line spectra experiment

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Bohr’s Model: Line Spectra

In the last study guide we learned that electromagnetic radiation has both wave and particle properties. In this study guide we will discuss how light and matter interact. The light we see from the sun or from a lightbulb is called white light because it consists of all the colors in the visible spectrum. It will produce a continuous spectrum of all colors from violet to red. There are no gaps in a continuous spectrum as can be seen in the figure below.

In fact, raindrops will disperse sunlight into a rainbow. Not all light sources will produce a continuous spectrum.

“Europa Rainbow” by Robert Couse-Baker is licensed with CC BY 2.0. To view a copy of this license, visit https://creativecommons.org/licenses/by/2.0/

If hydrogen gas is placed into a discharge tube, and a high voltage is applied, the result is 4 different colored lines. This is called a line spectrum which consists of radiation of specific wavelengths. The spectrum is not continuous and only has four different colors. The colors in the hydrogen emission spectrum correspond to wavelengths of 410.1, 434.1, 486.1, and 656.3 nm.

The line spectra below are for lithium and sodium. We see that sodium has two yellow lines that are typical of some street lights. When heated, elements will produce line spectra. Fireworks are made up of metal salts that when heated will produce light of different colors. Each element has its own line spectrum and a line spectrum can be used to identify different elements. The line spectrum is like a fingerprint and is specific for each element.

If you have ever let food boil over when cooking, you will notice the gas flame will turn a bright yellow-orange. This is due to the electrons in the sodium atoms gaining energy and then emitting the energy as yellow light. We will discuss this in more detail later in the study guide. For now we will focus on hydrogen.

Rutherford’s model stated the atom was made up of a dense positively charged nucleus with negatively charged electrons in the extranuclear region. There was a problem with this model. According to classical physics, if an electron was orbiting a nucleus, it would continuously lose energy in the form of electromagnetic radiation. The electron would spiral into the nucleus within about 10 -10 s. The stability of the atom could not be explained by this theory.

J.J. Balmer, in 1885, showed the wavelengths in the visible region of the hydrogen atom could be determined by the following equation:

\(\displaystyle \frac{1}{\lambda}\;=\;1.907\times\;10^7\;m^{-1}\Bigl(\frac{1}{2^2}\;-\;\frac{1}{n^2}\Bigr)\)

where n is an integer greater than 2. If we substitute 3 for n, we get 6.56 x 10 -7 m which is a wavelength of 656 nm. This wavelength corresponds to red light. Substituting 4, 5, and 6 for n will give the other three lines in the visible spectrum for the hydrogen atom.

A Danish physicist, Niels Bohr (1885 – 1962), used the work of Planck and Einstein to apply a model to explain the stability and the line spectrum of a hydrogen atom. Bohr’s postulates follow:

1. Energy Level Postulate: The electron can only have specific energy values in the hydrogen atom. These are called its stationary states (energy levels). In each state is a fixed circular orbit of the electron about the nucleus. The higher the energy level, the further the orbit is from the nucleus. The energy levels are called principal energy levels and are represented with a lowercase n, the principal quantum number . The value of n is an integer where n = 1, 2, 3, 4, … ∞. 2. The electron’s energy remains constant as long as it stays within a stationary state. 3. The electron can only change to another stationary state by absorbing or emitting a photon. The energy of the photon is equal to the differences in the energies of the two states.

Bohr derived the following formula for the energy levels in a hydrogen atom:

\(\displaystyle E\;=\;-\frac{R_H}{n^2}\;\;\;\;\;n\;=\;1,\;2,\;3,\;…\infty\) R H is a constant, in energy units, equal to 2.179 x 10 -18 J. The energies of the electron in the hydrogen atom can be calculated by putting in integer values for n; n = 1, 2, 3, 4, … ∞.

When an electron is in a higher energy level and falls to a lower energy level, light is emitted. This is shown in the figure below. The emission lines are for the electron in the hydrogen atom. The figure shows the Lyman Series (ultraviolet region), the Balmer Series (visible region) and the Paschen Series (infrared region). Bohr was able to predict all of the lines in the hydrogen atom spectrum including those in the ultraviolet and infrared region. When an electron is in a higher energy level and it falls to a lower energy level, the energy is emitted as light. We say the electron has undergone a transition .

Notice, in the figure above the energies have a negative sign. At n = ∞, the electron has been completely removed from the nucleus and is assigned zero energy. The ionization energy is the energy required when the electron has been completely removed from a hydrogen atom, at n = ∞. The further the electron is from the nucleus, the higher the energy. The closer the electron is to the nucleus, the lower the energy. The more negative the value of energy, the closer the electron is to the nucleus.

