A single-shot measurement of time-dependent diffusion over sub-millisecond timescales using static field gradient NMR
Teddy X. Cai, Nathan H. Williamson, Velencia J. Witherspoon, Rea Ravin, Peter J. Basser
aa r X i v : . [ q - b i o . Q M ] M a r A cce p t e d M a nu s c r i p t A single-shot measurement of time-dependent diffusion oversub-millisecond timescales using static field gradient NMR
Teddy X. Cai,
1, 2, a) Nathan H. Williamson,
1, 3
Velencia J. Witherspoon, Rea Ravin,
1, 4 and Peter J. Basser b) Section on Quantitative Imaging and Tissue Sciences, Eunice Kennedy Shriver National Instituteof Child Health and Human Development, National Institutes of Health, Bethesda, Maryland 20892,USA Wellcome Centre for Integrative Neuroimaging, Nuffield Department of Clinical Neurosciences, University of Oxford,Oxford, UK National Institute of General Medical Sciences, National Institutes of Health, Bethesda, Maryland 20892,USA Celoptics, Rockville, Maryland 20852, USA (Dated: 4 March 2021)
Time-dependent diffusion behavior is probed over sub-millisecond timescales in a single shot using an NMRstatic gradient, time-incremented echo train acquisition (SG-TIETA) framework. The method extends theCarr-Purcell-Meiboom-Gill (CPMG) cycle under a static field gradient by discretely incrementing the π -pulse spacings to simultaneously avoid off-resonance effects and probe a range of timescales ( − µ s).Pulse spacings are optimized based on a derived ruleset. The remaining effects of pulse inaccuracy areexamined and found to be consistent across pure liquids of different diffusivities: water, decane, and octanol-1. A pulse accuracy correction is developed. Instantaneous diffusivity, D inst ( t ) , curves (i.e., half of the timederivative of the mean-squared displacement in the gradient direction), are recovered from pulse accuracy-corrected SG-TIETA decays using a model-free, log-linear least squares inversion method validated by MonteCarlo simulations. A signal-averaged, 1-minute experiment is described. A flat D inst ( t ) is measured on puredodecamethylcyclohexasiloxane whereas decreasing D inst ( t ) are measured on yeast suspensions, consistentwith the expected short-time D inst ( t ) behavior for confining microstructural barriers on the order of microns. I. INTRODUCTION
As molecules diffuse, they interact with their surround-ings and “[feel] the boundary” , causing their ensembledisplacement behavior to be influenced by the morphol-ogy of the microenvironment. More specifically, long-range correlations such as confining barriers impart anonlinear time-dependence to the ensemble-averaged netmean-squared displacement, h r ( t ) i (in R ). This leadsto a time-dependent diffusion coefficient , D ( t ) = h r ( t ) i t ≡ t Z t ( t − t ′ ) tr ( D ( t ′ ) ) dt ′ , (1)where D ( t ′ ) = H ( t ′ ) h v ( t ′ ) v T (0) i ≡ ∂ t (cid:2) H ( t ) h r ( t ) r T (0) i (cid:3) is the causal velocity autocorrelation tensor, H ( t ) is theunit step function, and “tr” is the trace operation.Microstructural features can be inferred from the be-havior of D ( t ) at limiting short and long timescales (seeSen and Reynaud for review). At the short-time limit, D ( t ) exhibits universal behavior which depends on thebarrier surface-to-volume ratio, S/V , D ( t ) ≃ D (cid:20) − SV (cid:18) ℓ D √ π (cid:19)(cid:21) , t → , (2)where ℓ D = √ D t is the diffusion length scale and D ≡ D ( t ) | t =0 is the free diffusivity. As ℓ D increases, a) Electronic mail: [email protected] b) Electronic mail: [email protected] the barrier permeability, κ , may begin to affect D ( t ) . While ℓ D remains short, barriers appear flat to the smallfraction of nearby walkers that encounter them , in-troducing a linear κt term in Eq. (2), D ( t ) ≃ D (cid:20) − SV (cid:18) ℓ D √ π − κt (cid:19)(cid:21) , t ≪ τ D , (3)where τ D = ¯ a / (2 D ) is the time to diffuse across themean pore of size ¯ a = 6 V /S . Curvature and surface-relaxivity may also affect D ( t ) . Tortuosity principallyaffects the long-time D ( t ) and can be categorizedinto disorder classes with structural exponent, p . Thelong-time D ( t ) follows a p -dependent power law , D ( t ) ≃ D ∞ + const . · t − ϑ , t → ∞ , (4)where D ∞ = lim t →∞ D ( t ) and ϑ = ( p + 3) / .Diffusion-weighted (DW) nuclear magnetic resonance(NMR) methods are highly sensitive to h r ( t ) i , andprovide a powerful means to probe rich D ( t ) behaviorsand infer distinct microsctructural features. DW-NMRexperiments have been used to study the short- and long-time D ( t ) in porous media ranging from sedimentary rockto skeletal muscle. The NMR spin echo dephasing, however, is not simplywritten in terms of D ( t ) itself. Instead, the echo dephas-ing is often expressed in terms of the real part, ℜ , of theFourier transform of D ( t ) . From Eq. (1) , tr ( ℜ [ D ( ω )] ) D + Z ∞ ∂ t [ tD ( t )] e iωt dt. (5) cce p t e d M a nu s c r i p t tr ( ℜ [ D ( ω )] ) behaviors revealthe short- and long-time D ( t ) , respectively. The re-lationship between ℜ [ D ( ω )] and the ensemble echodephasing follows from a cumulant expansion of thephase expectation value and a Gaussian phase distri-bution approximation , yielding the normalized echointensity , I ( T ) I = exp (cid:18) − π Z ∞ F T ( ω ) ℜ [ D ( ω )] F ( ω ) dω (cid:19) , (6)where F ( ω ) is the truncated Fourier transform of F ( t ) atthe echo time, T , (i.e., F ( T ) = ), F ( ω ) = Z T F ( t ) e iωt dt, (7) F ( t ) = γ R t G ( t ′ ) dt ′ , G ( t ) is the gradient waveform,and γ is the gyromagnetic ratio. The assumed Gaussianphase approximation is valid for most relevant experi-mental cases (cf. Stepišnik ). Eqs. (6) and (7) show that spectral tuning of | F ( ω ) | results in narrow sampling of ℜ [ D ( ω )] in the gradientdirection, ˆ g . G ( t ) can be periodically time-modulated (e.g., by using a sinusoidal G ( t ) ) so that the spec-tral density of | F ( ω ) | concentrates near some frequency, ω F . In this case, I ( T ) /I becomes well-approximated by exp ( − b ( T ) × ˆ g T ℜ [ D ( ω F )] ˆ g ) , where b ( T ) = Z T | F ( t ) | dt ≡ π Z ∞ | F ( ω, T ) | dω. (8)Individual experiments become a point-wise sampling of ˆ g T ℜ [ D ( ω F )] ˆ g . This “temporal diffusion spectroscopy” approach is robust, but has limited time resolution be-cause it individually probes ω F . Furthermore, the short-est probe-able timescale (i.e., largest ω F ) is limited toabout 10 ms by the pulsed gradient hardware. Analternative approach is needed for the real-time studyof D ( t ) across timescales and to reach the information-rich, short-time regime ( . ms) in biological systems.Static gradient (SG) hardware permits extremely fastcycling of the effective gradient direction using radiofre-quency (RF) π -pulse trains and thereby provides accessto these timescales. Here, we extend the classicalCarr-Purcell-Meiboom-Gill (CPMG) experiment un-der an SG in the RF field to probe the time-varying,sub-millisecond diffusivity in one shot.
