Molecular dynamics parameter maps by 1H Hahn echo and mixed-echo phase-encoding MRI

Journal of Magnetic Resonance 227 (2013) 1–8

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Molecular dynamics parameter maps by 1H Hahn echo and mixed-echo phase-encoding MRI Dan E. Demco a,b,c,⇑, Ana-Maria Oros-Peusquens a, Lavinia Utiu b, Radu Fechete c, Bernhard Blümich d, Nadim Jon Shah a,e,⇑ a

Institute of Neuroscience and Medicine (INM 4), Research Centre Jülich, 52425 Juelich, Germany DWI RWTH Aachen University, Forckenbeckstraße 50, D-52074 Aachen, Germany Technical University of Cluj-Napoca, Memorandumului 28, R-400114 Cluj-Napoca, Romania d Institute for Technical and Macromolecular Chemistry, RWTH Aachen University, Worringenalle 1, D-52074 Aachen, Germany e Department of Neurology, Faculty of Medicine, JARA, RWTH Aachen University, Pauwelsstraße 30, D-52074 Aachen, Germany b c

a r t i c l e

i n f o

Article history: Received 22 June 2012 Revised 29 October 2012 Available online 19 November 2012 Keywords: Hahn and magic echo MRI Parameter maps encoded by molecular motions Residual dipolar couplings Filled elastomers

a b s t r a c t Residual dipolar couplings and averaged correlation time maps in soft matter were obtained by mixed echo phase-encoding solid imaging (MIPSI). Use of the mixed echo in soft matter NMR imaging experiments has two crucial advantages: the signal intensity is recovered with a weak incoherence losses, and second, the intervals during which the phase-encoding evolution due to the magnetic field gradients takes place can be chosen to be much larger than with all other spin echo experiments and hence, a higher special resolution can be achieved. The parameter maps are compared to those obtained by the Hahn-echo phase-frequency encoding method. For both MRI methods the density operator formalism is applied in the average Hamiltonian approximation to describe the encoding of the spin echoes by the molecular motions. The results of preliminary experiments are presented. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Homonuclear and heteronuclear dipolar couplings measured by NMR techniques represent an important source of information about structure and molecular dynamics in soft matter [1–3]. Using these quantities structure–dynamics–function relationship have been investigated for elastomeric materials as well as ordered tissues [3]. Volume-averaged, residual dipolar couplings (RDCs) can be measured using one-dimensional (1D) NMR methods based on the dipolar correlation effect in combination with the Hahn and solid echo [4,5], the stimulated echo [6], magic and mixed echoes [7,8], magnetization exchange [9] and cross-relaxation dynamics [10] that takes into account the solid-like and liquid-like contributions to the spin system response [11]. Model free access to the RDC via van Vleck moments is given by the accordion magic sandwich where a mixed echo pulse sequence of variable duration leads to a Hahn echo which samples the shape of the magic echo [12]. Furthermore, model free access is given by the analysis of multiple-quantum (MQ) build-up [13] and decay curves [14] measured in the initial time ⇑ Corresponding authors. Address: Institute of Neuroscience and Medicine (INM 4), Research Centre Jülich, 52425 Juelich, Germany. Fax: +49 241 233 01 (D.E. Demco), +49 2461 61 (N.J. Shah). E-mail addresses: [email protected] (D.E. Demco), [email protected] (N.J. Shah). 1090-7807/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmr.2012.11.005

regime of the excitation/reconversion periods. The advances of this method and applications to polymeric soft solids have been reviewed in Ref. [15]. The distributions of the RDCs in heterogeneous soft matter have been also obtained from longitudinal and transverse magnetization relaxation NMR [16,17] as well as double quantum (DQ) build-up curves [18]. Chemically site-selective RDCs of a sample can be elucidated by two-dimensional (2D) NMR spectroscopy using, for instance, 13C–1H heteronuclear residual dipolar encoded spinning sideband patterns [19], REDOR [20], NOESY/ MAS [21], 1H–1H DQ [22], and 13C–1H heteronuclear DQ [23] NMR spectroscopy. In the past MQ coherences, especially DQ coherences of 1H in dipolar-coupled systems, have been used for measurements of RDCs maps in soft matter [24]. Proton DQ-filtered MRI of bound water was measured for bovine and sheep Achilles tendon under mechanical stress [25]. These methods have a low signal-to noise ratio and hence a reduced spatial resolution as compared to methods using single-quantum (SQ) coherences. From this last class of methods an efficient MRI technique based on magic and mixed echoes have been demonstrated for solids with strong 1H dipolar couplings [26–29]. The magic-echo phase-encoding solid imaging (MEPSI) technique [26] has several advantages: (i) the signal intensity is recovered by the magic-echo with small coherences loses; (ii) proton residual second van Vleck moments and correlation times of slow molecular motions in soft matter can be measured

