Tracking Cellular Functions by Exploiting the Paramagnetic Properties of X‐Nuclei

Abstract

The term X‐nuclei summarises all nuclei (except protons) that occur in biological systems carrying a non‐zero nuclear spin. Significant involvement in physiological processes such as maintaining the transmembrane potential of living cells and energy metabolism make these nuclei highly interesting for nuclear magnetic resonance (NMR) experiments. In this chapter, a discussion of the basic physics of nuclei with a nuclear spin >1/2 is presented. On this basis, pulse sequences for the detection of multi quantum coherences (MQCs) are presented and explained. Information contained in the obtained MQC signal is linked to biophysical processes. Applications to study energy metabolism, oxygen consumption, and to track brain metabolites by means of X‐nuclei NMR are discussed as well as the use of functional phantoms, which can bridge the gap between basic biological research and NMR data interpretation.

1. Introduction

Apart from hydrogen (¹H) all nuclei carrying a non‐zero nuclear spin can be used for signal generation in nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI), and magnetic resonance spectroscopy (MRS) experiments. All these nuclei are referred to by the term X‐nuclei. Sodium (²³Na) is the X‐nucleus with the highest NMR sensitivity, and thus, it was already used for imaging 30 years ago [1]. In addition to ²³Na, there are other X‐nuclei, which can be used for MRI/MRS experiments. Together with potassium (³⁹K), ²³Na mainly determines the cell membrane potential. Chlorine (³⁵Cl) is the most abundant anion in the human body, and its intracellular and extracellular concentrations have a huge impact on cell volume regulation. Oxygen (¹⁷O) can give insight into the oxygen consumption of the tissue of interest. Phosphorus (³¹P) is heavily involved in the energy metabolism (e.g. in skeletal muscle). Carbon (¹³C) is the basis of all organic molecules and also enables the possibility to track brain metabolites. During the past years, several imaging techniques have been presented to exploit the signal of these nuclei [2].

The main question arising from the use of X‐nuclei in MRI/MRS experiments is data interpretation. In biological systems, various processes are always occurring simultaneously, making it difficult to link a specific effect to an underlying physiological process. Therefore, methods are needed to bridge the gap between phantom experiments, which are used to develop new measurement techniques, and in vivo experiments. This can be accomplished with functional phantoms [3], which provide a high degree of control over biological processes, and therefore lead to a better understanding of the recorded signals.

This chapter gives an overview on the natural abundant X‐nuclei, their physical properties, and physiological information, which can be obtained from NMR experiments with X‐nuclei. Especially, the physics of X‐nuclei with a nuclear spin >3/2 is discussed in detail with the goal to derive and understand enhanced pulse sequences for the observation of higher coherence orders. Applications for the spin 1/2 X‐nuclei ¹⁷O and ¹³C are also discussed. The chapter closes with the introduction of a MRI compatible bioreactor setup, which can be used to study the response of organotypic cell cultures to external stimulations.

2. NMR sensitivity

Every nucleus used for NMR experiments has different physical properties and thus exhibits a different sensitivity response to radiofrequency (RF) pulses. The NMR sensitivity of a nucleus is given by:

$S\propto C{\gamma }^{3}I(I+1)$E1

Here S is the sensitivity, C is the concentration of the nucleus, $\gamma$ is the gyromagnetic ratio, and I represents the nuclear spin of the nucleus. Admittedly, most X‐nuclei carry a nuclear spin > 1/2 which increases sensitivity. On the other hand, this effect is compensated by the much lower $\gamma$. Values for $\gamma$ (relative to the value for ¹H), nuclear spin, natural abundances, and mean values of in vivo concentrations for the X‐nuclei discussed here, are listed in Table 1. Since the nuclear spin and the gyromagnetic ratio are constants, the only variable in Eq. (1) is the concentration C. In contrast to ¹H, where the in vivo concentration reaches a molar (mol/l i.e. M) level, the concentrations of natural abundant X‐nuclei is in the range of mM (mmol/l). For instance, ²³Na is the most abundant X‐nucleus with the highest $\gamma$ value but compared to ¹H the sensitivity is approximately 20,000 times lower.

Nucleus	Nuclear spin	Rel.γ	Natural abundance [%]	Mean in vivo; concentration [mM]
^1H	$1/2$	$1.00$	$99.98$	88,0001
^13C	$1/2$	$0.25$	$1.11$	n. a.
^17O	$5/2$	$-0.14$	$0.037$	161
^23Na	$3/2$	$0.26$	$100.00$	402
^31P	$1/2$	$0.040$	$100.00$	751
^35Cl	$3/2$	$0.098$	$75.77$	363
^39K	$3/2$	$0.047$	$93.10$	1244

Table 1.

Nuclear spin values, relative gyromagnetic ratios, natural abundances, and mean values of in vivo concentrations of different X‐nuclei.

¹Abundance in the whole body.

²Tissue ²³Na concentration in human brain.

³Tissue ³⁵Cl concentration in human brain.

⁴Concentration in human calf muscle. Since ¹³C is mostly used for labelled precursors a value for in vivo concentration is lacking [4–8].

3. ²³Na, ³⁵Cl, and ³⁹K: interaction with proteins

The transmembrane potential mainly arises from concentration gradients of different ions between the intracellular and the extracellular compartments. Three X‐nuclei, ²³Na, ³⁵Cl, and ³⁹K, and their intracellular/extracellular distribution play a major role in generating this potential. All three nuclei have a nuclear spin equal to 3/2, which leads to a nuclear quadrupolar moment Q and a fast decay of the NMR signal.

