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Journal of Advanced Research
Survey article
A new perspective on the role of mathematics in medicine
Ahmed I. Zayed
Department of Mathematical Sciences, DePaul University, Chicago, IL 60614, USA
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
 The article gives a brief account of the
development of mathematics and its
relationship with practical
applications.
 This is an expository article that sheds
light on the role of mathematics in
medical imaging.
 It traces the development of CT scan
from infancy to the present.
 It reports on new advances in MRI
technology.
 Mathematical concepts explained in
non-technical terms.
The Radon transform is the mathematical basis of computer tomography.
a r t i c l e
i n f o
a b s t r a c t
Article history:
The aim of this expository article is to shed light on the role that mathematics plays in the advancement
Received 22 October 2018
Revised 25 January 2019
Accepted 26 January 2019
Available online 6 February 2019
of medicine. Many of the technological advances that physicians use every day are products of concerted
efforts of scientists, engineers, and mathematicians. One of the ubiquitous applications of mathematics in
medicine is the use of probability and statistics in validating the effectiveness of new drugs, or proce-
dures, or estimating the survival rate of cancer patients undergoing certain treatments. Setting this aside,
Keywords:
Computed tomography
CT scan
Radon transform
Magnetic Resonance Imaging (MRI)
there are important but less known applications of mathematics in medicine. The goal of the article is to
highlight some of these applications using as simple mathematical formulations as possible. The focus is
on the role of mathematics in medical imaging, in particular, in CT scans and MRI.
 2019 The Author. Published by Elsevier B.V. on behalf of Cairo University. This is an open access article
Compressed sensing
Introduction
gued but not totally surprised to know that most of them were not
aware of the role that mathematics has played in their fields
The subject of this expository article was motivated by discus-
whether in statistical analysis or even more importantly in the
sions I have had with some of my colleagues who are renowned
advancement of the technologies that they use every day. Mathe-
professors at medical and engineering schools in Egypt. I was intri-
matics for them is just an abstract and dry subject that you study
to become a teacher, or if you are lucky, you become a university
Peer review under responsibility of Cairo University.
professor in the faculty of science or engineering.
E-mail address: azayed@depaul.edu
2090-1232/ 2019 The Author. Published by Elsevier B.V. on behalf of Cairo University.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
50
A.I. Zayed/Journal of Advanced Research 17 (2019) 49–54
My target audience is physicians who do not have advanced
Ironically, in 1908 one of Hardy’s contributions to mathematics
background in mathematics but are interested in learning more
turned out to be useful in genetics and had a law named after him
about the role that mathematics plays in medical imaging. To this
‘‘Hardy’s Law.” It dealt with the proportions in which dominant
end, I have intentionally written the article as an expository article
and recessive Mendelian characters would be transmitted in a
and not as a research one. For those who would like to learn more
large mixed population. The law proved to be of central impor-
about the mathematical formulation involved, they may consult
tance in the study of Rh-blood-groups and the treatment of haemo-
the references at the end of the article for details.
lytic disease of the newborn.
Mathematics,
which
comes
from
the
Greek
Word
Here we should distinguish between two different but closely
‘‘la0hgla ¼ mathemata”, meaning subject of study, is one of the
related branches of mathematics: applied mathematics and pure
oldest subjects known to mankind. Its history goes back thousands
mathematics. Applied mathematics deals with real-world prob-
of years. Archaeological discoveries indicated that people of the Old
lems and phenomena and try to model them by equations and for-
Stone Age as early as 30,000 B.C could count. Mathematics, which in
mulas to better understand them and manage or predict them
the early days meant arithmetic and geometry, was invented to
more efficiently. Pure mathematics, on the other hand, and con-
solve practical problems. Arithmetic was needed to count livestock,
trary to the common belief, does not only deal with numbers.2 It
compute transactions in trading and bartering, and in making cal-
deals with abstract entities and tries to find relations between them
endars, while geometry was needed in setting boundaries of fields
and patterns and structures for them, and generalize them whenever
and properties and in the construction of buildings and temples.
