Quantum Dipole Interactions and Transient Hydrogen Bond Orientation Order in Cells, Cellular Membranes and Myelin Sheath: Implications for MRI Signal Relaxation, Anisotropy, and T1 Magnetic Field Dependence

Abstract

Purpose:

Despite significant impact on the study of human brain, MRI lacks a theory of signal formation that integrates quantum interactions involving proton dipoles (a primary MRI signal source) with brain intricate cellular environment. The purpose of the present study is developing such a theory.

Methods:

We introduce the Transient Hydrogen Bond (THB) model, where THB-mediated quantum dipole interactions between water and protons of hydrophilic heads of amphipathic biomolecules forming cells, cellular membranes and myelin sheath serve as a major source of MR signal relaxation.

Results:

The THB theory predicts the existence of a hydrogen-bond-driven structural order of dipole-dipole connections within THBs as a primary factor for the anisotropy observed in MRI signal relaxation. We have also demonstrated that the conventional Lorentzian spectral density function decreases too fast at high frequencies to adequately capture the field dependence of brain MRI signal relaxation. To bridge this gap, we introduced a stretched spectral density function that surpasses the limitations of Lorentzian dispersion. In human brain, our findings reveal that at any time point only about 4-to-7% of water protons are engaged in quantum encounters within THBs. These ultra-short (two-three-nanosecond), but frequent quantum spin exchanges lead to gradual recovery of magnetization towards thermodynamic equilibrium, i.e., relaxation of MRI signal.

Conclusion:

By incorporating quantum proton interactions involved in brain imaging, THB approach introduces new insights on the complex relationship between brain tissue cellular structure and MRI measurements, thus offering promising new tool for better understanding of brain microstructure in health and disease.

Introduction

Since introduction by Lauterbur [1], MRI plays an increasingly profound role in studying biological tissue properties in humans and animals in health and disease. Numerous MRI techniques have been developed to highlight different properties of biological tissue structure and functioning. Water dynamics and restrictions to water diffusion imposed by the cellular environment (mostly, cellular membranes and myelin sheath) form a basis of Diffusion MRI used in studying diseases such as stroke [2] and anisotropic properties of cellular organization [3]. Resting state functional MRI signal [4] provides insights into the large-scale brain circuit organization [5, 6]. Differences in T1 and T2 tissue relaxation properties allow highlighting tissue contrast in healthy and clinically pathological cases [7]. Magnetization transfer (MT) [8] (see also [9]) and chemical exchange [10] (see also [11]) experiments opened opportunities to use MRI for inferring information on the tissue-forming molecules (proteins, lipids, etc.). These approaches rely on knowledge of the biophysical mechanisms of MR signal relaxation properties (see [12] and recent review [13]).

In this paper we focus on biophysical mechanisms responsible for T1 relaxation of MRI signal. Initially, Koenig et al [14, 15] emphasized the role of myelin-building lipid bilayers in formation of MR signal relaxation in the brain. The authors suggested cholesterol as an important player in establishing gray-white matter contrast of MRI signal. In a follow-up papers, Kucharczyk et al [16] and Ceckler et al [17] underscored the role of hydroxyl groups in the hydrophilic heads of lipid bilayers. The role of lipids in forming GM/WM MRI contrast was also revealed through tissue clearing technique by Leuze et al [18]. The authors observed minimal T1 and mean diffusivity contrasts and even no contrast at all with T2 and MT MRI sequences when the lipids were removed from brain. The studies by Bryant et al [19–22] and Halle [23] emphasized the role of magnetic coupling of protons with motion-immobilized species and the role of water hydration layers on macromolecular surface [20] on T1 relaxation.

Yet, despite of all these important achievements, a theory of MR signal formation that effectively integrates quantum interactions involving hydrogen dipoles (a primary MRI signal source) with brain intricate cellular environment is still in the infant state. Hence, this paper is concerned with establishing a quantitative first-principle theory of brain tissue MRI signal T1 relaxation properties. Specifically, we demonstrate that hydrogen bonding of water molecules transiently trapped in hydrophilic heads of cells and myelin-forming lipid bilayers enables the quantum dipole-dipole spin exchange between protons of the molecules forming hydrophilic heads and water protons. Hence, the term Transient Hydrogen Bond (THB) model.

It is important to re-emphasize that only protons of tissue-forming molecules available for transient hydrogen bonding with water (such as residing in hydrophilic heads) contribute to the spin-exchange processes in THB model, while protons belonging to other parts of these molecules (e.g., the hydrocarbon cores of cellular membranes) do not. The microscopic theory of MR signal relaxation of the lipid-bound protons in the lipid chains due to their interaction with each other (Lateral Diffusion Model) was developed in our previous publication [24].

An important property of MR signal in biological tissue is its anisotropy [25]. While initial studies reported anisotropy of MR signal relaxation properties of tendon [26], skeletal muscle [27] and trabecular bone [28, 29] (see latest discussion in [30]), later publications reported on the anisotropy of MR signal phase related to myelinated axons microstructural anisotropy [31] and magnetic susceptibility anisotropy [32–34], as well as anisotropy of MR signal relaxation in brain WM [35–49].

In the THB approach, the relaxation of longitudinal $(M_z^W,M_z^H)$ magnetizations of water ($W$) and head-bound ($H$) protons is described by the master THB equations:

$$\begin{gathered} \frac{d{M}_z^W}{dt}=-\left[(1-n_{TR})\cdot r_{1W}+n_{TR}\cdot k({\mathbf{B}}_0)\right]\cdot \left(M_z^W-M_{z0}^W\right)+K({\mathbf{B}}_0)\cdot \left(M_z^H-M_{z0}^H\right) \\ \frac{d{M}_z^H}{dt}=-\left(r_{1H}+k({\mathbf{B}}_0)\right)\cdot \left(M_z^H-M_{z0}^H\right)+n_{TR}\cdot K({\mathbf{B}}_0)\cdot \left(M_z^W-M_{z0}^W\right) \end{gathered}$$ (1)

where $n_{TR}$ represents a fraction of water protons in the transient (hence, the subscript $TR$) hydrogen-bond state in the lipid heads, while field-strength and direction-dependent parameters $k({\mathbf{B}}_0)$ and $K({\mathbf{B}}_0)$ represent the diagonal and cross-relaxation exchange terms. The detail description of parameters in Eqs. (1) can be found in the main text, Eqs. (9). Equations (1) define bi-exponential character of T1 signal relaxation with parameters

$$\begin{gathered} R_{1W}=r_{1W}+n_{TR}\cdot k({\mathbf{B}}_0) \\ {R}_{1H}=r_{1H}+k({\mathbf{B}}_0) \end{gathered}$$ (2)

describing T1 relaxation of its two components in the main approximation with respect to the small parameter $n_{TR}$.

The THB equations describe relaxation of MRI signal due to quantum spin exchange processes that we derive based on the Bloembergen, Purcell and Pound theory of dipolar relaxation [50]. They are different from the commonly used Bloch-McConnell equations [51] treating spin cross-relaxation by means of direct “physical” exchange of protons. One of the important consequences of this difference is THB model prediction of the transient order in the orientations of connections between water protons and their THB counterparts. This order defines the anisotropy of quantum dipole interactions and, consequently, the anisotropy of signal relaxation. The orientation and organization of axons and lipid membranes in the myelin sheath is a secondary cause of relaxation anisotropy.

In the current paper we consider only the quantum dipole-dipole interactions and their contribution to tissue T1 relaxation. There are other important contributions (mainly, to T2 and T2*), especially at high ${\mathbf{B}}_0$ fields, related to the presence of iron in tissue (see [52], [53] and follow-up publications) and magnetic susceptibility effects; however, these effects are beyond the scope of the current study.