The nuclear quadrupolar moment is a measure of the deviation of the nuclear charge distribution from the spherical shape. Positive values of Q indicate, that the nuclear charge distribution takes the shape of a prolate spheroid while negative values indicate, that the nucleus takes the form of an inflate spheroid. The values for Q and the resulting shape of the discussed nuclei are listed in Table 2.

Nucleus	Quadrupolar moment Q [b]	Shape
^17O	−0.026^*	Inflate spheroid
^23Na	+0.10^*	Prolate spheroid
^35Cl	−0.10^*,†	Inflate spheroid
^39K	+0.049^*,†	Prolate spheroid

Table 2.

Quadrupolar moments and shapes of nuclei with nuclear spin >1/2.

^*Polarization or Sternheimer corrections incorporated.

^†Average value [9].

The quadrupolar moment also implicates an additional electrical field which, in turn, leads to an additional contribution to the potential energy of the nucleus. This means that the energy levels, which split due to the Zeeman effect,¹ will be shifted due to the quadrupolar interaction, ω_Q, which is related to the electrical field induced by the quadrupolar moment. A description of the electrical field can be formulated using of the electrical field gradient (EFG) tensor, which can be found in references [10, 11]. Figure 1 shows an illustration for the Zeeman effect in case of a spin 3/2 nucleus. On the left hand side of the figure, there is no external magnetic field and the energy levels are degenerate. Applying an external magnetic field nulls the degeneration and the energy levels split. In biological systems, the quadrupolar interaction can become time dependent. Therefore, it is useful to refer to the time averaged quadrupolar interaction ⟨ω_Q⟩. For a non‐zero external magnetic field and ω_Q = 0, this is shown in the middle of Figure 1. The right hand side of Figure 1 shows the case with an applied external magnetic field, where ω_Q ≠ 0. It can be seen that the inner transitions are shifted to lower energies while the outer transitions are shifted to higher energies. This leads to an alteration of the transition frequencies for the outer transitions.

Figure 1.

Zeeman effect in a spin 3/2 system. (Left) In the absence of an external magnetic field, the energy levels are degenerated. (Middle) Application of an external magnetic field leads (in the absence of a quadrupolar interaction) to an equidistant splitting of the energy levels. (Right) An additional quadrupolar interaction leads to a shift of the inner levels towards lower energy while the outer levels are shifted upwards. The end result is an alteration of the outer transitions.

The following paragraphs contain the description of the physical properties of signal generation and relaxation these nuclei. From this basis, dedicated pulse sequences for the observation of multi quantum coherences (MQCs) are derived. Applications of these sequences to phantoms are shown while the obtained data are analysed with the link to a physiological interpretation.

3.1. Quadrupolar relaxation

This section deals with the specific relaxation processes of quadrupolar nuclei. The detailed understanding of these processes requires a strong background in quantum physics, and the use of the irreducible tensor formalism which is beyond the scope of this book. For this reason, only the key results of the quantum mechanical description are presented in this chapter. The reader can find a more detailed description in references [10–12].

Taking this into account one can deduce the following two respective expressions for the real and imaginary parts (J_m and K_m) of the spectral density [11]:

$J_m\left(m{\omega }_0\right)=\frac{{\left(2\pi \right)}^2}{20}\frac{{\chi }^2{\tau }_c}{1+{\left(m{\omega }_0{\tau }_c\right)}^2}$E2

$K_m\left(m{\omega }_0\right)={\omega }_0\tau J_m\left(m{\omega }_0\right)$E3

In the latter equations, ω₀ is the resonance frequency, τ is the length of the excitation pulse, τ_c is the rotational correlation time, which is a measurement for the degree of freedom of a nucleus, χ represents the root mean square coupling constant, and m is an integer number, which will be discussed later. Since no other correlation time will be discussed in this chapter, the rotational correlation time will be referred only as correlation time.

If nuclei can move freely, like in isotropic liquids, the correlation time is rather small and in the range of ns. If the motion becomes more restricted, like it is the case during the interaction with macromolecules, such as proteins, the correlation time increases. The quadrupolar coupling constant ω_Q is accounted for in the spectral density via the parameter χ. In biological systems, there is an increased variability of different environments and simultaneously ongoing processes leading to local variations of ω_Q, which justifies the usage of the root mean square value χ.

The influence of τ_c and ω_Q on the spectral density was extensively discussed by Rooney et al. [13]. In their paper, they defined four different regimes for τ_c and ω_Q resulting in four different types of spectra which are shown in Figure 2. These four regimes are:

• Type a spectrum: ω_Qτ_c ≫ 1 and ω₀τ_c ≫ 1 and an additional macroscopic anisotropy in the sample (e.g. single or liquid crystals). Molecular motion is hardly present in this system which leads to distinct energy levels and three very narrow resonances comprising the central resonance and two symmetrical satellite resonances at higher and lower frequencies, respectively

• Type b spectrum: ω_Qτ_c ≫ 1 and ω₀τ_c ≫ 1 and a random distribution of the orientation of the EFG tensors (e.g. inhomogeneous powder). Similar to the case of type a, molecular motion is hardly present but the random distribution of the EFG tensors leads to a broadening of the energy levels. In contrast to type a, where the spectrum shows three sharp lines, the satellite transitions become broadened due to many different values for ω_Q, and form a powder spectrum

• Type c spectrum: ω_Qτ_c ≪ 1 and ω₀τ_c > 1 and restricted motion (e.g. due to the interaction with proteins). In this type, the molecular motion is much higher than in the latter two. As a result, the energy levels are continually modulated and the satellite transitions vanish. As it can be seen (second column from the left) in Figure 2 , the satellite resonances are completely vanished and the central resonance takes the shape of a Lorentzian which is also influenced by the orientation of the different EFG tensor orientations with respect to the external magnetic field (super‐Lorentzian)

• Type d spectrum: ⟨ω_Q⟩ = 0 and ω₀τ_c ≪ 1. This is the case in isotropic liquids. Due to rapid molecular motion, the time average of the quadrupolar interaction vanishes and the energy levels become sharp again. Moreover, there is no difference between the different transitions leading to a single, sharp resonance at ω₀

Figure 2.