From the law of Conservation of Energy, the final energy of the electron, E f plus the energy of the photon, hν, is equal to the initial energy of the electron, E i :

E f + hν = E i    (Equation 1)

Equation 1 can be rearranged to:

hν = E i – E f   (Equation 2)

Bohr obtained Balmer’s equation by substituting n i for the principle quantum number of the initial energy level, and n f for the principle quantum number of the final energy level into Equation 2.

\(\displaystyle E_i\;=\;-\frac{R_H}{n_i^2}\;\;\;and\;\;\;E_f\;=\;-\frac{R_H}{n_i^2}\)   and

\(\displaystyle h\nu\;=\;E_i\;-\;E_f\;=\;\Biggl (-\frac{R_H}{n_i^2}\Biggr )\;-\;\Biggl (-\frac{R_H}{n_f^2}\Biggr)\)   Rearranging the equation:

\(\displaystyle h\nu\;=\;-R_H\;\Biggl (\frac{1}{n_f^2}\;-\;\frac{1}{n_i^2}\Biggr )\)

We know that ν = c/λ and the equation can be rewritten as:

\(\displaystyle \frac{1}{\lambda}\;=\;-\frac{R_H}{hc}\Biggl (\frac{1}{n_f^2}\;-\;\frac{1}{n_i^2}\Biggr )\)

We calculate R H /hc as:

\(\displaystyle \frac{R_H}{hc}\;=\;-\frac{2.179\times\;10^{-18}\;J}{6.626\times\;10^{-34}\;J⋅s\;\times\;(2.998\times\;10^8\;m/s}\;=\;1.097\times\;10^7\;m^{-1}\)

This is the same constant as in the Balmer formula where n f = 2. This gives the four wavelengths in the visible region. If we change n f to equal 1, we get the lines in the ultraviolet series and when n f = 3, we get the wavelengths in the infrared series for the hydrogen atom.

We now ask how the electron in a hydrogen atom gets to a higher energy level. The electron must gain energy to get from a lower to a higher energy level. One way this can happen is through collisions between hydrogen atoms. If a collision is forceful enough, the electron in an atom can gain enough energy to transition to a higher energy level. The most common way is in a hot gas. A gas has a very high kinetic energy, and an electron can be excited to a higher state. The electron in a hydrogen atom usually resides in the lowest energy state, n = 1. When the electron gains energy it goes to a higher energy level, n, in a process called absorption . Eventually the electron will release the energy, as light, as it goes from a higher energy level to a lower energy level. This is called emission. Below is an absorption spectral series for the electron in a hydrogen atom.

A transition for an electron in the hydrogen atom from n = 1 → n = 3 would be absorption of energy. A transition from n = 4 → n = 2, would be emission of energy. Bohr was able to explain both emission and absorption for an electron in the hydrogen atom. Bohr’s postulates do not work well for atoms that have more than one electron. In multi-electron species, there are electron repulsions, and the energy levels cannot be determined by one simple equation. Although, in a multi-electron atom, if we know the wavelength of the emitted or the absorbed light, it can be related to the frequency and then finally to the energy differences between energy levels.

Worksheet: Bohr’s Model: Matter Waves

Watch the following video (Start at 32:27 minutes) for a discussion on the Bohr model and line spectra.

Exercise 1. Calculate the energy of the electron in the hydrogen atom if the electron is excited from n = 2 to n = 6. Calculate the wavelength when the electron transitions from n = 6 to n = 2.

Exercise 2. Indicate if energy is absorbed or emitted for the following electron transitions:

a) n = 2 → n = 7 b) n = 3 → n = 1 c) from an orbit of radius 1.46 Angstroms to an orbit with radius of 0.768 Angstroms.

Exercise 3. What is the energy, in kJ/mol, for an electron that transitions from n = 1 to n = ∞? This is the ionization energy for the hydrogen atom. Is the energy absorbed or emitted?

Exercise 4. Calculate the wavelengths for the three lines in the infrared region of the hydrogen spectrum.

Check Solutions/Answers to Exercises

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• 1.1 Electromagnetic Energy

Line Spectra

The paradox of discrete spectra.

In the late 1800’s, scientists struggled with a concerned of classical eletromagnetic theory: how atoms and molecules emit light. When you heat solids, liquids, or condensed gases sufficiently, they radiate excess energy as light. This light contains photos with a range of energies, creating a continues spectrum with an unbroken series of wavelengths. Most of the light from stars, including our sun, is produced in this way. You can observe all the colors of visible light in sunlight by using a prism to separate them.

