THE PHYSICS AND MATHEMATICS OF MRI

THE PHYSICS AND MATHEMATICS OF MRI

NUCLEAR MAGNETIC RESONANCE SPECTRA AND NUCLEAR MAGNETIC RESONANCE IMAGING WERE THE ORIGINAL NAMES OF THIS SPECTACULER TECHNOLOGY. The word “NUCLEAR” was lurking in the minds of the public reminding of the nuclear holocaust of World War II. The word “NUCLEAR” is dropped today. This technology is the happy brainchild of the wedding of two powerful human knowledge, PHYSICS AND MATHEMATICS.

 Isidor Isaac RabiThe Nobel Prize in Physics 1944

 Prize motivation: “for his resonance method for recording the magnetic properties of atomic nuclei”

 ISIDOR ISSAC RABI OF COLUMBIA UNIVERSITY NEW YORK USA

 American Physicist Isidor Issacs Rabi, won the Nobel prize in physics in 1944 for his discovery of nuclear magnetic resonance which is used in MRI. He is considered to be the founding father of MRI

NOBEL CITATION OF ISIDOR RABI

Protons and neutrons in nuclei act like small, rotating magnets. Atoms and molecules therefore align in a magnetic field. In 1938, Isaac Isidor Rabi passed a beam of molecules through a magnetic field. When the beam was exposed to radio waves, the direction of rotation could be changed, but only in certain stages, in accordance with quantum mechanics. When the atoms returned to their original positions, they emitted electromagnetic radiation with uniquely characteristic frequencies.

 The persons who furthered the theoretical foundations of this technology are Felix Bloch of Stanford University and Edward Mills Purcell of Harvard University.

EDWARD PURCEL   

FELIX BLOCH

FELIX BLOCH          EDWARD PURCELL

The Nobel Prize in Physics 1952 was awarded jointly to Felix Bloch and Edward Mills Purcell "for their development of new methods for nuclear magnetic precision measurements and discoveries in connection therewith"

In spite of the brilliant theoretical work of these Nobel Laureates it took four decades to fabricate a workable MRI machine. Finally in 1989 a medical doctor Ramond V.Damadian produced full body scanning MRI machine. Now the MRI machines are installed in hospitals all over the world saving thousands of lives.

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RAMOND V. DAMADIAN WITH THE FIRST FULL BODY MRI SCANNER HE INVENTED IN 1989

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Magnetic resonance imaging (MRI) uses the body’s natural magnetic properties to produce detailed images from any part of the body. For the purpose of imaging the hydrogen nucleus (a single proton) is used because of its abundance in water and fat in the body. We can draw a similarity between hydrogen proton to the planet earth spinning on its axis with a north south pole. This makes it to behave like small bar magnets. Under normal circumstances these hydrogen proton bar magnets spin in the body with their axes randomly aligned.

When the body is placed in a strong magnetic field like the MRI scanner the protons axes all line up. This uniform alignment creates a magnetic vector oriented along the axis of the MRI scanner. MRI scanners come in different magnetic field strength usually between 0.5 and 1.5 TESLA

When an additional energy in the form of radio waves is added to the magnetic field the magnetic vector is deflected. This deflection is called a wobbling or precession like wobbling of child’s toy top. This wobbling resonates with radio frequency applied.

The strength of the magnetic field can be altered electronically from head to toe using a series of gradient electric coils. By altering the local magnetic strength by small increment different slices of the body protons will resonate at frequencies of the radio waves applied.

Here the resonance means the absorbance of applied radio frequency by precessed proton. By this resonance the precessed proton goes to a higher energy state according to

Quantum theory and resonance spectroscopy.

When the radio frequency source is switched off the excited protons return to their original state, emitting the absorbed radio frequency. It is this emitted radio frequency is used to create the MR images. The emitted radio waves are received by the coils used around the body part under examination, this means that coils function like signal receiving aerial or antennae,

When the radio frequency is switched off different tissues in different stages will release the absorbed radio frequency in different ways. The release of absorbed frequency is called relaxation. The time taken for the protons to fully relax is measured in two ways. The first is the time taken for the magnetic vector to return to its resting state and the second is the time needed for the axial spin to return to its resting state. The first one is called T1 relaxation and second is called T2 relaxation.

An NMR examination is made up of a series of pulse sequences. Different tissues containing fat and water have different relaxation time and can be identified separately.

Most diseases manifest themselves by an increase in water content of the tissue.

RESONANCE - A VERY IMPORTANT WORD IN THE ACRONYM

(MRI) IN THE FORM OF LETTER “R”

Resonance in physics is defined as “A phenomenon in which an external force or a vibrating system forces another system around it around it to vibrate with greater amplitude at a specific frequency of operation. The examples are oscillation in pendulum in swings in play grounds. What causes resonance to occur. That is when the frequency of an object matches with the frequency of an externally vibrating object then resonance occurs.