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D.E. Demco et al. / Journal of Magnetic Resonance 227 (2013) 1–8

that leads to contrast parameters that might have a significant potential for the characterization of tissue. This new feature of the MEPSI is discussed in the present work. (iii) The time interval during which the phase-encoding takes place can be chosen to be much longer than with ordinary solid-echo, Hahn-echo and freeinduction signals. This is a very crucial condition because of the ‘‘broadening’’ effect of dipolar interactions in soft matter. The aim of this paper is to discuss the performance of Hahn and magic echoes (mixed echo) phase-encoding solid imaging method (hereafter MIPSI) on the characterization of local structure and molecular dynamics of soft matter which in our particular case is represented by a phantom made from a stack of elastomers with different types of fillers. We show that maps encoded by the RDCs and correlation times can be obtained by exploiting several MR images acquired with different free evolution times of MIPSI scheme. This work is a continuation of the MEPSI techniques that has been applied to samples with strong dipolar interactions [26] mainly for revealing spin density spatial distributions and not for generating residual dipolar couplings and correlation time maps of soft solids. The results are compared to that of Hahn-echo based MRI taking into account the molecular motion encoding evaluated for a multispin system response in the presence of dipolar couplings.

Fig. 1. Idealized radiofrequency and field-gradient pulse scheme of the mixed echo phase-encoding solid imaging (MIPSI) without slice selection. The magic-echo pulse sequence starts with an excitation pulse of single-quantum coherences followed by the magic sandwich ðp=2Þy ðaÞx ðaÞx ðp=2Þy of duration 4s. A one-dimensional scheme used phase-encoding gradients GX that are active during the free evolution windows. The gradients for 2D MR images could be applied for the duration of the pulse sequence or only in the free-evolution windows. When t = 6s a mixed echo is formed that refocused all homogeneous and heterogenous spin interaction. A slice selection was made by a spin-look composite pulse sequence [26] working in the presence of a magnetic field gradient oriented in the Z direction applied before the p/2 excitation pulse (not shown).

2. Experimental 2.1. MRI phantom A series of ethylene propylene diene monomer (EPDM) samples with different fillers content were prepared at Degusa (Evonik) Germany. The samples were not cross-linked. The EPDM polymer (KELTAN 512) was mixed with the filler and drew out a rubber sheet on a mill. A summary of the filled EPDM properties can be found in Ref. [16]. The EPDM elastomer with carbon-black filler (N121) has the concentration of 60 phr (parts of filler per hundred parts of polymer by weight). Carbon black N121 filler with paramagnetic properties has semi-reinforcing power and a particle size in the range of 45–73 nm. The measured surface area was found to be 40.1 m2 g1. The second filled EPDM sample employed for making part of the MRI phantom was using precipitated calcium carbonate filler denoted PrecarbÒ 400 with concentration of 20 phr. This filler is diamagnetic and exhibits a small surface area of 8.0 m2 g1. The MRI phantom was composed of two slabs of EPDM with carbon black fillers and EPDM with PrecarbÒ 400 fillers. The length of both samples was 40 mm and the average cross-sections were 5  7 mm2 for EPDM/N121 and 6  12 mm2 for EPDM/PrecarbÒ 400. Both filled elastomers were positioned face-to-face on the side with the largest surface in a glass NMR tube with an inner diameter of 20 mm. 2.2. Magnetic resonance image pulse sequences The pulse sequence shown in Fig. 1 was used for generation of mixed echoes in the presence of phase encoding field gradients applied along the X and Y directions [26]. Although this scheme was discussed in Ref. [26], it was not applied to generate NMR parameter maps. Slice-selection was applied before the pulse sequence shown in Fig. 1. It consists of spin-lock pulse sequence with a flip-back pulse [26] applied in the presence of a magnetic field gradient oriented along Z-direction. The same slice selection was applied for the case of MR images using Hahn echo. Free evolution periods are denoted by s. The maximal field gradient was 0.0809 T/m with 32 phase encoding equidistant steps and the switching time was 100 ls.