The four possible regimes for ²³Na spectra according to Rooney et al. Figure drawn from reference [13]. Copyright permission by John Wiley & Sons Ltd (license no. 3824201099423).

As it can be seen in Figure 2, nuclei with a nuclear spin > 1/2 can exhibit more than just one resonance. Therefore, transitions equal multiple times of the resonance frequency are possible. These transitions are known as multi quantum coherences (MQCs). In the case of spin 3/2 nuclei, single quantum (SQ) coherences can be observed, which are normally used for proton MRI and two MQCs, double quantum (DQ) coherences and triple quantum (TQ) coherences, respectively. In order to understand the relaxation properties of SQ coherences, a homogeneous and isotropic environment with only one compartment is assumed. The detailed discussion of higher coherences will be treated separately.

3.1.1. Longitudinal relaxation

As known from ¹H‐NMR the specific time constant T₁ for the longitudinal relaxation can be determined by usage of an inversion recovery (IR) sequence. After an initial 180° pulse follows an evolution period called inversion time (T_i). Observable magnetization is then generated by an additional 90° excitation pulse.

If the time between the last pulse and the acquisition is kept minimal, the acquired signal can be formulated according to reference [11]:

$S\left(S_0,T_i\right)=S_0\left(1-\frac{2}{5}\left(e^{-R_1^0{T}_i}+4e^{-R_2^0{T}_i}\right)\right)$E4

with S₀, the signal which can be obtained in the case T_i = 0, and with the relaxation rates

$R_1^0=2J_1$E5

$R_2^0=2J_2$E6

While the relaxation rates $R_1^0$ and $R_2^0$ are given by the spectral densities J₁ and J₂ [11]:

$J_1=\frac{{\left(2\pi \right)}^2}{20}\frac{{\chi }^2{\tau }_c}{1+{\left({\omega }_0{\tau }_c\right)}^2}$E7

$J_2=\frac{{\left(2\pi \right)}^2}{20}\frac{{\chi }^2{\tau }_c}{1+{\left(2{\omega }_0{\tau }_c\right)}^2}$E8

Here, it should be pointed out that the integer m, which was mentioned in the definition of the spectral densities (see Eqs. (2) and (3)), takes the values of 1 and 2. Theoretically, a biexponential relaxation curve which contains a fast and a slow relaxing component can be observed. However, it is very difficult to observe a biexponential T₁ relaxation, especially in biological tissue. Nevertheless, for ³⁵Cl a biexponential T₁ relaxation has been observed in vivo [14, 15].

It is common to express the relaxation rates with their inverse value, which leads to two different time constants:

$T_{1f}=\frac{1}{R_1^0}$E9

$T_{1s}=\frac{1}{R_2^0}$E10

T_1f depicts the time constant for the fast, and T_1s for the slow relaxing component. In the extreme narrowing limit ω₀τ_c ≪ 1 (i.e. isotropic liquid), it follows that T_1f = T_1s. Therefore, the relaxation becomes monoexponential with the time constant T₁ and the signal equation simplifies to

$S\left(S_0,T_i\right)=S_0\left(1-2e^{\frac{-T_i}{T_1}}\right)$E11

3.1.2. Transverse relaxation

To measure the characteristic time constant T₂ for the longitudinal relaxation, spin echo (SE) sequences are commonly used. These sequences begin with an initial 90° excitation pulse. Subsequent to this pulse a 180° inversion pulse is placed in the middle of an evolution interval called echo time (T_E).

A more detailed version of the following description for the elicited SE signal can be found in reference [11]. The SE signal takes the form:

$S\left(S_0,T_E\right)=S_0\frac{1}{5}\left(3e^{-R_1^1{T}_E}+2e^{-R_2^1{T}_E}\right)$E12

S₀ is the signal intensity when T_E is minimal and the relaxation rates are represented by:

$R_1^1=J_0+J_1$E13

$R_2^1=J_1+J_2$E14

In this case, the spectral density with m = 0 equals

$J_0=\frac{{\left(2\pi \right)}^2}{20}{\chi }^2{\tau }_c$E15

where $J_1$ and $J_2$ are the same spectral densities defined in Eqs. (7) and (8). Expressing the relaxation rates with their inverse and leads to

$T_{2f}=\frac{1}{R_1^1}$E16

$T_{2s}=\frac{1}{R_2^1}$E17

Similar to the case of T₁ relaxation, the T₂ relaxation is biexponential and consists of a fast relaxing component with relaxation time T_2f and a slow relaxing component with a relaxation time T_2s. In the extreme narrowing limit, the relaxation times become equal and lead to a monoexponential relaxation. In contrast to longitudinal relaxation, the two components of the transverse relaxation can be measured straight forward. Examples can be found in references [14, 15].

3.1.3. Rotational correlation time and its influence on relaxation times

As referred to in the discussion of longitudinal and transverse relaxation time, the rotational correlation time τ_c, and the root mean square value of the quadrupolar interaction constant χ appear in each of these two processes. The relaxation rates of longitudinal and transverse relaxation can be used to determine τ_c and χ in a model with a single compartment. However, since the biexponential behaviour can be observed much easier in transverse relaxation, the relaxation rates $R_1^1$ and $R_2^1$ are used here to determine τ_c and χ. Additionally, the substitution x = (ω₀τ_c)² is used.