possible. The British philosopher, logician, and mathematician, Ber-
The Ancient Egyptians, Babylonians, and Mayan Indians of Cen-
trand Russell.3 (1872–1970) described mathematics in a philosoph-
tral America developed their own number systems and were able
ical and somewhat sarcastic way as.
to solve simple equations. While the Ancient Egyptians’ number
system was decimal, i.e., counting by powers of 10, the Babylo-
nian’s system used powers of 60, and the Mayans’ system used
powers of 20. The decimal system is the most commonly used sys-
‘‘Mathematics maybe defined as the subject in which we never
know what we are talking about, nor whether what we are say-
ing is true.”
tem nowadays.
In those early civilizations deriving formulas and proving results
Nevertheless, there is a plethora of examples of useful and prac-
were not common. For example, the Ancient Egyptians knew of and
tical applications that came out of the clouds of abstract mathe-
usedthefamousPythagoreanTheoremforright-angletriangles,but
matics, such as Hardy’s work mentioned above and the work of
did not provide proof of it. A more striking example is the formula
John Nash (1928–2015)4 on Game Theory which earned him the
for the volume of a truncated pyramid, which was inscribed on the
Nobel Prize in Economics in 1994.
Moscow Papyrus1 but without proof [1], so how the Ancient Egyp-
In the next sections we will see two other examples of ideas
tians obtained that formula is still a mystery.
from pure mathematics that turned out to be useful in medicine.
The nature of mathematics changed with the rise of the Greek
I will try not to delve into technicality and keep the presentation
civilization and the emergence of the Library of Alexandria where
as simple and non-technical as possible, but some mathematical
Greek scholars and philosophers came to pursue their study and
formulations will be introduced for those who are interested, but
contribute to the intellectual atmosphere that prevailed in Alexan-
which the non-expert can skip. This approach may lead me to give
dria. The master and one of the most genius minds of all times was
loose interpretations or explanation of some facts for which I apol-
Euclid who taught and founded a school in Alexandria (circa 300 B.
ogize. The focus will be on two techniques in medical imaging.
C.). He wrote a book ‘‘Elements of Geometry,” also known as the
‘‘Elements,” which consisted of 13 volumes and in which he laid
Mathematics and Computed Tomography (CT) scan
the foundation of mathematics as we know it today.
In Euclid’s view, mathematics is based on three components:
The most common application of mathematics in medicine and
definitions, postulates, and rules of logic, and everything else, lem-
pharmacology is probability and statistics where, for example, the
mas, propositions, and theorems are derived from these compo-
effectiveness of new drugs or medical procedures is validated by
nents. The notion of seeking after knowledge for its own sake,
statistical analysis before they are approved, for example, in the
which was completely alien to older civilizations, began to emerge.
United States by the Food and Drug Adminstration (FDA). But in
As a result, the Greeks transformed mathematics and viewed it as
this short article I will try to shed light on other applications of
an intellectual subject to be pursued regardless of its utility.
pure mathematics, in particular, on two recent technological
This new way of thinking about mathematics has become the
advances in the medical field that probably would not have existed
norm and continued until now. The nineteenth and twentieth cen-
without the help of mathematics.
turies witnessed the rise of abstract fields of mathematics, such as
The first story is about CT scan, known as computed tomography
abstract algebra, topology, category theory, differential geometry,
scan, or sometimes is also called CAT scan, for computerized axial
etc. Mathematicians focused on advancing the knowledge in their
tomography or computer aided tomography. The word tomography
fields regardless of whether their work had any applications. In
fact, some zealot mathematicians bragged that their work was
intellectually beautiful but had no applications. A prominent repre-
sentative of this group was the British mathematician, Godfrey
Harold Hardy (1877–1947), one of the most renowned mathemati-
cians of the twentieth century, who once said [2].
is derived from the old Greek word ‘‘solo1 ¼ tomos”, meaning
‘‘slice or section” and ‘‘cqa0/x ¼ grapho”, meaning ‘‘to write.”.
