${\mathbf{R}}_{1_{\mathit{\rho }}}$ relaxation functions for weak, intermediate and strong collision models, revisited for frequency swept RF pulses

Abstract

In this work, we evaluated the expressions derived for describing rotating frame longitudinal ${\text{T}}_{1\rho }$ relaxations in the case of homonuclear dipole-dipole interactions spanning the entire range of strong, intermediate and weak collision regimes (SCR, ICR and WCR, respectively). First, the integral-based expressions for ${\text{R}}_{1\rho }\equiv 1/{\text{T}}_{1\rho }$ in the complete range of correlation times were represented through the summation of effective spectral density function terms. Then, the formalism for continuous wave spin-lock ${\text{T}}_{1\rho }$ was expanded to the time-dependent ${\text{T}}_{1\rho }\left(\text{t}\right)$ during radiofrequency (RF) irradiation delivered with amplitude and frequency modulated hyperbolic secant (HS) pulses, and the effective spectral density function terms were propagated through the HS pulses. The derived expressions are in good agreement with the formalism obtained previously for dipole-dipole relaxations in the weak collision case during HS pulses. In addition, the derived theory describes well the spin-lock ${\text{T}}_{1\rho }$ relaxation data of polyacetal (Delrin) in the SCR. In addition, we provided an application of the derived theory to spin-lock off-resonance ${\text{T}}_{1\rho }$ relaxation data of glycogen. Because the developed formalism is valid for the entire motional regime, from ultra-slow to fast motions, its applicability in vivo is plausible.

Introduction

In this work we expand the general formalism originally developed by Wolf et al. [1–3] for the longitudinal rotating frame relaxation due to homonuclear dipole-dipole interactions between like spins $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ considered for the weak, intermediate and strong collision regimes (WCR, ICR and SCR, respectively), to treat physical cases applicable in vivo in Magnetic Resonance Imaging (MRI). The general formalisms describing each condition, WCR, ICR and SCR separately had been developed already in the early days of MR. Wolf et al. [2] utilized the seminal theoretical work by Jacquinot and Goldman [4] dealing with the problem of the derivation of longitudinal rate expressions applicable in the solid state arena. Through a great deal of detailed derivations, Wolf et al. arrived at an integral-based expression for ${\text{R}}_{1\rho }\equiv 1/{\text{T}}_{1\rho }$ valid over the entire range of correlation times for the off-resonance spin lock (SL) continuous wave (CW) case. The predominant method of development of ${\text{R}}_{1\rho }$ numerical calculations in protein dynamics characterization, and in some in vivo cases, follows from the formalism developed by Blicharski following Jones [5, 6]. This method of derivations is specifically focused on WCR, strictly meaning that the local transverse relaxation time constant of the lattice is much greater than the lifetime of the considered particle between jumps to other sites. Here, we have adapted the formalism of Wolf et al. [2] from integral-based relationships to effective spectral density functions. We thus demonstrate that the newly developed relaxation functions can be easily utilized in routine calculations of ${\text{R}}_{1\rho }$ in the regime of intermediate and slow motion, i.e. with correlation times, ${\tau }_{\text{c}}$, slower than 10⁻⁶ s and up to 10⁻¹ s.

Tissue contrast in MRI is topic of extensive research. Yet, detailed understanding of the relaxation contrasts in vivo is challenged by multiple experimental obstacles and complexities of relaxation processes in the living sample. This is obvious, since the contrast observed in MRI is unavoidably linked to a pulse sequence designed for its measurement, and thus it reflects the different weighting of the techniques. Isolation of specific relaxation pathways that define the MRI contrast in vivo is challenging, and requires independent quantitative relaxation assessments using a broad arsenal of MRI modalities and pulse sequence architectures, including ${\text{T}}_1,{\text{T}}_2,{\text{T}}_1\rho ,{\text{T}}_2\rho ,{{\text{T}}_2}^{*}$ and diffusion weighted pulse sequences. Several distinct water proton pools can be identified in the living sample, e.g., semi-solid pool associated with the bound protons forming cellular composition, free tumbling pool with water molecules in the extracellular space, or pool of buried water with restricted mobility [7–9]. Therefore, the description of relaxation in vivo generally requires correlation functions other than just a single mono-exponential, because spectral densities with a single correlation time cannot adequately describe the complexity of the system [8, 10–12]. It had been demonstrated that the unique information related to a hydrogen bond orientation order could be deciphered through the free–precession ${\text{T}}_1$ relaxation, with the detection of fast and slow relaxing pools [7, 9, 12, 13]. Despite that, however, conventional ${\text{T}}_1$ and ${\text{T}}_2$ measurements with the settings used for applications in vivo are unable to detect relaxations in the semi-solid pools outside of the WCR. Determination of tissue fundamental parameters warrants the development and application of methods that can tackle slow motion in tissue, and in particular at high magnetic fields, e.g., spin lock and adiabatic ${\text{T}}_{1\rho }$ and ${\text{T}}_{2\rho }$ [14–17], or their derivative method that comprises ${\text{T}}_{1\rho }$ and ${\text{T}}_{2\rho }$ relaxations, entitled RAFFn (Relaxations Along a Fictitious Field in the Rotating frame of rank n, RAFFn) [18, 19].

Despite the recent active development and applications of quantitative MR relaxometry using frequency swept (FS) RF pulses [14–16, 18–39], the description of ${\text{T}}_{1\rho }$ relaxations outside of the WCR in the slow-intermediate motional regimes during modulated RF pulses has not been presented so far. Therefore, in this work we developed a general formalism for the description of rotating frame longitudinal relaxations during FS modulated RF pulses in a wide spectrum of motions, from ultra-fast to ultra-slow. We thus approached this problem in multiple steps. First, we generalized rotating frame longitudinal relaxation expressions by expanding the integral-based expressions of Wolf et al. [2] into a summation of effective spectral density terms. Second, we expanded the formalism of Wolf et al. [2] derived in a broad range of motional regimes for off-resonance SL to the case of time-dependent ${\text{R}}_{1\rho }\left(\text{t}\right)$ during amplitude and frequency modulated RF pulses of the hyperbolic secant (HS) family. To validate our derivations, we confirmed our results derived in the WCR with the solution documented previously by Blicharski, using an instantaneous approximation that propagates the relaxation expressions for ${\text{R}}_{1\rho }$ through the modulation functions of the HS pulse, as originally described in [16, 35, 40]. In addition, for the SCR, we performed a theoretical fitting using the derived expressions to the data obtained by De Luca et al. for the ${\text{R}}_{1\rho }$ measurements of polyacetal (Delrin) obtained under continuous-wave (CW) spin lock (SL) conditions [41]. Because the formalism is valid for the entire range of motional regimes, from ultra-slow to fast motions, we anticipate that it will be useful for in vivo applications.