II. THEORYA. Time-dependent signal representation
To begin, an alternative signal representation isused. The echo attenuation is related to the sta-tionary position autocorrelation tensor, R ( t, t ′ ) = FIG. 1. Example G ( t ) calculation. (a) Radiofrequency (RF)pulses in a static gradient of amplitude g . (b) G ( t ) , shifted G ( t + s ) (red, dashed), and F ( t ) . (c) Corresponding G ( t ) . G ( t ) is assumed to be zero outside of t = [0 , τ ] . h r ( t ) r T ( t ′ ) i ≡ R ( | t − t ′ | ) – again assuming a Gaussianphase distribution – by I ( T ) I = exp − γ Z T Z T G T ( t ) R ( t, t ′ ) G ( t ′ ) dtdt ′ ! , (9)Eq. (9) can be rewritten according to Ning et al. byintegrating along the level set of t − t ′ . For unidirectionalencoding, i.e., G ( t ) = || G ( t ) || and F ( t ) = γ R t G ( t ′ ) dt ′ , I ( T ) I = exp − Z T C ( t ) D inst ( t ) dt ! , (10)where D inst ( t ) is the instantaneous diffusivity along thegradient direction ˆ g , D inst ( t ) := ∂ t " ˆ g T R ( t ) ˆ g ≡ ∂ t " h [ r ( t ) · ˆ g ] i , t > , (11)(dropping ˆ g as implied) and C ( t ) is the cumulative inte-gral of the γG ( t ) autocorrelation function, C ( t ) = Z t G ( t ′ ) dt ′ G ( t ′ ) = γ Z T G ( t ′ ) G ( t ′ + s ) ds, (12)schematized in Fig. 1. The attenuation becomes a sam-pling of D inst ( t ) weighted by C ( t ) , similar to how Eq. (6)describes a sampling of ℜ [ D ( ω )] by | F ( ω ) | . B. Recasting the problem
Eq. (10) is advantageous compared to Eq. (6) be-cause C ( t ) is simple for general, non-periodic G ( t ) . Theill-posed inverse problem of finding D inst ( t ) from manytrivial G ( t ) is tractable. For a train of N echoes refocus-ing at times T n , C n ( t ) can be evaluated for each inter-echoattenuation I ( T n ) /I ( T n − ) ( T = 0 ); i.e., each pair of ad-jacent echoes can be treated as an independent spin echodiffusion measurement. The problem is then recast as aweighted and regularized log-linear least squares (LLS)inversion by discretizing the time domain into K bins ofvariable width, ∆ t ( k ) : arg min X || W / ( AX − B ) || + λ || Γ X || , (13) cce p t e d M a nu s c r i p t A = R ∆ t (1)0 C ( t ) dt . . . R ∆ t ( K )∆ t ( K − C ( t ) dt ... ... R ∆ t (1)0 C N ( t ) dt . . . R ∆ t ( K )∆ t ( K − C N ( t ) dt , where X consists of time-interval D inst ( t ) averages, X = t (1) R ∆ t (1)0 D inst ( t ) dt ... t ( K ) − ∆ t ( K − R ∆ t ( K )∆ t ( K − D inst ( t ) dt , and B T = − ln (cid:2) I ( T ) /I . . . I ( T N ) /I ( T N − ) (cid:3) . Theregularization matrix, Γ , is chosen to consist of first andsecond-order finite difference matrices, reflecting an apriori assumption of the smoothness and concavity of D inst ( t ) . The choice of N is dictated by when the echosignal decays to the noise floor. The choice of K and ∆ t ( k ) is more arbitrary. As a preliminary heuristic, K should be similar in magnitude to N , and ∆ t ( k ) shouldbe chosen such that (1) D inst ( t ) does not vary greatlyover any interval and (2) the C n ( t ) integrals that com-prise the entries of A are appreciable. Considering thebehavior of Eqs. (3) and (4), ∆ t ( k ) should start outsmall and may gradually lengthen. The norm is weightedby a proportionality of the signal-to-noise ratio (SNR);since B consists