Sunlight also contains UV light (shorter wavelengths) and IR light (longer wavelengths) as shown on the spectra in the previous page , detectable by instruments but not the human eye. Incandescent (glowing) solids, like tungsten filaments, emit light with all visible wavelength. You can often approximate these continuous spectra with blackbody radiation curves at suitable temperature, such as those shown in the previous page .

Discrete Light

In contrast, light can also appear as discrete or line spectra with narrow line widths, as shown in FIgure 2. Exciting a gas at low partial pressure using an electrical current, or heating it, it emits these lines spectra. Fluorescent light bulbs and neon signs operate in this way (Figure 1). Each element displays its own characteristic set of lines, as do molecules, although their spectra are generally much more complicated.

Emission Lines and Balmer’s Equation

Each emission line has a single wavelength of light, which implies that the light emitted by a gas consists of a set of discrete energies. For example, an electric current passes through a tube containing hydrogen gas at low pressure, the H 2 molecules break apart into separate H atoms, creating a blue-pink glow. This light in a line spectrum shows four visible wavelengths when passed through a prism, see in Figure 2.

In the late nineteenth century, scientists found discrete spectra in atoms and molecules extremely puzzling, as classical electromagnetic theory predicted only continuous spectra. So, in 1885, Johann Balmer derived an empirical equation linking four visible wavelengths of hydrogen atoms to whole integers. That equation is the following one, in which k is a constant:

The Rydberg Formula

Other discrete lines for the hydrogen atom were found in the UV and IR regions. Johannes Rydberg generalized Balmer’s work and developed an empirical formula that predicted all of hydrogen’s emission lines, not just those restricted to the visible range, where, n 1 and n 2 are integers, n 1 < n 2 , and R ∞ is the Rydberg constant (1.097 × 10 7 m −1 ).

Even in the late nineteenth century, scientists measured the wavelengths of hydrogen with great precision in spectroscopy. This high accuracy implied that they could also determine the Rydberg constant very precisely. That such a simple formula as the Rydberg formula could account for such precise measurements seemed astounding at the time. Ultimately, it was the eventual explanation for emission spectra by Neils Bohr in 1913 that convinced scientists to abandon classical physics and spurred the development of modern quantum mechanics.

• Line Spectra
• Photoelectric Effect
• Observations of Photoelectric Effect
• Failure of Classical Wave Theory
• Understanding Photoelectric Effect
• Einstein's Photoelectric Equation
• Wave Particle Duality
• Bohr Model of The Atom
• Energy Level Diagram For Hydrogen
• Coolidge X-Ray Tube
• Features of X-ray Spectrum
• Properties of Lasers
• Exciting The Atom
• How Lasers Works
• Helium-Neon Laser
• Heisenberg's Uncertainty Principle
• The Schrodinger Equation And Wave Function
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• Reflection And Transmission
• Scanning Tunnelling Microscope

Emission Line Spectrum

The atom is first excited by a colliding electron. When the atom goes back to its ground state, either directly or via intermediate energy levels, photon of only certain frequencies are emitted due to the discrete energy levels. Hence only certain frequencies of light are observed, forming the emission spectrum, which is discrete bright coloured lines on a dark background.

Steps to obtain emission line spectrum:

• Gases such as hydrogen or neon are placed in an discharge tube at low pressure.
• A high voltage of several kilo-volts is applied across the cathode and anode of the discharge tube.
• The gas atoms become excited by the collision with the electrons passing through the tube.
• When the gas atoms fall to a lower energy level, the excess energy is emitted as electromagnetic radiation(photon) with a specific frequency. The frequency f of the emission line is dependent on the difference between the high and low energy levels. ΔE = hf
• Only certain frequency lines are present in the spectrum as only certain high to low energy level transitions are possible within the atom.

The emitted light are analyzed with a spectrometer and discrete bright lines in a dark background are observed.

The well-defined separation of lines is experimental evidence for the existance of separate or ‘quantized’ energy levels in the atom. No two gases give the same exact line spectrum.

If the gases used are not at low pressure, there will be a continuous range of colours . At high pressure, tightly packed gas atoms or molecules will be vibrating, rotating or colliding with each other, such that many more energy levels will be created. Hence, there will be no separated/isolated lines of definite frequency.