A Bruker AV300 NMR spectrometer operating at a 1H frequency of 300 MHz was used in combination with a micro-imaging unit with a birdcage coil of 25 mm inner diameter. The 90°-pulse length, the dwell time and the recycle delay were 9 ls, 20 ls, and 3 s, respectively. The number of scans was 32. The magic sandwich [26] has long radio-frequency pulses of strength 28 kHz that is larger that the linewidth of the elastomer phantom which is several 100 Hz. The MR images had a field-of-view of FOV = 15  15 mm2, 32  32 pix2 with zero-filled to 128  128 pix2. In this case the achieved resolution is 117 lm/pixel in both directions X and Y perpendicular on ~ B0 . The total experimental time for 2D MR image with MIPSI and Hahn-echo [30] schemes were about 7 h and ½ h, respectively. 3. Molecular motion encoded parameter maps 3.1. Molecular motion encoded Hahn echo MRI The space localized response of the spin system to the Hahn echo pulse sequence (p/2)x–s–(p)y–s-Hahn echo [30], encoded by the molecular motions can be evaluated using the density operator formalism and the results are presented in Appendix A. The normalized Hahn echo decay given by the Eq. ( A15) is valid for a multispin system within the limit of the exponential correlation function and for hM 2 i1=2 sc  1. Furthermore, the Hahn echo decay is encoded by the dipolar couplings as well as by fluctuations of the dipolar interactions characterized by the correlation time. This can be seen from Eq. ( A15) by using a relationship with better convergence properties, i.e.,        hM 2 i 2s s ð2sÞ2 exp hM 2 is2c exp  SHE ð2sÞ  exp  þ2 1 ; sc sc 2

ð1Þ

where the first and second factors of the product describe the solidlike and liquid-like behavior of the polymer network, respectively. Phase encoded Hahn echo signal induced by the magnetic field gradient GX can easily obtained by introducing the wave vector kX  cGX s and taking into account the action of the gradient propagator. From Eq. (1) the one-dimensional image encoded by segmental dynamics is given by

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D.E. Demco et al. / Journal of Magnetic Resonance 227 (2013) 1–8

SHE ðkX Þ 

I

Dð~ rÞ exp fhM 2 ð~ rÞif ðs; sc ð~ rÞÞg cosðkX XÞdX;

ð2Þ

sample

where Dð~ rÞ is the local spin density. The segmental dynamics is describe by the voxel localized function

(



f ðs; sc ð~ rÞÞ ¼ s2c ð~ rÞ exp 





)

2

2s s þ2 sc ð~rÞ sc ð~rÞ

Table 1 Volume-averaged residual second van Vleck moments hM2i measured from the Hahn echo and the mixed echo decays (Fig. 2) for EPDM filled elastomers that comprise the phantom used in the MRI experiments.

þ2

s 1 ; sc ð~rÞ

ð3Þ

In the above equations the residual second van Vleck moment and correlation time of a voxel is denoted by hM 2 ð~ rÞi and sc ð~ rÞ, respectively. The above relationship that describes the 1D localized signal of Hahn echo can be generalized for the 2D and 3D cases by introducing the factor cosð~ k ~ rÞ in the integral kernel of Eq. (2) (vide infra).

a

Hahn-echo amplitude

Hahn echo decay EPDM/N121/60phr 9 2 2 = 1.2x10 rad /s -6

τ c = 1.9x10 s

0.2 0.0 0

1

Hahn echo Mixed echo

1.2  109 1.1  109

4.9  108 4.8  108

The errors are of the order of 5%.