Methods

In this paper, we use the classical BPP theory of relaxation based on the dipolar interaction between spins [50] in the formalism similar to that proposed in [54] and [55]. The Hamiltonian of a two-spins system in the external magnetic field ${\mathbf{B}}_0$ consists of the Zeeman term ${\mathbf{H}}_0$ and the term ${\mathbf{H}}_{dip}$ describing the dipole-dipole interactions between two spins, ${\mathbf{s}}_1$ and ${\mathbf{s}}_2$:

$$\begin{gathered} \mathbf{H}={\mathbf{H}}_0+{\mathbf{H}}_{dip} \\ {\mathbf{H}}_0=-\hslash \cdot \omega \left(s_{1z}+s_{2z}\right),{\mathbf{H}}_{dip}={\left(\hslash \gamma \right)}^2\cdot \frac{{\mathbf{s}}_1{\mathbf{s}}_2-3\cdot ({\mathbf{s}}_1\mathbf{n})\cdot ({\mathbf{s}}_2\mathbf{n})}{r^3},\mathbf{r}=r\mathbf{n} \end{gathered}$$ (3)

where $\hslash$ is the Planck constant, $\omega =\gamma B_0$ is the Larmor frequency, $\gamma$ is the gyromagnetic ratio, $\mathbf{\text{r}}$ is the radius-vector between two spins, $\mathbf{n}=\mathbf{r}/r=(\sin \theta \cdot \cos \phi ,\sin \theta \cdot \sin \phi ,\cos \theta )$ (the polar axis along ${\mathbf{B}}_0$). The Hamiltonian ${\mathbf{H}}_{dip}$ can be presented in terms of the second-degree spherical harmonics $F^{(m)}$ and combinations of spin operators $A^{(m)}$:

$${\mathbf{H}}_{dip}=\hslash \cdot \sum _{m=-2}^2{F}^{(m)}\cdot A^{(m)}$$ (4)

$$F^{(0)}=\frac{1-3\cdot {\cos }^2\theta }{r^3},F^{(\pm 1)}=\frac{\sin \theta \cdot \cos \theta }{r^3}\cdot e^{\mp i\phi },F^{(\pm 2)}=\frac{{\sin }^2\theta }{r^3}\cdot e^{\mp 2i\phi }$$ (5)

$$\begin{gathered} A^{(0)}=\hslash {\gamma }^2\cdot s_{1z}{s}_{2z}-\frac{1}{4}\cdot \hslash {\gamma }^2\cdot \left(s_{1+}{s}_{2-}+s_{1-}{s}_{2+}\right) \\ {A}^{(1)}={\left(A^{(-1)}\right)}^{+}=-\frac{3}{2}\cdot \hslash {\gamma }^2\cdot \left(s_{1z}{s}_{2+}+s_{1+}{s}_{2z}\right), \\ {A}^{(2)}={\left(A^{(-2)}\right)}^{+}=-\frac{3}{4}\cdot \hslash {\gamma }^2\cdot s_{1+}{s}_{2+} \end{gathered}$$ (6)

In the system under consideration, all interacting spins can be divided into two pools: water protons (denoted in the paper by the index $W$) and protons of tissue-forming molecules in the hydrophilic heads available for transient hydrogen bonding with water (denoted in the paper by the index $H$, ($H$-protons)). In both the pools, particles have the same spins $s=1/2$. While there is a frequency shift, $\text{Δ}\omega$ (on the order of a few ppm) between water and bound protons, it does not affect our results because hydrogen-bonding time, ${\tau }_H$ (nanosecond-range), is several orders of magnitude smaller than $1/\text{Δ}\omega$ (millisecond-range). However, the dynamics and environments in the two pools are substantially different.

Biophysical Transient Hydrogen Bond (THB) model

Cellular membranes (a.k.a. plasma membrane, lipid bilayers) are the major parts of biological tissue structure. They separate intracellular machinery from extracellular fluid encompassing all cells and cellular processes such as dendrites, dendritic spines, etc. These membranes are abundant in both WM, where they form multilamellar structures of myelin sheath around axons, and in GM, where membranes cover cells along with their numerous processes. As we show below, the THB model equally-well describes T1 relaxation in both, GM, and WM.

A schematic structure of lipid bilayer with adjacent water compartments is shown in Figure 1. It consists of a hydrophobic hydrocarbon core, hydrophilic heads intermixed with water molecules, and adjacent water layers.

Figure 1. A schematic structure of bilayer membrane (modified with permission from [56]).

Bold lines are used for the headgroups and the glycerol moiety of the lipid, thin lines - for the hydrocarbon chains, and the spheres - for the water molecules.

While the central part of lipid bilayers is hydrophobic and the water concentration in this part of the membrane is very low (about 10⁻⁴ of its concentration in the surrounding space), the external parts of membranes, i.e., lipid heads, are hydrophilic. Molecular dynamic simulations demonstrated that about 25% of cellular membranes outer part is occupied simultaneously by $H$-protons and water protons [56, 57]. This offers a favorable environment for hydrogen bonding between $H$-protons in lipid heads and free water protons [58, 59].

Let $N_W$ represent a total number of water protons in a unit volume, with a fraction $n_{TR}$ representing water protons in transient hydrogen-bond state in the hydrophilic heads. Let $N_H$ represent a total number of $H$-protons in hydrophilic heads in the unit volume, participating in the dipole-dipole interaction with water protons. Hence:

$$n_{TR}\cdot N_W=N_H$$ (7)

It is important to emphasize that $N_H$ does not represent the total number of chemically-bound protons in lipid bilayers, but only those residing in polar head groups such as OH, NH, or carbonyl groups (hence, the term $N_H$) allowing close contact with water protons from hydrogen-bonded water molecules. The chemically-bound protons in the vicinity of hydrophobic lipid tails provide a minor contribution to the water-lipid interaction because they cannot form hydrogen bonds.

We further introduce magnetization vectors (i.e., magnetic moments per unit volume) of two pools:

$$\begin{gathered} {\mathbf{M}}^W=N_W\cdot \hslash \cdot \gamma \text{Sp}\left(\tilde{\rho }(t){\mathbf{s}}^W\right) \\ {\mathbf{M}}^H=N_H\cdot \hslash \cdot \gamma \text{Sp}\left(\tilde{\rho }(t){\mathbf{s}}^H\right) \end{gathered}$$ (8)

where $\tilde{\rho }(t)$ is the density matrix in the interaction representation [55].

Note that at any given moment, only the fraction $n_{TR}$ of water protons interacts with available head protons, while $\left(1-n_{TR}\right)$ water protons interact only with neighboring water protons.