Theory

We utilize expressions obtained by Wolf et al. [2], and we expand them for applications to probe slow motion regime (SMR). For convenience, we start from the general expression of off-resonance ${\text{R}}_{1\rho }$ in the wide motional regime case, as follows [2]:

$$R_{1\rho }\left(t,{\tau }_c,T_m\right)=\frac{1}{2tr\left({H^2}_{0\rho }(t)\right)}\left(tr\left(A^{\left(0\right)2}\right){\frac{\partial G\left(t,{\tau }_c\right)}{\partial t}}_{t=0}\right)+\sum _{p=-2}^2\sum _{q=-2}^2\frac{tr\left(A^{\left(p\right)T}\cdot A^{\left(q\right)}\right)}{2tr\left({H^2}_{0\rho }\left(t\right)\right)}{a}^{\left(p\right)}(t)a^{\left(q\right)}(t)\underset{0}{\overset{\infty }{\int }}\frac{{\partial }^2}{{\partial }^2\tau }G\left(\tau ,{\tau }_c\right)\text{exp}\left(-\frac{\tau }{T_m}\right)\text{exp}\left(i{\omega }_{eff}(p,t)\tau \right)d\tau$$ (1,a)

and

$${\omega }_{eff}\left(p,t\right)=p{\omega }_{eff}\left(t\right).$$ (1,b)

Here, ${\mathit{A}}^{\left(0\right)}$ is a 0-rank tensor, ${\mathit{A}}^{\left(0\right)}={\mathit{A}}^{\left(0\right)\text{T}}$ whose components are defined below. Similarly, ${\mathit{A}}^{\left(\text{p}\right)}$ is a p rank tensor with $p$ equal to ±1, ±2. The same p is used in Eq (1,b) for the definition of $p{\omega }_{\mathit{\text{eff}}}\left(t\right)$. Here, the symbol $T$ is used to denote the matrix operation of taking the transpose of a square matrix. ${H^2}_{0\rho }\left(t\right)$ is the Hamiltonian. Notably, the tensors terms in Eq (1) are orthonormal under the trace operation:

$$\text{tr}\left(A^{\left(p\right)T}\cdot A^{\left(p\right)}\right)=1,\left(\text{tr}\left(A^{\left(q\right)T}\cdot A^{\left(q\right)}\right)=1,tr\left(A^{\left(q\right)T}\cdot A^{\left(p\right)}\right)=0,tr\left(A^{\left(p\right)T}\cdot A^{\left(q\right)}\right)=0.\right.$$ (1.c)

The relations in Eq (1,c) imply that the double sum over p and q in Eq (1,a) becomes a single summation in the variable p. We also note that $a^{\left(p\right)}\left(t\right)$ and $a^{\left(q\right)}\left(t\right)$ are trigonometric functions that specify the coefficients weighting the tensor expansion of the dipolar two coupled spin $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ of the isotropic interaction Hamiltonian (see definitions of tensors in Appendix I); $\text{G}\left(\tau ,{\tau }_{\text{c}}\right)$ is the definition of the correlation function chosen for the spin system of interest. Here, ${\tau }_{\text{c}}$ is the rotational correlation time for the dipolar interaction, and ${\text{T}}_{\text{m}}$ is the transverse spin-spin relaxation time constant of the lattice environment, which, following the definition of Wolf et al., defines the time required for the spin system to achieve equilibrium in the rotating frame, or the time required for thermal mixing of dipolar and Zeeman heat reservoirs [2, 4, 42].

Next, we define the terms necessary to evaluate Eq (1) for the definition of ${\text{R}}_{1\rho }$ for a single two proton system such as a water molecule. Eq (1) is a generalization of Eq (2.58) of Wolf et al. [2] for the time dependent RF modulation functions of interest using an instantaneous approximation, where the relaxation functions are calculated at each point in time during the RF waveforms [16]. Here, ${\omega }_1\left(t\right)$ is defined as the amplitude modulating function, and $\Delta \omega \left(t\right)$ is the frequency modulation function of the RF pulse, given as follows:

$${\omega }_1\left(t\right)={\omega }_1^{max}\mathit{\text{sech}}\left(\beta \left(2t/T_p-1\right)\right),$$ (2)

$$\Delta \omega \left(t\right)={\omega }_{RF}\left(t\right)-{\omega }_c=A\mathit{\text{tanh}}\left(\beta \left(2t/T_p-1\right)\right).$$ (3)

Following standard notation, the angle between the quantization axis in the first rotating frame (FRF) $z$’ and the time dependent effective magnetic field ${\mathit{B}}_{\mathit{\text{eff}}}\left(t\right)={\gamma }^{-1}{\omega }_{\mathit{\text{eff}}}\left(t\right)$ is defined as follows:

$$\theta (t)=\mathit{\text{arctan}}\left(\frac{{\omega }_1\left(t\right)}{\Delta \omega \left(t\right)}\right),$$ (4,a)

and the time dependent effective frequency ${\omega }_{\mathit{\text{eff}}}\left(t\right)$ is given by:

$${\omega }_{eff}\left(t\right)=\sqrt{{\omega }_1^2\left(t\right)+\Delta {\omega }^2\left(t\right)}.$$ (4,b)

Here, $\beta$ is a truncation factor, $A$ is the amplitude of the frequency sweep in rad/s with the bandwidth of the pulse being $BW=2A,T_p$ is the pulse duration, $t\in [0,Tp],{\omega }_{\text{c}}$ is carrier frequency (the center frequency in the bandwidth of interest), and ${{\omega }_1}^{\text{max}}$ is the maximum value of ${\omega }_1\left(t\right)$. Following Wolf et al. [2], the effective frequency of the rank $p$ is given by ${\omega }_{\mathit{\text{eff}}}(p,t)$ in $exp\left(i{\omega }_{\mathit{\text{eff}}}(p,t)\tau \right)$ (Eq (1)). For details, the reader is referred to Supplementary Materials, Appendix I, which summarizes the list of matrix representations of rank 2 spherical tensors used in Eq (1). We next call attention to the fact that Eq (1) can be expanded upon rewriting the second term, as follows:

$$R_{1\rho }\left(t,{\tau }_c,T_m\right)=\left(\frac{1}{2tr\left({H^2}_{0\rho }\left(t\right)\right)}tr\left(A^{\left(0\right)2}\right){\frac{\partial G\left(t,{\tau }_c\right)}{\partial t}}_{t=0}|+{R^{\left(0\right)}}_{1\rho }\left(t,{\tau }_c,T_m\right)+{R^{\left(1\right)}}_{1\rho }\left(t,{\tau }_c,T_m\right)+{R^{\left(2\right)}}_{1\rho }\left(t,{\tau }_c,T_m\right)\right),$$ (5)

where the summation in the expression for $p=0$ indicates:

$${R^{\left(0\right)}}_{1\rho }\left(t,{\tau }_c,T_m\right)=\frac{tr\left(A^{\left(0\right)}\cdot A^{\left(0\right)}\right)}{2tr\left({H^2}_{0\rho }\left(t\right)\right)}4{f^{\left(0\right)}}_{\theta }\left(t\right)\underset{0}{\overset{\infty }{\int }}\frac{{\partial }^2}{{\partial }^2\tau }G\left(\tau ,{\tau }_c\right)\text{exp}\left(-\frac{\tau }{T_m}\right)d\tau .$$ (6)