Emission Line Spectrum of hydrogen

Emission Line Spectrum of iron

Absorption Line Spectrum

White light is used to excite the atoms. Those incident photons whose energies are exactly equal to the difference between the atom’s energy levels are being absorbed. Since the energy levels are discrete, only photons of certain frequencies are absorbed. As these frequencies of light are now missing, they account for the dark lines in the absorption spectrum, which is discrete dark line on a continuous spectrum

Steps to obtain absorption line spectrum:

• Gases such as hydrogen are placed in a tube.
• White light is passed through the tube.
• The atoms of the gas absorb light of the same wavelengths which they can emit, and then re-radiate the same wavelengths almost immediately but in all directions. Hence, the parts of the spectrum corresponding to these wavelengths appear dark by comparison with the other wavelengths not absorbed.

Absorption Line Spectrum of hydrogen

Distinguish between emission and absorption line spectra

In the case of an emission spectrum, the atom is first excited by a colliding electron. The colliding electron must have kinetic energy greater than or equal to the difference between energy levels of the atom. When the atom goes back to its ground state, either directly or via the intermediate energy levels, photon of only certain frequencies are emitted due to the discrete energy levels. Hence only certain frequencies of light are observed, forming the emission spectrum, which is discrete bright coloured lines on a dark background.

In the case of an absorption spectrum, white light is used to excite the atoms. Those incident photons whose energies are exactly equal to the difference between the atom’s energy levels are being absorbed. SInce the energy levels are discrete, only photons of certain frequencies are absorbed. As these frequencies of light are now missing, they account for the dark lines in the absorption spectrum, which is discrete dark line on a continuous spectrum.

Back To Quantum Physics And Lasers (A Level Physics)

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• Published: 14 September 2024

Quantitative separation of CEST effect by R ex -line-fit analysis of Z-spectra

• Gang Xiao 1   na1 ,
• Xiao-Lei Zhang 2   na1 ,
• Si-Qi Wang 3 ,
• Shi-Xin Lai 3 ,
• Ting-Ting Nie 4 ,
• Yao-Wen Chen 3 ,
• Cai-Yu Zhuang 2 ,
• Gen Yan 5 &
• Ren-Hua Wu 2  

Scientific Reports volume  14 , Article number:  21471 ( 2024 ) Cite this article

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• Mathematics and computing
• Medical research
• Molecular medicine

The process of chemical exchange saturation transfer (CEST) is quantified by evaluating a Z-spectra, where CEST signal quantification and Z-spectra fitting have been widely used to distinguish the contributions from multiple origins. Based on the exchange-dependent relaxation rate in the rotating frame ( R ex ), this paper introduces an additional pathway to quantitative separation of CEST effect. The proposed R ex -line-fit method is solved by a multi-pool model and presents the advantage of only being dependent of the specific parameters (solute concentration, solute‐water exchange rate, solute transverse relaxation, and irradiation power). Herein we show that both solute‐water exchange rate and solute concentration monotonously vary with R ex for Amide, Guanidino, NOE and MT, which has the potential to assist in solving quantitative separation of CEST effect. Furthermore, we achieve R ex imaging of Amide, Guanidino, NOE and MT, which may provide direct insight into the dependency of measurable CEST effects on underlying parameters such as the exchange rate and solute concentration, as well as the solute transverse relaxation.

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Introduction.

Chemical exchange saturation transfer (CEST)-magnetic resonance imaging (MRI) is a new contrast enhancement technique that indirectly measures molecules with exchangeable protons and exchange-related properties, providing high detection sensitivity 1 , 2 , 3 , 4 . In practice, the saturation transfer effects of CEST-MRI are often assessed and quantified using a Z-spectra where the water signal is plotted as a function of the applied saturation frequency. For in vivo CEST-MRI, proper parameter quantifications demand careful measurement of the CEST effects (uncontaminated and with sufficient SNR) such as solute concentration and solute‐water exchange rate, thus rendering quantitative CEST a challenging task 5 .

Theoretically, the CEST parameter quantification through Z-spectra fitting demonstrated by Bloch-McConnell (BM) equations could provide a feasible approach, and yet there are the common problems of slow operation speed and converging to local optimal solution. Nevertheless, scholars spent their efforts and carried on studies of CEST quantification in another way. The symmetric magnetization transfer ratio (MTR asym ) that calculated from a Z-spectra is the most used CEST quantification method 6 , 7 , 8 . However, MTR asym is confounded by several types of contamination, including direct saturation (DS), semisolid macromolecular magnetization transfer (MT) and nuclear Overhauser effect (NOE) 9 , 10 .