Results

According to the above consideration of interacting pools and following the Abragam’s formalism [55] (see also [50, 54]) of the dipole-dipole interactions, it can be shown that in the second order of perturbation theory with respect to ${\mathbf{H}}_{dip}$, the longitudinal ($z$) components of $W$- and $H$- magnetizations, Eqs. (8), satisfy the set of coupled differential equations:

$$\begin{gathered} \frac{d{M}_z^W}{dt}=-\left[(1-n_{TR})\cdot r_{1W}+n_{TR}\cdot k\right]\cdot \left(M_z^W-M_{z0}^W\right)+K\cdot \left(M_z^H-M_{z0}^H\right) \\ \frac{d{M}_z^H}{dt}=-\left(r_{1H}+k\right)\cdot \left(M_z^H-M_{z0}^H\right)+n_{TR}\cdot K\cdot \left(M_z^W-M_{z0}^W\right) \end{gathered}$$ (9)

where $H$ are the magnetizations of the $W$- and $H$- pools at the thermal equilibrium. The diagonal and cross-relaxation parameters in Eqs. (9) are as follows:

$$\begin{gathered} r_{1W}=\frac{9}{8}\cdot {\left(\hslash {\gamma }^2\right)}^2\cdot \left(J_{WW}^{(1)}(\omega )+J_{WW}^{(2)}(2\omega )\right)+\text{Δ}{r}_{1W}(a) \\ {r}_{1H}=\frac{9}{8}\cdot {\left(\hslash {\gamma }^2\right)}^2\left(J_{HH}^{(1)}(\omega )+J_{HH}^{(2)}(2\omega )\right)(b) \\ k=\frac{9}{16}\cdot {\left(\hslash {\gamma }^2\right)}^2\left(\frac{1}{9}{J}_{WH}^{(0)}(0)+2J_{WH}^{(1)}(\omega )+J_{WH}^{(2)}(2\omega )\right)(c) \\ K=\frac{9}{16}\cdot {\left(\hslash {\gamma }^2\right)}^2\left(\frac{1}{9}{J}_{WH}^{(0)}(0)-J_{WH}^{(2)}(\omega )\right)(d) \end{gathered}$$ (10)

In Eqs. (10), $\omega =\gamma B_0$, the term $K$ describes contributions to water proton relaxation from other than the $W$-$W$ and $W$-$H$ interactions (e.g., water protons interaction with iron, oxygen, proteins that are not parts of cellular membranes and myelin sheath). Importantly, the diagonal ($k$) and cross-relaxation ($K$) exchange parameters are different from each other. While ordinarily spectral densities at $\omega =0$ affect transverse relaxation only, in the THB model the exchange $W$-$H$ interactions at $\omega =0$ also contribute to the longitudinal relaxation.

The spectral densities $J_{WW}^{(m)},J_{HH}^{(m)},J_{WH}^{(m)}$ ($m=0,1,2$, except of $J_{HH}^{(0)}$, see below) are the Fourier transforms of correlation functions $G_j^{(m)}(\tau )$:

$$\begin{gathered} J_j^{(m)}(\omega )=2\cdot \underset{0}{\overset{\infty }{\int }}d\tau G_j^{(m)}(\tau )\cdot \cos (m\omega \tau ),j=WW,HH,WH \\ {G}_j^{(m)}(\tau )={\left[\overline{F^{(m)}(t)\cdot F^{(m)*}(t+\tau )}\right]}_j \end{gathered}$$ (11)

The upper bar means averaging over the dynamic trajectories of the interacting dipoles.

The correlation functions $G_{WW}^{(m)}(\tau )$ and $G_{HH}^{(m)}(\tau )$ correspond to interacting dipoles in the $W$- and $H$- pools, while the correlation functions $G_{WH}^{(m)}(\tau )$ describe correlations between dipoles from different pools. Herein, we focus on the role of spin-exchange interaction between $W$- and $H$- pools. A theory of spin-exchange processes in water can be found elsewhere [50, 55].

General solutions to Eqs. (9) are the bi-exponential functions:

$$\begin{gathered} M_z^W(t)=M_0^W+ \\ +\left(M_z^W(0)-M_0^W\right)\cdot \left[\frac{(R_{1H}-R_{1W}+Y_1)}{2Y_1}\cdot \exp \left(-{\lambda }_1^{(-)}t\right)+\frac{(R_{1W}-R_{1H}+Y_1)}{2Y_1}\cdot \exp \left(-{\lambda }_1^{(+)}t\right)\right]+ \\ +\left(M_z^H(0)-M_0^H\right)\cdot \frac{K}{Y_1}\cdot \left[\exp \left(-{\lambda }_1^{(-)}t\right)-\exp \left(-{\lambda }_1^{(+)}t\right)\right] \end{gathered}$$ (12)

$$\begin{gathered} M_z^H(t)=M_0^H+ \\ +\left(M_z^H(0)-M_0^H\right)\cdot \left[\frac{(R_{1H}-R_{1W}+Y_1)}{2Y_1}\cdot \exp \left(-{\lambda }_1^{(+)}t\right)+\frac{(R_{1W}-R_{1H}+Y_1)}{2Y_1}\cdot \exp \left(-{\lambda }_1^{(-)}t\right)\right]+ \\ +\left(M_z^W(0)-M_0^W\right)\cdot \frac{n_{TR}\cdot K}{Y_1}\cdot \left[\exp \left(-{\lambda }_1^{(-)}t\right)-\exp \left(-{\lambda }_1^{(+)}t\right)\right] \end{gathered}$$ (13)

where $M_z^W(0)$ and $M_z^H(0)$ are the initial (after RF pulse) values of the longitudinal magnetizations in the $W$- and $H$- pools,

$$\begin{gathered} R_{1W}=(1-n_{TR})\cdot r_{1W}+n_{TR}\cdot k \\ {R}_{1H}=r_{1H}+k \end{gathered}$$ (14)

and

$${\lambda }_1^{(\pm )}=\frac{1}{2}\cdot \left(R_{1W}+R_{1H}\pm Y_1\right),Y_1={\left[{\left(R_{1W}-R_{1H}\right)}^2+4\cdot n_{TR}\cdot K^2\right]}^{1/2}$$ (15)

Correlation functions

For the case of water proton transiently trapped in hydrophilic heads, the dynamics is similar to a Kramers problem [60], i.e., the escape of a particle over a potential barrier subject to a thermal motion [61]. Hence, the temporal decay of the probability to still find water molecule within a potential well can be given by an exponential function:

$$p(t)=\exp (-t/\tau )$$ (16)

where $\tau$ is a hydrogen bond lifetime that follows temperature ($T$) dependence per an Arrhenius activation with activation energy $E_a$ [62],

$$\tau \propto \exp (E_a/T)$$ (17)

Accordingly, Eqs. (16) and (17) lead to the exponential correlation function:

$$\langle F^{(m)}(t_0+t)\cdot F^{(m)*}(t_0)\rangle =\langle {\left|F(t_0)\right|}^2\rangle \cdot \exp (-t/\tau )$$ (18)

For the exponential dependence of the correlation functions $G_{WH}^{(m)}(\tau )$ on $\tau$ (Eqs. (11), (18)), the spectral densities $J_{WH}^{(m)}(\omega )$, Eq. (11), have the Lorentzian form,

$$J_{WH}^{(m)}(\omega )=2G_{WH}^{(m)}(0)\cdot \frac{\tau }{1+{\left(\omega \cdot \tau \right)}^2}$$ (19)

Our analysis below suggests that the Lorentzian spectral density function cannot adequately describe magnetic field dependence of T1 relaxation, the feature also reported in the studies of the dynamics of protein solutions [63]. To address this issue, we introduce a “stretched” spectral density function:

$$J_{WH}^{(m)}(\omega )=2G_{WH}^{(m)}(0)\cdot \frac{\tau }{1+{\left(\omega \cdot \tau \right)}^{2b}}$$ (20)

With parameter $b<1$, it “stretches” the dispersion to higher frequencies as compared with the Lorentzian form in Eq. (19). As we see below, utilizing this function in the framework of the THB model, makes it possible to describe experimental data in the human brain extremely well.