The factor 4 originates from considering two interacting particles as given by Eq (2.58) in Wolf and Jung [2], and the integration is over t. Note that the factor of 4 is cancelled by the denominator of the definition of the function ${f^{\left(0\right)}}_{\theta }\left(t\right)$ in Eq (10) below. For detailed derivations, the reader is referred to Supplementary Material, Appendix II. The relaxation terms ${{\text{R}}^{\left(1\right)}}_{1\rho }\left(\text{t},{\tau }_{\text{c}},{\text{T}}_{\text{m}}\right)$ and ${{\text{R}}^{\left(2\right)}}_{1\rho }\left(\text{t},{\tau }_{\text{c}},{\text{T}}_{\text{m}}\right)$ are given in Eq (7) and Eq (8):

$${R^{\left(1\right)}}_{1\rho }\left(t,{\tau }_c,T_m\right)=\frac{tr\left(A^{\left(1\right)T}\cdot A^{\left(1\right)}\right)}{2tr\left({H^2}_{0\rho }\left(t\right)\right)}16{f^{\left(1\right)}}_{\theta }\left(t\right)\underset{0}{\overset{\infty }{\int }}\frac{{\partial }^2}{{\partial }^2\tau }G\left(\tau ,{\tau }_c\right)\text{exp}\left(-\frac{\tau }{T_m}\right)\text{cos}\left({\omega }_{eff}\left(t\right)\tau \right)d\tau ,$$ (7)

$${R^{\left(2\right)}}_{1\rho }\left(t,{\tau }_c,T_m\right)=\frac{tr\left(A^{\left(2\right)T}\cdot A^{\left(2\right)}\right)}{2tr\left({H^2}_{0\rho }(t)\right)}16{f^{\left(2\right)}}_{\theta }\left(t\right)\underset{0}{\overset{\infty }{\int }}\frac{{\partial }^2}{{\partial }^2\tau }G\left(\tau ,{\tau }_c\right)\text{exp}\left(-\frac{\tau }{T_m}\right)\text{cos}\left(2{\omega }_{eff}\left(t\right)\tau \right)d\tau .$$ (8)

Note that the cosine term in Eq (7,8), with the multiplicative factor of 2 in the argument of Eq (8), originates from the exponential term of Eq (1) (see tensor definitions in Supplementary Material, Appendix III). We note that in Eqs (7,8) the following relation was used:

$$\left|tr\left[T_2^{\left(p\right)T}\cdot T_2^{\left(q\right)}\right]\right|=\delta (p,q)$$ (9)

Here $\delta (p,q)$ is the Kronecker’s delta-function. Notably, the Eqs (7,8) are of the form of cosine transform with the limits of integrations from 0 to $\infty$. The following terms $f_{\theta }^{\left(p\right)}\left(t\right)={\left(a^{\left(p\right)}\left(t\right)\right)}^2$ were defined:

$$f_{\theta }^{\left(0\right)}\left(t\right)=\frac{1}{4}{\left(1-3{\text{cos}}^2\left(\theta \right(t\left)\right)\right)}^2,$$ (10)

$${f^{\left(1\right)}}_{\theta }\left(t\right)=\frac{1}{16}{\text{sin}}^2\left(2\theta \right(t\left)\right),$$ (11)

$${f^{\left(1\right)}}_J\left(t\right)=\frac{1}{4}{\text{sin}}^4\left(\theta \right(t\left)\right).$$ (12)

Equations (10–12) are formulated with the algebraic manipulations that led from Eq (1) to Eqs (6–8). The term ${f^{\left(0\right)}}_{\theta }\left(t\right)$ defined in Eq (10) was obtained from the variables in Eq (1), i.e., $a^{\left(p\right)}\left(t\right)a^{\left(q\right)}\left(t\right)$ with $\text{p}=\text{q}=0$. Continuing this logical progression, the meaning of ${f^{\left(1\right)}}_{\theta }\left(t\right)$ in Eq (11) corresponds to $\text{p}=1,\text{q}=-1$. Similarly, for ${f^{\left(2\right)}}_{\theta }\left(t\right)$, in the expression $a^{\left(p\right)}\left(t\right)a^{\left(q\right)}\left(t\right),\text{p}=2,\text{q}=-2$, as in Eq (12).

In this work we utilized a single monoexponential correlation function and we introduced the integrals of effective spectral density terms to derive an expression for effective ${\text{R}}_{1\rho }$. We accomplished this by integrating by parts twice in Eqs (6–8). In fact, the terms in Eq (1) could be rewritten as a summation of relevant rank $p$ terms given in Eqs (6–8), where the transformations by parts were carried out (see Appendix II, III and IV in Supplementary Material).

The expressions of the longitudinal and transverse rotating frame relaxation rates, ${\text{R}}_{1\rho }$ and ${\text{R}}_{2\rho }$, due to dipolar relaxations between like spins for stationary case in a WCR were derived by Blicharski [5], and subsequently were extended to the description of dipolar relaxations during adiabatic and non-adiabatic FS pulses within a time dependent framework [15, 16, 19, 29, 36]. To directly utilize the formalism developed in this work, we used the equations Eqs (IV.15) and (IV.16) in Appendix IV of Supplementary Material for comparing with the Blicharski’s result derived in the WCR (Eq (IV.1)), ${\text{R}}_{1\text{p}}\left(\text{t},{\tau }_{\text{c}},{\text{T}}_{\text{m}}\right)$, with a simple exponential correlation function. As it is demonstrated in Appendices III and IV, the dominant terms of Eq (1) that contribute to the Blicharski’s expression are:

$$R_{1\rho }\left(t,{\tau }_c,T_m\right)={R^{\left(1\right)}}_{1\rho }\left(t,{\tau }_c,T_m\right)+{R^{\left(2\right)}}_{1\rho }\left(t,{\tau }_c,T_m\right).$$ (13)

Here we define:

$${R_{1\rho }}^{\left(1\right)}\left(t,{\tau }_c,T_m\right)=R_D\frac{9}{80}16{f^{\left(1\right)}}_{\theta }\left(t\right)J_{eff}\left({\omega }_{eff}\left(t\right)\right),$$ (14)

$${R^{\left(2\right)}}_{1\rho }\left(t,{\tau }_c,T_m\right)=R_D\frac{9}{80}16{f^{\left(2\right)}}_{\theta }\left(t\right)J_{eff}\left(2{\omega }_{eff}\left(t\right)\right).$$ (15)

The functions ${f^{\left(1\right)}}_{\theta }\left(t\right)$ and ${f^{\left(2\right)}}_{\theta }\left(t\right)$ are given in Eqs (11) and Eq (12), respectively. The spectral density functions used here were derived in Supplementary Material, Appendix IV, and the effective correlation time is defined as follows (Eq. (IV.5)):

$${\tau }_{eff}=\frac{{\tau }_c{T}_m}{\left(T_m+{\tau }_c\right)}.$$ (16)

For the present purposes, we chose an exponential correlation function of the form [5]:

$$G\left(\tau ,{\tau }_c\right)=\text{exp}\left(-\frac{\tau }{{\tau }_c}\right).$$ (17)

The value $R_D$ reflects the squared dipolar coupling for a pair of spins $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ under the point-dipole approximation:

$$R_D=\frac{{\gamma }^4{\hslash }^2}{r^6},$$ (18)

where $r$ is the distance between the two protons, $\gamma$ is the gyromagnetic ratio of a proton, $\hslash$ is Planck’s constant divided by $2\pi$. Examining the denominator of Eq (16), one can see that for ${\text{T}}_{\text{m}}\gg {\tau }_{\text{c}}$ the denominator is effectively ${\text{T}}_{\text{m}}$. The spectral densities were found by simplifying the terms in Eqs (IV.15) and (IV.16) in Appendix 4 using the fact that, for the model considered here, ${\text{T}}_{\text{m}}\gg {\tau }_{\text{c}}$ when the correlation times are within the range between 10⁻¹¹ and 10⁻⁶ s.