HERE IN OUR CASE FREQUENCY OF THE OBJECT IS FREQUENCY OF PRECESSED PROTON AND THE FREQENCY OF EXTERNALLY VIBRATING OBJECT IS THE RADIO WAVES CREATED.

I wish to dramatize the villainous character of resonance

You may ask how and why? It is like this. The infra-red rays produced after the interaction of sunlight with various parts of plant earth, when it tries to escape into earth’s upper atmosphere is blocked by greenhouses gases like carbon dioxide and methane. This is because of vibration frequency of infrared rays and greenhouse gases match and leads to resonance. The blocked infrared rays are reflected back to earth leading to increase in earths atmospheric temperature. This increase in atmospheric temperature results in the melting of glaciers in the mountain leading to unexpected floods and rain in different parts of planet earth.

THE VALUABLE EXTENSION OF RESONANCE IS SPECTROSCOPY. SPECTROSCOPY HELPED IN PYSICS AND CHEMSTRY TO IDENTIFY THE NATURE OF MOLECULES AND ELEMENTS.

This is by exciting the electrons surrounding the nucleus exciting with the ultraviolet through visible to infrared of the electromagnetic spectrum. The range of electromagnetic spectrum can be manipulated with a prism or glass grating inside an instrument called spectrophotometer. This instrument is available today in the high school physics and chemistry laboratory.

When we come to MRI, we have to handle the nucleus itself. This requires a special high energy electromagnetic radiation called RADIOFREQUENCY WAVES (RF) to excite the protons to the high energy state. How we have come to the conclusion that radio waves are required to excite protons needs a special mathematical treatment.

As shown in the figure above the hydrogen atom or proton posses a positive charge and spins. Because of these two characters according to quantum theory, it has MAGNETIC MOMENT AND ANGULAR MOMENTUM.

GYROMETRIC RATIO

In physics the gyrometric ratio is ratio of its magnetic moment to its angular momentum and it is denoted by the Greek symbol GAMMA. In SI units it is radian per second per TESLA. TESLA being the unit of magnetic strength.

The term GYRO in the gyromagnetic ratio comes from the Greek word which means TURN. It refers to the rotational or gyric property of angular momentum.

DERIVATION OF GYROMAGNETIC RATIO

Let us take into consideration the Proton. It is subatomic particle with a (+) charge and spins like a top. Any subatomic particle with a charge and spin produces a magnetic field. Let us consider such protons which are abundant in our body in water content and various tissue components and they rotate around the circle as shown in the diagram. Proton spin is a quantum mechanical concept and not a classical rotation like a spinning top. It is quantized and can take only specific values. Further the hydrogen can be likened to the planet earth spinning in its own axis with a NORTH SOUTH pole behaving like small bar magnets. Under normal circumstances the hydrogen proton bar magnet spin in the BODY with them with their axes RANDOMLY aligned. When the body is placed in a strong magnetic field such as MRI scanner the proton axes all line up. This uniform alignment creates a magnetic vector oriented along the axis of MRI scanner. MRI SCANNER comes in different field strength usually between 0.5 and 1.5 TESLA. (After the Serbian genius electrical engineer NIKOLAI TESLA)

BOLTZMAN DISTRIBUTION IN MRI

In MRI a low energy proton spin has greater magnetization because when placed in a strong magnetic field more protons align themselves with the field representing a lower energy state, resulting in a net magnetic moment that can be detected, whereas high energy spins oppose the field leading to a smaller net magnetization due to cancellation of opposing magnetic moments

KEY POINTS to understand this concept

Proton spin and magnetic field

Protons behave like tiny magnets with a spin meaning they can align either with (parallel) or against (antiparallel) the external magnetic field

Energy levels

When placed in a magnetic field proton in alignment with field (parallel) are considered to be in lower energy state while those aligned against field are in a higher energy state

How does this relate to MRI imaging

Radio frequency pulse

By applying a radiofrequency pulse at the specific resonant frequency of the proton (LARMOR FREQENCY) the low energy protons can FLIPPED to higher energy state creating a detectable signal (quantum mechanical principle).

SIGNAL STRENGTH

THE STRENGTH OF THE SIGNAL IS DIRECTLY PROPORTIONAL TO THE NET MAGNETIZATION WHICH IS PRIMARILY DETERMINED BY THE EXCESS OF PROTONS IN THE LOWER ENERGY STATE.

In MRI the strength of the signal is directly related to the BOLTZMANN DISTRIBUTION, which means that a stronger magnetic field leads to a larger population difference between the “spin up” and “spin down” states of the nuclei (protons) resulting in a stronger MRI SIGNAL. This essentially means the BOLTZNANN EQUATION determines the ratio of these spin states based on the applied magnetic field strength and temperature, allowing for a better signal detection in high field MRI

The equation is written as shown above

BOLTZMANN DISTRUTION EQUATION WAS DEVELOPED BY THE AUSTRIAN THEORETICAL PHYSICIST LUDWIG BOLTZMANN IN 1872.