The parameters $G_{WH}^{(m)}(0)$ depend on the average length $d_H$ of the vectors $\mathbf{\text{r}}$ in Eq. (3) between water protons and their counterparts in lipid heads (see Eqs. (5) in the Method Section). Using Eqs. (11), factor $G_{WH}^{(m)}(0)$. became:

$$\begin{gathered} G_{WH}^{(m)}(0)=\frac{1}{d_H^6}\cdot g_m,m=0,1,2 \\ {g}_0=\langle {(1-3\cdot {\cos }^2\theta )}^2\rangle ,g_1=\langle {\sin }^2\theta \cdot {\cos }^2\theta \rangle ,g_2=\langle {\sin }^4\theta \rangle \end{gathered}$$ (21)

where the functions $g_m$ depend on the orientation distribution of vectors $\mathbf{r}$ with respect to the external magnetic field ${\mathbf{B}}_0$. The angle brackets in Eqs. (21) mean averaging over orientations of vectors $\mathbf{r}$ in the bilayers and over the bilayers’ orientations with respect to ${\mathbf{B}}_0$. The specific expressions of the functions $g_m$ for an individual bilayers and axonal bundles are provided below (Eqs. (30), (32)). Hence,

$$J_{WH}^{(m)}(\omega )=\frac{2\tau }{d_H^6}\cdot g_m\cdot L(\omega \tau ),L(\omega \tau )=\frac{1}{1+{\left(\omega \cdot \tau \right)}^{2b}}$$ (22)

By making use of Eqs. (21)-(22), the spin-exchange parameters in Eqs. (10) can be written as follows:

$$\begin{gathered} k=\frac{3}{4}\cdot \lambda \cdot \left(\frac{1}{9}{g}_0\cdot L(0)+2g_1\cdot L(\omega \tau )+g_2\cdot L(2\omega \tau )\right); \\ K=\frac{3}{4}\cdot \lambda \cdot \left(\frac{1}{9}{g}_0\cdot L(0)-g_2\cdot L(2\omega \tau )\right) \end{gathered}$$ (23)

where

$$\lambda =\frac{3}{2}\cdot \frac{\tau }{d_H^6}\cdot {\left(\hslash {\gamma }^2\right)}^2$$ (24)

The parameter $\lambda$ provides a general scale factor characterizing contributions of the dipole-dipole spin exchange interactions between THB-trapped protons in hydrophilic heads to the MR signal relaxation. To give a sense of this scale, we note that for $d_H$ = 1 Å and $\tau$ = 1 ns, the parameter $\lambda$ = 846 (sec⁻¹). It increases linearly with $\tau$ and decreases as $d_H^{-6}$ with the distance $d_H$.

Spin-exchange effects on T1 relaxation

The general theory of T1 relaxation due to the spin-exchange effects is described above. Herein, to elucidate the specific results, we note that the parameter $n_{TR}$ defining the fraction of water protons transiently trapped by hydrogen bonding in hydrophilic heads, is small. In this case, in the leading approximation with respect to $n_{TR}$ and assuming that $k>>K$ (will be justified below), we arrive at the following simplified equations:

$$\begin{gathered} M_z^W(t)=M_0^W+\left(M_z^W(0)-M_0^W\right)\cdot \left[\exp \left(-R_{1W}\cdot t\right)+n_{TR}\cdot {\left(\frac{K}{R_{1H}}\right)}^2\cdot \exp \left(-R_{1H}\cdot t\right)\right]+ \\ +\left(M_z^H(0)-M_0^H\right)\cdot \left(\frac{K}{R_{1H}}\right)\cdot \left[\exp \left(-R_{1W}\cdot t\right)-\exp \left(-R_{1H}\cdot t\right)\right] \end{gathered}$$ (25)

$$\begin{gathered} M_z^H(t)=M_0^H+\left(M_z^H(0)-M_0^H\right)\cdot \left[\exp \left(-R_{1H}\cdot t\right)+n_{TR}\cdot {\left(\frac{K}{R_{1H}}\right)}^2\cdot \exp \left(-R_{1W}\cdot t\right)\right]+ \\ +\left(M_z^W(0)-M_0^W\right)\cdot n_{TR}\cdot \left(\frac{K}{R_{1H}}\right)\cdot \left[\exp \left(-R_{1W}\cdot t\right)-\exp \left(-R_{1H}\cdot t\right)\right] \end{gathered}$$ (26)

$$R_{1W}=r_{1W}+n_{TR}\cdot k,R_{1H}=r_{1H}+k$$ (27)

with exchange parameters $k$ and $K$ defined in Eqs. (23). Since $n_{TR}\ll 1$, the relaxation rate parameters satisfy relationship $R_{1W}\ll R_{1H}$. It means that measuring T1 relaxation time parameter with sufficiently long times, $t\gg 1/R_{1H}$, will result in the same $T_1=1/R_{1W}$ for both water and lipid-bound protons signals. Though the amplitude of the $R_{1W}$ term in the lipid-bound proton signal is much smaller than the amplitude of the $R_{1H}$ term.

To test THB model, we use previously published data by Rooney et al [64] and Wang et al [65] that provided detail T1 measurements in human subjects for a broad range of magnetic fields (the data are summarized in Table 1).

Table 1. R₁ relaxation rate-parameter as a function of magnetic field.

R₁ data (1/sec) obtained from inversion recovery measurements of Rooney et al [64] (a) and Wang et al [65] (b) for several brain regions. In [64], WM and GM were selected from Frontal WM and GM. In [65], WM data were selected from splenium corpus callosum. WM and GM data from [65] at 0.55T were excluded per the authors’ comment on their low-resolution effect in these two regions. All values of R₁ were found by means of mono-exponential fitting of experimental data, except for R₁ at 3T and 7T for which we used the values corresponding to the slow decaying component of the bi-exponential fit in [65].

B0 (T)	WM	cGM	Caudate	Thalamus	Putamen	Glob pall	Ref.
0.2	2.77	1.57	1.80	1.92	1.91	2.43	(a)
0.55			1.41	1.55	1.50		(b)
1	1.80	0.97	1.11	1.24	1.23	1.60	(a)
1.5	1.53	0.82	0.97	1.12	1.03		(b)
1.5	1.52	0.84	0.92	1.03	1.02	1.34	(a)
2.89	1.11	0.64	0.75	0.84	0.79		(b)
4	1.00	0.58	0.66	0.69	0.69	0.87	(a)
6.98	0.76	0.49	0.57	0.63	0.59		(b)
7	0.82	0.47	0.57	0.60	0.59	0.74	(a)

For fitting THB model to the data in Table 1, we used Eq. (27) for $R_{1W}$ with the parameter $k$ defined in Eq. (23) with the stretched shape, Eq. (22), and the coefficients $g^{(m)}$ corresponding to the uniform orientation distribution of vectors $\mathbf{\text{r}}$, Eq. (34). The results are presented in Figure 2 (for brevity, in all the Figures we denote the parameter $R_{1W}$ as $R_1$) and Table 2A. By using data for the fast T1 component available from Wang et al [65], our theoretical equations for $R_{1H}$ and $R_{1W}$, Eq. (14), and expression for $\lambda$, Eq. (24), we can also estimate the parameters $n_{TR},\lambda ,d_H$. The results are presented in Table 2B.

Figure 2. Comparison of the THB model with experimental T1 data.

Figure shows results of fitting the theoretical THB model expression for $R_{1W}$, Eq. (27), to the data in Table 1. The model with the uniform orientation distribution of vectors $\mathbf{r}$, Eq. (34), is used. Symbols are experimental data; lines are the fitting curves.

Table 2. Parameters of the biophysical THB model.

(A) Results of fitting the theoretical expression for $R_{1W}$, Eq. (27), to the data in Table 1. Numbers represent fitting parameters ± fitting errors. (B) Detail biophysical parameters of THB model. Data represent estimates of parameters $n_{TR},\lambda ,d_H$ using data for the fast T1 component [65] (not available for Globus Pallidum), the theoretical equations for $R_{1H}$ and $R_{1W}$, Eq. (14), and the expression for $\lambda$. Eq. (24). In both cases, A and B, the uniform orientation distribution of vectors $\mathbf{r}$, Eq. (34), is used.

