Validity extension of stimulated echoes to imaginary signals arising for double diffusion encoding of closed pores
Kerstin Demberg1, Frederik Bernd Laun2, Peter Bachert1, and Tristan Anselm Kuder1

1Medical Physics in Radiology, German Cancer Research Center (DKFZ), Heidelberg, Germany, 2Institute of Radiology, University Hospital Erlangen, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany


We extend diffusion-weighted acquisitions using the stimulated echo acquisition mode (STEAM) to imaginary signals. In classical diffusion encoding, only real signals arise; however, for special applications of double diffusion encoding (DDE), complex signals arise, as shown here for diffusion pore imaging, a method that allows determining the shape of arbitrary closed pores filled with an NMR-detectable medium. It is shown that the phase information of complex signals is preserved under application of stimulated echoes: We show the analytically derived signal for DDE with STEAM and use Monte Carlo simulations for validation. Consequently STEAM can be employed for diffusion pore imaging.


Diffusion pore imaging enables the measurement of the average shape of arbitrary closed pores in an imaging volume element. Examples of such pores could in principle comprise cells in biological tissue or oil containing cavities in porous rocks. Measuring the Fourier transform of the pore space function is realized with non-conventional gradient profiles: Either by applying a long-narrow gradient profile1-5 or by combination of q-space imaging with double diffusion encoded (DDE) measurements6-8. Both methods enable measuring imaginary signals containing the pore shape information, which gets lost in classical diffusion encoding. While the first approach is limited in its sequence design, the second is highly flexible and allows incorporating stimulated echoes to reach the long-time limit exploiting the longer T1-relaxation time compared to T2. To the authors’ knowledge, only real signals have been acquired with stimulated echoes so far.

The present study presents the analytically derived signal for DDE-based pore imaging with stimulated echo acquisition mode (STEAM) demonstrating that the phase information is preserved for stimulated echoes. Validation using Monte Carlo simulations is presented and appropriate choice of RF-pulse phases is investigated.

Methods 1: Analytical derivation of STEAM signals

The signals arising for the sequences depicted in Fig. 1(a)-(d) are derived by sequentially applying building blocks for a) short gradient pulses and b) RF-pulse sequences generating stimulated echoes:

a) A gradient pulse $$$\boldsymbol{G}_i$$$ of duration $$$\delta$$$ induces a phase factor $$$e^{iφ_i}$$$ to spins resting at position $$${\boldsymbol{x}}_i$$$ during gradient application with $$$φ_i=-γδ{\boldsymbol{G}}_i{\boldsymbol{x}}_i$$$. If the pore’s center of mass $$$\boldsymbol{x}_{\text{cm}}$$$ is translated relative to the origin, additional phase shifts $$$ϑ$$$ occur.

b) If a pulse sequence $$$\left(90_{α_1}^°-τ-90_{α_2}^°\right)$$$ is applied to a transverse magnetization $$$e^{iφ}$$$ ($$$α_j$$$= phase of 90° pulse), the first pulse $$$90_{α_1}^°$$$ stores the real part as z-magnetization which is returned to the transversal plane by $$$90_{α_2}^°$$$ resulting in9 $$${\text{Re}}\left(-e^{i(φ-α_1)}\right)\,e^{iα_2}=-\frac{1}{2}\,\left(e^{i(φ-α_1+α_2)}+e^{i(-φ+α_1+α_2)}\right)$$$. Definitions of coordinate system and phases in Fig.$$$~$$$1(e).

