diff --git a/1 Introduction to qMRI/01-intro.md b/1 Introduction to qMRI/01-intro.md index 41cdb43..86f3ba7 100644 --- a/1 Introduction to qMRI/01-intro.md +++ b/1 Introduction to qMRI/01-intro.md @@ -7,13 +7,13 @@ authors: - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada --- -This section starts by explaining the distinction between MRI and quantitative MRI (qMRI), which is of essence to the central premise of this MOOC. +This section starts by explaining the distinction between MRI and quantitative MRI (qMRI), which is of essence to the central premise of this mOOC. Next, it aims at delivering an intuitive understanding of how MRI works by using cartoons, simulations and example applications, all introduced in the context of overarching concepts from physics and everyday life. ::: {admonition} See also :class: seealso -For a more theoretical introductory explanation, the reader is referred to [Nishimura](https://en.wikipedia.org/wiki/Dwight_Nishimura) [-@Nishimura:1996uc]. +For a more theoretical introductory explanation, the reader is referred to [@Nishimura:1996uc]. ::: After covering the basics of MRI, the relationship between data acquisition and parameter estimation will be explained based on two basic qMRI applications: _T_{sub}`1` and _T_{sub}`2` mapping. diff --git a/1 Introduction to qMRI/02-million_dollar.md b/1 Introduction to qMRI/02-million_dollar.md index 03adc51..2d39ab6 100644 --- a/1 Introduction to qMRI/02-million_dollar.md +++ b/1 Introduction to qMRI/02-million_dollar.md @@ -23,7 +23,7 @@ We will start answering this question by looking at the most prominent use cases In the clinics, MRI stands out as one of the most preferred imaging methods, because it can generate detailed images with superb soft tissue contrast, without using ionizing radiation or cutting open the human body. Surprisingly, MRI scanners have also been extensively used in food science to study soft tissue. For example, several studies used MRI to observe how moisture migrates towards the center of jellybeans over time [@Troutman:2001vk; @Ziegler:2003th] Be it in diagnostic radiology, or in food science, it is the superior soft tissue contrast that makes MRI appealing. In routine diagnostic readings, the radiologists browse through MR images to capture abnormalities that may be resolved by conventional MRI contrasts, i.e. `T1-` or `T2-weighted` images. As a result, the detection of pathological patterns depends on a radiologists’ visual assessment, which is then transferred to a written report – _a narration of observations_ – such as: - > _T_{sub}`2` hyperintense appearance in the left parieto-occipital lobe sug- gests hemorrhagic infarction [Fig. %sf](#intFig1). + > _T_{sub}`2` hyperintense appearance in the left parieto-occipital lobe suggests hemorrhagic infarction [Fig. %sf](#intFig1). Here, the word `hyperintense` implies a relative comparison. [Fig. %se](#intFig1) illustrates that cropping the tumorous region away from the image removes the basis of comparison and makes the hyperintense appearance irrelevant. This is because the pixel brightness of conventional MR images is assigned using an arbitrary scale consisting of shades of gray. Due to the lack of a calibrated measurement scale, conventional MRI is considered to be qualitative. @@ -37,7 +37,7 @@ An illustrative comparison between the conventional and quantitative MRI (qMRI). Using the same MRI scanner, it is possible to assign meaningful numbers to the images and this approach turns out to be the most common MRI method in food engineering [@Mariette:2012tv; @Ziegler:2003th]. [](#intFig1) illustrates the added value of quantitative MRI (qMRI) when applied to a sample familiar to everyone: a jellybean. The moisturization map indicates that the jellybean has formed a crispy shell while remaining chewy at the center, which is the desired texture [Fig. %sb](#intFig1). Given that the level of chewiness is determined by a threshold on a standardized measurement scale, a randomly selected part of the image can be still characterized by comparing selected pixel values against the established threshold [](#intFig1). This feature of qMRI offers an objective insight into how the texture of this soft confection changes over time, which would help determine its best before date ([](#intFig1), prognosis). -The ability to reveal what underpins the appearance of visually similar samples is yet an- other powerful feature of qMRI. In a [Bean-Boozled](https://en.wikipedia.org/wiki/Jelly_Belly) challenge, which is a Russian roulette of jellybean flavors, tasty flavors are mixed with