First, the ratio a₁ of the two relaxation rates can be calculated as

$a_1=\frac{R_1^1}{R_2^1}=\frac{\left(2+x\right)\left(1+4x\right)}{2+5x}$E18

Rearranging Eq. (18) for τ_c leads to:

${\tau }_c=\frac{1}{{\omega }_0}\frac{\sqrt{-9+5a_1\pm \sqrt{25a_1^2-58a_1+49}}}{8}$E19

Knowledge of τ_c can then be used to calculate the value for χ. Herein, the first step is to compute the difference b₁ of the relaxation rates:

$b_1=R_1^1-R_2^1=\frac{4{\pi }^2}{5}{\chi }^2{\tau }_c\frac{x}{1+4x}$E20

Solving for χ leads to,

$\chi =\frac{1}{2\pi }\sqrt{\frac{5b_1\left(1+4x\right)}{{\tau }_{c}x}}$E21

In order to study the behaviour of the relaxation times under the influence of τ_c, a constant value for χ is assumed. From Eq. (19) follows that τ_c depends on the resonance frequency and thus on the magnetic field strength, leading to a dependence of all relaxation times on the magnetic field strength. Figure 3 shows the simulated behaviour of the product of T₁ and T₂ with χ being dependent on τ_c for magnetic field strengths of 9.4 T and 21.1 T. For the longitudinal relaxation, the assumed monoexponential behaviour leads to a single longitudinal relaxation time constant T₁. In case of the transverse relaxation, a biexponential behaviour was assumed, which leads to a fast relaxing (T_2f) and a slow relaxing (T_2s) component of the transverse relaxation time constant T₂. The value for τ_c, where the product ω₀τ_c = 1, is indicated with the dotted red lines for both field strengths. At increasing correlation times, it can be clearly seen that the fast transverse relaxation time is continuously decreasing. Time constants for longitudinal relaxation as well as the constants for slow transverse relaxation first decrease at an increasing correlation time. Around the area ω₀τ_c = 1, the relaxation times start to increase again. This clarifies why the value for longitudinal relaxation times in solids is of the order of seconds, whereas it is in the range of milliseconds for liquids.

Figure 3.

Dependence of relaxation times on the correlation time. Longitudinal as well as the slow components of transverse relaxation times initially decrease at increasing correlation times. Above the extreme narrowing limit, these relaxation times increase again. Short components of transverse relaxation times fall continuously at increasing correlation times.

3.1.4. Multi quantum coherences

As mentioned earlier, X‐nuclei with a nuclear spin > 1/2 can exhibit MQCs. The theoretical description of MQCs is very complex and requires a background in quantum mechanics, a formulation that is outside the scope of this book. However, very extensive descriptions can be found in references [11, 16, 17].

MQCs can be a valuable tool for providing physiological information in MRI experiments. X‐nuclei such as ²³Na, ³⁵Cl, and ³⁹K, which are heavily involved in physiological processes possess a nuclear spin equal to 3/2, and can therefore be used to generate MQCs. Figure 4 shows all possible coherences in a spin 3/2 system. SQ and DQ coherences can be found in liquids as well as in environments with restricted motion. The most specific coherence is the TQ coherence. This coherence can only be found above the extreme narrowing limit ω₀τ_c ≥ 1, which can only be reached when there is at least temporary binding. In biological systems, this is realized by the interaction with macromolecules, such as proteins. For that reason, the intensity of the TQ signal can give insight in the amount of interaction between ions and proteins. Therefore, the sequences presented subsequently, exclusively deal with the generation and detection of SQ and TQ coherences.

Figure 4.

Multi quantum coherences in a spin 3/2 system. (Black) single quantum (SQ) transitions occur between neighbouring levels. (Blue) Double quantum (DQ) transitions skip one energy level. (Red) triple quantum (TQ) transitions skip two energy levels.

If pulsed excitation is used, like in common NMR spectrometers or MRI scanners, it is not possible to record MCQs directly. Instead of direct excitation with a single RF pulse, several excitation pulses need to be applied to generate a signal which includes MCQs. Typically, the signal generated by MCQs is much weaker than signal generated SQ coherences. As a result, one has to apply filter techniques to suppress contributions from unwanted coherences. There are two ways to filter different coherences: either through the application of gradient pulses, or by cycling the phases of applied pulses and/or the receiver.

Usage of gradient pulses is associated with the advantage of short total measurement times but the signal intensity is reduced by a factor of two. If phase cycling is used, the sequence must be repeated several times, each time with a different set of pulses and/or receiver phases. If one deals with TQ coherences, the avoidance of signal loss is often more important than saving scanning time. For this reason, phase cycling is the preferred method in this chapter. Nevertheless, the reader can find some examples for filtration with gradient pulses in reference [18].

A common method for exciting and detecting TQ coherences is a pulse scheme consisting of three 90° excitation pulses, as illustrated in Figure 5. As it can be seen, all pulses have the same amplitude, while the phases of the first two pulses can vary. The time period between the first and the second pulse is called evolution time τ_Evo, and it is typically within the range of ms. Between the second and the third pulse, there is another short delay called mixing time, τ_Mix, which is typically in the range of several μs. In the case of TQ filtration, Φ′ is set to 90° while

is cycled through the values of 30°, 90°, 150°, 210°, 270°and 330°. In addition to this, the receiver phase is altered between 0° and 180° after each subsequent scan and the phase of the third pulse is constantly equal to 0°. During the second pulse of the phase cycle, each coherence will accumulate the phases linearly, according to the number of energy levels which where skipped, i.e. DQ coherences will accumulate the phase with a factor one, and TQ coherences with a factor of two, respectively. In the literature, this is referred to as triple quantum filtered T₂ (TQF‐T₂) experiment, or simply, a TQF experiment.