To further boost CEST specificity, Z-spectra fitting has been successfully applied to differentiate the contributions from multiple origins 11 , such as multiple‐pool Lorentzian fit 12 , 13 , 14 , the Lorentzian difference (LD) analysis 15 , 16 , and three‐point method 11 , 12 . For a specific solute along with overlapping signals from nearby pools, the LD analysis that employs a single Lorentzian line may not provide an accurate reference signal 10 , 16 . The same problem would still occur for a three‐point method that relies on two nearby signals as a reference. The multiple‐pool Lorentzian fit strongly relies on assumption that each CEST signal can be approximated as a Lorentzian lineshape 17 .

Recent advancements in the quantification methods of CEST and NOE techniques have significantly improved their application in biomedical imaging, particularly in the context of brain tumors detection 18 , 19 , 20 , 21 , 22 . For example, Glang et al. proposed a deep neural network with uncertainty quantification that can efficiently and accurately predict Lorentzian parameters from CEST MRI spectra, providing fast and reliable CEST contrast image reconstruction while indicating prediction trustworthiness 18 . Cui et al. proposed a new method termed as 2π-CEST to reduce the contribution from APT in detecting NOE, offering a more accurate strategy than the conventional asymmetric analysis 20 . The study concludes that NOE (− 3.5 ppm) serves as a highly sensitive MRI contrast for imaging membrane lipids in the brain, with lipids being the primary contributor to NOE (− 3.5 ppm) signals, rather than proteins, explaining variations in signals between tumors, gray matter, and white matter 21 .

Theoretically, the CEST specificity through Z-spectra rely on the pool size, exchange rate and relaxation time, as described by BM equations. Particularly, the exchange-dependent relaxation rate in the rotating frame ( R ex ) that solved from the BM equations by an eigenspace approach, operates independently of non‐specific tissue parameters and depends on specific parameters (solute concentration, solute‐water exchange rate, solute transverse relaxation and irradiation power), therefore it is able to make the CEST more specific 2 , 4 , 16 .

In this paper, a voxel-wise R ex -line-fit method is developed to improve the reliability of Z-spectra fitting and investigate the potential of quantitative separation (Fig.  1 ), in which the simulation of a 5-pool model is used to complement the program capabilities. Our study first elucidates the relationship between R ex and parameters such as solute concentration, solute‐water exchange rate and T 2,s . Then the R ex imaging of Amide, NOE (− 3.5 ppm), Guanidino and MT is achieved by our method. Finally, we apply the R ex -line-fit to study CEST effect in a brain tumor model, and the performance of this method in fitting quality is evaluated.

Flow chart of data processing steps of R ex based approach.

Materials and methods

Exchange-dependent relaxation rate in the rotating frame ( r ex ).

The resulting solution for the Z-spectra can be described by the monoexponential decay of the z-magnetization as a function of time with the rate R 1ρ 2

where P zeff is the projection factor on z-axis of the effective frame, t sat is saturation time.

In the case of steady-state, the resulting solution for the Z-spectra at each offset \(\Delta \omega\) simplifies to 23 , 24

where \(Z^{{{\text{ss}}}}\) is the steady-state condition, \(R_{1w}\) denotes the longitudinal relaxation rate of water, and \(\theta = \tan^{ - 1} \left( {{{\omega_{1} } \mathord{\left/ {\vphantom {{\omega_{1} } {\Delta \omega }}} \right. \kern-0pt} {\Delta \omega }}} \right)\) where \(\omega_{1} = \gamma B_{1}\) is the amplitude of the RF field. The \(R_{{{\text{eff}}}}\) which describes the relaxation of free water in the rotating frame can be approximated by

Further, the \(R_{{{\text{ex}}}}\) at a particular off-resonant frequency \(\Delta \omega\) for a general exchanging pool i is 2

where \(f_{i}\) is a fraction of the total proton for the i th pool, \(k_{i}\) is its exchange rate with water in Hz, \(R_{{\text{2,i}}}\) is its transversal relaxation rate, \(\delta \omega_{i}\) is its frequency offset in Hz, \(\Delta \omega_{i}\) is the difference in Larmor frequency between pool i and water, and the full width half maximum \(\Gamma_{i}\) is

Multiple-pool Lorentzian-line-fit

To estimate CEST effects from individual components, we performed the multiple-pool Lorentzian fitting of Z spectra using a non‐linear optimization algorithm 25 :

Equation ( 7 ) represents a Lorentzian line with central frequency offset from water ( \(\Delta_{i}\) ), peak FWHM ( \(W_{i}\) ), and peak amplitude ( \(A_{i}\) ). The value of N is the number of fitted pools; S is the measured signal on the Z-spectra; and S 0 is the non-irradiation control signal.

