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The aim of this expository article is to shed light on the role that mathematics plays in the advancement 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 procedures, or estimating the survival rate of cancer patients undergoing certain treatments. Setting this aside, 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.

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Contents lists available at ScienceDirect

Journal of Advanced Research

journal homepage: www.elsevier.com/locate/jare

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

under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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.

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