Figure 5.

Three‐pulse scheme for excitation and detection of MCQs. Pulse phases are indicated with Φ and Φ′. The time delays τ_Evo and τ_Mix represent evolution and mixing time. Data acquisition is indicated with ADC which refers to the analogue‐to‐digital converter.

The actual signal is obtained by complex addition of all subsequent scans. Table 3 shows the phases accumulated by all coherences during the TQF experiment. As it can be seen from the values listed in Table 3, the contribution from SQ and DQ coherences cancel each other out while the phases of the TQ contributions are constant. After addition, the TQ signal can be expressed with the following equation [11]:

$S\left(a,{\tau }_{Evo}\right)=\frac{15}{16}\frac{\sqrt{6}}{5}\left(e^{-R_1^1{\tau }_{Evo}}-e^{-R_2^1{\tau }_{Evo}}\right)$E22

The relaxation rates $R_1^1$ and $R_2^1$ were defined in Eqs. (13) and (14). To avoid the influence of the acquisition time of the generated free induction decay (FID), a Fourier transform (FT) can be performed along the acquisition time domain. The dependence of τ_Evo is then found at ω₀ along the direction of τ_Evo.

SQ	DQ	TQ
${30}^{\circ }$	${150}^{\circ }$	${270}^{\circ }$
${90}^{\circ }$	${90}^{\circ }$	${270}^{\circ }$
${150}^{\circ }$	${30}^{\circ }$	${270}^{\circ }$
${210}^{\circ }$	${330}^{\circ }$	${270}^{\circ }$
${270}^{\circ }$	${270}^{\circ }$	${270}^{\circ }$
${330}^{\circ }$	${210}^{\circ }$	${270}^{\circ }$

Table 3.

Accumulated phases of the different coherences during the TQF experiment.

Figure 6.

TQF signal of ²³Na at different agarose concentrations as a function of τ_Evo at 9.4 T. Increasing agarose content leads to an increased interaction between ²³Na and its environment, therefore, the TQF signal also increases. Additionally, both relaxation times become shorter, leading to a more rapid rise and decay of the signal.

Situations with ω₀τ_c ≥ 1 can be easily simulated using solutions and the addition of agarose. The more agarose the sample contains, the higher the correlation time. In order to simulate different in vivo situations the agarose concentration can be varied from 1 to 7.5%. From this data, the correlation time can be extracted according to the one compartment model. This phantom data can help to interpret in vivo data regarding to the degree of ion binding. Figure 6 shows the TQF signals of ²³Na in samples with different agarose concentration and 154 mM ²³Na concentration in dependence of the evolution time τ_Evo recorded at 9.4 T. All curves were normalized after acquisition and fitted according to the signal equation. With increasing agarose levels, one can clearly see an increase in the signal‐to‐noise ratio (SNR) indicating that the interaction of the ²³Na ions with their environment is also increasing. Additionally, the curves exhibit faster rise and decay times with increasing agarose, which is equivalent to an increase in both relaxation rates and a decrease in both relaxation times, respectively.

As previously mentioned, the degree of binding will lead to an increase in TQ contributions.

If one wants to record dynamic processes, such as changes in the amount of free water or ion concentrations, a reference is needed. In in vivo experiments, it is often difficult to work with an external reference which underlies variations in transmit and receive fields. It would be more accurate to generate a spectrum with an intrinsic reference. In case of ¹H‐MRS or ³¹P‐MRS, a spectrum with multiple peaks arises from different binding partners of the nucleus under observation. Unfortunately, X‐nuclei with nuclear spin > 1/2 hardly have permanent binding partners. In most cases, the binding is related to interactions with macromolecules and cannot be seen in conventional spectroscopy.

SQ	DQ	TQ
${90}^{\circ }$	${270}^{\circ }$	${90}^{\circ }$
${90}^{\circ }$	${90}^{\circ }$	${90}^{\circ }$
${135}^{\circ }$	$0^{\circ }$	${225}^{\circ }$
${135}^{\circ }$	${180}^{\circ }$	${225}^{\circ }$
${180}^{\circ }$	${90}^{\circ }$	$0^{\circ }$
${180}^{\circ }$	${270}^{\circ }$	$0^{\circ }$
${225}^{\circ }$	${180}^{\circ }$	${135}^{\circ }$
${225}^{\circ }$	$0^{\circ }$	${135}^{\circ }$
${270}^{\circ }$	${270}^{\circ }$	${270}^{\circ }$
${270}^{\circ }$	${90}^{\circ }$	${270}^{\circ }$
${315}^{\circ }$	$0^{\circ }$	${45}^{\circ }$
${315}^{\circ }$	${180}^{\circ }$	${45}^{\circ }$
$0^{\circ }$	${90}^{\circ }$	${180}^{\circ }$
$0^{\circ }$	${270}^{\circ }$	${180}^{\circ }$
${45}^{\circ }$	${180}^{\circ }$	${315}^{\circ }$
${45}^{\circ }$	$0^{\circ }$	${315}^{\circ }$

Table 4.

Phases of different coherences by applying triple quantum filtered time proportional phase increment (TQTPPI) spectroscopy with DQ suppression.