A	WM	cGM	Caudate	Thalamus	Putamen	Glob. Pall.
$n_{TR}\cdot \lambda$ (s−1)	4.43 ± 0.37	3.19 ± 1.26	2.82 ± 0.31	2.90 ± 0.26	2.99 ± 0.29	3.45 ± 0.32
$\tau$ (ns)	2.51 ± 0.19	4.96 ± 3.52	3.49 ± 0.49	2.75 ± 0.26	3.06 ± 0.35	2.47 ± 0.18
b	0.56 ± 0.08	0.42 ± 0.18	0.57 ± 0.09	0.58 ± 0.08	0.56 ± 0.08	0.68 ± 0.13
$r_{1W}$ (s−1)	0.23 ± 0.11	0.08 ± 0.22	0.27 ± 0.07	0.27 ± 0.07	0.24 ± 0.07	0.37 ± 0.10
B
$n_{TR}$	0.07	0.04	0.03	0.04	0.04
$\lambda /\tau$ (1/s/ns)	25.74	15.96	29.35	29.46	27.56
$d_H$ (Å)	1.58	1.71	1.55	1.55	1.56

Orientation dependence of spin-exchange in individual bilayers

T1 data in Table 2 were analyzed for MRI signals averaged across large ROIs, where multiple orientations of bilayers and axons contribute to the measured signal, resulting in Eqs. (34). However, experimental data [49, 66] in WM regions with preferable orientations of axonal bundles show the anisotropic behavior of T1 relaxation. Within THB theory this issue is addressed by analyzing behavior of the correlation functions in THBs in axonal geometry (Figure 3).

Figure 3. A schematic structure of magnetic field orientation with respect to bilayer membrane.

(A), and axonal bundle (shown as a bundle of cylinders in B). Picture of bilayer membrane is adapted from https://commons.wikimedia.org/w/index.php?curid=3032610.

In lipid bilayers, the angle $\theta$ between vector $\mathbf{\text{r}}$ and the external field ${\mathbf{B}}_0$ can be expressed in terms of the angles {$\alpha ,\beta ,\psi$}, where $\alpha$ is an angle between the normal $\mathbf{\text{n}}$ to the bilayer’s surface and (Figure 3A), $\beta$ and $\psi$ are the polar and azimuthal angles in the spherical coordinate system related to the bilayer, with the polar axis parallel to $\mathbf{\text{n}}$:

$$\cos \theta =\left(\cos \beta \cdot \cos \alpha -\sin \beta \cdot \sin \psi \cdot \sin \alpha \right)$$ (28)

It is reasonable to assume that the directions of vectors $\mathbf{r}$ in the bilayers distributed symmetrically in-plane, i.e., satisfying the cylindrical symmetry with respect to the normal $\mathbf{\text{n}}$. Hence, Eqs. (21) should be averaged with respect to the azimuthal angle $\psi$ in Eq. (28). By further introducing the function $P(\beta )$ characterizing the distribution of the polar angle $\beta$, the averaged $g_m$ for bilayers become functions of the angle $\alpha$ between the normal $\mathbf{\text{n}}$ to the bilayer and ${\mathbf{B}}_0$ (Figure 3A):

$$g_{bilayer}^{(m)}(\alpha )\equiv \frac{1}{4\pi }\cdot \underset{0}{\overset{2\pi }{\int }}d\psi \underset{0}{\overset{\pi }{\int }}d\beta \cdot \sin \beta \cdot P(\beta )\cdot g_m(\alpha ,\beta ,\psi )$$ (29)

After straightforward calculations, the functions $g_m(\alpha )$ can be presented in the form:

$$g_{bilayer}^{(m)}(\alpha )=h_0^{(m)}+h_1^{(m)}\cdot {\cos }^2\alpha +h_2^{(m)}\cdot {\cos }^4\alpha$$ (30)

Importantly, parameters $h_{0,1,2}^{(m)}$ (see Appendix) are independent from the orientation of the bilayer with respect to ${\mathbf{B}}_0$ and depend only on the orientation distribution of vectors $\mathbf{r}$ in the bilayers with respect to the normal $\mathbf{n}$.

Orientation dependence of spin-exchange in WM axonal bundles

In a myelin sheath, lipid bilayers form concentric circular layers around an axon. Hence, for myelin sheaths, Eqs. (29) (with definitions in Eq. (48)) should be further averaged with respect to the bilayers’ directions around the axon. Let $\vartheta$ be an angle between the axon’s (cylinder) axis and ${\mathbf{B}}_0$ (Figure 3B). Then the angle $\alpha ={\mathbf{B}}_0\wedge \mathbf{n}$ is given by

$$\cos \alpha =-\sin \vartheta \cdot \cos \varphi ,\varphi \in \left[0,2\pi \right]$$ (31)

Averaging over $\varphi$, we obtain for WM axonal bundles (ab):

$$g_{ab}^{(m)}(\vartheta )={\tilde{h}}_0^{(m)}+{\tilde{h}}_1^{(m)}\cdot {\cos }^2\vartheta +{\tilde{h}}_2^{(m)}\cdot {\cos }^4\vartheta$$ (32)

The parameters ${\tilde{h}}_{0,1,2}^{(m)}$ (see Appendix) are independent from the axonal orientation with respect to ${\mathbf{B}}_0$ and depend only on the orientation distribution of vectors $\mathbf{r}$ inside the bilayers forming myelin sheaths.

The uniform orientation distribution of vectors $\mathbf{r}$ (with respect to the bilayer normal or the axonal axis) corresponds to $P(\beta )=1$. In this case,

$$\langle \left({\cos }^2\beta -1/3\right)\rangle =0,\langle {\left({\cos }^2\beta -1/3\right)}^2\rangle =4/45$$ (33)

and functions $g_{bilayer}^{(m)}(\alpha )$ and $g_{ab}^{(m)}(\vartheta )$ are independent from bilayers or axonal orientations with respect to magnetic field ${\mathbf{B}}_0$:

$$g_{bilayer}^{(0)}(\alpha )=g_{ab}^{(0)}(\vartheta )=\frac{4}{5},g_{bilayer}^{(1)}(\alpha )=g_{ab}^{(1)}(\vartheta )=\frac{2}{15},g_{bilayer}^{(2)}(\alpha )=g_{ab}^{(2)}(\vartheta )=\frac{8}{15}$$ (34)

For such a uniform distribution, all spin-exchange parameters are also independent of bilayers and axonal orientations with respect to ${\mathbf{B}}_0$, leading to the isotropy of longitudinal relaxation. However, if orientation distributions are not uniform, T1 relaxation becomes anisotropic. To define a non-uniform distribution of $\beta$, we need to know specific details of the orientation distribution function (ODF) $P(\beta )$. As an example, one can use Watson distribution [67]:

$$P(\beta ,\eta )=\sqrt{\frac{4\eta }{\pi }}\cdot \frac{1}{\text{erfi}(\sqrt{\eta })}\cdot \exp \left(\eta \cdot {\cos }^2\beta \right)$$ (35)

where $\text{erfi}(\mathrm{...})$ is the imaginary error function.