For the relevant range of correlation times of 10⁻¹¹ to 10⁻⁶, using Eq (16), the sum of Eqs (14 and 15) simplifies as follows:

$$R_{1\rho }\left(t,{\tau }_c,T_m\right)=R_D\frac{9}{5}{f_{\theta }}^{\left(1\right)}\left(t\right){\tau }_{eff}\left({\tau }_c,T_m\right)/\left(1+{\left({\tau }_{eff}\left({\tau }_c,T_m\right){\omega }_{eff}\left(t\right)\right)}^2\right)+R_D\frac{9}{5}{f_{\theta }}^{\left(2\right)}\left(t\right){\tau }_{eff}\left({\tau }_c,T_m\right)/\left(1+{\left(2{\tau }_{eff}\left({\tau }_c,T_m\right){\omega }_{eff}\left(t\right)\right)}^2\right)$$ (19)

For the details see also SM IV, Eq (IV.17). Using the instantaneous approximation, the averaged ${\text{R}}_{1\rho }$ rate constant during the AFP pulse is given by the time integral normalized to the duration of the pulse ${\text{T}}_{\text{p}}$, as follows:

$${R_{1\rho }}^{\mathit{\text{Norm}}}\left({\tau }_c,T_m\right)=\frac{1}{T_p}\left(\underset{0}{\overset{T_p}{\int }}\left(R_D\frac{9}{5}{f_{\theta }}^{\left(1\right)}\left(t^{\prime }\right){\tau }_{eff}\left({\tau }_c,T_m\right)/\left(1+{\left({\tau }_{eff}\left({\tau }_c,T_m\right){\omega }_{eff}\left(t^{\prime }\right)\right)}^2\right)d{t}^{\prime }\right.\right.\left.+\underset{0}{\overset{T_p}{\int }}\left(R_D\frac{9}{5}{f_{\theta }}^{\left(2\right)}\left(t^{\prime }\right){\tau }_{eff}\left({\tau }_c,T_m\right)/\left(1+{\left(2{\tau }_{eff}\left({\tau }_c,T_m\right){\omega }_{eff}\left(t^{\prime }\right)\right)}^2\right)\right)d{t}^{\prime }\right).$$ (20)

For, ${\text{T}}_{\text{m}}\gg {\tau }_{\text{c}}$, we find that:

$${\tau }_{eff}\simeq {\tau }_c.$$ (21)

Using Eqs (10–15), the Eq (13) can be rewritten after straightforward analysis as:

$$R_{1\rho }\left(t,{\tau }_c\right)=R_D\frac{9}{80}{\text{sin}}^2\left(2\theta \left(t\right)\right)\frac{{\tau }_c}{\left(1+{\left({\tau }_c{\omega }_{eff}\left(t\right)\right)}^2\right)}+R_D\frac{9}{20}{\text{sin}}^4\left(\theta \left(t\right)\right)\frac{{\tau }_c}{\left(1+4{\left({\tau }_c{\omega }_{eff}\left(t\right)\right)}^2\right)}.$$ (22)

Under the assumption that ${\tau }_{\text{c}}^2{{\omega }_{\text{eff}}}^2\left(\text{t}\right)\ll 1$, the denominator in Eq (20) reduces effectively to 1, leading to the final approximate expressions of ${R^{\text{WCR}}}_{1\rho }$:

$${R^{\text{WCR}}}_{1\rho }\approx R_D\frac{9}{80}{\text{sin}}^2\left(2\theta \left(t\right)\right){\tau }_c+R_D\frac{9}{20}{\text{sin}}^4\left(\theta \left(t\right)\right){\tau }_c,$$ (23)

and to the expression of ${\text{R}}_{1\rho }^{\text{SCR}}$ under the assumption that ${\tau }_{\text{c}}\gg {\text{T}}_{\text{m}}$:

$${R^{SCR}}_{1\rho }\approx R_D\frac{9}{80}{\text{sin}}^2\left(2\theta \left(t\right)\right)\frac{T_m}{\left(1+{\left(T_m{\omega }_{eff}\left(t\right)\right)}^2\right)}+R_D\frac{9}{20}{\text{sin}}^4\left(\theta \left(t\right)\right)\frac{T_m}{\left(1+4{\left(T_m{\omega }_{eff}\left(t\right)\right)}^2\right)}.$$ (24)

Notably, by using basic trigonometric relationship $\text{sin}\left(2\theta \right(t\left)\right)=2\text{sin}\left(\theta \right(t\left)\right)\text{cos}\left(\theta \right(t\left)\right)$ in Eq (23), one can obtain Eq (24). It should be also noted that for the SCR conditions, when the ${\tau }_{\text{c}}\gg {\text{T}}_{\text{m}}$, from Eq (16) it directly follows that the ${\tau }_{\text{eff}}={\text{T}}_{\text{m}}$. Thus, Eq (24), which is presented as the SCR expression for ${\text{R}}_{1\rho }$, should contain ${\text{T}}_{\text{m}}$ and not ${\tau }_{\text{c}}$. The dependencies of ${\tau }_{\text{eff}}$ on ${\tau }_{\text{c}}$ and ${\text{T}}_{\text{m}}$ are presented in Fig. 3. Note that these dependencies are substantially different in the WCR as compared to ICR and SCR. It could be noticed that the behavior of ${\tau }_{\text{eff}}$ in different regimes directly implies the dependence of the averaged ${\text{R}}_{1\rho }$ rate constants in these regimes, shown in Fig. 4. Eq (24) is one of the central results of this paper, which expresses the ${\text{R}}_{1\rho }$ dependence on the effective frequency and ${\text{T}}_{\text{m}}$ in the SCR.

Figure 3. Effective correlation times in the WCR, ICR and SCR.

The dependence of the effective correlation time ${\tau }_{\text{eff}}$ on ${\tau }_{\text{c}}$ and ${\text{T}}_{\text{m}}$ in three regimes, WCR, ICR and SCR. For the calculations, Eq (16) was used. The 3D plots exhibit substantial differences of the ${\tau }_{\text{eff}}$ for fast and slow correlation times in different regimes as defined by the ${\text{T}}_{\text{m}}$.

Figure 4. ${{\text{R}}_{1\rho }}^{\text{Norm}}$ as a function of ${\tau }_{\text{c}}$ and ${\text{T}}_{\text{m}}$ in WCR, ICR and SCR.

Dependencies on $T_{\text{m}}$ and ${\tau }_{\text{c}}$ of the ${{\text{R}}_{1\rho }}^{\text{Norm}}$ rate constants, integrated during the HS (time BW product R=20) pulse and normalized to the pulse duration. For the calculations, ${{\omega }_1}^{\text{max}}/\left(2\pi \right)=2\text{kHz}$ and ${\text{T}}_{\text{p}}=4\text{ms}$ were used. For the calculations Eq (20) was used. The relaxation rates were calculated by integrating instantaneous solutions for ${\text{R}}_{1\rho }\left(\text{t}\right)$ during the AFP HS pulse, and normalizing the integral to the pulse time duration ${\text{T}}_{\text{p}}.\text{r}=1.53\acute{Å}$ s was used.