The solution to this problem is provided by multi quantum (MQ) spectroscopy. A very promising sequence to generate more resonance peaks from different MQCs in a single spectrum is referred to with the term triple quantum filtered time proportional phase increment (TQTPPI) [19]. The sequence diagram is basically the same shown in Figure 5. The only difference is the applied phase cycle and the fact that τ_Evo is incremented after each step in the phase cycle. Since DQ coherences also occur in liquids, a suppression of these contributions can help to simplify the signal and can be achieved by application of a 16 step phase cycle starting with Φ = 90°, for the first pulse. The second pulse carries the variable phase Φ and the constant phase Φ′ = 90° and the phase of the third pulse is, like the receiver phase, equal to zero. In order to suppress DQ coherences, each step in the phase cycle has to be repeated twice, the second time with an additional 180° phase on the middle pulse. Incrementing Φ and τ_Evo is then performed after each second step in the phase cycle. Typical values for Φ and τ_Evo are 45° and 100μs. Table 4 shows the resulting phases of all occurring coherences after application of the 16 step phase cycle. It can be seen that in the case of SQ and TQ contributions, the phases of two subsequent scans are equal, while the two subsequent phases for DQ contributions differ by 180°. DQ suppression can then be achieved by the addition of the two subsequent scans.

Figure 7.

TQTPPI spectra of ²³Na at 9.4 T with different concentrations of agarose. The frequency increases from right to left. All curves were normalized with respect to the SQ resonance. At increasing agarose content (shown along the y‐axis), the TQ contribution becomes more pronounced.

Generating the desired signal includes two steps: First, a FT has to be performed in the acquisition domain. Second, a second FID is generated by taking the points at ω₀ in the evolution time domain. This second dimension FID can be therefore turned into a spectrum by the application of an additional FT. Only the experimental results of this sequence are shown. Interested readers can find a very detailed derivation of the signal equation in reference [17].

Figure 7 shows TQTPPI spectra with DQ suppression of ²³Na obtained at 9.4 T. Different agarose concentrations are located along the y‐axis. The first peak from the right is the SQ resonance which is located at 1.25 kHz. Accordingly, the TQ resonance appears at a frequency of 3.75 kHz which is exactly three times the SQ frequency. All of the spectra are normalized to the maximum value of the SQ resonance. One can clearly see the gain in TQ signal at increasing agarose concentrations. It is possible to consider the SQ peak as an internal reference, to calculate the area under both peaks and to build the ratio TQ/SQ. Observing the TQ/SQ ratio allows the conduct of dynamic studies of changes in the motional freedom induced by changes in the binding of the nucleus under investigation. If applied to biological tissue, this can give valuable information about the amount of bound ions and their interaction with proteins in the tissue.

3.2. ³¹P: Energy metabolism

³¹P spectroscopy has found its way into basic and clinical research due to the fact that ³¹P is involved in energy metabolism. Energy consuming processes, such as the maintenance of the membrane potential, use the hydrolysis of adenosine triphosphate (ATP) as the energy source. The reaction is in accordance to:

$ATP+H_{2}O\to ADP+P_i$E23

Figure 8.

³¹P spectrum of a resting, arterially perfused cat soleus muscle. One can clearly see the phosphocreatine (PCr) resonance. The α, β, and γ resonances of adenosine triphosphate (ATP) are found at positive values for δ, while the inorganic phosphate (Pi) resonates at negative δ values. Contributions of sugar phosphate (SP) are also found at negative δ values. Figure drawn from reference [20]. Copyright permission by The American Physiological Society (license no. 3871910770507).

It can be seen that adenosine diphosphate (ADP) and inorganic phosphate (Pi) are produced during this reaction. In addition to the resonances of ATP, ADP and Pi one can also detect ³¹P resonances from phosphomonoesters and diesters. Figure 8 shows an example for a ³¹P spectrum recorded from a resting, arterially perfused cat soleus muscle [20]. One can clearly see the three resonances of ATP, the phosphocreatine (PCr) resonance, and the resonance line of Pi. Small contributions from sugar phosphate (SG) are also visible.

An overview of resonances of detectable metabolites is shown in Table 5 [4]. As it can be seen from Table 5, the chemical shift ($\delta$) of the different metabolites covers a wide range. By convention, the PCr resonance is set as internal reference (δ = 0.00 ppm).

Metabolite	Chemical shift
Adenosine monophosphate (AMP)	$6.33$
Adenosine diphosphate (ADP)	$-7.05 (\alpha )$; $-3.09 (\beta )$
Adenosine triphosphate	$-7.52 (\alpha )$; $-16.26 (\beta )$; $-2.48 (\gamma )$
Dihydroxyacetone phosphate	$7.56$
Glucose‐1‐phosphate	$5.15$
Glucose‐6‐phosphate	$7.20$
Inorganic phosphate (Pi)	$5.02$
Phosphocreatine (PCr)	$0.00$
Phosphoenolpyruvate	$2.06$
Phosphoryl choline	$5.88$
Phosphoryl ethanolamine	$6.78$
Nicotinamide adenine dinucleotide (NADH)	$-8.30$

Table 5.

Detectable metabolites by ³¹P spectroscopy with their chemical shift.

In ³¹P applications, it is not only possible to measure the concentration of a metabolite, it is also possible to derive rate constants based on changes of the concentrations of the PCr, Pi, and ATP. To influence the concentrations of PCr, Pi, and ATP, it is common to acquire the data while a volunteer is exercising and during the recovery period after the exercise. In the literature, one can find a variety of studies of the energy metabolism of human calf muscles. An extensive review is presented in reference [21].