Medical imaging is about taking pictures and seeing inside of
the human body without incisions or having to cut it to see what
is inside. What is an image? or more precisely what is a black
‘‘I have never done anything ‘‘useful”. No discovery of mine has
2 The branch of mathematics that deals with numbers and their properties is called
made, or is likely to make, directly or indirectly, for good or ill,
the least difference to the amenity of the world.”
Number Theory.
3 Bertrand Russell was a writer, philosopher, logician, and an anti-war activist. He
discovered a paradox in Set Theory which was named after him as Russell’s Paradox.
He received the Nobel Prize in Literature in 1950.
4 John Nash was considered a mathematical genius. He received his Ph.D. from
Princeton University in 1950 and later became a professor of mathematics at MIT. He
1 It is called the Moscow Papyrus because it reposes in the Pushkin State Museum
suffered from mental illness in midst of his career and was the subject of the movie ‘‘A
of Fine Arts in Moscow.
Beautiful Mind”.
A.I. Zayed/Journal of Advanced Research 17 (2019) 49–54
51
and white digital image? A digital image or a picture is a collection
question was the starting point for Allen Cormack, one of the inven-
of points, called pixels and is usually denoted by two coordinates
tors of the CT scanner.
ðx;yÞ, and each pixel has light intensity, called gray level, ranging
In 1956, Allen Cormack, a young South African physicist, was
from white to black.
appointed at the Radiology Department at the Groote Schuur hos-
Mathematically speaking, a black and white picture is a func-
pital, the teaching hospital for the University of Cape Town’s med-
tion fðx;yÞ that assigns to each pixel some number corresponding
ical school. This hospital later became the site of the world’s first
to its gray level. In the 1920s, pictures were coded using five dis-
heart transplant. Cormack took on himself, as one of his first duties
tinct levels of gray resulting in low quality pictures. Nowadays,
at the new job, the task of finding a set of maps of absorption coef-
the number of gray levels is an integer power of 2, that is 2k for
some positive integer k. The standard now is 8-bit images, that is
28 ¼ 256 levels of gray, with 0 for white and 255 levels or shades
of gray. An image with many variations in the gray levels tends
to be sharper than an image with small variations in the gray scale.
The latter tends to be dull and washed out [3].
One of the oldest techniques used in medical imaging is X-rays
where the patient is placed between an X-ray source and a film
sensitive to X-ray energy, but in digital radiography the film is dig-
itized or the X-rays after passing through the patient are captured
ficients for different sections of the human body.
The results of the task would make X-ray radiotherapy treat-
ments more efficient. He soon realized that what he needed to
complete his task was measurements of the absorption of X-rays
along lines in thin sections of the body. Since the logarithm of
the ratio of incident to emergent X-ray intensities along a given
line is just the line integral of the absorption coefficient along that
line, the problem mathematically was equivalent to finding a func-
tion fðx;yÞ from the values of its integrals along all or some lines in
the plane [4].
by a digital devise. The intensity of the X-rays changes as they pass
‘‘This struck me as a typical nineteenth century piece of math-
through the patient and fall on the film or the devise. Another med-
ematics which a Cauchy5 or a Riemann6 might have dashed off
ical application of X-ray technology is in Angiography where an
in a light moment, but a diligent search of standard texts on anal-
X-ray contrast medium is injected into the patient through a
ysis failed to reveal it, so I had to solve the problem myself,” says
catheter which enhances the image of the blood vessels and
Cormack. ‘‘I still felt that the problem must have been solved, so I
enables the radiologist to see any blockage. X-rays are also used
contacted mathematicians on three continents to see if they
in industry and in screening passengers and luggage at airports.
knew about it, but to no avail” adds Cormack [4].