In this study, a five-pool model of Lorentzian-line-fit including Amide at 3.5 ppm ( L 1 ), Guanidyl/Amine at 2.0 ppm ( L 2 ), Water at 0 ppm ( L 3 ), MT at − 2.4 ppm ( L 4 ), and NOE at − 3.5 ppm ( L 5 ) was performed to estimate CEST effects from individual components.

In vivo MR imaging

All animal care and experimental procedures comply with the National Research Council Guide for the Care and Use of Laboratory Animals. All animal experiments were approved by the Ethics Committee of Shantou University Medical College (Approval ID: SUMC2022-204) and conducted in accordance with the ARRIVE guidelines.

For our study, we used 8-week-old male SD rats (Beijing Vital River Laboratory Animal Technology Co., Ltd.) weighing approximately 250 g to establish a tumor-bearing model. In this study, three rats were prepared, where two rats were excluded from the present study due to tumor modeling failure that could not be used during data analysis. To implant the rat glioma C6 cells, a 10 µL suspension containing approximately 2 × 10 6 cells was injected into the right basal ganglia of the rats using a Hamilton syringe and a 30-gauge needle. After two weeks of tumor cell implantation, the rats underwent MRI. During the MRI procedure, the rats were anesthetized with a mixture of isoflurane and O 2 at a rate of 1 L/min. Anesthesia induction was achieved using 4.0% isoflurane, followed by maintenance anesthesia with 2.0–3.0% isoflurane. To monitor the breath rate, a respiratory probe was utilized throughout the MRI experiments. The rats' respiration rate and body temperature during the 7 T scan were maintained at approximately 60–70 breaths per minute and 38.5–39.5 °C, respectively.

MRI was conducted using a 7T horizontal bore small animal MRI scanner (Agilent Technologies, Santa Clara, CA, USA) equipped with a surface coil (Timemedical Technologies, China) for both transmission and reception. The positioning of the rat was carefully done to ensure that the tumor was accurately centered within the magnetic field. Imaging parameters were as follows: B 1  = 1 µT, repetition time (TR) = 6000 ms, echo time (TE) = 40 ms, array = frequency offsets, slice thickness = 2 mm, field of view (FOV) = 64 × 64 mm, matrix size = 64 × 64, spatial resolution = 1 × 1 mm, averages = 1. An echo planar imaging (EPI) readout sequence was employed to acquire CEST images, utilizing continuous wave (CW) RF irradiation on the scanners. The saturation time was set to 5.0 s, with 49 frequency offsets evenly distributed from − 6 to 6 ppm relative to the resonance frequency of water.

To assess the performance of the proposed R ex -line-fit, simulated Z-spectra are created using 5-pool system. The R ex fitting is conducted by using a non‐linear least square constrained optimization algorithm and referencing the pool parameters 1 , 26 , 27 in Table 1 . Pseudo-code of our method for R ex imaging and Z-spectra fitting is shown Table 2 . The proposed R ex fitting is compared experimentally to AREX 28 and multiple-pool Lorentzian fit 25 . The AREX is a reduced form of R ex and follows a Lorentzian function 28 , so the same parameters listed in Table 2 is used. Table 3 lists the boundaries of the multiple-pool Lorentzian fit 25 .

Parameter separation

It is worthwhile to evaluate the correlation between R ex and parameters (solute‐water exchange rate k s , solute concentration f s and solute transverse relaxation T 2,s ), which may assist in elucidating the R ex specificity and separating CEST parameters. For Amide, NOE (− 3.5 ppm), Guanidino and MT, Fig.  2 illustrates the correlations between parameters ( k s , f s ) and R ex . The surface plots demonstrate the dependence of R ex on k s and f s , where R ex is linear monotonically increasing with parameters ( k s , f s ) for the Amide, NOE (− 3.5 ppm) and Guanidino. For MT, its R ex is linear monotonically increasing with solute concentration f s , while a nonlinear monotonically increasing relationship between R ex and solute‐water exchange rate k s is observed. It should be noted that the R ex of Guanidino depicts a monotonically increasing trend first and then decrease corresponding to k s , while its R ex follows a monotonically increasing pattern in respect to f s . In addition, R ex is nonlinear monotonically increasing with T 2,s for the Amide, NOE (-3.5 ppm) and Guanidino, while R ex shows a trend of slight monotonic decrease corresponding to T 2,s for MT, as illustrated in Fig.  3 . In addition, Fig.  4 illustrates an example of R ex changing with k s and R 2,s (1/ T 2,s ) for Amide, where R ex follows a increasing pattern in respect to R 2,s at different k s and R ex shows a trend of decrease corresponding k s at different R 2,s .