The fact that the chemical shift of many ³¹P resonances is dependent on the intracellular pH and magnesium concentration, leads to another interesting application, namely, the in vivo measurement of pH values. This can be achieved using the Henderson‐Hasselbach equation:

$pH=pK+log\left(\frac{\delta -{\delta }_{HA}}{{\delta }_A-\delta }\right)$E24

where pK is the equilibrium constant for the acid‐base equilibrium between A and HA. The chemical shifts of the protonated and dissociated forms of the molecule under observation are expressed by δ_HA and δ_A, respectively. Determining pH is achieved by measuring the chemical shift δ between PCr and Pi. With literature values, Eq. (24) takes the form [22]:

$pH=6.803+log\left(\frac{\delta -3.22}{5.73-\delta }\right)$E25

3.3. ¹⁷O: Oxygen consumption

Supplying living cells with oxygen is of crucial importance for their viability. This can be seen by the huge impact a shortage of oxygen (hypoxia) has on, for example, brain functions. In order to access the production of NMR visible H₂¹⁷O, the paramagnetic properties of ¹⁷O can be used. There are two ways to use ¹⁷O in magnetic resonance (MR) experiments.

Firstly, with direct detection of the ¹⁷O resonance, and secondly, by use of the change in the ¹H relaxation times due to the coupling between ¹⁷O and ¹H. Direct and indirect detections suffer from the low natural abundance of ¹⁷O of 0.037%. Together with low values for γ, this leads to a sensitivity which is a factor of 1.7 × 10⁵ lower than for ¹H. To overcome this obstacle, techniques for increasing the ¹⁷O concentration are highly valuable. For this purpose, setups, like those listed in reference [23], for the inhalation of ¹⁷O enriched gas which can increase the ¹⁷O concentration above the natural abundance, are continuously developed.

As mentioned, a concentration increase can be reached by the inhalation of ¹⁷O enriched gas. After inhalation, the oxygen binds to haemoglobin due to lung exchange and is transported to the brain via the vascular system. As long as the oxygen gas is bound to haemoglobin, it is practically NMR invisible. In this state, the rotational motion is very slow leading to a very low value for τ_c and therefore to a rapid transverse relaxation. Through cerebral oxygen metabolism (CMRO₂), NMR observable ¹⁷O enriched water is produced according the following equation:

$4{\text{H}}^{+}+3{\text{e}}^{-}+{}^{17}{\text{O}}_2\to 2{\text{H}}_2{}_{}{}^{17}\text{O}$E26

According to references [10] and [24], the relaxation rates R₁ and R₂ for the ¹⁷O nucleus in tissue water within the extreme narrowing limit can be estimated as follows:

$R_2=\frac{1}{T_2}\cong R_1=\frac{1}{T_1}=1.056{\chi }^2{\tau }_c$E27

where $\chi$ is the root mean square coupling constant introduced in Eq. (2). Under coupling to ¹⁷O, the transverse relaxation rate R_2,H of the ¹H nucleus can be estimated as follows:

$R_{2,H}=\frac{1}{T_{2,H}}\approx \frac{1}{T_{2,H}^O}+\frac{35}{12}P\tau J^2$E28

The relaxation time $T_{2H}^O$ denotes the transverse relaxation time for ¹H when it is bound to ¹⁶O, P is the molar fraction for H₂¹⁷O which is equivalent to the ¹⁷O enrichment factor, τ is in this case the characteristic proton exchange lifetime and J² is the scalar ¹⁷O‐¹H coupling constant. Hopkins et al. have shown that only the transverse and not the longitudinal relaxation is influenced by the presence of ¹⁷O enriched water [25].

3.4. ¹³C: Brain metabolites

Like all organic compounds, brain metabolites are consisted of carbon atoms and protons. Therefore, for MRS experiments, it is sensible to consider the carbon isotope ¹³C, which carries a nuclear spin equal to 1/2. From Table 1 follows that the value for γ is just 25% of the proton value. Given the low natural abundance of 1.1%, it also has a relatively low NMR sensitivity. If a carbon spectrum is recorded at natural abundance, it will be dominated by the resonances of free fatty acids. Nevertheless, due to the high spectral range of approximately 200 ppm, the spectral resolution of carbon spectra is outstanding. The carbon resonances can be categorized in four groups, which are shown in Table 6 [4]:

Resonance of nuclei in different CH groups			Resonance of nuclei adjacent to hydroxyl groups	Resonance of nuclei in lipids	Resonance of nuclei in carbonyl groups
$\text{δ}\approx 25–60$ ppm			$\text{δ}\approx 60–100$ ppm	$\text{δ}\approx 20-50$ ppm and $\text{δ}>120$ ppm	$\text{δ}>150$ ppm
${\text{CH}}_3$ groups	${\text{CH}}_2$ groups	$\text{CH}$ groups
$\text{δ}<25$ ppm	$\text{δ}\approx 25-45$ ppm	$\text{δ}\approx 45-60$ ppm

Table 6.

Chemical shifts of different ¹³C moieties.

There are three techniques to increase the sensitivity of the ¹³C nucleus. One is the application of cryogenic coils where the RF coil is actively cooled by coolant, for example, liquid nitrogen. This reduces the electrical resistance of the coil and increases the signal strength. Another way to increase sensitivity is the application of heteronuclear broadband decoupling. This method also simplifies the interpretation of the spectrum since the scalar coupling between ¹³C and ¹H is lifted. Application of ¹³C‐labelled precursors does not only increase sensitivity. Furthermore, it enables the tracking of brain metabolites, such as glycogen and neurotransmitters [26].

Some examples of detectable metabolites and the chemical shifts of the single carbon atoms are listed in Table 7 [4]. A very good example for the application of ¹³C spectroscopy lies in the observation of glycogen regulation. This might have tremendous impact in the understanding of pathologies such as diabetes mellitus. Tracking the metabolic pathways provides a unique information which can only be acquired with the use of ¹³C spectroscopy.