nauseous look-alikes [@Gambon:2015uq]. For example, a green jellybean may taste like lime (tasty) or lawn clippings (nauseous) in the Bean-Boozled game [](#intFig2). Therefore, no matter how experienced the player is, the chances of picking up a lime-flavored bean is as good as tossing a coin. Conventional MR images of a handful of green jellybeans do not offer a distinguishing feature, but only reveal their structure. As a result, the chances of making an unfortunate choice remain the same [](#intFig2). +The ability to reveal what underpins the appearance of visually similar samples is yet another powerful feature of qMRI. In a [Bean-Boozled](https://en.wikipedia.org/wiki/Jelly_Belly) challenge, which is a Russian roulette of jellybean flavors, tasty flavors are mixed with nauseous look-alikes [@Gambon:2015uq]. For example, a green jellybean may taste like lime (tasty) or lawn clippings (nauseous) in the Bean-Boozled game [](#intFig2). Therefore, no matter how experienced the player is, the chances of picking up a lime-flavored bean is as good as tossing a coin. Conventional MR images of a handful of green jellybeans do not offer a distinguishing feature, but only reveal their structure. As a result, the chances of making an unfortunate choice remain the same [](#intFig2). ```{figure} ./img/int_fig2.jpg :label: intFig2 diff --git a/1 Introduction to qMRI/03-journey.md b/1 Introduction to qMRI/03-journey.md index db24b83..90f229c 100644 --- a/1 Introduction to qMRI/03-journey.md +++ b/1 Introduction to qMRI/03-journey.md @@ -28,11 +28,11 @@ Once the mass and charge of the hydrogen particles were known, nobody suspected #### Picking up a hydrogen atom in the quantum realm: The perfect match -The hydrogen atom comes online only when standing up under the influence of a magnetic field (Figure 2.3a,b). The chances of getting a response from the hydrogen firstly depends on whether our message kindles just the right amount of excitement for it to switch between those hyper-finely separated energy levels (Figure 2.3c). Although it almost seems like the hydrogen is sidestepping a conversation, all it takes is finding the right wavelength to meet this first requirement. Six years after the introduction of the spin concept, Rabi and Breit finally discovered that to resonate with hydrogen’s energy levels, we need to send our messages in the radiofrequency (RF) range of the electromagnetic spectrum (Breit and Rabi, 1931). +The hydrogen atom comes online only when standing up under the influence of a magnetic field ([](#intFig3)a,b). The chances of getting a response from the hydrogen firstly depends on whether our message kindles just the right amount of excitement for it to switch between those hyper-finely separated energy levels ([](#intFig3)). Although it almost seems like the hydrogen is sidestepping a conversation, all it takes is finding the right wavelength to meet this first requirement. Six years after the introduction of the spin concept, Rabi and Breit finally discovered that to resonate with hydrogen’s energy levels, we need to send our messages in the radiofrequency (RF) range of the electromagnetic spectrum (Breit and Rabi, 1931). -However, not all hydrogen atoms behave the same. We need to be familiar with the peculiarities of the hydrogen we are in touch with. There are two key attributes: where is the hydrogen from, and in which energy state it is when our resonating message is delivered (Figure 2.3c). The last nuance to get on the same wavelength is finding the right angle to approach it. If we meet all the requirements, we will see the hydrogen getting excited and responding to us within a certain RF bandwidth. +However, not all hydrogen atoms behave the same. We need to be familiar with the peculiarities of the hydrogen we are in touch with. There are two key attributes: where is the hydrogen from, and in which energy state it is when our resonating message is delivered ([](#intFig3)c). The last nuance to get on the same wavelength is finding the right angle to approach it. If we meet all the requirements, we will see the hydrogen getting excited and responding to us within a certain RF bandwidth. -For the imaging of the human body using MRI, we will be mostly communicating with the hydrogen from the water (Figure 2.4). In general, water hydrogens are more easy-going because their electron spin states are balanced, so we are only concerned with the energy levels emerging from their nuclei. This is why we call this pick-up line the nuclear magnetic resonance (NMR). +For the imaging of the human body using MRI, we