The correlation between R ex and parameters ( k s , f s ) for Amide, NOE (− 3.5 ppm), Guanidino and MT. For each subplot, the red line denotes the contour.

The correlation between R ex and T 2,s for Amide, NOE (− 3.5 ppm), Guanidino and MT.

An example of R ex changing with k s and R 2,s (1/ T 2,s ) for Amide.

R ex , Lorentzian, and AREX imaging

Herein, we conduct an experiment of R ex , Lorentzian and AREX imaging for Amide at 3.5 ppm, Guanidino at 2.0 ppm, MT at − 2.4 ppm and NOE at − 3.5 ppm, where each pixel of imaging is obtained by computing the peaks of R ex and Lorentzian decompositions. Figure  5 illustrates the R ex , Lorentzian and AREX imaging of these CEST effects. The region of pseudo color image overlaid on anatomy image is the region of interest (ROI), where the region of tumor region is marked with solid red contour and the solid red contour denotes the contralateral region. Visually, the R ex shows different structure distributions on the R ex imaging for Amide, Guanidino, MT and NOE, this is different from Lorentzian and AREX.

The R ex , Lorentzian and AREX imaging of Amide at 3.5 ppm, Guanidino at 2.0 ppm, MT at − 2.4 ppm and NOE at − 3.5 ppm. The solid red contour denotes the tumor region and the dashed red contour is the contralateral region.

Figure  6 illustrate the Z-spectra fitting from the tumor region and the contralateral region using the R ex , Lorentzian and AREX approach, respectively. The results show that the satisfied accuracy and consistency are obtained by the proposed R ex -line-fit and it displays great agreement and follows the same tendency as the actual measurements. Table 4 lists the mean value and standard deviation of residual between the considered fitting approach and the experimental Z-spectra for the tumor region and its contralateral region.

Z-spectra fitting and decomposition from the tumor region and its contralateral region using the R ex , Lorentzian and AREX approaches.

We further applied the linear regression analysis 29 to assess the general performance of the R ex , Lorentzian and AREX approach using the whole ROI data of CEST images at 49 frequency offsets. Figure  7 displays the R ex , Lorentzian and AREX for fitting CEST signal by plotting the linear regression lines between the experimentally acquired data and the fitting. The excellent performance of our R ex method is confirmed by the scatter and linear regression lines, resulting in a very high coefficient of determination ( R 2  = 0.9937).

Linear regression analysis of the R ex , Lorentzian and AREX fitting when they compared with the experimentally acquired data within the ROI.

The exchange-dependent relaxation R ex is an important parameter for CEST effects and can be used to determine the exchange rate k s of the exchangeable protons with concentration fraction f s and transverse relaxation T 2,s . In this study, we proposed a method that can support reliable quantitative separation of CEST effect by R ex . This is important because specificity of in vivo CEST effect is challenging due to careful measurement of the CEST effects. Nevertheless, some discussions should be made as follows.

In the Eq. ( 4 ) of R ex , the first term named ‘ k i -term’ is the dominant factor, which comprises the product of peaks for water pool (‘a-peak’) and CEST pool (‘b-peak’), respectively. The ‘ R 2,i -term’, together with the ‘ \({\text{b - peak}}\) ’, denotes the exchange dependent relaxation that yields an off-resonant peak in CEST. So the R ex turns into two peaks, but unlike Lorentzian and AREX lineshape that gives only one peak. In fact, the effect of water T1 and T2 relaxation time on Lorentzian shape is described by formula in Ref. 17 . In contrast, R ex excludes water T1 and T2 contributions, which serves as a tool for calculating the CEST signal, offering a more representative depiction of chemical exchange processes than traditional CEST analysis methods 30 , 31 . As a reduced form of R ex , AREX is a Lorentzian function 28 , excluding the water pool (‘a-peak’), unlike the complete Rex expression 2 , 28 . It is interesting to study the ‘a-peak’ imaging, which will be presented in our next work.