  LUDWIG EDUARD BOLTZMANN

          (1844-1906)

   UNIVERSITY OF VIENNA, AUSTRIA

Boltzmann Equation explained

Now I shall explain the importance and implication of this equation in the working of MRI

Nupper/Nlower  = e (-delta E/KT)

We want to get rid of the function “e”

For this we have to multiply both sides of the equation with inverse function of “e” that is “ln”, so “ln e” cancels out. Now we can write the equation as

Ln (N upper/N lower) = - delta E / kT

Now when you increase -delta E the fraction on the left side DECEASES because of the negative sign on delta E that means the denominator (N lower) increases. This means the population of proton in the low energy level INCREASES. When you increase the temperature T, the fraction INCREASES, that means the population of protons in the upper energy INCREASES. In the lower

The increase of population of protons in the lower energy levels increases the parallelly oriented proton population which in turn increases the magnetization and magnetic vector as shown in the figure below. This leads  to very good and clear imaging signal. 

HOW IT WORKS IN PRACTICE

ALIGNMENT OF SPINS

When a patient is placed inside an MRI scanner, the strong alignment magnetic field aligns MOST OF THE HYDROGEN NUCLEI (PROTONS) creating a net magnetization along the field direction.

Boltzmann distribution further tells that at room temperature the number of spins in the lower N, slightly outnumber the number in the upper-level N.

KEY POINT ABOUT BOLTZMANN DISTRIBUTION IN MRI

Spin alignment

When placed in a magnetic field the magnetic moments of protons can align either parallel or antiparallel to applied field

Energy difference

The parallel alignment is a lower energy state so according to Boltzmann’s distribution slightly more spins will be in this state at thermal equilibrium.

Signal strength

This small difference in spin population creates a net magnetization which is the source of the MRI signal

Field strength

Increasing the magnetic field strength (BO) further separates the energy levels of the spin states leading to larger population difference and a stronger MRI signal

LARMOR FREQUENCY

It is the rate at which a charged particles like proton magnetic moment processes or wobble around an external magnetic field. It is a fundamental phenomenon in physics and is used in NMR and NMRI

THE LARMOR FREQENCY IS CALCULATED USING THE LARMOR FREQUENCY EQUATION

W =YB

W = Larmor frequency in megahertz (MHz)

Y = the gyrometric static magnetic field ratio in MHz per TESLA(T)

B = is the strength of the static magnetic field in TESLA (T)

The Larmor frequency is specific to each nucleus and depends on the magnetic field it experiences  

THE LARMOR FREQUENCY IS NAMED AFTER JOSEPH LARMOR, an IRISH PHYSICIST who lived from 1857 to 1942. He occupied the prestigious LUCASIAN PROFESSOR OF MATHEMATICS at Cambridge University. In 1897 Larmor calculated that the frequency of precession or wobbling of protons was directly proportional to the strength of the applied magnetic field.

LARMOR FREQUENCY EQUATION

Precession (wobbling)

 THE GYROMETRIC RATIO FOR PROTON IN MRI IS 42.58 MHz/TESLA

PRECESSION (WOBBLING)

Larmor frequency describes how fast a charged particle’s magnetic moment circles an external magnetic field. It is named after JOSEPH LARMOR and is crucial in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI)

JOSEPH LARMOR

(1857-1942)

14th LUCASIAN PROFESSOR OF MATHEMATICS AT THE UNIVERSITY OF CAMBRIDGE

  Explanation

THE LARMOR FREQUENCY IS THE ANGULAR FREQUENCY AT WHICH UNPAIRED PROTONS PRECESS OR WOBBLES AROUND A STRONG MAGNETIC FIELD. IT IS SPECIFIC TO EACH NUCLEUS AND DEPENDS ON THE MAGNETIC FIELD IT EXPERIENCES.

 RESONANCE

This is a very important word in MRI (MAGNETIC RESONANCE IMAGING)

IN MRI THE PROTON PRECESSION FREQUENCY IS EXACTLY EQUAL TO LARMOR FREQENCY. ONLY THEN RESONANCE WILL TAKE PLACE AND IT IS CRUCIAL REQUIREMENT FOR MRI IMAGING. WHEN RESONANCE TAKES PLACE SOME OF THE PROTONS IN THE LOWER ENERGY STATE ABSORB THE LARMOR FREQENCY RADIO WAVES AND GOES TO A HIGHER ENERGY LEVEL WHICH VERY TEMPERORY (QUANTUM MECHANICAL PRINCIPLE). WHEN LARMOR FREQENCY WAVES ARE SWITCHED OFF PROTONS WHICH ABSORBED THE LARMOR FREQUENCY RADIO WAVES SHED THESE WAVES COME TO TO THEIR GROUND STATE. THE SHED RADIO WAVES ARE ACOLLECTED BY RADIO ANTENNAE SURROUNDING THE PATIENT AND SENT TO A COMPUTER AS MRI SIGNAL OF THE PATIENT.