2

3

Hahn echo Mixed echo a

EPDM/N121/60 phr sc (s)a

sc (s)a

1.9  106 1.7  106

7.2  10–6 4.1  106

The phase encoding induced by the gradient GX can be obtained from the results presented in Ref. [26] and molecular motion encoding of the mixed echo from the computations presented in Ref. [7]. The effect of molecular dynamics on the mixed magic echo was discussed using the density operator formalism as in the case of Hahn echo (see above). The following assumptions were employed [7]: (i) series development of the quantum-mechanics propagator up to the term proportional to (6s)2 can be made in the time regime 6s < ð-d Þ1 , where -d is the strength of the residual dipolar coupling; (ii) the dipolar autocorrelation function is

1.0

(c) mixed echo decay EPDM/N121 60 phr 9 2 2 = 1.1x10 rad /s

0.8 0.6

-6

τc = 1.7x10 s

0.4 0.2 0.0 0.0

4

0.5

1.0

Hahn-echo amplitude

Hahn echo decay EPDM/Precarb400/20phr 8 2 2 = 4.9x10 rad /s -6

τ c = 7.2x10 s

0.4 0.2 0.0 0

1

2

τ [ms]

3

4

normlaized mixed echo intensity

(b)

1.0

0.6

1.5

2.0

2.5

3.0

τ [ms]

τ [ms]

0.8

EPDM/Precarb 400/20 phr

The errors are of the order of 5%.

normlaized mixed echo intensity

(a)

1.0

0.4

EPDM/Precarb 400/20 phr hM2i (rad2/s2)a

Decay

We describe the spin evolution in a voxel for an idealized MRI experiment using the MIPSI scheme shown in Fig. 1 [26,27]. In our particular MIPSI scheme, the magnetic field gradient is active only during the free evolution periods of duration ð0; sÞ and ðs; 5sÞ. This pulse sequence generates a mixed echo composed of magic and Hahn echoes meaning that the homogeneous and inhomogeneous spin interactions are refocused. At this point we note that the decoupling between the spatial and motional encoding of Hamiltonians, and thus the NMR signal for both MIPSI and Hahn-echo MR images, is valid only in the approximation of secular fluctuating dipolar Hamiltonian (see Eq. ( A2)). This is no longer valid if the single-quantum and double-quantum terms of dipolar Hamiltonian are taken into account whereby the changes in the phase gradient will also affect the spin evolution under dipolar interactions.

0.6

EPDM/N121/60 phr hM2i (rad2/s2)a

Table 2 Volume-averaged correlation times sc measured from the Hahn echo and the mixed echo decays (Fig. 2) for EPDM filled elastomers that comprise the phantom used in the MRI experiments.

3.2. Molecular motion encoded MIPSI

0.8

Decay

1.0

(d)

0.8

mixed echo decay EPDM/Precarb400 20 phr 8 2 2 = 4.8x10 rad /s

0.6

-6

τc = 4.1x10 s

0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

τ [ms]

Fig. 2. Hahn echo and mixed echo decays for the filled elastomers EPDM/N121 (a and c), and EPDM/PrecarbÒ 400 (b and d), respectively. The data were fitted (solid lines) with Eqs. (3) and (8) for Hahn echo decays and with Eqs. (5) and (8) for mixed echo from which the volume-averaged second van Vleck moments hM 2 i and correlation time sc were obtained and reported in Tables 1 and 2. The errors of the fits are below 1%.

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taken to be described by an exponential function, i.e., expft=sc ð~ rÞg, where sc ð~ rÞ is the local correlation time; (iii) the inequality hM 2 ð~ rÞi1=2 sc ð~ rÞ  1 is fulfilled for all voxels. Finally for a 1D MIPSI experiment (Fig. 1) with the phase encoding gradient GX and wave vector kX  cGX 2s we obtain

SMIPSI ðkX Þ 

I

Dð~ rÞf1  hM 2 if ðs; sc ð~ rÞÞg cosðkX XÞdX;

ð4Þ

sample

where           6s 5s 9 4s s 3s 13 :  3exp  þ exp  þ 3exp  þ  f ðs; sc Þ ¼ s2c exp  sc sc 4 sc sc sc 4 ð5Þ

Furthermore, we can introduce the ad hoc exponential approximation

SMIPSI ðkX Þ 

I

Dð~ rÞ expfhM 2 if ðs; sc ð~ rÞÞg cosðkX XÞdX:

ð6Þ

sample

This result is similar to that obtained above for the MR images based on Hahn echo but the molecular motion encoding function Eq. (3) is different from that given by Eq. (6). Transverse relaxation processes described by a single correlation time are rarely found in elastomers and biological objects. In such cases a distribution of the correlation times must be taken into account as discussed in Ref. [7]. For the MIPSI scheme with two phase-encoding gradients the spin response signal is given by *