will be mostly communicating with the hydrogen from the water ([](#intFig4)). In general, water hydrogens are more easy-going because their electron spin states are balanced, so we are only concerned with the energy levels emerging from their nuclei. This is why we call this pick-up line the nuclear magnetic resonance (NMR). ```{figure} ./img/int_fig3.jpg :label: intFig3 @@ -41,7 +41,7 @@ For the imaging of the human body using MRI, we will be mostly communicating wit a) A hydrogen atom has one proton and one electron, with each particle is assumed to have two possible spin states for simplicity (up or down). In the absence of a magnetic field, the atom is at a random orientation (offline) and shows a single energy level (the red line). b) Under a uniform magnetic field (B0), the atom is aligned with the field (online) and shows multiple energy levels (the colored lines). These energy levels are in quantum superposition: all the levels simultaneously exist and we can only determine one state upon observation. This behaviour of the energy levels is represented by the Schrödinger’s cat2. c) Once it is online, the hydrogen atom can be contacted through a communication line that operates within the radiofrequency (RF) range. ``` -Figure 2.4 shows that the hydrogen from water has two energy levels under a uniform magnetic field: low energy and high energy. If the resonating message flips the hydrogen’s energy state from higher (red) to lower (cyan) state, the result is a “radio silence” (Figure 2.4a), i.e. the hydrogen gives up a small amount of excess energy instead of a response. But if the resonating message elevates the hydrogen’s mood from lower (red) to the higher (cyan) state, the hydrogen gets excited (Figure 2.4b). After the message is delivered, we will finally get a response as the excitement quickly fades away. This final process is termed “relaxation”, which is of essence to the MRI contrast, because the message carries information about where that hydrogen is from. +[](#intFig4) shows that the hydrogen from water has two energy levels under a uniform magnetic field: low energy and high energy. If the resonating message flips the hydrogen’s energy state from higher (red) to lower (cyan) state, the result is a “radio silence” ([](#intFig4)a), i.e. the hydrogen gives up a small amount of excess energy instead of a response. But if the resonating message elevates the hydrogen’s mood from lower (red) to the higher (cyan) state, the hydrogen gets excited ([](#intFig4)b). After the message is delivered, we will finally get a response as the excitement quickly fades away. This final process is termed “relaxation”, which is of essence to the MRI contrast, because the message carries information about where that hydrogen is from. ```{figure} ./img/int_fig4.jpg :label: intFig4 @@ -52,20 +52,20 @@ a) After receiving the call, the hydrogen atom from water switches from the high The tone in the response shifts slightly if the hydrogen is from non-water molecules, e.g., fat or protein. This slight difference is caused by the amount of negative energy a hydrogen is surrounded by, which interferes with the contribution of hydrogen’s electron to its energy state, namely shielding. The more negative energy around the hydrogen, the shorter the response. It appears that the positive effects of being near water on the energy levels tran- scends scales, from humans themselves (Cracknell et al., 2016) to the hydrogen atoms that make up them (Lawrence and McDonald, 1966). -So far we looked at the quantum-level interactions between the hydrogen atom and RF energy, and tied it with the NMR phenomenon. However, in reality, we don’t have access to observe NMR effects at such a fine-grained level; because no appropriate instrumentation exists, and quantum-jitters make such instrumentation nearly impossible (Erkintalo, 2021). For example, we cannot detect the uncertainty of a single hydrogen atom’s energy levels. The best we can do is to visualize the concept using metaphorical illustrations, such as using Schrödinger’s cat to imply the quantum state of the hydrogen’s energy levels in Figure 2.3b. Until we observe the consequences (whether the hydrogen will respond to our resonant message or not), all the energy levels are assumed to be in quantum superposition, even for only two energy states of proton spins as shown in Figure 2.4. To achieve observational accuracy, we need to move from the uncertainties of the energy levels to a probability of getting a response to our resonant message, which is what we will look at in the following section. +So