of log ratios, the appropriate weightsmatrix, W , is an N × N matrix of signal differences, i.e., I ( T n − ) − I ( T n ) . In this way, D inst ( t ) can be estimatedfrom a single echo train with varied C n ( t ) .The motivating question of this Communication is asfollows: Can a DW-NMR method probe the time-varyingdiffusivity in real-time? With Eq. (13) in mind, the ques-tion can be separated into two parts: (1) How can C n ( t ) ofvarious time sensitivities be produced in one echo train?(2) How can every echo be made accurate? The answerto the first part follows from a well-known DW-NMRprotocol. As mentioned, the SG-CPMG experiment canproduce rapid effective gradient oscillation characterizedby a triangle wave F ( t ) with ω F = ( π/ τ ) rad/s, where τ is the spacing between π -pulses. This SG-CPMG ω F (up to tens of kHz ) exceeds that which is attainablewith oscillating or pulsed field gradient (PFG) methods(up to ∼ Hz). As a result, the SG-CPMG method isuniquely able to probe the short-time diffusion regime insmall ( ¯ a . µ m) structures. Stimulated echoesrepresent another candidate DW-NMR method, but areill-suited to single-shot, multi-echo acquisitions due tothe signal loss inherent to π/ -pulses.Varying the spacing of the π -pulses in an SG-CPMGstyled acquisition is thus the preferred method to pro-duce various C n ( t ) in one shot. Others have exploredthe concept of modifying SG-CPMG pulse spacings tomeasure time-varying diffusion, but ultimately retaineda π -pulse train with repeated spacing. We extend suchmethods by modifying every π -pulse spacing. Spacings FIG. 2. Example SG-TIETA sequence. (a) Timings: m j = { , , , , } , τ = 4 δ , and δ = 14 µ s = 1 dash. π -pulses occurat t n and direct echoes (blue lines) form at T n . Magentaline indicates timing behavior: T n = t n + h n , where h n isthe normalized | F ( t ) | /γg “height” at t n , given by h = τ and h n = 2 τ + m n δ − h n − for n > . (b) Direct echo F ( t ) andother coherence pathways that refocus (red, dash-dot) or donot refocus (gray, dotted). Relative values of h n are indicated. can be incremented to retain signal. We choose the spac-ing between π -pulses to take the form: τ + m j δ , where j indexes the π -pulse-to-pulse spacing, m j ∈ N , and δ is a unit time increment. We term this discrete spacingmethod the SG, time-incremented echo train acquisition(SG-TIETA), e.g., Fig. 2. For SG-TIETA, C n ( t ) = γ g ( t (cid:0) − t + 2 h n (cid:1) ≤ t ≤ h n t (cid:0) t − h n (cid:1) + 2 h n h n ≤ t ≤ h n , (14)where h n is the n th peak of | F ( t ) | /γg . According toEq. (14), the n th inter-echo interval probes D inst ( t ) overthe time interval [0 , h n ] , with a peak at h n . With thecore experimental method described, we turn towards theproblem making each echo accurate. An initial step is toisolate the direct echo pathway. C. Isolating the direct echo pathway in the time domain