SMIPSI ð~ kÞ / tsample Dð~ rÞ expfhM 2 if ðs; sc ð~ rÞÞg cosð r ~ rÞdXdY;

ð7Þ

j, and ~ r ¼ X~i þ Y~ j. where ~ k ¼ kX~i þ kY~

Fig. 3. Residual second van Vleck moment hM 2 i (a) and correlation time sc (b) maps for the elastomer phantom obtained from Hahn echo images with different echo times s = TE/2 = 1–4 ms. These maps were generated by fits of the signal decay of each pixel using Eq. (2) with the molecular motion encoding function f given by Eq. (3). The EPDM/ N121 and EPDM/PrecarbÒ 400 elastomers correspond to the left- and right-hand side of each maps, respectively. A slice selection was obtained by a spin-look composite pulse sequence [26] applied in the presence of a magnetic field gradient oriented in the Z direction.

D.E. Demco et al. / Journal of Magnetic Resonance 227 (2013) 1–8

4. Results and discussion 4.1. Volume-averaged residual dipolar couplings and correlation times The volume-averaged residual second van Vleck moment and correlation time for the elastomers that comprise the MRI phantom were measured separately from the decays of the Hahn echo and mixed echoes. These decays are shown for Hahn and mixed echoes in Fig. 2a–d, respectively. The fit of the experimental decays for the Hahn echo for EPDM/ N121 and EPDM/Precarb samples (Fig. 2a and b, respectively) were made using the normalized decay function

Sspin echo ð2sÞ exp fhM 2 if ðs; sc Þg  ; Sspin echo ð2s0 Þ expfhM 2 if ðs0 ; sc Þg

ð8Þ

where the function f ðs; sc Þ is given by Eq. (3) and the initial decay time is s0 ¼ 4 ls. The same relationship was used for the fit of the mixed echo decays (Fig. 2c and d) where the function f ðs; sc Þ is now given by Eq. (5) and the initial decay time is s0 ¼ 20 ls. The values obtained from these fits for the volume averaged molecular dynamics parameters hM2 i and sc for the employed filled polymer networks are given in Tables 1 and 2, respectively. We note that the values of hM2 i and sc are slightly larger for those obtained from Hahn echo decays compared to those of mixed echo decays. This can be explained by the time windows of NMR experiments that allow more efficient molecular motion averages of dipolar couplings for mixed echo than for the Hahn echo. 4.2. Residual dipolar couplings and correlation time maps by Hahn echo MRI Two-dimensional images were obtained using the Hahn echo method for different values of the s parameter in the range of 1– 6 ms. For s > 2.55 ms the signal of the EPDM/N121 elastomer is no longer detected due to short transverse relaxation in the presence of the carbon black paramagnetic fillers. These images are encoded by the spin density, the residual dipolar couplings, and the correlation times. The maps of hM2 i and sc can be obtained by using the NMR Hahn echo decay given by Eq. (2) for each pixel. The resulting

5

parameter maps are shown in Fig. 3. The Hahn echo signal in each pixel is reduced in intensity due to faster relaxation induced by the n o solid-like contribution exp  hM22 i ð2sÞ2 (see Eq. (1)) that leads to images and hence parameter maps with lower resolution due to lower S/N compared to the mixed echo method. 4.3. Residual dipolar couplings and correlation time maps by MIPSI Two-dimensional MR images were obtained by the MEPSI and MIPSI methods with slice selection [26] for elastomer phantom and are shown in Fig. 4. In both cases the free evolution window, where encoding gradients are applied, is s  297 ls and all other parameters are identical. As expected, the intensities of the pixels in the MR image obtained using MIPSI are higher compared to that of the MEPSI image. Therefore, it is evident that the refocusing of the magnetic field inhomogeneity by the Hahn echo, present for the MIPSI technique, is taking place. These MR images are encoded by the spin density and parameters of the chain dynamics. The 2D images obtained using the MIPSI method for different values of s in the range 297–537 ls were measured for our elastomer phantom taken as an example of soft matter. Hence, the mixed echo time 6s takes values in the range of 1.8–3.2 ms. The decrease of the pixel intensity as a function of the mixed echo time is described by an equation similar to Eq. (7) where function f is given by Eq. (5) and s0 = 297 ls. An image processing algorithm was written in C that allows one to fit the signal of each pixel as a function of echo time and generates the residual second van Vleck moment hM2 i and correlation time sc maps (Fig. 5). Finally, we mention that the decay of mixed echo is lower as compared to that of Hahn echo due to the refocusing of the residual dipolar Hamiltonian Hd by the magic echo. Therefore, the signal in each pixel is larger than that of Hahn echo images and the contrast is improved as seen by the hM 2 i and sc maps (Fig. 5). 4.4. Contrast and resolution of the parameter maps obtained by MIPSI and Hahn echo methods The contrast and resolution of the residual dipolar couplings reflected in the values of hM 2 i and the correlation times sc maps are different when Hahn echo (Fig. 3) and mixed echo (Fig. 5) methods