The uniform distribution corresponds to $\eta =0$, $<{\cos }^2\beta >=1/3$ and $<\beta >=1$. The predominantly parallel or antiparallel alignments of vectors $\mathbf{r}$ with the normal $\mathbf{\text{n}}$, the “long cigars”-like distribution ($<\beta >\approx 0\text{or}\pi$, $<{\cos }^2\beta >\approx 1$), corresponds to $\eta >>1$. The predominantly perpendicular alignment of vectors $\mathbf{r}$ with respect to the normal $\mathbf{\text{n}}$, the “pancake”-like distribution ($<\beta >\approx \pi /2,<{\cos }^2\beta >\approx 0$) corresponds to large negative $\eta <0$, $\left|\eta \right|>>1$. The comparison with experimental data (see Discussion section) suggests that the “long-cigar” case is realized in WM, i.e., transient hydrogen bonds have predominantly parallel or antiparallel alignments of vectors $\mathbf{r}$ with the normal $\mathbf{\text{n}}$. In this case, the orientation relationships for correlation functions can be reduced to Eq. (36) in parallel bilayers and to Eq. (37) for axonal bundles in WM (see Discussion section).

The results of simulations based on different types of the dipole-dipole connections orders in THB are presented in Figure 4.

Figure 4. Contribution of dipole-dipole interactions to longitudinal (T1) relaxation of MR signal in White Matter axonal bundles.

The THB model-predicted angular dependence of $R_{1W}$, Eq. (27), with parameters for WM in Table 2. Three types of dipole connection orders in bilayers wrapping around axons are shown: “long cigars” (connections are parallel to the bilayer’s normal, $\beta =0$), “pancakes” (connections are in the bilayer’s plane, $\beta =\pi /2$), and the uniform distribution of $\beta$. Results are shown for $B_0=1,5\text{T},3\text{T},\text{and}7\text{T}$.

Discussion

Inherent differences in MR signal relaxation properties between diverse biological tissues and their conditions (i.e., healthy vs. diseased) result in multiple applications of MRI in biology and medicine. Since inception of MRI, numerous papers have been devoted to understanding basic mechanisms relating MR signal relaxation properties to underlying microstructure of biological tissues. Such a knowledge allows improved MRI techniques better outlying tissue anatomy (e.g., gray matter vs. white matter) and pathology (e.g., tumors, lesions, etc.). This matter, however, is far from being resolved due to the enormous complexity of biological tissues and relatively low resolution of MRI, where hundreds (if not thousands) cells and subcellular structures are usually present in each imaging voxel. Hence, a spectrum of cellular characteristics may underlie various attributes of MRI signals. Therefore, theoretical models elucidating the relaxation properties of MRI signals should consider the prominent features of tissue microstructure that are relevant to the specific inquiries MRI experiments aim to tackle.

Within this context, this paper places its primary focus on the longitudinal relaxation of MRI signal in the brain and its relationship to one of the major parts of cellular structure – lipid bilayers forming cellular and myelin membranes. The biophysical theoretical model, introduced in this paper, emphasizes the role of hydrogen bonds (THBs), forming within hydrophilic heads of bilayers, as a pivotal factor in defining the longitudinal relaxation properties of MR signals. THBs allow strong quantum dipole-dipole interactions between the protons in water molecules and the protons of tissue-building molecules.

During MR experiments, water molecules randomly walk (diffuse) through multiple tissue environments, spending a portion of travelling time being transiently attached by hydrogen bonds to $H$-protons in lipid heads of cellular and myelin bilayers. Herein we use the theory proposed by Bloembergen, Purcell, and Pound [50] to describe a contribution of quantum spin exchange between water and $H$-protons in THBs.

It is important to emphasize that not all protons of tissue-building molecules, but only those structurally available for close transient contact with water, contribute to the magnetic spin exchange, hence to the relaxation properties of measured water MR signal relaxation. These protons represent a sub-population of protons residing in the hydrophilic heads. Magnetic properties of these protons are strongly affected by interaction with water as they always have a transiently hydrogen-bonded water proton counterpart to interact with. Water protons, on the other hand, are affected much less as they spent only a part of their “travelling” time in the hydrogen-bond-connected state.

Transient Hydrogen Bond model vs. Bloch-McConnell approach

The Bloch-McConnell equations permeate MR signal modeling from the original theory of MT effects developed by Morrison et al [9] to recent modeling of MT contributions to MR signal relaxation (see, e.g., [68] and references therein). The Bloch-McConnell equations result from merging the Bloch equations, describing T1 and T2 relaxation of magnetization in a single-component media, with the particle exchange mechanism proposed by McConnell [51] for treating chemical reactions. Importantly, this mechanical exchange does not depend on magnetic properties of exchanging particles. While such an approach is adequate for treating magnetization exchange due to diffusion of water molecules between cellular compartments (e.g., intracellular, extracellular, trapped between myelin layers), it is not adequate for treating magnetization exchange due to the quantum spin-spin interactions [69].

In the THB approach, the relaxations of longitudinal $\left(M_z^W,M_z^H\right)$ magnetizations of water ($W$) and head ($H$) protons are described by Eqs. (9). Several features are important for our analysis that distinguish Eqs. (9), describing relaxation due to quantum spin exchange processes in the THB model, from the Bloch-McConnell exchange model describing spin cross-relaxation by means of direct “physical” exchange of protons. The most important feature is that the exchange terms in the THB model, diagonal ($k$) and cross-relaxation ($K$), depend on the strength and orientation of the magnetic field ${\mathbf{B}}_0$ with respect to geometry of transient hydrogen bonds, Eqs. (23)-(24), thus leading to anisotropy of T1 (and T2, not considered in this paper) tissue relaxation properties (Figure 4). In contrast, in the Bloch-McConnell exchange model, these parameters depend only on the kinetic properties of water protons exchange in different chemical environments and are not related to tissue magnetic properties. In the THB model, the diagonal exchange term ($k$) and its cross-relaxation counterpart ($K$) are different from each other, while in the Bloch-McConnell equations all the exchange parameters are the same. Furthermore, the sign of the term $K$ in Eq. (9) depends on the relationship between the strength of $B_0$ and the time that water molecules transiently spend in the hydrogen-bonded state. Higher magnetic fields and longer transient times favor a positive value for $K$. All these properties are illustrated in Figure 5.

Figure 5. THB defined exchange terms dependence on magnetic field strength.

Left panel: the diagonal ($k$) and cross-relaxation ($K$) exchange parameters as functions of the magnetic field $B_0$ (A) and of the dimensionless parameter $\omega \cdot \tau$, (B). Middle panel: the parameters $k$ and $K$ as functions of $B_0$ in axonal bundles oriented parallel (C) and perpendicular (D) to ${\mathbf{B}}_0$. Right panel: the parameters $k$ and $K$ as functions of $B_0$ in individual membranes with the normal $\mathbf{\text{n}}$ oriented parallel (E) and perpendicular (F) to ${\mathbf{B}}_0$. The “long-cigar” distribution of the THB ($\beta =0$) is assumed in (C-F). Vertical axes in all plots are normalized to $\lambda$, Eq. (24), and the WM parameters from Table 2 are used.

According to Figure 5, the exchange parameter $K$ becomes small and even crosses zero around $B_0$ of 1–4 T (depending on the fiber or bilayer orientation with respect to ${\mathbf{B}}_0$). Since $K$ is responsible for the bi-exponential behavior of the longitudinal relaxation (see Eq. (25)), identifying the bi-exponential behavior in the region of small $K$ could be challenging. This reduction of $K$ can explain the results of Wang et al [65], who reported the mono-exponential behavior of water signal at $B_0=0.55\text{T}$ and 1.5T, while the bi-exponential behavior was detected at $B_0=3\text{T}$ and 7T.

Note that the THB-defined Eqs. (9) are similar in structure to those derived by Solomon [54], however, the specific parameters in Eqs. (9) reflect the specific features of our THB model. It should also be noted that comparison between the Solomon [54] and the Bloch-McConnell models was previously provided in [70] (see also [71, 72]) mostly focusing on their analogy. Herein we focused on their important differences.