Ignoring the effects of magnetic susceptibility, surface-relaxivity , and spin-spin ( T ) relaxation (for the mean-time), the predominant source of extraneous signalbehavior is off-resonance coherence transfer pathways(CTPs). When the bandwidth of Larmor precessionfrequencies spanned by an SG exceeds the bandwidthof π -pulses, every pulse is slice-selective and excites allCTPs. For SG-CPMG measurements, the number ofrefocused off-resonance CTPs grows exponentially with N ( ∼ N ) , resulting in significant deviations from theexpected echo attenuation. Phase cycling remedia-tion schemes require ∼ N steps and are thus infeasible.Unconventional approaches such as time-based avoidanceof CTPs become necessary. An SG aids these time- cce p t e d M a nu s c r i p t τ sep , is shortened. Specifically, τ sep is constrainedby τ sep ≥ τ p , where τ p = 2 π/ ( γg ∆ z ) is the length of the π -pulse and ∆ z is the resulting slice thickness. The echowidth under an SG is on the order of τ p such that thisconstraint can be understood as avoiding undesired echooverlap. For hard π -pulses, τ p is on the order of ∼ µ s.In the interest of acquiring an accurate direct echoCTP, we should ask: What choice of m j , τ , and δ sep-arates off-resonance CTPs from the direct echo CTP by ≥ τ p ? Consider that CTPs are piece-wise linear func-tions in F ( t ) . The three magnetization states for a spin- / nuclei (in shorthand: M ∈ { + , − , } ) correspondto + γg , − γg , and slopes, respectively (see Fig. 2b).Refocusing of undesired signal occurs when the summeddifference between an off-resonance F ( t ) and the directecho F ( t ) , P ∆ F ( t ) , equals 0. Rules for m j , τ , and δ aredeveloped. Singly stimulated echoes (e.g., +0 − ) arisedue to two h n matching. Thus:(i) Absolute F ( t ) heights, or h n , may not be repeated.Next, consider CTPs that see the initial π/ -pulse.These CTPs can alter P ∆ F ( t ) /γg by one of { , , } × ( − j − (2 τ + m j δ ) . Accounting for P ∆ F ( t ) = 0 withup to four non-zero terms:(ii) m j and m j ± ∆ j with odd ∆ j may not be the same.(iii) Twice any m j may not equal the sum of m j +∆ j and m j − ∆ j for even ∆ j .(iv) Any two m j with even j may not equal the sum ofany two m j with odd j .(v) Twice any m j with even j may not equal the sumof any two m j with odd j , and vice versa.Another class of off-resonance CTPs is associated withthe introduction of transverse magnetization from spinsthat incorrectly see π -pulses as initial π/ -pulses. TheseCTPs emerge at the time of π -pulses and thus invariablystart with P ∆ F ( t ) containing an odd multiple of τ (i.e., τ + 2 jτ ). Choosing τ and δ such that ( τ mod δ ) = δ/ ensures that P ∆ F ( t ) ≥ δ/ . Incorporating τ p :(vi) δ and τ satisfy ( τ mod δ ) = δ/ and δ > τ p .Note that ( τ mod δ ) = 0 results in the refocused CTPsin Fig. 2b. The sequence in Fig. 2 does, in fact, satisfyrules (i–v). This limited ruleset (i–vi) may be sufficientto ostensibly avoid off-resonance effects considering thatsingly stimulated echoes are known to be the most signif-icant contributor to SG-CPMG off-resonance effects .The generation of m j that satisfies rules (i–v) is dis-cussed in the Supplementary Material (SM) Section I.Python code is provided. A solution for τ , δ , and m j ,used throughout, is τ = 49 µ s , δ = 14 µ s ,m j = (cid:26) . . . (cid:27) , (15) which gives time sensitivity over t ∼ − µ s, h n =
49 63 77 105 91 147 119 133175 203 189 245 217 161 231 329259 301 273 287 . . . µ s . III. EXPERIMENTAL SETUP
NMR measurements were performed at B = 0 . (proton ω = 13 .