far we looked at the quantum-level interactions between the hydrogen atom and RF energy, and tied it with the NMR phenomenon. However, in reality, we don’t have access to observed NMR effects at such a fine-grained level; because no appropriate instrumentation exists, and [quantum-jitters](wiki:Quantum_fluctuation) make such instrumentation nearly impossible (Erkintalo, 2021). For example, we cannot detect the uncertainty of a single hydrogen atom’s energy levels. The best we can do is to visualize the concept using metaphorical illustrations, such as using [Schrödinger’s cat](wiki:Schrödinger%27s_cat) to imply the quantum state of the hydrogen’s energy levels in [](#intFig3)b. Until we observe the consequences (whether the hydrogen will respond to our resonant message or not), all the energy levels are assumed to be in quantum superposition, even for only two energy states of proton spins as shown in [](#intFig4). To achieve observational accuracy, we need to move from the uncertainties of the energy levels to a probability of getting a response to our resonant message, which is what we will look at in the following section. #### Finding the perfect quantum match is not practical, but there are plenty of fish in the sea -We can harness the benefits of NMR without having a complete theoretical understanding of the underlying quantum interactions (see proton spin crisis (Siegel, 2017)), because these effects simply smooth over at the macroscale. At this point we will leave the online dating analogy behind, because even at the smallest macroscopic scale, “there are plenty of fish in the sea” (Figure 4a). At the level where an NMR measurement is technically feasible, we will be concerned with a large pool of hydrogen atoms. Here, the individual behaviour of particles becomes useless for characterizing the system in aggregate. This concept of integrating over microscopic details to achieve a compact and useful system description is coarse-graining (Figure 2.5). +We can harness the benefits of NMR without having a complete theoretical understanding of the underlying quantum interactions (see proton spin crisis (Siegel, 2017)), because these effects simply smooth over at the macroscale. At this point we will leave the online dating analogy behind, because even at the smallest macroscopic scale, “there are plenty of fish in the sea” ([](#intFig4)a). At the level where an NMR measurement is technically feasible, we will be concerned with a large pool of hydrogen atoms. Here, the individual behaviour of particles becomes useless for characterizing the system in aggregate. This concept of integrating over microscopic details to achieve a compact and useful system description is coarse-graining ([](#intFig5)). ```{figure} ./img/int_fig5.jpg :label: intFig5 :align: center -At the macro-scale, the quantum mechanical behaviour of individual hydrogen atoms is averaged over, i.e., coarse-grained. Figure 2 illustrates a fine-grained description of the behaviour of a single hydrogen atom in absence and presence of a magnetic field. After coarse graining, the representation simplifies to an arrow passing through the center of the proton, which is aligned parallel (i.e., spin-up, low energy) or antiparallel (i.e., spin-down, high energy) with the magnetic field (blue), or at random if the field is absent (pink). Follow- ing coarse-graining, the terms spin, proton, nuclei or hydrogen can be used interchangeably. +At the macro-scale, the quantum mechanical behaviour of individual hydrogen atoms is averaged over, i.e., coarse-grained. [](#intFig2) illustrates a fine-grained description of the behaviour of a single hydrogen atom in absence and presence of a magnetic field. After coarse graining, the representation simplifies to an arrow passing through the center of the proton, which is aligned parallel (i.e., spin-up, low energy) or antiparallel (i.e., spin-down, high energy) with the magnetic field (blue), or at random if the field is absent (pink). Follow- ing coarse-graining, the terms spin, proton, nuclei or hydrogen can be used interchangeably. ``` -Figure 2.5 shows how the representation of a hydrogen atom is changed after coarse-graining microscopic details of spin interactions. From this point onward in this document, hydrogen will be illustrated as shown in Figure 2.5, and used interchangeably with the terms spin, proton and nuclei. To refer to the energy level associated with a single hydrogen atom, we will describe the magnetic moment (Figure 2.6a): If the proton was merely a rigid body, its rotation about an imaginary axis that passes