Now, let’s consider $T_m\simeq {\tau }_c$. In this case, we are in the ICR, and from Eq (16) we see that:

$${\tau }_{eff}\simeq \frac{{\tau }_c}{2}.$$ (25)

Therefore, after some straightforward algebra Eq (20) leads to:

$${R^{ICR}}_{1\rho }\simeq R_D\frac{9}{80}{\text{sin}}^2\left(2\theta \right(t\left)\right)\frac{{\tau }_c/2}{\left(1+{\left(\left({\tau }_c/2\right){\omega }_{\mathit{\text{eff}}}\left(t\right)\right)}^2\right)}+R_D\frac{9}{20}{\text{sin}}^4\left(\theta \right(t\left)\right)\frac{{\tau }_c/2}{\left(1+{\left({\tau }_c{\omega }_{\mathit{\text{eff}}}\left(t\right)\right)}^2\right)}$$ (26)

We now note that this expression exactly reproduces the relevant dominant terms in Eq (31) by Blicharski [5] without taking into account $\text{J}\left({\omega }_0\right)$ and $\text{J}\left(2{\omega }_0\right)$ terms (See also Eq (IV24) in SM):

$${R^{WCR}}_{1\rho }=\frac{9}{20}{R}_D\left({\text{sin}}^2\theta (t){\text{cos}}^2\left(\theta \left(t\right)\right)\frac{{\tau }_c}{\left(1+{\left({\tau }_c{\omega }_{eff}\left(t\right)\right)}^2\right)}+{\text{sin}}^4\left(\theta \left(t\right)\right)\frac{{\tau }_c}{\left(1+4{\left({\tau }_c{\omega }_{eff}\left(t\right)\right)}^2\right)}\right).$$ (27)

It should be pointed out that there is a factor of $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ difference in the definitions of Eq (26) and (27). To detail the argument, Eq (26) is Eq (19) with ${\tau }_{\mathit{\text{eff}}}=\frac{\tau }{2}$, given the definition of ${\tau }_{\mathit{\text{eff}}}$ as specified in Eq (16). We call to the reader’s attention that Eq (23) exactly agrees with our Eq (IV.1) in Appendix 4 for Blicharski’s result [5].

Results and Discussion

Relaxation theory as applied to the interpretation of experimental results in the purview of bio-physical magnetic resonance has focused on utilizing a theoretical formalism which is within the WCR because the majority of considered systems were in aqueous environments where the range of effective correlation times is in the fast case between 10⁻¹² and 10⁻⁶ s. Seminal contributions by Jones and Blicharski [5, 6] described the weak collision case for ${\text{R}}_{1\rho }$ relaxations, and the work pioneered by Ailion and Slichter [43–45] and Wolf et al. [1–3] provided the description of the weak, intermediate and strong collision cases. Although these fundamental contributions were dedicated to the description of each motional regime of ${\text{R}}_{1\rho }$ separately, the generalization explicated by Wolf et al. in [2] allowed to treat each of these regimes using one general formalism. With the ascendance of rotating frame MRI methodologies based on AM and FM RF pulses, the need to interpret in vivo tissue contrasts emphasizing slow motion has become an urgent issue. Therefore, in this work we aimed to address the problem of evaluating the adiabatic ${\text{R}}_{1\rho }$ relaxations during HS pulses in the wide range of motional regimes by applying the general formalism of Wolf et al. [1, 2].

For the convenience of the reader, we detail the operational time constant areas of validity of the different regimes of motions, as follows:

$$\begin{gathered} \mathit{\text{Weak}}:T_m>>{\tau }_c \\ \mathit{\text{Intermediate}}:T_m\approx {\tau }_c \\ \mathit{\text{Strong}}:{\tau }_c>>T_m \end{gathered}$$ (28)

The effort, demonstrated in this paper, has emphasized the development of ${\text{R}}_{1\rho }$ equations valid from the fast to intermediate and slow-motion arena, and had been extended from the CW SL RF irradiation for ${\text{R}}_{1\rho }$ to the time dependent case through the utilization of the amplitude and frequency modulated adiabatic RF pulses, i.e., adiabatic ${\text{R}}_{1\rho }$ technique [16, 35]. To substantiate the validity of the developed theory, first we provide detailed comparison with the canonical results by Blicharski [5]. In Fig. 1a and Fig.1b, the 3D plots of ${\text{R}}_{1\rho }$ in the WCR during the HS pulse as a function of rotational correlation time are presented. For the calculations shown in Fig. 1a, we used Eq (31) by Blicharski [5], adapted to the time dependent case of ${\text{R}}_{1\rho }\left(\text{t}\right)$ relaxations during the HS pulse [16, 35, 46]. For the analysis, we applied the instantaneous approximation by propagating the relaxation functions through each point in time during the durations of the pulse [16]. The formalism used was for ${\omega }_0\gg {\omega }_{\text{eff}}\left(\text{t}\right)$, where ${\mathit{\omega }}_0$ is the Larmor frequency of the proton, and ${\omega }_{\text{eff}}\left(\text{t}\right)$ is defined in Eq (4.b). The results of ${\text{R}}_{1\rho }\left(\text{t},{\tau }_{\text{c}},{\text{T}}_{\text{m}}\right)$ shown in Fig. 1b were obtained using Eq (19). The agreement between plots in Fig. 1a and Fig. 1b is rather remarkable. The detailed derivation of the appropriate expressions for simulating the Blicharski’s results for the fast motion are presented in Supplementary Materials (see Eq (III-15) and Eq (III-33)).

Figure 1. Calculations of ${\text{R}}_{1\rho }$ during the HS1 pulse, obtained in the WCR and ICR with different formalisms and for different ranges of correlation times.

Calculated relaxation rate constants due to dipole-dipole interactions during the HS1 pulse in the weak (a, b, c, e) and intermediate (d) collision regimes. For the calculations ${{\omega }_1}^{\text{max}}/\left(2\pi \right)=2\text{kHz}$, and time BW product R=20 were used. (a) Calculations were done using Eq (31) in Blicharski [5]; (b,c,e) calculations were conducted using Eq (23) derived in WCR following Wolf et al. [2]. The calculations using Eq (27) result in similar values as with Eq (23); (e) Eq (26) was used for the calculations in ICR, and $\text{r}=1.53\acute{Å}$ was used.

The calculated ${\text{R}}_{1\rho }$ relaxation rates due to homonuclear dipole-dipole interactions in the WCR, ICR and SCR during the HS pulse are shown in Fig. 2. For the calculations, Eq (19) was used. The 3D plot in Fig. 2a represents relaxation rates in the WCR, since ${\text{T}}_{\text{m}}\gg {\tau }_{\text{c}}$ up to a factor of 10, and the 3D plot shown in the Fig. 2b represents ${\text{R}}_{1\rho }$ in both WCR and ICR since it includes the range where ${\text{T}}_{\text{m}}\gg {\tau }_{\text{c}}$ and ${\text{T}}_{\text{m}}\approx {\tau }_{\text{c}}$. To evaluate ${\text{R}}_{1\rho }$ in the WCR and ICR, we considered ${\text{R}}_{1\rho }$ relaxations when ${\text{T}}_{\text{m}}=10\text{ms}$ and 1 ms (Fig. 2 b,c). The relations for the slow range of correlation times are plotted in Fig. 2, where the expressions given by Eqs. (24, IV-29) provide the ${\text{T}}_{\text{m}}$ dependence of ${\text{R}}_{1\rho }$ in the SCR ($R_{1\rho }^{\text{SCR}}$). The 3D plot shown in Fig. 2c represents $R_{1\rho }$ in the WCR and ICR, and the 3D plot in Fig. 2d includes the entire range of motional regimes since for ${\text{T}}_{\text{m}}=100\mu \text{s}$ all three regimes WCR $\left({\text{T}}_{\text{m}}\gg {\tau }_{\text{c}}\right)$, ICR $\left({\text{T}}_{\text{m}}\approx {\tau }_{\text{c}}\right)$ and SCR $\left({\text{T}}_{\text{m}}\ll {\tau }_{\text{c}}\right)$ are covered by the range of correlation times between 10⁻⁶ and 10⁻² s. We assert that the derived expression for ${\text{R}}_{1\rho }\left(\text{t},{\tau }_{\text{c}},{\text{T}}_{\text{m}}\right)$ given by Eq (19) is a plausible candidate for modeling the transition between ICR to SCR using a correlation function given in Eq (17). This is the central results of the paper, and to our knowledge, though quite straightforward, has not appeared in the field of biomedical MR so far. This serves to lend validation to the formalism developed in this work.