79 MHz ) using a PM-10 NMR MOUSEsingle-sided permanent magnet (Magritek, AachenGermany) and a Kea 2 spectrometer (Magritek, Welling-ton, New Zealand). The decay of the magnetic fieldwith distance from the magnet produces a strong SGof g = 15 . / m (
650 kHz / mm ). Measurements used ahome-built test chamber and a × mm solenoid RF coiland RF circuit. Additional information concerning theexperimental setup can be found in Williamson et al. The SG-TIETA pulse program was written in ProspaV3.22 by modifying the standard CPMG sequence. SeeSM Section IV for details.For measurements on twice-distilled water, decane(Sigma-Aldrich, St. Louis, MO, USA.), 1-octanol(Sigma-Aldrich), and dodecamethylcyclohexasiloxane(D6) kinematic Viscosity = 6.6 cSt @ 25°C (Gelest, inc.Morrisville, PA, USA.), the liquids were transferred to2 cm glass capillary sections (1.1 mm OD, Kwik-Fil™,World Precision Instruments, Inc., Sarasota, FL, USA)and the capillaries were sealed with a hot glue gun.For measurements on yeast (
S. cerevisiae ), 1.72 g ofdry yeast was mixed in 10 ml of tap water and stored ina 50 ml tube with the lid screwed on loosely to allow gasto escape. After three days (72 hours), the yeast was re-suspended and samples were taken for NMR experimentsand for cell density measurement. The density of the firstsample (yeast . × cells/mlusing a hemocytometer. The remaining yeast was cen-trifuged, the pellet was re-diluted to 2X the initial con-centration, and a second sample (yeast ® Implant Mem-brane sections (500,000 Dalton molecular weight cut off,1 mm outer diameter, SpectrumLabs, Waltham, MA,USA) and the membranes were sealed by pinching themembrane with heated forceps. Yeast was kept from dry-ing out by performing measurements immediately uponfilling and sealing the capillary and by lining the insidewith wet tissue paper to increase humidity.Experiments were performed at ambient temperature.Sample temperature was monitored with a fiber opticsensor (PicoM Opsens Solutions Inc., Québec, Canada).The average temperatures of the samples during thecourse of the experiments were 25 ◦ C , 22 ◦ C , 22 ◦ C , 23 ◦ C ,and 24 ◦ C for the water, decane, octanol-1, D6, and yeast,respectively. cce p t e d M a nu s c r i p t FIG. 3. LLS inversion on Monte Carlo simulated data. (a)Simulated h [ r ( t ) · ˆ g ] i ( D = 2 . µ m / ms ) for free diffusion(black) and restricted geometries (see Fig. S1). P C n ( t ) (a.u.)for Eq. (15) is overlaid, omitting the first two echoes. (b)Echo decays simulated from same color curves in (a) for γg =4 . µ m − ms − . Insets show B with W / for the blackcurve and A for ∆ t ( k ) = { , , , , , , , } µ s.(c) D inst ( t ) from the gradient of h [ r ( t ) · ˆ g ] i curves in (a) (solidlines) compared to X inverted from decays in (b) with addedGaussian noise (SNR = 25). Err. bars = ± SD from 100replications. Initial X guesses were D for the first point and D ∞ (dashed lines, { . , . , . } µ m / ms ) for remainingpoints. λ = 2 × − , selected manually. See Eq. (S1) for Γ . IV. RESULTS AND DISCUSSION
As an initial proof-of-principle, the LLS inversiondescribed in Eq. (13) was performed on noisy SG-TIETA decays generated using the timings in Eq. (15)and h [ r ( t ) · ˆ g ] i curves from Monte Carlo simulations ,shown in Fig. 3. Further details and MATLAB code areprovided in SM Section II. Inverted X values are shownto be accurate and robust to noise. No systematic errorsother than potential over-regularization are observed.Echo-to-echo accuracy remains experimentally difficultto achieve, however. Consider the signal decay due to T and pulse inaccuracy effects. Relaxation may beignored if (2 τ + m j δ ) ≪ T ∀ j . Diffusion-weighted T values for each sample, as shown in Table I, indi-cate that this condition holds for all pure liquids stud-ied here. However, the yeast is described by a distribu-tion of T with a 5% water population with T similarto max { τ + m j δ } = 539 µ s . The instantaneous dif-fusion measurements of yeast may be slightly weightedby T relaxation. See SM Section IV.D for