But more modern and sophisticated machines than X-ray
machines are the CT scanners which produce 3-dimensional
images of organs inside the human body. How do they work and
what is the story behind them?
The first CT scanner invented by Allen Cormack and GGodfrey
Hounsfield in 1963 had a single X-ray source and a detector which
moved in parallel and rotated during the scanning process. This
technique has been replaced by what is called a fan-beam scanner
in which the source runs on a circle around the body firing a fan (or
a cone) of X-rays which are received after they pass through the
body by an array of detectors in the form of a ring encircling the
patient and concentric with the source ring. The process is
repeated and the data is collected and processed by a computer
to construct an image that represents a slice of the object. The
object is slowly moved in a direction perpendicular to the ring of
detectors producing a set of slices of the object which, when put
together, constitute a three-dimensional image of the object.
Recall that a black and white image is just a function fðx;yÞ
defined on pixels. In a standard college calculus course, students
are taught how to integrate functions and, with little effort, they
can integrate functions along straight lines. The integral of a func-
tion along a straight line in some sense measures a weighted aver-
age of the function along that line. But a more interesting and
much more challenging mathematical problem is the inverse
problem, that is suppose that we know the line integrals of a func-
tion fðx;yÞ along all possible straight lines, can we construct
fðx;yÞ?
When an X-ray beam passes through an object lying perpendic-
ularly to the beam path, the detector records the attenuation of the
beam through the ray path which is caused by the tissues’ absorp-
tion of the X-rays. What the detector records is proportional to a
line integral along that path of the function fðx;yÞ that represents
the X-ray attenuation coefficient of the tissue at the point ðx;yÞ. If
we rotate the beam around the object, the detector will measure
A few years later, Cormack immigrated to the United States and
became a naturalized citizen. Because of the demands of his new
position, he had to pursue his problem part time as a hobby. But
by 1963 he had already found three alternative forms of solutions
to the problem and published his results. He contacted some
research hospitals and groups, like NASA, to see if his work would
be useful to them but received little or no response.
Cormack continued working on some generalizations of his
problem, such as recovering a function from its line integrals along
circles through the origin. Because there was almost no response to
his publications, or at least that was what he thought, Cormack felt
somewhat disappointed and forgot about the problem for a while.
By a mere accident, Cormack discovered that his mathematical
results were a special case of a more general result by Johann
Radon, in which Radon introduced an integral transform and its
inverse and showed how one could construct a two-dimensional
function fðx;yÞ from its line integrals. Even more, he showed how
one can reconstruct an n-dimensional function from its integrals
over hyper-planes of dimension n  1. That integral transform is
now called the Radon transform. The transform and its inverse
are the essence of the mathematical theory behind CT scans.
As is often the case with many beautiful and significant mathe-
matical discoveries, the Radon transform was discovered and went
unnoticed for very many years. And when it was rediscovered, it
was rediscovered independently by several people in different
fields. The Radon transformation without doubt is one of the most
versatile function transformations. Its applications are numerous
and its scope is immense. Chief among its applications are com-
puted tomography (CT) and nuclear magnetic resonance (NMR).
Not only the transform, but also its history deserves a great deal
of attention.
the line integrals of the function fðx;yÞ from all possible directions.
Now the following question immediately arises: can we construct
fðx;yÞfrom its line integrals? Since fðx;yÞrepresents, in some sense,
5 Augustin Louis Cauchy (1789–1857) was one of the leading French mathemati-
cians of the 19th century who contributed significantly to several branches of
mathematics, in particular, to mathematical analysis.
the image of the cross section of the object, that question is equiv-
6 Bernhard Riemann (1826–1866) was one of the best German mathematicians of
alent toasking whether we can constructthe image of the crosssec-
his era. Many mathematical concepts and results were named after him, such as
tion of the object from the data that the detector compiled. This
Riemann integrals, Riemann surfaces, and the famous Riemann Hypothesis which is
still an open question.