For DDE with STEAM [Fig. 1(d)], the sequence $$90_{α_0}^°-{\boldsymbol{G}}_1/2-\left(90_{α_1}^°-τ-90_{α_2}^°\right)-{\boldsymbol{G}}_2-\left(90_{α_3}^°-τ-90_{α_4}^°\right)-{\boldsymbol{G}}_3/2-{\text{acq}}_{α_{\text{a}}}$$ gives $$\frac{1}{4}\,e^{i(α_0-α_1-α_2+α_3+α_4)}\,e^{i\left(\frac{φ_1}{2}-φ_2+\frac{φ_3}{2}\right)}+\frac{1}{4}\,e^{i(α_0-α_1+α_2-α_3+α_4)}\,e^{i\left(\frac{φ_1}{2}+φ_2+\frac{φ_3}{2}+2ϑ\right)}\\+\frac{1}{4}\,e^{i(-α_0+α_1-α_2+α_3+α_4)}\,e^{i\left(-\frac{φ_1}{2}-φ_2+\frac{φ_3}{2}-ϑ\right)}+\frac{1}{4}\,e^{i(-α_0+α_1+α_2-α_3+α_4)}\,e^{i\left(-\frac{φ_1}{2}+φ_2+\frac{φ_3}{2}+ϑ\right)},$$ with $$$ϑ=-γδ\boldsymbol{G}_2\boldsymbol{x}_{\text{cm}}$$$. The signal is calculated by averaging over all random walkers of all pores. $$$\delta\rightarrow0$$$ and diffusion time $$$T\rightarrow\infty$$$ allows averaging independently over each $$$\boldsymbol{x}_i$$$ with $$$e^{iφ_i}$$$ resulting in $$$\int_{Pore}\rho(\boldsymbol{x}_i )e^{-i\boldsymbol{qx}_i}d\boldsymbol{x}_i=\tilde{\rho}(\boldsymbol{q})$$$ with the pore space function $$$\rho(\boldsymbol{x})$$$, its Fourier transform $$$\tilde{\rho}(\boldsymbol{q})$$$ and q-space vector $$$\boldsymbol{q}=\gamma\boldsymbol{G}_2\delta$$$. Averaging over $$$\boldsymbol{x}_{\text{cm}}$$$ introduces the Fourier transform of the pore center of mass distribution $$$\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$$$ as prefactors (see results).

Methods 2: Monte Carlo simulations

Monte Carlo simulations were performed realizing RF-pulses with rotation matrices. Pores of equilateral triangular shape were chosen with 10$$$~$$$ µm edge length and 100,000 random walkers per pore.

Three 2D distributions of the pores centers of mass were simulated: 1) rock or tissue sample: 10,000 pores randomly distributed in 1x1 mm; 2) large unordered phantom: 1100 pores randomly distributed in 5x5$$$~$$$mm having dimensions comparable to the phantom studied in 10; 3) small grid phantom: 9 pores on a square grid, distances of 10$$$~$$$µm between centers of mass.

Additional parameters: diffusion coefficient 1$$$~$$$µm²/ms, $$$\delta$$$=2.88$$$~$$$ms, G=3.87$$$~$$$T/m, $$$T$$$=150$$$~$$$ms.

Results 1: STEAM signals

The STEAM DDE signal is composed of four terms:


Analogously for the STEAM q-space imaging signal:


The asterisk marks the complex conjugate. First terms correspond to the respective spin echo signals, corresponding to the desired pore imaging signal, reduced by a factor of 2 for each storage in the longitudinal direction, for $$$\alpha_{0,...,4}=0$$$. The additional undesired terms are modulated with the Fourier transform of the center of mass distribution $$$\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$$$.

Results 2: Validation with simulations

In Fig.$$$~$$$2-4, spin echo and STEAM signals were simulated for the vertical gradient direction with $$$\alpha_i=0$$$ for all 90°-pulses. For the q-space and DDE signal, analytical calculations (light-colored lines) for the respective $$$\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$$$ are shown and for fewer sampling points Monte-Carlo spin echo (darker-colored lines) and STEAM signals (dots). For the rock/tissue sample (Fig.$$$~$$$2), Monte Carlo simulations fit very well with the analytical derivations and pore images show clearly discernible triangles. For a reduced number of pores, the amplitude of $$$\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$$$ does not drop as low and the STEAM signal can deviate from the spin-echo result if a sampling point hits occasionally an outlier of $$$\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$$$, see Fig.$$$~$$$3 for a phantom similar to 5,10-12. If the pore number is decreased further, the contribution of the three undesired terms in $$$S_{\text{DDE}}(\boldsymbol{q})$$$ gets stronger [Fig.$$$~$$$4(a)-(d)].