Fig. 4. Two-dimensional MR images of the elastomer phantom obtained by the MEPSI (left-hand side) and MIPSI (right-hand side) techniques with the same pulse sequence parameters. The phase encoding gradients in the X and Y directions are applied for times s = 296.5 ls (Fig. 1). A slice selection of about 3 mm was obtained by a spin-look composite pulse sequence [26] applied in the presence of a magnetic field gradient oriented in the Z direction. The EPDM/N121 and EPDM/PrecarbÒ 400 elastomers correspond to the left- and right-hand side of each maps, respectively.

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D.E. Demco et al. / Journal of Magnetic Resonance 227 (2013) 1–8

Fig. 5. Residual second van Vleck moment hM 2 i (a) and correlation time sc (b) maps obtained by the MIPSI images. The phase encoding gradients are applied in the time range s = 297–537 ls. These maps were generated by fits of the signal decay of each pixel using an equation similar to Eq. (8) but with the molecular motion encoding function f given by Eq. (5). The EPDM/N121 and EPDM/PrecarbÒ 400 elastomers correspond to the left- and right-hand side of each maps, respectively.

are used. There are several factors responsible for these differences one being related to the refocusing of static dipolar interactions that implicitly lead to larger signal-to-noise (S/N) ratio and longer phase-encoding windows for MIPSI. This is especially evident in Fig. 5a and b obtained by MIPSI method as compared to the parameter maps generating by Hahn echo technique (Fig. 3). The differences in the filler distribution reflected in the hM 2 i, and sc maps are less pronounced for maps shown in Fig. 3. This is related to the presence of the solid-like contribution of the homonuclear dipolar couplings to the Hahn-echo decay given by n o exp  hM22 i ð2sÞ2 , (see Eq. (1)) that leads to images and hence parameter maps with lower resolution due to lower S/N compared to the mixed echo method.

Another important factor that differentiate the elastomers of the phantom and finally leads to different contrast in parameter maps is related to the type, content, and distribution of the fillers. In the case of paramagnetic fillers like carbon black N121 the shorter longitudinal relaxation time of the polymer network segments will increase the resolution for the EPDM sample (left-hand side in Fig. 5a) due to larger signal-to-noise ratio compared to the case of diamagnetic fillers such as calcium carbonate PrecarbÒ 400 (see right-hand side in Fig. 5a). The differences in the filler distribution reflected in the hM 2 i, and sc maps are less pronounced for maps shown in Fig. 3 obtained by Hahn-echo method. The presence of the solid-like contribution to the Hahn-echo decay originating from the proton–electron static dipolar interactions will lead to a faster decay of the echo intensity.

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D.E. Demco et al. / Journal of Magnetic Resonance 227 (2013) 1–8

5. Conclusions

DHd ð~ r; tÞ ¼ Hd ð~ r; tÞ  Hd ð~ rÞ ¼

The general advantage of mixed echo phase-encoding techniques is a dramatic increase of the accessible encoding time so that soft matter can be imaged with high spatial resolution using operational gradients. Classes of soft matter include inter alia polymer networks, connective and brain tissues. Maps of molecular motion parameters represented by the residual second van Vleck moment hM2 i and the correlation time sc were generated using the MIPSI method. A series of images with different mixed echo times were produced and the decay of the NMR signal of each pixel was fitted by a molecular motion encoding relationship. This procedure is based on the model of fluctuating dipolar interactions in a multispin system and was applied for an elastomer phantom as an example of soft matter. The pulse sequence discussed in this work is the simplest version of mixed-echo phase-encoding imaging sequences. This pulse sequence may also be combined with multiple-quantum filters, and faster readouts. The same imaging procedure should also work with quadrupolar nuclei such as 2H and 23Na. Maps of molecular motion parameters obtained using MIPSI were compared with those obtained by the Hahn-echo phase-frequency encoding technique. To obtain hM 2 i and sc maps using the latter method, a theoretical treatment of the Hahn echo decay induced by fluctuating dipolar interactions was developed in the approximation of the average Hamiltonian theory for a network of dipolar coupled spins. The NMR time windows of MR imaging techniques using Hahn echo and mixed echo are different and hence the maps of molecular motion parameters show different contrast.