relaxationmeasurement methods and a T distribution analysis foryeast T , pulse inaccuracy cannot be ig-nored. Inaccuracy effects may be described using an n - dependent pulse accuracy factor, A p ( n ) . The signal isthen corrected as I ( T n ) /I × [1 / Q nl =1 A p ( l ) ] , assumingtotal avoidance of off-resonance CTPs via rules (i–vi). TABLE I. Relaxation times.Sample T [ms] T [ms]water 3250 decane 1340 octanol-1 440 D6 788 yeast yeast Calibration A p ( n ) values were approximated from de-cays of pure liquids, shown in Fig. 4. Echo amplitudeswere calculated as the sum of all real signal points withinthe echo window, which was set to 16 µ s. All decays werenormalized to the first echo and each repetition consistedof 32 summed (i.e., signal-averaged) scans. To elucidatethe expected A p ( n ) behavior, we consider the spatial, i.e.,slice effects. The bandwidth/slice excited by refocusing π -pulses has inconsistent frequency content such that A p ( n ) < . With each pulse, spins which rotate by anglesother than π do not refocus until only a stable, centralslice remains. In the time domain, this slice-thinning andloss of frequency content is expected to broaden the echowidth, which we experimentally verify in Fig. 5. Basedon the evolution of the echo shape, A p ( n ) should sharplyincrease then taper. This behavior is observed in Fig.4c and is consistent across liquids of vastly different D .Similar A p ( n ) values were also obtained for τ = 77 µ s(see SM Section III, Fig. S5), further supporting that A p ( n ) is independent of the diffusion weighting.Another pre-processing step is designed to mitigate theeffects of early A p ( n ) variability and to ensure the non-negativity of B and W entries. A piece-wise linear fit ofadjacent ln ( I ( T n ) /I ) points vs. b is performed, specify-ing (1) an intercept with [ b, ln ( I ( T n ) /I ) ] = [0 , , (2) noslope exceeds D , and (3) the piece-wise slopes decreasemonotonically (i.e., D inst ( t ) has non-negative concavity).Altogether, SG-TIETA decays are analyzed in five steps:(1) summing × , (2) normalization to the first echo, (3) I ( T n ) /I × [1 / Q nl =1 A p ( l )] correction, (4) a constrainedlog b domain fit to the repetition(s), and, finally, (5) theLLS inversion. This pipeline was applied to SG-TIETAdecays of D6 and yeast using the 1-octanol and water A p ( n ) values obtained in Fig. 4c, respectively.Results are summarized in Fig. 6. The D inst ( t ) forD6 – with D = 0 . ± . µ m / ms – is expectedlyflat, thus validating the A p ( n ) correction. The D inst ( t ) for yeast are the key results of this Communication. Forcomparison, the short-time D inst ( t ) predicted by Eq. (3)(i.e., d [ tD ( t )] /dt ) is plotted for several S/V and κ values.Fig. 6b indicates that ¯ a ≃ µ m and that a doublingof the cell density approximately halves ¯ a . For yeast’s ≈ µ m spherical diameter , ¯ a is calculated as 42 and 21 cce p t e d M a nu s c r i p t FIG. 4. SG-TIETA A p ( n ) calibration using pure liquids. (a)Observed SG-TIETA decays compared to exp ( − bD ) . D was measured (see SM Section IV) in legend order as . ± . , . ± . , . ± . µ m / ms . Err. bars = ± SD for25, 38, 70 repetitions, respectively. (b) Decay vs. exp ( − bD ) ratio truncated at n = 12 , , . Inset shows cubic splinefits. (c) A p ( n ) approximated using adjacent fitted ratios. µ m at these cell densities, suggesting contributions from sub -cellular length scales. This ensemble ¯ a estimate of ≃ µ m ( ≡ V /S ≃ µ m) is within the range of estimates( ¯ a ∼ − µ m) reported in previous PFG and SG DW-NMR studies of similar yeast densities. Several factors may contribute to the discrepancy be-tween the experimental and the predicted short-time D inst ( t ) . On the numerical side, over-regularization mayartificially flatten D inst ( t ) at short times, as shown in Fig.3. Values of X are also plotted inexactly at the midpointsof P k ∆ t ( k ) . On the theoretical side, Eq. (3) does notinclude a term for the curvature, which may be signifi-cant at these timescales for