FOR A TYPICAL CLINICAL MRI MACHINE WITH A MAGNETICFIELD STRENGTH OF 1.5 TESLA THIS FREQENCY IS 64 MHz

The pictures shown above of electromagnetic spectrum are self-explanatory and are amenable to image formation. But the radio frequency used in MRI requires a special treatment because of their long wavelength

In MRI the shed energy from the excited protons is collected by the coil antenna surrounding patient and send to a computer close by. The computer carries a special algorithm of a mathematical technique FOURIER TRANFORMATION which converts radio wave frequencies to visible images of the parts of the body scanned.

 THE COMPUTER AFTER RECEIVING THE MRI SIGNAL PROCESS THE SIGNAL TO VISUAL MRI IMAGES USING A MATHEMATICAL TECHNIQUE CALLED FOURIER TRANSFORMATION (FT)

JEAN BAPTISTE JOSEPH FOURIER (1768-1830)

 WHAT IS FOURIER TRANSFORM (FT)

Fourier transform is a mathematical technique that allows an MR signal now considered to be a sinusoidal wave (sine wave) be broken up into its components of DIFFERENT FREQUENCIES PHASES AND AMPLITUDES.  This remarkable mathematical feat was developed by JEAN BAPTISTE JOSEPH FOURIER (1768-1830) a French mathematician and physicist.

FOURIER SHOWED THAT ANY PERIODIC SIGNAL (st) CAN BE WRITTE AS THE SUM OF THE SINE WAVES WITH VARIOUS AMPLITUDES FREQUENCIES AND PHASES.

where ai's  are amplitudes, ϕi's are phase shifts, and ω is the fundamental frequency. The higher order frequencies 2ω, 3ω, etc. are called harmonics. For example, the Fourier expansion of a square wave can be written as

The time domain signal of the square wave, s(t), is shown on the left. The so-called frequency domain representation, S(ω), is shown on the right. S(ω) is called the Fourier transform of s(t). In general S(ω) is a complex-valued function composed of harmonic frequencies, phases, and their amplitudes obtained from the Fourier expansion.

	Fourier transformation is the mathematical procedure connecting s(t) and S(ω). If s(t) is specified, S(ω) may be computed, and vice versa. The equations require some knowledge of complex numbers and calculus to make sense, but don't worry if you don't understand them.:

The equation on the left is the Fourier Transform. That on the right is the inverse Fourier

Transform.

.	Fourier Transform Pairs. Only the real part of the transform is shown.

 he equation on the left is the Fourier Transform. That on the right is the inverse Fourier

Transform.

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 t

he two waves at the same point in time

The two waves at the same point in time

For the benefit of the readers of this blog I have given here the illustration of the various parts of the electromagnetic spectrum, the sinusoidal waves, the amplitude, phase and frequency associated with sinusoidal waves.

Before closing down a philosophical thought came to my mind. MRI technology is the perfect ground to express the wave particle duality of the electromagnetic radiation. To explain the behaviors of protons under the influence of the strong magnet and spew out the extra energy absorbed by the protons we have to invoke QUATUM THEORY of electromagnetic radiation and once the radio frequency leaves MRI machine and enters the computer to process the image, we have to invoke WAVE THEORY of electromagnetic radiation through FOURIER TRANSFORMATION. This in turn reminds me a couplet from KATHO UPANISHAD

“ADVAITHAM PARAMARTHO HI DWAITHAM THATHBHETH UCHAITE”

MEANING THEREBY THE NON-DUALITY IS THE ULTIMATE REALITY AND DUALITY EMNATES OUT OF NON DUALTY.

IN OUR CASE THE ELECTROMAGNETIC SPECTRUM IS ULTIMATE REALITY AND THE DUALITY IS QUANTUM AND WAVE THEORY OF LIGHT OTHERWISE POPULARLY KNOWN AS WAVE PARTICLE THEORY OF ELECTRO MAGNETIC SPECTRUM.

With the advent of Artificial Intelligence (AI) we can anticipate great improvement in MRI imaging.

Before I close down, I wish to express my heartfelt thanks to our daughter Sukanya Subramanian our grand daughters Ananya Subramanian and Meghna Hari Krishna for their help in the explanation of Boltzmann Distribution Equation

Professor K.A.Balasubramanian

Ph.D. (IARI-New Delhi), Ph.D. (Imperial College, London),

DIC (Imperial College, London)

Denver, Colorado, USA

October 23rd, 2024