through its center would create a small angular momentum aligned with that axis. Given that the proton is a charged particle, it also creates a microscopic magnetic moment (μ) as a result of this rotation, which is a vector in the same direction (Figure 2.6a). The ratio between the angular momentum and the magnetic moment yields the gyromagnetic ratio (γ), which is 42.59 MHz for the hydrogen at 1 Tesla (T) magnetic field. This frequency at which the spins rotate is commensurate with the magnetic field strength. The product of the gyromagnetic ratio and the field strength (γB0) yields the Larmor frequency, at which the RF energy must be delivered to achieve nuclear resonance. +[](#intFig5) shows how the representation of a hydrogen atom is changed after coarse-graining microscopic details of spin interactions. From this point onward in this document, hydrogen will be illustrated as shown in [](#intFig5), and used interchangeably with the terms spin, proton and nuclei. To refer to the energy level associated with a single hydrogen atom, we will describe the magnetic moment ([](#intFig6)a): If the proton was merely a rigid body, its rotation about an imaginary axis that passes through its center would create a small angular momentum aligned with that axis. Given that the proton is a charged particle, it also creates a microscopic magnetic moment (μ) as a result of this rotation, which is a vector in the same direction ([](#intFig6)a). The ratio between the angular momentum and the magnetic moment yields the gyromagnetic ratio (γ), which is 42.59 MHz for the hydrogen at 1 Tesla (T) magnetic field. This frequency at which the spins rotate is commensurate with the magnetic field strength. The product of the gyromagnetic ratio and the field strength (γB0) yields the Larmor frequency, at which the RF energy must be delivered to achieve nuclear resonance. ```{figure} ./img/int_fig6.jpg :label: intFig6 @@ -74,7 +74,7 @@ Figure 2.5 shows how the representation of a hydrogen atom is changed after coar a) There are millions of protons (i.e., spin or nuclei) even at the smallest macro- scopic unit volume relevant to the MR imaging of the human body. Each individual proton exhibits an infinitesimally small magnetic moment (μ). b) Without B0, the protons in a spin pool exhibit random alignment. c) In presence of B0, the spins are aligned with the magnetic field (parallel or antiparallel), giving rise to a net magnetization (M). ``` -At the macroscale, we will be concerned with millions of protons at once (Figure 2.6a), even for a unit volume of 1mm3. Absent an external magnetic field, magnetic moment vectors are oriented at random (Figure 2.6b). When a magnetic field is applied, they are aligned either parallel (spin-up, low energy) or antiparallel (spin-down, high energy) with the applied field (Figure 2.6c). Given the vast abundance of these spins, the relevant question becomes: which spin configuration is dominant? According to the second law of thermodynamics, if the spin system is at thermal equilibrium, i.e., no energy enters or leaves the system, the entropy of that spin system increases (Carnot et al., 1899). This omnipresent tendency toward disorder favors low energy (Ferris, 2019). Therefore, the spin system tends to have slightly more low-energy hydrogen atoms (spin-up, parallel). Although the difference is as small as 40 per million protons (Webb, 2016), a net magnetic magnetization (Figure 2.6c) can be observed by a real-world NMR experiment. +At the macroscale, we will be concerned with millions of protons at once ([](#intFig6)a), even for a unit volume of 1mm3. Absent an external magnetic field, magnetic moment vectors are oriented at random ([](#intFig6)b). When a magnetic field is applied, they are aligned either parallel (spin-up, low energy) or antiparallel (spin-down, high energy) with the applied field ([](#intFig6)c). Given the vast abundance of these spins, the relevant question becomes: which spin configuration is dominant? According to the second law of thermodynamics, if the spin system is at thermal equilibrium, i.e., no energy enters or leaves the system, the entropy of that spin system increases (Carnot et al., 1899). This omnipresent tendency toward disorder favors low energy (Ferris, 2019). Therefore, the spin system tends to have slightly more low-energy hydrogen atoms (spin-up, parallel). Although the difference is as small as 40 per million protons (Webb, 2016), a net magnetic magnetization ([](#intFig6)c) can be observed by a real-world NMR experiment. ```{figure} ./img/int_fig7.jpg :label: intFig7 @@ -83,5 +83,5 @@ At the macroscale, we will be concerned with millions of protons at once (Figure The measurement instrumentation is concerned with the relevant degrees of free- dom and an effective theory that explains how these coarse-grained variables respond to the perturbations of the measurement system. Some fundamental components of an MRI mea- surement include a uniform magnetic field generator (i.e., a magnet), an RF transmitter (Tx, yellow), an RF receiver (Rx, purple) and an analog-to-digital converter (ADC). ``` -To perform a real-world measurement, an effective theory is needed to describe how the coarse-grained features of the targeted system (Figure 2.6) changes upon interaction with the measurement instrumentation (Figure 2.7). The effective theory of relaxation for the bulk magnetization M was described by Felix Bloch in 1957, laying one of the cornerstones to bring MRI to reality (Bloch, 1957). A basic instrumentation to study the relaxation of a spin system is depicted in Figure 6c, including a uniform magnetic field (B0) generator, an RF transmission system tuned to the Larmor frequency and an RF receiver coil followed by an analog-to-digital converter (ADC). The following section describes how Bloch equations explain the macroscopic behaviour of the net magnetization and how MRI scanners make use of this effective theory to create images. +To perform a real-world measurement, an effective theory is needed to describe how the coarse-grained features of the targeted system ([](#intFig6)) changes upon interaction with the measurement instrumentation ([](#intFig7)). The effective theory of relaxation for the bulk magnetization M was described by Felix Bloch in 1957, laying one of the cornerstones to bring MRI to reality (Bloch, 1957). A basic instrumentation to study the relaxation of a spin system is depicted in [](#intFig6)c, including a uniform magnetic field (B0) generator, an RF transmission system tuned to the Larmor frequency and an RF receiver coil followed by an analog-to-digital converter (ADC). The following section describes how Bloch equations explain the macroscopic behaviour of the net magnetization and how MRI scanners make use of this effective theory to create images. diff --git a/1 Introduction to qMRI/04-encode.md b/1 Introduction to qMRI/04-encode.md index 7d9b50d..5fdf309 100644 --- a/1 Introduction to qMRI/04-encode.md +++ b/1 Introduction to qMRI/04-encode.md @@ -22,7 +22,7 @@ Following an excitation of the spin system by an RF energy at the Larmor frequen \frac{\mathrm{d}M }{\mathrm{d} t}=\gamma{M\times B_{0}}-\frac{M_{x}\hat{\textbf{i}} + M_{y}\hat{\textbf{j}}}{T2} -\frac{(M_{z} - M_{0})\hat{\textnormal{\textbf{k}}}}{T1} ``` -where `M0` is the initial magnetization of the spin system and `T1` and `T2` are the time constants for the longitudinal and relaxational components of relaxation. The first term of the Equation 2.1 is precessional, and the last two terms are the relaxational components of the Bloch equation. Recall that in thermal equilibrium, the net magnetization is aligned with the applied magnetic field (Figure 2.6c), where the longitudinal component is the net magnetization (`M` = `Mz` = `M0`) and the transverse component equals zero (`Mx` = `My` = `0`). In this case, the last two terms of the equation vanish, leaving the precessional component of the equation. When the phenomenological Equation 2.1 is solved for the longitudinal (`Mz`) and the transverse (`Mxy`) components of the macroscopic magnetization, the explicit solutions are given by: +where `M0` is the initial magnetization of the spin system and `T1` and `T2` are the time constants for the longitudinal and relaxational components of relaxation. The first term of the [](#eq1) is precessional, and the last two terms are the relaxational components of the Bloch equation. Recall that in thermal equilibrium, the net magnetization is aligned with the applied magnetic field ([](#intFig6)c), where the longitudinal component is the net magnetization (`M` = `Mz` = `M0`) and the transverse component equals zero (`Mx` = `My` = `0`). In this case, the last two terms of the equation vanish, leaving the precessional component of the equation. When the phenomenological [](#eq1) is solved for the longitudinal (`Mz`) and the transverse (`Mxy`) components of the macroscopic magnetization, the explicit solutions are given by: ```{math} :label: eq2 @@ -33,13 +33,13 @@ M_{z}(t) = M_{z}(0)e^{\frac{-t}{T1}} + M_0(1-e^{\frac{-t}{T1}}) M_{xy}(t) = M_{xy}(0)e^{\frac{-t}{T2}} ``` -Note that Equation 2.2 describes an exponential recovery for