 d ð~ DH r; tÞ ¼

vX oxel

*

 ijd ð r ÞT ij2;0 ; dxijd ð~ r; tÞx

 ijd =-ijd is the normalized fluctuating residual dipolar where dxijd  Dx couplings. To simplify the notation in the following we shall omit the explicit dependence on ~ r. Moreover, it is assumed that these fluctuations are independent of specific spin pairs and reflect the overall segmental motions as in the case of polymer networks. Hence, from Eq. ( A4) one obtains vX oxel

 ijd T ij2;0 : x

DHd ðtÞ ¼ dxd ðtÞ

b y ðpÞHd ðtÞ ¼ Hd ðtÞ from Eq. Taking into account the relationship P ( A1) we obtain



qð2sÞ ¼ exp i

Z 2s s

  Z s vX oxel b d ðtÞdt exp i b d ðtÞdt H H Iy : 0

 b y ðpÞ b d ðtÞdt P H s  Z s  b d ðtÞdt P b x ðp=2Þqð0Þ; H  exp i

1 2s

Z 2s

DHd ðtÞdt;

ðA1Þ

b q  OqO1 and where the Liouville operator is defined by O 1 OO = 1. The hard pulse propagator is Px ðhÞ ¼ expfi#Ix g, where # ¼ p=2 or # ¼ p. Initially, the spin system is in thermodynamical equilibrium in the static magnetic field and the density operator P is qð0Þ / kv oxel Iz . For the localized residual dipolar Hamiltonian we can write

Hd ð~ rÞ ¼

vX oxel

 ijd ð~ x rÞT ij2;0 ;

ðA2Þ

i>j

where  ijd ð~ rÞ is the residual dipolar coupling for (ij) spin pair of the ~ r voxel and T ij2;0 is the irreducible tensor operator describing the spin

x

part of the secular dipolar couplings [7]. The dipolar Hamiltonian fluctuating around the residual part given by Eq. ( A2) is described by

ðA7Þ

0

where the two terms on the right-hand side of Eq. ( A7) do not commute for a multispin system. Finally, in this approximation from the above equations we can write

b

)  2 vX oxel 1 b H d ð2sÞ2 þ . . . Iy : 2 k

ðA8Þ

  This equation is valid within the limit Hd þDHd ðtÞ2s  1, where kð. . .Þk is the norm of the dipolar Hamiltonians. The normalized Hahn echo signal SHE ðtÞ  SðtÞ=Sð0Þ at t = 2s, can be written as

Sð2sÞ ¼ Sð0Þ

0

ðA6Þ

k

One possibility to evaluate the density operator at the maximum of Hahn echo is to use the averaged dipolar Hamiltonian, i.e.,

(

Z 2s

ðA5Þ

i>j

qð2sÞ / 1  i H d 2s 



ðA4Þ

i>j

Appendix A

qð2sÞ ¼ exp i

ðA3Þ

 ijd ð~ where Dx r; tÞ is the localized fluctuating dipolar coupling for (ij) spin pair. To simplify the computations, only the zero-quantum part of the fluctuating dipolar Hamiltonian is considered in the above equation. Single-quantum and double-quantum dipolar terms have little influence on the spin echo decay in the slow motion regime. In order to evaluate the spin echo decays encoded by molecular dynamics we shall rewrite Eq. ( A3) as

Hd  H d þ

The space localized response of the spin system to the Hahn echo pulse sequence (p/2)x–s–(p)y –s-Hahn echo, encoded only by molecular motions can be evaluated using the density operator qð~r; 2sÞ  qð2sÞ under the chain of canonic quantum–mechanical transformation