Figure 2. ${\text{R}}_{1\rho }$ relaxation rate constants in the WCR, ICR and SCR during the HS1 pulse.

Calculated relaxation rates due to homonuclear dipole-dipole interactions in the WCR, ICR and SCR during the HS1 pulse. For the calculations, ${{\omega }_1}^{\text{max}}/\left(2\pi \right)=2\text{kHz}$, and time BW product R=20 were used. (a) the 3D plot represents relaxation rates in the WCR since ${\text{T}}_{\text{m}}\gg {\tau }_{\text{c}}$ up to factor of 10; (b) The 3D plot represents relaxation rate constants in both WCR and ICR since it covers both ranges where ${\text{T}}_{\text{m}}\gg {\tau }_{\text{c}}$ and ${\text{T}}_{\text{m}}\approx {\tau }_{\text{c}}$; (c) 3D representation of ${\text{R}}_{1\rho }$ in the WCR and ICR; (d) For ${\text{T}}_{\text{m}}=100\mu \text{s}$ all three regimes WCR $\left({\text{T}}_{\text{m}}\gg {\tau }_{\text{c}}\right)$, ICR $\left({\text{T}}_{\text{m}}\approx {\tau }_{\text{c}}\right)$ and SCR $\left({\text{T}}_{\text{m}}\ll {\tau }_{\text{c}}\right)$ are covered by the range of correlation times between 10⁻⁶ and 10⁻² s. For the calculations, Eq (19) was used, and $\text{r}=1.53\acute{\AA }$ was used.

In Fig. 3, the dependencies of the effective correlation time ${\tau }_{\text{eff}}$ on ${\tau }_{\text{c}}$ and ${\text{T}}_{\text{m}}$ in three regimes, WCR, ICR and SCR are presented. For the calculations, Eq (16) was used. The 3D plots exhibit substantial differences of ${\tau }_{\text{eff}}$ in correspondence of correlation times in slow vs fast motional regimes as defined by ${\text{T}}_{\text{m}}$. Fig. 4 represents the dependence of the relaxation rates ${{\text{R}}^{\text{Norm}}}_{1\rho }$ on ${\text{T}}_{\text{m}}$ and ${\tau }_{\text{c}}$ in three different regimes, WCR, ICR and SCR. The 3D plots of the ${{\text{R}}^{\text{Norm}}}_{1\rho }$ were calculated by integrating the instantaneous ${\text{R}}_{1\rho }\left(\text{t}\right)$ during the AFP HS pulse, and normalizing to the pulse duration ${\text{T}}_{\text{p}}$ using Eq (20). The noticeable differences between rate constants obtained in three different regimes shown in Fig. 4 are determined by the differences in the ${\tau }_{\text{eff}}$ dependencies on ${\text{T}}_{\text{m}}$ and ${\tau }_{\text{c}}$ (Fig. 3). In Fig. 4a and Fig. 4c, we demonstrate that the theory predicts that rate constants have similar magnitude in the WCR and SCR, while they have different dependencies on ${\text{T}}_{\text{m}}$. On the other hand, Fig. 4b predicts an increase of ${{\text{R}}^{\text{norm}}}_{1\rho }$ in the ICR with ${\tau }_{\text{c}}$ and ${\text{T}}_{\text{m}}$. It is important also to note that the 3D plots exhibit opposite behavior for fast and slow correlation times, which depend on the pulse’s parameters settings. As it is shown in Fig. 4a, because the maximum of the correlation time ${\tau }_{\text{c}}={10}^{-1}$ s and the ${\omega }_{\text{eff}}~{1.4}^{*}{10}^4\text{rads}/\text{sec}$, there is a decrease of ${\text{R}}_{1\rho }$ because the denominator in Eq (19) increases faster than the numerator with the increase of ${\text{T}}_{\text{m}}$ or ${\tau }_{\text{c}}$, while this is opposite in Figs. 4 b,c.

Representative examples of the ${{\text{R}}^{\text{Norm}}}_{1\rho }$ dependencies on ${\text{T}}_{\text{m}}$ for a fixed ${\tau }_{\text{c}}={10}^{-5}\text{s}$ in three different regimes WCR, ICR and SCR are presented in Fig. 5. In the WCR, the relaxations are determined by the internal motion, while the spin dynamics characterized by ${\text{T}}_{\text{m}}$ has a negligible effect on relaxations (Fig. 5a). In the SCR when ${\tau }_{\text{c}}\gg {\text{T}}_{\text{m}}$ (Fig. 5c), the relaxation process is dominated by the spin dynamics of the rigid lattice dipolar operators, and the internal motion dependent on ${\tau }_{\text{c}}$ can be ignored. In the ICR, the relaxation rate constants are significantly ${\text{T}}_{\text{m}}$ dependent at fixed ${\tau }_{\text{c}}$, and they increase for larger ${\text{T}}_{\text{m}}$ values (Fig. 5b) because relaxations are governed by both spin interactions with the rigid lattice and the internal motion.

Figure 5. ${{\text{R}}^{\text{Norm}}}_{1\rho }$ as a function of ${\text{T}}_{\text{m}}$ in the WCR (a), ICR (b) and SCR (c).

Dependence of the ${{\text{R}}^{\text{Norm}}}_{1\rho }$ on ${\text{T}}_{\text{m}}$ for ${\tau }_{\text{c}}={10}^{-5}\text{s}$. For the calculations, ${{\omega }_1}^{\text{max}}/\left(2\pi \right)=2\text{kHz}$ and ${\text{T}}_{\text{p}}=4\text{ms}$ were used. The relaxation rates were calculated by integrating instantaneous solutions for ${\text{R}}_{1\rho }\left(\text{t}\right)$ during the AFP HS pulse, and normalizing to the pulse time duration ${\text{T}}_{\text{p}}$. For the calculations, Eq (20) was used, and $\text{r}=1.53\acute{Å}$ was used.

To validate the expression derived for ${\text{R}}_{1\rho }$ in the SCR, we display in Fig. 6 a theoretical fitting of the experimental ${\text{T}}_{1\rho }$ measurements of polyacetal (Delrin) performed by De Luca et al. using continuous wave (CW) spin-locking (SL) irradiation at the RF power 40 kHz [41]. Specifically, in Fig. 6a the natural logarithms (ln) of the SIs measured at different spin-locking times are re-plotted (see Fig. 4 in De Luca et al. [41]) to obtain the slope for estimating the time constant ${\text{T}}_{1\rho }$ of the polyacetal, which was found here to be ${\text{T}}_{1\rho }~4.3\text{ms}$. It is plausible that multiple decaying components could contribute. In fact, a multi-exponential SI decay may originate from different effects, e.g., anisotropy of dipolar interactions which may lead to orientation dependence of the rate constants, or from the interference between different relaxation mechanisms such as dipole-dipole interactions and chemical shift anisotropy [47]. However, since there is no evidence for a multiexponential evolution of the spin ensemble from the experimental data, and the linear function provides a good fit to the ln(SI) dependence on the locking times, we chose a monoexponential function for fitting the experimental data. The fitting to the SIs of Delrin presented in Fig. 4 of [41] using Eq (24) with its dependence on ${\text{T}}_{\text{m}}$ in the SCR resulted in relaxation time constant ${\text{T}}_{1\rho }=4.3\text{ms}$, which is in a good agreement with the ${\text{T}}_{1\rho }$ obtained from the experimental data by De Luca et al. [41]. Using Eq (24) for ${\text{R}}_{1\rho }$ in the SCR, we determined the ${\text{T}}_{\text{m}}~79.0\text{ns}$.