THB model explains ${\text{B}}_0$ field dependence of longitudinal (T1) relaxation of MR signal

As demonstrated in Figure 2, the THB model quantitatively explains the dependence of T1 relaxation on the strength of magnetic field in WM and several GM regions. To compare our prediction of R1 dependence on $B_0$, Eq. (27), we used experimental data from Rooney et al [64] and Wang et al [65] that span a range of magnetic fields $B_0$ from 0.2T to 7T. A slight (about 0.13 sec⁻¹) increase of $r_{1W}$ in Globus Pallidum is most likely caused by the presence of high concentration of iron in this structure [73] and is consistent with the previously reported relationships between R1 and tissue iron content [74]. The results of the THB model are also in agreement with the previously reported bi-exponential behavior of T1 signal [65, 75] and shed new light on the nature of the so-called fast and slow components previously treated in the framework of the Bloch-McConnell equations.

By applying the THB model to experimental data in the human brain, we were able to determine biologically important parameters characterizing water interaction with bilayers forming cellular membranes and myelin sheaths (Table 2). We also identified hydrogen bonds with the average lifetime of about 3 ns as a source of the T1 relaxation mechanism governing the dipole-dipole interaction between water protons transiently trapped by these hydrogen bonds and chemically bound protons in the vicinity of these bonds. This result is consistent with reports on the presence of a slow component in 10-nanosecond time scale in aqueous solutions of biological systems [76]. Our results also showed that, during this interaction, an average distance between water protons and their chemically bound counterparts is about 1.6Å.

Transient Hydrogen Bond Order is a primary source of the MR signal relaxation anisotropy in WM

The THB theory shows that the effect of dipole interactions between transiently hydrogen-bonded water protons and their proton counterparts chemically bound in lipid heads, strongly depend on the transient orientation order of the dipole-dipole connections with respect to the magnetic field in individual bilayers, Eq. (30), or bilayers wrapped around axons, Eq. (32). In the hypothetical case of uniformly distributed orientations of dipole-dipole connections (absence of the THB order), the coefficients $h_{1,2}^{(m)}$ in Eq. (30), and coefficients ${\tilde{h}}_{1,2}^{(m)}$ in Eq. (32) are equal to zero, and there would be no anisotropy in MRI signal relaxation (Eqs. (34)). As a result, MRI signal relaxation parameters would be independent from the axonal orientation with respect to ${\mathbf{B}}_0$. Apparently, this is not the case in the brain tissue – experimental data [49, 66] show that MRI signal relaxation parameters depend on the orientation of axons with respect to ${\mathbf{B}}_0$. Hence, the transient dipole connections are not uniformly oriented but form a certain THB orientation order.

Moreover, in experimental reports [36–49], the relaxation rate parameters for axons oriented parallel to ${\mathbf{B}}_0$ are always smaller than for the axons perpendicular to ${\mathbf{B}}_0$. Comparing these result to the THB-predicted angular dependences in Figure 4, we can conclude that the sources of the relaxation anisotropy are the transient hydrogen bonds residing in the bilayers wrapped around axons with the dipole connections mostly parallel to the bilayers normal ($\beta \approx 0$, the cigars-like distribution with the cigars’ main axes perpendicular to the main axonal axis). In this case, the general expressions for the parameters $g^{(m)}(\alpha )$, Eqs. (30) and (32), can be simplified for parallel bilayers:

$$\begin{gathered} g_{bilayers}^{(0)}(\alpha )={\left(1-3\cdot {\cos }^2\alpha \right)}^2 \\ {g}_{bilayers}^{(1)}(\alpha )={\sin }^2\alpha \cdot {\cos }^2\alpha \\ {g}_{bilayers}^{(2)}(\alpha )={\sin }^4\alpha \end{gathered}$$ (36)

and for axonal bundles in WM:

$$\begin{gathered} g_{ab}^{(0)}(\vartheta )=\frac{1}{8}\cdot \left(11-30\cdot {\cos }^2\vartheta +27\cdot {\cos }^4\vartheta \right) \\ {g}_{ab}^{(1)}(\vartheta )=\frac{1}{8}\cdot \left(1+2\cdot {\cos }^2\vartheta -3\cdot {\cos }^4\vartheta \right) \\ {g}_{ab}^{(2)}(\vartheta )=\frac{1}{8}\cdot \left(3+2\cdot {\cos }^2\vartheta +3\cdot {\cos }^4\vartheta \right) \end{gathered}$$ (37)

The idea of the THB order introduced in our theoretical approach is consistent with results of [77] demonstrating that the lipid carbonyl groups served as efficient hydrogen bond acceptors allowing the water hydrogen bond network to reach, with its (up-oriented) O–H groups, into the headgroup of the lipid (see also [78]).

The THB theory presented above explains all major features of T1 signal anisotropy in the brain previously discovered by Kauppinen et al [49, 66]: a broad peak centered around 40–50⁰ at 3T and 7T, and a smaller R1 for WM fibers parallel to ${\mathbf{B}}_0$ as compared to the fibers perpendicular to ${\mathbf{B}}_0$. One should note that our data in Figure 4 correspond to an actual rotation of axons around ${\mathbf{B}}_0$, while experimental data are often obtained by measuring the fiber-to-field angle (${\theta }_{FB}$) with ${\theta }_{FB}$ determined by means of diffusion tensor imaging. This might not provide the same results as in a rotational experiment due to variation in the WM structure across the brain. Nevertheless, all salient features of T1 signal anisotropy reported in [49, 66] are well explained by the THB theory.

We should also note that, in addition to bilayers, another potential source of anisotropy could be THB of water with proteins forming the intra-axonal microtubules. Analysis similar to provided above shows that experimental data suggesting stronger T1 relaxation in the WM bundles perpendicular to ${\mathbf{B}}_0$ as compared to those parallel to ${\mathbf{B}}_0$ can be explained by the hydrogen bonds transiently residing in the intra-axonal microtubules with the dipole connections perpendicular to the axonal axis (pancake-like distribution).

Myelin Water

Figure 4 and Table 2 represent the results for the tissue-specific longitudinal relaxation rate parameter for experimental conditions allowing water molecules to travel over multiple cellular environments, i.e., intracellular, extracellular, trapped between bilayers in myelin sheath, etc. This is the case for typical T1 measurements with inversion recovery or saturation recovery sequences with long repetition times, on the order of seconds (as in [64, 65]). However, T1 measurements are often performed with the sequences based on short repetition times that allow separation of MR signals from different cellular compartments [79]. Also, experiments with stacked bilayers [80–82] have a potential to measure the signal generated by water trapped between bilayers.

In myelin, the lipid bilayers form the concentric structure with multiple layers with water trapped between them (so called myelin water). The analysis of multiple lipid bilayers [57] showed that there are on average 25 (range 9–35) water molecules per lipid molecule with 25–30% of them residing in the hydrophilic head. Hence, for myelin water:

$$n_{TR}(myelinwater)\approx 0.25-0.3$$ (38)

Comparing this estimate with estimate for average $n_{TR}$ of about 0.07 in WM (Table 2), we can predict that the dependence of myelin water $R_1$ on the strength of ${\mathbf{B}}_0$ and orientation of WM axonal bundles with respect to ${\mathbf{B}}_0$ are similar to those displayed in Figure 4 but scaled by a factor about 4. These dependencies are summarized in Figure 6A. Figure 6B illustrates the $R_1$ behavior that would be expected in water trapped between stacks of parallel bilayers. The angular dependencies for all $B_0$ reveal a profound minimum with the position about 45⁰ at low $B_0$, shifting to the magic angle with $B_0$ increases.