*

 ijd ð r ; tÞT ij2;0 ; Dx

i>j

Acknowledgment R.F. and D.E.D. acknowledge the financial support from Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project PN-II-ID-PCE-2011-3-0544.

vX oxel

*

+ Tr Iy qð2sÞ ; TrfI2y g

ðA9Þ

where the symbol hð. . .Þi denotes the average over the statistical ensemble of the fluctuations of dipolar interactions and the average over the internuclear azimuthal angles in a powder sample. It is obvious that hHd i ¼ 0 and therefore, from Eqs. ( A8) and ( A9) for the normalized Hahn echo amplitude, one finally obtains

   b 2 Tr ð Hð2sÞIy Þ2 SHE ð2sÞ  1 þ

TrfI2y g

s2    

ðA10Þ

The above equation describes the Hahn echo decay in the time regime 2s < ð-d Þ1 , where -d is the strength of the residual dipolar couplings. For many polymer networks and biological objects this corresponds to the millisecond regime. The Hahn echo amplitude decay can be obtained from Eqs. ( A7) and ( A10) in terms of the autocorrelation function of the dipolar interaction fluctuations, i.e.,

8

D.E. Demco et al. / Journal of Magnetic Resonance 227 (2013) 1–8

SHE ð2sÞ  1 



hM 2 i 4s2 þ 2

Z 2s 0

dt

0

Z 2s

 00 dt hdxd ðt0 Þdxd ðt 00 Þi ;

ðA11Þ

0

where the cross-correlation terms have been neglected. That means that dipolar fluctuations of three or more spins are less correlated than those of the spin pairs. The residual second van Vleck moment is given by

*

*

*  +

2  b Tr H d Iy  TrfI2y g

M2

1

ðA12Þ

The dipolar autocorrelation function is supposed to be described by a stationary random process, i.e., hdxd ðt 0 Þdxd ðt 00 Þi  cðjt00  t0 jÞ. In the simple approximation of an exponential correlation function one can write

cðjtjÞ ¼ expfjtj=sc g; 00

ðA13Þ

0

where t  t  t and sc is the average correlation time of the molecular motions. The integrals in Eq. ( A11) can be easily evaluated using the method described in Ref. [6], and finally we obtain Z 2s 0

dt

0

Z 2s 0

   2s 00 dt hdxd ðt0 Þdxd ðt 00 Þi ¼ 2s2C exp  þ2

sc



s 1 ; sc

ðA14Þ

and

   2s s2 þ2 2þ2 SHE ð2sÞ  1  hM 2 is2c exp 

sc

sc



s 1 : sc

ðA15Þ

The normalized Hahn echo decay given by the above equation is valid for a multispin system within the limit of the exponential correlation function and for hM 2 i1=2 sc  1. References [1] J.-P. Cohen Addad, NMR and fractal properties of polymeric liquids and gels, Prog. NMR Spectrosc. 25 (1993) 1–316. [2] G. Navon, H. Shinar, U. Eliav, Y. Seo, Multiquantum filters and order in tissues, NMR Biomed. 14 (2001) 112–132. [3] D.E. Demco, S. Hafner, H.W. Spiess, in: V.M. Litvinov, P.P. De (Eds.), Multidimensional NMR Techniques for the Characterization of Viscoelastic Materials, Handbook of Spectroscopy of Rubbery Materials, Rapra Technology Ltd., Shawbury, 2002 (and References therein). [4] J. Collignon, H. Sillescu, H.W. Spiess, Pseudo-solid echoes of proton and deuteron NMR in polyethylene melts, Colloid Polym. Sci. 259 (1981) 220–226. [5] P.T. Callaghan, E.T. Samulski, Molecular ordering and the direct measurements of weak proton–proton dipolar interactions in a rubber network, Macromolecules 30 (1997) 113–122. [6] R. Kimmich, NMR: Tomography: Diffusiometry, Relaxometry, Springer, Berlin/ Heidelberg/New York, 1997 (and references therein). [7] R. Fechete, D.E. Demco, B. Blümich, Chain orientation and slow dynamics in elastomers by mixed magic-Hahn echo decays, J. Chem. Phys. 118 (2003) 2411–2421. [8] S. Sturniolo, K. Saalwächter, Breakdown in the efficiency factor of the mixed magic sandwich echo: a novel NMR probe for slow motions, Chem. Phys. Lett. 516 (2011) 106–110.

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