Figure 6. Theoretical fit of the relaxation data measured on Delrin.

(a) Natural logarithm (ln) of the SI decay measured on a polyacetal (Delrin) phantom using spin–lock ${\text{T}}_{1\rho }$ by De Luca et al. [41]. The plot was obtained by taking ln of the magnetization measured in ${\text{T}}_{1\rho }$ experiments (see Fig. 4 in [41]). The slope for estimation of the time constant ${\text{T}}_{1\rho }$ of the polyacetal was found here to be ${\text{T}}_{1\rho }~4.2\text{ms}$. (b) Fitting to the experimental ${\text{R}}_{1\rho }$ measurements of the polyacetal (Delrin) using Eq (24) derived in the SCR resulted in ${\text{T}}_{1\rho }=4.3\text{ms}$ and ${\text{T}}_{\text{m}}=79.0\text{ns}$. For the simulations, the amplitude of the SL RF irradiation was 40 kHz.

Subsequently, Eq (26) in the ICR was used to extract the correlation time and ${\text{T}}_{\text{m}}$, using:

$$\frac{1}{T_m}\left({\omega }_0,{\tau }_c\right)=\frac{3}{20}{R}_D\left(3J\left(0\right)+5J\left({\omega }_0\right)+2J\left(2{\omega }_0\right)\right)$$ (29)

The determined value of ${\tau }_{\text{c}}$ was ${\tau }_{\text{c}=}0.117\text{s}$, and the corresponding ${\text{T}}_{\text{m}}$ was found to be ~ 10⁻⁹s. Such a value of ${\text{T}}_{\text{m}}$ violates the assumption ${\text{T}}_{\text{m}}={\tau }_{\text{c}}$ that we used in the data analysis, and thus the ICR could be ruled out. Similar argument was used for the fast correlation times of the ICR. Here we found that, in the ICR regime, the correlation time ${\tau }_{\text{c}}=5.37*{10}^{-8}$ that best fits the data results in ${\text{T}}_{\text{m}}~2.5\text{ms}\left({\text{R}}_{\text{m}}=432.9{\text{s}}^{-1}\right)$. The difference between ${\text{T}}_{\text{m}}$ and ${\tau }_{\text{c}}$ indicates the condition WCR, i.e., ${\tau }_{\text{c}}\ll {\text{T}}_{\text{m}}$, which lead us to conclude that the ICR could be ruled out also for the fast correlation times.

The conventional continuous wave spin lock experiment is used in main stream applications of protein dynamics characterization. It is primarily applicable in the motional regimes characterized by the range of correlation times from ${\tau }_{\text{c}}~{10}^{-11}$ to 10⁻⁶ s, and is typically treated using the WCR formalism. The treatment developed in this contribution extends the range of validity of the conventionally used theory derived in WCR from 10⁻⁶ s to 10⁻¹ s and greater. RF pulses with a broad variety of amplitude and frequency modulation functions are also frequently used in MR. They commonly operate with RF peak amplitudes and bandwidths in the kHz range. These pulse parameters result in spectral densities with predominant sensitivity to the range of correlation times in the microsecond-millisecond timescale. Additionally, because amplitude and frequency pulse modulation functions are time-variant, the effective frequency ${\omega }_{\text{eff}}\left(\text{t}\right)$ is time dependent as well, which results essentially in multiple off-resonance spin-locking acquisitions during the RF pulses, e.g., using adiabatic ${\text{T}}_{1\rho }$ and ${\text{T}}_{2\rho }$, and non-adiabatic RAFFn methods [14–16, 18, 19, 35]. Recent developments included utilization of the ultra-short and zero-echo time pulses sequences as a readout for quantitative MRI relaxometry, which combined with the use of short FS pulses in sub-millisecond time scale provide a possibility to probe fast relaxing spin packets in tissue [48, 49]. Therefore, the ability of rotating frame modalities based on frequency swept RF pulses to probe slow motions warrants the theoretical description of relaxations in intermediate/slow motional regimes, initial development of which was the ultimate goal of this work.

The formalism developed here based on the work by Wolf and Jung [2] fundamentally treats the case of a conventional exponential correlation function. The advantage of this proposed treatment is that researchers in the bio-NMR community are very familiar with its use and can more easily follow the theoretical formalism that we developed to extend the range of analytical formalism for ${\text{R}}_{1\rho }$ to the entire useful range of correlation times, from ${\tau }_{\text{c}}~{10}^{-11}$ to 10⁻¹ and greater. The jump model of Wolf et al. [1, 2] is more difficult to utilize in regard to the correlation function used, which characterizes the number of jumps and the lifetime of the particular state. Here, we utilized a different approach using a more familiar exponential correlation function, which allowed more intuitive formalism valid in a broad range of motional regimes.

The theory offered in this work could be useful for the description of relaxations in a broad range of motional regimes. Below, we present another example of the analysis of relaxation data of glycogen available from the measurements conducted by Desvaux et al. [50]. In this work the ¹H protons (1–4) relaxations of glycogen, the critically important energy storage in mammals, were studies. Desvaux et al. using off resonance spin lock ${\text{T}}_{1\rho }$ have demonstrated that relaxations of glycogen cannot be described adequately by one single correlation time in the range 6.4*10⁻⁹ s. Interestingly, it was found that the ${\text{R}}_{1\rho }$ of glycogen were independent on external magnetic field in the range between 300 MHz and 500 MHz. In addition, no dependence of the rate constants on RF power of spin-lock pulses with the amplitudes of 2 kHz and 5 kHz was observed (see Fig. 3 in [50]). These observations suggest that the relevant correlation times are within the range ~10⁻⁹ s and ~10⁻⁸ s and in the microseconds-milliseconds time scale. In this work we conducted modelling of the experimental data of glycogen obtained by Desvaux et al. [50] (Fig. 7). The fitting of the SIs shown in Fig. 3 of Desvaux et al. [50] was performed using Eqs (19) as a function of angles of the locking fields and effective frequencies for one duration of the spin locking pulses. In agreement with Desvaux et al., the monoexponential decay function resulted in the insufficient fitting quality, while the bi-exponential function with the rate constants derived in the WCR and SCR allowed to obtain a good fit to the experimental data. The modelling using bi-exponential decay was conducted as follows:

$$SI=A_0\left(\text{exp}\left(-{R^{\left(f\right)}}_{1\rho }{T}_{sl}\right)+\text{exp}\left(-{R^{\left(s\right)}}_{1\rho }{T}_{sl}\right)\right)$$ (30)

The time of spin locking ${\text{T}}_{\text{sl}}$ was set to 3 s [50], the indexes indicate fast ${R^{\left(f\right)}}_{1\rho }$ and slow ${R^{\left(s\right)}}_{1\rho }$ motions, and ${\text{A}}_0$ is the pre-exponential amplitude. Eq (30) used for the analysis includes two exponential terms different in their dependencies on the correlation times. Each rate constant $R_{1\rho }$ has two independent variables, the correlation time and ${\text{T}}_{\text{m}}$. The results of the modelling using Eq (19) indicated the contributions of two motional regimes with correlation times summarized in Table 1, where the mean values and standard errors of the fitting parameters are shown. ${\text{T}}_{\text{m}}$ was determined by Eq (29). Inspection of Table 1 allowed us to conclude that the system could be described in two regimes, (i) in the WCR with ${\tau }_{\text{c}}\ll {\text{T}}_{\text{m}}$, and (ii) in the SCR with ${\tau }_{\text{c}}\gg {\text{T}}_{\text{m}}$ (Eq (28)). As we have shown in Fig. 4a and Fig. 4c (see the discussion above), the theory predicts that the relaxation rate constants have similar values in the WCR and SCR, while they have different dependencies on ${\tau }_{\text{c}}$ and ${\text{T}}_{\text{m}}$. Notably, the relaxation rate constants are dependent on ${\tau }_{\text{eff}}$ which is symmetrical in respect to ${\tau }_{\text{c}}$ and ${\text{T}}_{\text{m}}$ (Eq (16). Due to such symmetry in the extreme cases of fast and slow regimes the theory predicts similar values of the rate constants. The results of our analysis characterize probable relaxation processes that describe local and global tumbling, or anisotropy of dipolar interactions [47]. We note that our analysis should be interpreted with caution as they are limited to evaluation of rapid internal and global motions without considering variability in molecular weight of glycogen, and without considering other relaxation pathways such as exchange or cross-relaxation. Such investigations would require application of different additional methodologies, e.g., Chemical Exchange Saturation Transfer (CEST) [51] or Nuclear Overhauser Enhancement [52, 53], and are outside of the scope of the current presentation.