For experiments, a smoothing of the STEAM signals might occur due to inhomogeneity of the gradient field thus effectively averaging over a small span of q-values so that the influence of the three undesired terms in $$$S_{\text{DDE}}(\boldsymbol{q})$$$ goes unnoticed. But if $$$\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(\boldsymbol{q})$$$ does not drop to zero fast enough, phase cycling can be utilized to remove them: Following Tab.$$$~$$$1(b), the combination $$$\text{add}\,0\,(0\,0)\,(0\,0)\,0-\text{add}\,0\,\left(\frac{π}{2}\,\frac{π}{2}\right)\,\left(\frac{π}{2}\,\frac{π}{2}\right)\,0-\text{substract}\,0\,(0\,0)\,\left(\frac{π}{2}\,\frac{π}{2}\right)\,0-\text{subtract}\,0\,\left(\frac{π}{2}\,\frac{π}{2}\right)\,(0\,0)\,0$$$ results in pure $$$e^{i\left(\frac{φ_1}{2}-φ_2+\frac{φ_3}{2}\right)}$$$-term, yielding the desired spin echo signal [Fig.$$$~$$$4(e)-(g)].


The phase information of complex signals is preserved under application of stimulated echoes enabling diffusion pore imaging with that approach.


Financial support by the DFG (grant no. KU 3362/1-1) is gratefully acknowledged.


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Fig. 1: Schemes of the RF and gradient pulse sequences: (a) q-space imaging. (b) q-space imaging with STEAM. It is assumed that in the time interval $$$\tau$$$ during which the magnetization resides in the z-direction the transverse magnetization is dephased sufficiently to not contribute to the signal. (c) Double diffusion encoding (DDE) sequence where the two diffusion encodings are applied with anti-parallel wave vectors without temporal separation. (d) DDE with STEAM. $$$\alpha_i$$$ are the phases of the respective pulses, $$$\alpha_{\text{a}}$$$ the acquisition phase, $$$\boldsymbol{G}_i$$$ and $$$\delta$$$ the gradient amplitudes and durations, and $$$T$$$ the diffusion time. (e) Coordinate system.

Fig. 2: Results for the rock/ tissue sample for the vertical gradient direction. All 90°-pulse phases were 0. (a) q-space signal. (b) DDE signal. Calculated: Signals calculated analytically at high sampling rate. Simulated: Monte Carlo results for spin echo acquisition and STEAM. Real parts in blue, imaginary in red. (c) Fourier transform of the pore distribution with high sampling used for the calculated STEAM signals and at low sampling rate (20 q-values) used in the simulations. (d) One half of q-space was sampled with q-space imaging and DDE from which the pore images were reconstructed via an iterative approach7.

Fig. 3: Results for the large unordered phantom for the vertical gradient direction. All 90°-pulse phases were 0. Rest analog to Fig. 2. For the equilateral triangular shape, imaginary signal parts occur in gradient directions where the projected random walks are not distributed axially symmetric as shown for the projection onto the vertical gradient direction in (b).

Fig. 4: Results for the small grid phantom for the vertical gradient direction with 3x3 pores. In (a)-(d) all 90°-pulse phases were 0. Depiction in (a)-(d) analog to Fig. 2. (e) Phase cycling result for q-space imaging and (f) double diffusion encoding and (g) the effect on the pore image. The analytically calculated signal in (a)-(b) represents the Monte Carlo signal very well: Strong influence of the undesired STEAM-terms (three for $$$S_{\text{DDE}}(q)$$$, one for $$$S_{\text{QSI}}(q)$$$), which drop with $$$\tilde{P}_{\boldsymbol{x}_{\text{cm}}}(q)$$$, prevents measuring the exact pore shape in (d). This can be solved through phase cycling, see (g) and Tab.1.

Tab. 1: Phase cycling table for (a) q-space imaging with one 90° RF pulse pair [Fig. 1(b)] and for (b) double diffusion encoding (DDE) with two 90° RF pulse pairs [Fig. 1(d)]. $$$\alpha_0$$$ is the excitation pulse phase and $$$\alpha_{\text{a}}$$$ is the acquisition phase. For q-space imaging, the combination "$$$\text{subtract}\,0\,(0\,0)\,0$$$ and $$$\text{add}\,0\,\left(\frac{π}{2}\,\frac{π}{2}\right)\,0$$$" results in pure component A, which is the compensated component resulting in the spin echo signal $$$|\tilde{\rho}(\boldsymbol{q})|^2$$$. For the DDE phase cycling scheme, please refer to the discussion section.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)