sub-cellular water. Consider,also, the confounding effects of T , surface relaxation, theGaussian phase approximation, and the assumption ofecho number translation invariance (i.e., that each C n ( t ) is presumed to start from t = 0 ). Echo number transla-tion invariance does not hold for spatially heterogeneousmicroenvironments. If different water pools exhibit vary-ing decay rates, the relative signal contributions will de-pend on the echo number, n . Indeed, spatial heterogene-ity and the resulting weighting towards slowly decayingwater pools at larger n may explain the convergence ofthe yeast X values. These yeast results are thus non-quantitative. Nonetheless, the sensitivity of SG-TIETAto apparent microstructural features is clear. V. CONCLUSIONS
We have developed a real-time protocol to measuretime-varying diffusion. Inversion for D inst ( t ) from exper-imental SG-TIETA decays is demonstrated. An approx-imately 1-minute ( × repetition time of s) experimentis described. In contrast to conventional temporal diffu- FIG. 5. Comparison of the direct echo SG-TIETA (blue) andCPMG (orange) attenuation and echo shape for 1-octanolwith τ = TE = 98 ms. Echo shapes show the real (solidlines) and imaginary (dotted lines) signal normalized by thearea under the real signal curve in a 16 µ s window. TheCPMG echo width decreases with n and stabilizes around n = 3 , consistent with the approach to asymptotic behav-ior described by Hürlimann & Griffin. In contrast, the SG-TIETA echo width increases and stabilizes around n = 3 ,consistent with the direct echo CTP being preferential to theon-resonance signal. See Figs. S7 and S8 for all echo shapesand an exemplar echo decay, respectively. sion spectroscopy methods, the single-shot nature of SG-TIETA permits true signal averaging in order to improveSNR. A post hoc A p ( n ) correction is proposed to improvethe quantitative accuracy of the method. To support the cce p t e d M a nu s c r i p t FIG. 6. SG-TIETA decays and inverted X for D6, yeast,and water. (a) Decays analyzed as described in the text.See SM Section II for fitting procedures. Err. bars = ± N = 34 , , , , respectively. Note erratic early A p ( n ) behavior. (b) X solutions. Inversion parameters were iden-tical to Fig. 3 other than ∆ t ( k ) for D6. Initial guessesof D = 2 . , D ∞ = { . , . } µ m / ms for yeast and D = D ∞ = 0 . µ m / ms for D6 were provided. Zoomedplot compares short-time D inst ( t ) plotted up to t < . × τ D . validity of this correction, we present preliminary evi-dence in the observed echo shape behavior and in the con-sistency of A p ( n ) values across different diffusion weight-ings. Regarding potential applications, SG-TIETA atthis g can probe porous media microstructure on mi-cron length scales, i.e., over sub-millisecond timescales.SG-TIETA can also be used to study phenomena as-sociated with other long-range correlations, e.g., poly-mer dynamics , the glass transition , and high Pècletfluxes driven by flagella , which likewise exhibit time-dependence in this sub-millisecond range. The methodscontained in this Communication may open new avenuesof research within DW-NMR. SUPPLEMENTARY MATERIAL
In the supplementary material, we include sectionscontaining (I) Python code to generate m j , (II) represen-tative MATLAB code for the Monte Carlo simulations,fitting procedures, and LLS inversion, (III) a replicationof the analysis in Figs. 4 and 6 using SG-TIETA decaysfor τ = 77 µ s, and (IV) additional NMR experimen-tal methodology, which includes all echo shapes and astitched echo decay for 1-octanol. ACKNOWLEDGMENTS
The authors would like to thank Dr. Dan Benjaminiand Dr. Michal Komlosh for helpful discussions concern-ing numerical programming and time-based avoidance ofunwanted coherence pathways, respectively.TXC, VW, RR, and PJB were supported by the IRPof the NICHD, NIH. TXC is a graduate student inthe NIH-Oxford-Cambridge Scholars Program. NHWwas funded by the NIGMS PRAT Fellowship Award
DATA AVAILABILITY
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