them from serious disease.
Although this procedure carried a slight risk of dying, at least according to the
statistical work of Daniel Bernoulli, using this approach, smallpox epidemics ceased
to be a problem at least in England, by the end of the 19th century.
Another important aspect of Daniel Bernoulli's work during his time in Basel that
proved important in the development of mathematical physics was his acceptance of
many of Newton's theories and his use of those together with the tools coming from
the more powerful calculus of Leibniz.[1]
[3] Daniel worked in Basel on mechanics and again used the principle of conservation
of energy, which supported Newton's basic equations.
In 1782, Daniel Bernoulli died, at the age of 82 years, in Basel.[1] Daniel together with his father, Johann, his uncle, Jakob, and their close friend
Leonard Euler were among the most outstanding scientists and mathematicians in Europe
in the 17th and 18th centuries. Their achievements still influence our daily lives
and practices, even in the operating room. What heroes!