Mz to return the equilibrium after excitation. On the other hand, Equation 2.3 states that the transverse magnetization follows an exponential decay, quickly converging to zero. +Note that [](#eq2) describes an exponential recovery for Mz to return the equilibrium after excitation. On the other hand, [](#eq3) states that the transverse magnetization follows an exponential decay, quickly converging to zero. ### Analogical explanation using a defibrillator -The relationship between the longitudinal and transverse components of magnetization in an NMR experiment can be understood through the analogy of how a defibrillator’s capacitor charges and discharges. As soon as the paramedic hits the shock ⚡️ button, the capacitor abruptly empties to deliver an immediate and strong jolt to the patient (Figure 2.8a). At this moment, the paramedic's focus is on how the energy dissipates across the patient's body, which lies in the transverse plane (`Mxy`). The time it takes from the start of the shock until 37% of the energy remains (`1/e = 0.37`) corresponds to the `T2` time constant of `Mxy`. This process is very brief, similar to the sound of a click. +The relationship between the longitudinal and transverse components of magnetization in an NMR experiment can be understood through the analogy of how a defibrillator’s capacitor charges and discharges. As soon as the paramedic hits the shock ⚡️ button, the capacitor abruptly empties to deliver an immediate and strong jolt to the patient ([](#intFig8)). At this moment, the paramedic's focus is on how the energy dissipates across the patient's body, which lies in the transverse plane (`Mxy`). The time it takes from the start of the shock until 37% of the energy remains (`1/e = 0.37`) corresponds to the `T2` time constant of `Mxy`. This process is very brief, similar to the sound of a click. -After delivering the shock, the capacitor must recharge to the desired level to be ready for the next shock (Figure 2.8b). The time required for the capacitor to reach 63% of its total charge capacity (`1-1/e`) corresponds to `T1` time constant of the `Mz`. This recharging process is slower and is often accompanied by a rising whine or whirring sound, indicating the gradual buildup of energy. +After delivering the shock, the capacitor must recharge to the desired level to be ready for the next shock ([](#intFig8)b). The time required for the capacitor to reach 63% of its total charge capacity (`1-1/e`) corresponds to `T1` time constant of the `Mz`. This recharging process is slower and is often accompanied by a rising whine or whirring sound, indicating the gradual buildup of energy. ```{figure} ./img/int_fig8.jpg @@ -62,7 +62,7 @@ With that said, I wish all our readers to be unfamiliar with the practical appli The use of the defibrillator by a paramedic highlights the distinct nature of `T1` and `T2` relaxation times in the context of energy dissipation and recovery in a repetitive process. However, it lacks the measurement aspect of an NMR experiment. -To complete the analogy, we will design a calibration setup to measure the energy delivered to the patient’s torso. To measure this indirectly, a loop will be placed under the stretcher and the current induced in the loop as a result of the delivered shock to the patient’s body will be recorded (Figure 2.9). The signal observed at the end of each shock corresponds to the free induction decay (FID) in an NMR experiment. +To complete the analogy, we will design a calibration setup to measure the energy delivered to the patient’s torso. To measure this indirectly, a loop will be placed under the stretcher and the current induced in the loop as a result of the delivered shock to the patient’s body will be recorded ([](#intFig9)). The signal observed at the end of each shock corresponds to the free induction decay (FID) in an NMR experiment. ```{figure} ./img/int_fig9.jpg :label: intFig9 @@ -76,7 +76,7 @@ A hypothetical calibration setup: To have a measure of the energy delivered to t Remember that in a spin system, a hydrogen atom precesses at its Larmor frequency (`γB0`). Following an excitation pulse, `Mxy` can be observed, because the on-resonance RF energy nudges all the spins toward rotating synchronously (i.e., in-phase). As they fall out of phase (i.e., dephased), the measured signal fades out. ::: -If the biochemical composition of the excited volume varies spatially, the measured FID will be a summation of slightly varying frequency components. For example, a hydrogen atom from the water has a longer response than a hydrogen from the fat (`T2fat