Figure 6. THB-predicted R₁ relaxation of water trapped between bilayers (myelin water).

The dependence of R₁ on the external magnetic field $B_0$ (A) in WM bundles (bilayers wrapped around axons) for the angle $\vartheta =0^{°},{45}^{°},\text{and}{90}^{°}$ between the axonal axis and ${\mathbf{B}}_0$; (B) in a stack of parallel bilayers for $B_0=1.5\text{T},3\text{T},\text{and}7\text{T}$; $\alpha$ is the angle between the bilayers’ normal $\mathbf{\text{n}}$ and ${\mathbf{B}}_0$. In both panels, the long-cigar dipole connection order is assumed ($\beta =0$, dipole connections are mostly oriented parallel to the bilayers’ normal).

Previously we have also demonstrated that the “Hokey Pokey”–like dance of water molecules in myelin sheath from aqueous phase to just beyond the lipid surface (i.e., lipid heads) can explain a correct sign and anisotropy properties of myelin water signal phase [83] reported in [34, 84].

Stretched dispersion function describes correlation of dipole interactions in THB model

As we demonstrated, the often-explored Lorentzian spectral density function $\tau /\left[1+{(\omega \tau )}^2\right]$ cannot adequately describe behavior of T1 relaxation in the brain in a broad range of magnetic fields (from 0.2T to 7T). To remedy this situation, we have introduced a new “stretched” spectral density function $\tau /\left[1+{(\omega \tau )}^{2b}\right]$ that for b<1 “stretches” dispersion to higher frequencies as compared with the Lorentzian form. We further demonstrated that in most brain tissues the parameter b is indeed less than 1 (see Table 2) and is equal on average to 0.56 with only slight variation across different brain structures (STD=0.08).

It can be further demonstrated that the stretched dispersion function can result from a distribution of correlation times characterizing the Lorentzian dispersion:

$$\frac{\tau }{1+{(\omega \tau )}^{2b}}=\underset{0}{\overset{\infty }{\int }}d{\tau }^{\prime }{f}_b(\tau |{\tau }^{\prime })\cdot \frac{{\tau }^{\prime }}{1+{(\omega {\tau }^{\prime })}^2}$$ (39)

where distribution function $f_b(\tau |{\tau }^{\prime })$ is

$$f_b(\tau |{\tau }^{\prime })=\frac{2}{\pi \tau }\cdot \left[\frac{{({\tau }^{\prime }/\tau )}^{2b+1}\cdot \sin \pi b}{1+2\cdot {({\tau }^{\prime }/\tau )}^{2b}\cdot \cos \pi b+{({\tau }^{\prime }/\tau )}^{4b}}\right]$$ (40)

with $\tau$ being the average lifetime:

$$\tau =\underset{0}{\overset{\infty }{\int }}d{\tau }^{\prime }{f}_b(\tau |{\tau }^{\prime })\cdot {\tau }^{\prime }$$ (41)

In summary, the introduction of the THB model represents an important advancement in comprehending MRI signal relaxation complexities within the human brain GM and WM cellular environment. We highlight the role of transient hydrogen bonds (THBs) that modulate quantum dipole interactions between water and protons of tissue-building molecules in cells, cellular and myelin membranes. Notably, the THB theory predicted the existence of a hydrogen-bond-driven transient structural order of dipole-dipole connections within THBs as a major factor for the anisotropy observed in MRI signal relaxation. Thus, THBs emerge as key determinants of tissue-specific field dependence and anisotropy in MRI signal relaxation.

By establishing equations governing spin exchange within THBs based on the principles of quantum dipolar relaxation theory, we have demonstrated a critical departure from the conventional Lorentzian frequency dispersion function that inadequately captures the magnetic field dependence of brain MRI signal relaxation. Instead, we proposed a stretched (extended) dispersion function that accommodates higher frequencies, surpassing the limitations of Lorentzian dispersion.

In the application of the THB theory to human brain data, our findings revealed the THBs lifetime about 3 nanoseconds and an average dipole distance of approximately 1.6Å for hydrogen bonds forming MRI signals. Our work not only aligns the quantum interactions with cellular intricacies in brain imaging but also introduces new insights into relationships between tissue microstructure and MRI measurements. By revealing the salient features of relaxation dynamics, the THB theory addresses existing gaps in our understanding of MRI signal formation within the brain and holds promise for developing new research and clinical protocols.

Appendix

Coefficients in Eq. (30):

$$h_0^{(0)}=\frac{4}{5}-\frac{3u}{2}+\frac{27w}{8},h_1^{(0)}=9u-\frac{135w}{4},h_2^{(0)}=-\frac{15u}{2}+\frac{315w}{8}$$ (42)

$$h_0^{(1)}=\frac{2}{15}-\frac{3w}{8},h_1^{(1)}=-\frac{u}{2}+\frac{15w}{4},h_2^{(1)}=\frac{5u}{6}-\frac{35w}{8}$$ (43)

$$h_0^{(2)}=\frac{8}{15}+\frac{u}{2}+\frac{3w}{8},h_1^{(2)}=-u-\frac{15w}{4},h_2^{(2)}=-\frac{5u}{6}+\frac{35w}{8}$$ (44)

Coefficients in Eq. (32):

$${\tilde{h}}_0^{(0)}=\frac{4}{5}+\frac{3u}{16}+\frac{81w}{64},{\tilde{h}}_1^{(0)}=\frac{9u}{8}-\frac{405w}{32},{\tilde{h}}_2^{(0)}=-\frac{45u}{16}+\frac{945w}{64}$$ (45)

$${\tilde{h}}_0^{(1)}=\frac{2}{15}+\frac{u}{16}-\frac{9w}{64},{\tilde{h}}_1^{(1)}=-\frac{3u}{8}+\frac{45w}{32},{\tilde{h}}_2^{(1)}=\frac{5u}{16}-\frac{105w}{64}$$ (46)

$${\tilde{h}}_0^{(2)}=\frac{8}{15}-\frac{5u}{16}+\frac{9w}{64},{\tilde{h}}_1^{(2)}=\frac{9u}{8}+\frac{45w}{32},{\tilde{h}}_2^{(2)}=-\frac{5u}{16}+\frac{105w}{64}$$ (47)

where $u$ and $v$ are defined by Eqs. (48):

$$\begin{gathered} u=\frac{1}{2}\cdot \underset{0}{\overset{\pi }{\int }}d\beta \cdot \sin \beta \cdot \left({\cos }^2\beta -\frac{1}{3}\right)\cdot P(\beta ) \\ w=\frac{1}{2}\cdot \underset{0}{\overset{\pi }{\int }}d\beta \cdot \sin \beta \cdot \left[{\left({\cos }^2\beta -\frac{1}{3}\right)}^2-\frac{4}{45}\right]\cdot P(\beta ) \end{gathered}$$ (48)

For ODF in Eq. (35), the functions $u=u(\eta )$ and $w(\eta )$ are as follows:

$$\begin{gathered} u(\eta )=1+\frac{3}{2\eta }-\frac{3\cdot \exp (\eta )}{\sqrt{\pi \eta }\cdot \mathrm{erfi}\left(\sqrt{\eta }\right)} \\ w(\eta )=\frac{1}{5}+\frac{3}{\eta }+\frac{27}{4{\eta }^2}+\frac{3\cdot \left(2\eta -9\right)\cdot \exp (\eta )}{2{\eta }^{3/2}\cdot \mathrm{erfi}\left(\sqrt{\eta }\right)} \end{gathered}$$ (49)