Figure 7. Relaxation analysis of glycogen.

Signal Intensity dependence (dots) on the angle between effective frequency and the axis of quantization z’ during spin-lock off-resonance experiment of the Oyster type IX glycogen solutions, reported by Desvaux et al. [50]. The superimposed solid line is the analytical fitting of the experimental data using a bi-exponential decay function (see Eq (30)) with the parameters summarized in Table 1. Here, $\text{r}=1.53\acute{Å}$ was used for modeling.

Table 1.

Fitting parameters of ${\text{R}}_{1\mathit{\rho }}$ relaxations of Glycogen

Regime	SCR: ${\tau }_{\text{c},\text{s}}\left(\text{s}\right)$	WCR: ${\tau }_{\text{c},\text{f}}\left(\text{s}\right)$	${\text{A}}_0$	${\text{T}}_{\text{m}}\left({\tau }_{\text{c},\text{s}}\right)\left(\text{s}\right)$	${\text{T}}_{\text{m}}\left({\tau }_{\text{c},\text{f}}\right)\left(\text{s}\right)$
WCR/SCR	(2.02±0.1) × 10−2	(0.18±0.02) × 10−9	3.01±0.042	2.5*10−9	0.021

The results of simulations in the Table 1 are mean of the fitting parameters and the standard errors. The analyses were performed with the rate constants derived in the WCR and SCR. The ${\text{T}}_{\text{m}}$ values were calculated from the mean values of the correlation times. The analyses were performed using Eqs (19, 30).

An additional example of the applicability of proposed theory is to tissue ${\text{R}}_{1\rho }$ relaxations of water protons in vivo. In our previous study we conducted detailed analysis of adiabatic ${\text{T}}_{1\rho }$ relaxations in the human brain at 4T, and we determined multi-exponential signal intensity decay using adiabatic pulses with different modulation functions of the Hyperbolic Secant family [16]. In this work, the rotational correlation times of two water populations were found to be ${\tau }_{\text{c}}=5.6\text{ns}$ and 76 ps for fast and slow decaying components of the water resonance, respectively. Using these correlation times and the distance between protons $1.58\acute{Å}$, the estimated values of ${\text{T}}_{\text{m}}$ are ~0.01 s and ~0.240 s for fast and slow components, respectively, that suggest that the water populations in the normal brain tissue appear in WCR.

Molecular dynamics of proteins and in vivo system is typically characterized by multiple correlation times describing global and local motional regimes. Usually, the range of the correlation times describing local motion is in the nanosecond time scale or shorter, while the global motion is characterized by correlation times in the millisecond-microseconds time scale. In this work, we formed a general model of relaxations that covers the entire scale of motions, from ultra-slow to fast. We provided the theoretical description of the WCR, SCR and ICR for the homonuclear dipole-dipole relaxations between like $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ spins. We then validated our model by comparing with result by Blicharski’s in the WCR, and we evaluated our derived SCR for different ranges of the ${\tau }_{\text{c}}$ and ${\text{T}}_{\text{m}}$. We next extended our description to the ICR for the fast and slow ranges of the ${\tau }_{\text{c}}$ with the calculated ${\text{R}}_{1\rho }$ values as a function of ${\tau }_{\text{c}}$ and ${\text{T}}_{\text{m}}$. We foresee that the developed here formalism could be extended to a more complex motional conditions with multiple correlation times.

To attain the results presented in Supplementary Material, Appendix IV, it was necessary to derive in detail effective spectral density expansions of the integral results of Eq (2.58) of Wolf et al. [2]. The development presented in Supplementary Materials; Appendix III is needed when the correlation function considered for the system of interest is not straightforward but of a complicated algebraic form. The results obtained in our work are sufficiently general to form the basis for deriving relations of pertinent relaxation functions of interest to the MR community, such as ${\text{R}}_1,{\text{R}}_2,{\text{R}}_{2\rho }$. In addition, these formulations could be extended to heteronuclear dipolar or chemical shift anisotropy (CSA) relaxation mechanisms, as well as to magic-angle spinning technique. These treatments are left, however, for future work. Finally, given the growing interest in the solid-state NMR community to characterize protein dynamics in slow to intermediate time scales during RF irradiation, the necessity of detailed theoretical formalism to interpret experimental data became quite salient [54, 55].

Conclusions

In this work, we present the detailed analytical treatment of rotating frame longitudinal relaxations due to homonuclear dipole-dipole interactions between like spins in a broad range of motional regimes. The relaxation functions were derived for the time-dependent case of ${\omega }_{\text{eff}}\left(t\right)$ resulting from the modulated amplitude and frequency of the adiabatic HS pulse. We anticipate, that the developed formalism will be useful for understanding of MRI relaxation contrasts in vivo, particularly those obtained with zero echo time techniques which allow to probe very fast relaxing spins [48]. We foresee that the initial steps of the theoretical development initiated in this work will find continuation in applications to the analysis of the quantitative relaxations data, which include assessment of, e.g., dysmyelination processes in vivo where molecular motional regime is in the milliseconds (and slower) time scales (see [37] and references therein), quantification of myocardium infarction by MRI [56], and determination of potential biomarkers of Alzheimer’s disease (AD) such as Amyloid $-\beta \left(\text{A}\beta \right)$ as an earliest hallmark of AD [57], accumulation of iron loads during Parkinson’s disease [34] and utility of cardiac MRI in calcification assessments [58].

Supplementary Material

1

Appendix I: Definitions of second rank spherical tensors

Appendix II: Partial integration to evaluate p=0 case

Appendix III: Derivations of the ${{\text{R}}^{\left(1\right)}}_{1\rho }\left(\text{t},{\tau }_{\text{c}},{\text{T}}_{\text{m}}\right)\text{and}{{\text{R}}^{\left(2\right)}}_{1\rho }\left(\text{t},{\tau }_{\text{c}},{\text{T}}_{\text{m}}\right)$ with the p=1,2

Appendix IV: Derivation of the $R_{1\rho }^{\text{SCR}}\text{and}{R}_{1\rho }^{\text{WCR}}$ Functions

Research Highlights.

• The general expression for ${\text{R}}_{1\rho }$ in the weak, intermediate and strong collision regimes was expanded to a summation of spectral densities-like terms

• The expressions for off-resonance ${\text{R}}_{1\rho }$ were expanded to time-dependent functions

• The developed theory had been validated in weak and strong collision regimes

Data Availability

All calculations included in this work were conducted using Mathematica 12 and 13 software packages, and the programs/code can be provided by the authors upon reasonable request.