In the R ex imaging, the tumor regions marked with solid red contour show reduced values in correspondence with the contralateral regions (Fig.  5 ). In practice, the exchange rate k i can be determined by analyzing the CEST signal as a function of pH: k amide  = 5.57 × 10 pH–6.4 , k guanidyl  = 5.57 × 10 pH–6.4 32 . An intuitive explanation is that the reduced exchange rate with lower pH in and around the tumor region causes the lowering of the R ex peak values (Fig.  6 ). In fact, the mechanism behind tumors is more complex compared with the clear process of stroke. Particularly, the R ex mechanism is considered that many factors (the exchange rate k s , the concentration fraction f s and transverse relaxation T 2,s ) participated in this process (Eq.  4 ). To some extent, the R ex imaging shows the specificity for different chemical groups, because different structure distributions on the R ex imaging for Amide, Guanidino, MT and NOE are obtained, this is different from multiple-pool Lorentzian, AREX and T 1 map (Fig.  5 ). Nevertheless, some multi-pool models have been applied to tumor research, such as Refs. 33 , 34 .

In this study, we have made some meaningful explorations and obtained promising research results. We first determined whether parameters (solute‐water exchange rate and solute concentration) and R ex have a monotonous relationship for Amide at 3.5 ppm, Guanidino at 2.0 ppm and NOE at − 3.5 ppm (Figs. 2 , 3 , 4 ). With this knowledge in mind, this makes it possible to isolate some part parameters by extending previous approaches, where numerical simulations of R ex can be used to obtain saturation parameters for CEST effect.

We further implemented R ex as a novel model to provide high-accuracy CEST fitting and decomposition where multiple CEST saturation transfer pools are present. The proposed R ex -line-fit avoids specific selection of tissue parameters and minimizes operator bias, enabling adaptive fitting and decomposition for reliable estimation of CEST effects. The accuracy of R ex -line-fit was first validated by the test of in vivo mouse, which revealed that R ex method provided a near-perfect approximation to the experimentally acquired Z-spectra (Table 4 , Figs. 6 and 7 ).

As an improvement method that only is dependent of the specific parameters (solute concentration, solute‐water exchange rate, solute transverse relaxation, and irradiation power), our R ex -line-fit can provide a simple, robust and more accurate approach for approximating CEST and further serve for quantitative separation of CEST effect. More in vivo validations and at the clinical field strength will be performed in the future.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

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The study was supported by the National Natural Science Foundation of China (Grant/Award Number: 82020108016), the Grant for Key Disciplinary Project of Clinical Medicine under the Guangdong High-Level University Development Program (Grant/Award Number: 002-18120302), the Functional Substances in Medicinal Edible Resources and Healthcare Products (Grant/Award Number: 2021B1212040015), the 2021 Medical Research Foundation of Guangdong (Grant/Award Number: 202011102275838), the 2021 Grant for Key Science Technology and Innovation Project under the Guangdong Jieyang Development Program (Grant/Award Number: 210517084612609), and the Medical Health Science and Technology Project of Shantou (Grant/Award Number: 2022-88-16).

Author information

These authors contributed equally: Gang Xiao and Xiao-Lei Zhang.

Authors and Affiliations

School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China

Department of Radiology, Second Affiliated Hospital of Shantou University Medical College, Shantou, 515041, China

Xiao-Lei Zhang, Cai-Yu Zhuang & Ren-Hua Wu

College of Engineering, Shantou University, Shantou, 515063, China

Si-Qi Wang, Shi-Xin Lai & Yao-Wen Chen

Department of Radiology, Hubei Cancer Hospital, Tongji Medical College, Huazhong University of Science and Technology, Wuhan, 430079, China

Ting-Ting Nie

Department of Radiology, Second Affiliated Hospital of Xiamen Medical College, Xiamen, 361021, China

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GX: Conceptualization, Methodology, Software; X.-L.Z: Methodology, Formal analysis, Investigation, Validation, Visualization, Writing – original draft; S.-Q.W: Formal analysis, Data curation; S.-X.L: Formal analysis, Data curation; T.-T.N: Resources, Data curation; Y.-W.C: Investigation, Supervision; C.-Y.Z: Resources, Data curation; GY: Investigation, Supervision, Writing – review & editing; R.-H.W: Funding acquisition, Project administration, Writing – review & editing.

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Correspondence to Gen Yan or Ren-Hua Wu .

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Xiao, G., Zhang, XL., Wang, SQ. et al. Quantitative separation of CEST effect by R ex -line-fit analysis of Z-spectra. Sci Rep 14 , 21471 (2024). https://doi.org/10.1038/s41598-024-72141-4

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Received : 15 January 2024

Accepted : 04 September 2024

Published : 14 September 2024

DOI : https://doi.org/10.1038/s41598-024-72141-4

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Crash of a Tupolev TU-104B in Omsk

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To calculate the distance between Bucharest and Omsk, the place names are converted into coordinates (latitude and longitude). The respective geographic centre is used for cities, regions and countries. To calculate the distance the Haversine formula is applied.