Metabolite	Atom no.
	1	2	3	4	5	6
$\gamma$‐aminobutyric acid (GABA)	182.3	35.2	24.6	40.2	–	–
Glutamate	175.3	55.6	27.8	34.2	182.0	61.4
Glycogen	100.5	–	74.0	78.1	72.1	61.4
N‐acetylaspartylglutamic acid (NAA)	179.7	54.0	40.3	179.7	174.3	22.8

Table 7.

Chemical shifts (in ppm) of detectable carbon metabolites.

3.5. Functional phantoms and data interpretation

As shown in Section 3.1.4, the main advantage of MQ spectroscopy lies in its capability of recording signals from different coherences simultaneously. However, in order to connect the observed effect to a specific physiological response often remains difficult. In an in vivo experiment one has to face the fact that physiological parameters, such as the pH value, temperature, and ion concentrations are hardly, or even not at all accessible. What can be done is, of course, a set of experiments consisting of experiments under pathological conditions and experiments under standard conditions.

A solution to this problem may be found by conducting basic biological experiments, which are carried out on organotypic cell cultures. In contrast to in vivo experiments, organotypic cell cultures in microbioreactors provide a high degree of control over the experiment. Specific reactions can be initiated by adding drugs directly to the cell culture.

The principal approach of combining organotypic cell culture experiments in microbioreactors with MRI‐detection techniques was impressively shown by Gottwald et al. [3]. In their study, Gottwald et al. used a MRI compatible bioreactor to perform contrast enhanced perfusion ¹H‐MRI. The setup contained the MRI compatible reactor, which in turn contains a perfused three dimensional cell culture on a chip (3D‐KITChip), an external perfusion system with medium supply, and a gas mixing station. It could be shown that the perfusion characteristics are nearly independent of the flow rates, and the system could be completely washed out from applied drugs or contrast agent. From this, it follows that every drug applied in the system will be washed out, allowing the cell culture to reach its initial state again. Basically, this system can serve as a functional phantom for the development and testing of new MR sequences, as well as for recording the specific response of cells to different drugs under various physiological conditions.

The bioreactor with a medium reservoir and a peristaltic pump is shown in Figure 9a. A cross section of the reactor with the flux of cell culture medium is depicted in Figure 9b. One can see that the medium enters the bioreactor housing from below the chip and is then pumped through the pores of the chip, and therefore through the tissue residing in the microcavities, to the compartment above the tissue. The medium finally exits the bioreactor to the right side. By the use of this perfusion technique, the cells can be ideally supplied with medium and can be cultured organotypically for weeks. This setup can then be used to record MQ spectra from the cell culture. An example ²³Na‐TQTPPI spectrum without cells, recorded with a custom built ²³Na surface coil at 9.4 T, is shown in Figure 9c. The SQ resonance is shown on the right side of the spectrum. It is not surprising that the SQ resonance is extremely large compared to the rest of the spectrum, since this resonance contains the complete amount of free ions. Higher frequencies are displayed from right to left. It should be noted that despite the DQ suppression, the DQ resonance was not suppressed completely. This is related to imperfection in the excitation pulses arising from the inhomogeneous pulse profile of the surface coil. A zoomed section (shown by the red box) of the spectrum is depicted in Figure 9d. As one can see, there is no TQ resonance at the expected frequency.

Figure 9.

(a) Bioreactor with medium reservoir and peristaltic pump. (b) Cross section of the bioreactor with perfusion direction. (c) With custom built surface coil recorded complete ²³Na‐TQTPPI spectrum of the bioreactor without cells at 9.4 T. (d) Zoomed section (red box in c) of the spectrum without cells. The spectrum shows no significant TQ contribution. (e) Zoomed section of a ²³Na‐TQTPPI spectrum with cells. The TQ contribution is clearly present.

The situation changes critically when the experiment is repeated with an active cell culture. In Figure 9d, the zoomed section of a ²³Na‐TQTPPI spectrum is shown from an experiment with a cell culture. One can clearly see the TQ resonance at the expected frequency. Compared to the other resonances, the TQ contribution is extremely small, but the quality of the gained information is revealed if one takes the dimensions of the experiment into account. In the very best case scenario, the entire fraction of the cell culture is approximately 1.2% of the complete volume under investigation. It is obvious that the TQ signal arises from this small fraction which proves that the TQTPPI spectroscopy is a very sensitive tool.

3.6. Conclusions

Based on their involvement in physiological processes such as energy metabolism, generation of action potentials and cell volume regulation, NMR experiments on X‐nuclei can provide valuable physiological information.

On the one hand, there are nuclei with a nuclear spin equal to 1/2 which can be measured by means of pulse sequences known from ¹H‐NMR. For instance, ³¹P spectra can be generated by usage of simple single pulse sequences. Despite the relative low NMR sensitivity, the information obtained by the application of such simple sequences is always related to the biological background of the nucleus under investigation. On the other hand, for the exploitation of the full potential of spin 3/2 X‐nuclei, their unique (quantum) physical properties must be utilised. Sequences capable to generate and record MQCs can be used to study changes in the motional freedom of the nuclei. In living systems, the motional freedom can be influenced by a change in ion binding or by morphological changes of the environment of the nucleus. Therefore, the analysis of MQCs in biological tissue can provide additional information about protein activity or changes in cell volume.

It is of high importance to link changes in the observed signal to the underlying physiological processes. The usage of functional phantoms is a very promising way to establish that link. Since the available magnetic field strength is continuously increasing, the relative low NMR sensitivity of these nuclei may no longer be a drawback in the future.