Guilhem Pages
email: g.pages@mmb.usyd.edu.au

Philip W Kuchel

NMR Methods for the Fast Recording of Diffusion

Keywords: Diffusion experiment, diffusion ordered spectroscopy, DOSY, echo train, fast diffusion, lineshape fitting, multiple coherence pathways, single scan BPPSTE, bipolar stimulated echo, CPMG, Carr-Purcell-Meiboom-Gill, DDF, distant dipolar field, DOSY, diffusion ordered spectroscopy, FID, free induction decay, MMME, multiple modulation multiple echoes, NMR, nuclear magnetic resonance, PFG, pulsed field gradient, PFGMSE, pulsed-field gradient multiple spinecho, S/N, signal to noise ratio

The time taken to complete an NMR diffusion experiment is typically ~10 minutes. For systems that rapidly evolve conventional pulsed field gradient spin-echo (PGSE) experiments cannot be used to obtain reliable estimates of molecular mobility. Modern 'fast-diffusion' experiments provide a means of obtaining this information and thus open up new vistas for the application of PGSE NMR. In this paper we review the various advantages and disadvantages of these methods.

Introduction

Pulsed field gradient spin-echo nuclear magnetic resonance (PGSE NMR) experiments are a method of choice for studying diffusional mobility of molecules in living and inanimate systems. The method is non-invasive and is able to measure in a few minutes the diffusion coefficients of solutes and solvents in complex mixtures [1][START_REF] Momot | Concepts in Magnetic Resonance[END_REF][3][4][5]. In PGSE NMR the signal intensity S(g) of the molecule of interest is measured as a function of the magnitude of the magnetic field gradient pulses that are used. The diffusion coefficient D is estimated from the data by regressing onto them the following Stejskal-Tanner equation [6]:

where S 0 denotes the signal intensity without gradients, γ is the nuclear magnetogyric ratio, g is the gradient amplitude, δ is the gradient pulse duration, and Δ' is the 'corrected' diffusion time. In the simplest pulse sequences Δ' is simply the interval between the two field gradient pulses, or more generally Δ' = (Δ -δ/3).

One well-known extension to data acquisition and processing using PGSE NMR is diffusion ordered spectroscopy (DOSY) [7]. The final output is a 2-dimensional contour map in which one axis records diffusion coefficients and the other chemical shifts. Thus one dimension indicates a physical property and not, as is usually the case, information on magnetic interactions between nuclei.

New pulse sequences and processing methods have been developed to study a diverse variety of samples from: discrete to polydisperse samples; chemical to biological ones; unrestricted to restricted ones or those with anisotropic diffusion; and stagnant to rapidly flowing mixtures. Several reviews have been written describing these applications [1,3,[8][9][10].

Even if the total time of the experiment is short, it may be too long to interrogate systems in which diffusion coefficients change rapidly. Such systems include biological ones with enzyme-catalyzed reactions, or those in rapidly flowing fluids in industrial processes. The first fast-diffusion NMR pulse sequence was described in 1969 [11]; and recently a few more methods have been published that can measure a diffusion coefficient in less than 1 minute. We review these techniques in this paper.

Pulse sequences to measure diffusion coefficients very rapidly have also been developed to handle the specific requirements of grossly inhomogeneous magnetic fields [12], and for use with laser-excited hyper-polarized gas of relatively short half-life [13]. Because these methods are highly specialized we will not discuss with them in any detail. We have classified the remainder of the methods into two classes: (1) those from which the diffusion coefficient is extracted from data obtained by using only one or two gradient pulses; and (2) those in which diffusion is elicited by a train of gradient pulses. The latter we analyze and discuss first.

Diffusion measured using a train of gradient pulses

Saving time by decreasing phase cycling

By keeping to the traditional approach with PGSE NMR experiments the only recourse to saving time is to decrease the number of transients in each phase cycle. In the bipolar-pulse stimulated-echo (BPPSTE) experiment [14], the phase cycling of the 180° pulses requires a EXORCYCLE [15] with 16 (or optimally 64) transients in order to record only the spatially encoded magnetization. This EXORCYCLE enforces a zeroorder coherence during the delay Δ.

The approach developed by Morris et al. [16] is based on suppression of unwanted coherence-transfer pathways using deliberately unbalanced pulse pairs (Fig. 1). The magnitude of the two pulses in a pair have the ratio 1:1+α. During the experiment, α is held constant and the appropriately transformed Stejskal-Tanner equation is:

2 2 2 2 2 2 1 ( ) 6 2 ( ) (0) g S g e S α α γ δ δ τ - - - Δ + + = ( 2
)
where τ denotes the delay between the two components of the bipolar magnetic field gradient pulses. The choice of the value of α depends on BB 1 inhomogeneity and signal-tonoise ratio and is usually ~ 0.2. Additional gradient pulses can be used in B to refocus the lock signal and to dephase unwanted coherences. Spoil gradients can also be added to the BPPLED pulse sequence. RF denotes the radio-frequency time train and Gz the gradient pulse time train. δ is the gradient duration, g is the gradient amplitude, Δ is the diffusion time (midpoints of the two diffusion encoding periods), τ is the time between the midpoints of the antiphase field gradients, and α is the unbalancing factor of the gradient pulses.

Series of gradient echoes

Another way to minimize the acquisition time of a PGSE experiment is to record within a single transient the echoes that are modulated by diffusion. There are four pulse sequences that are based on this principle.

The first two pulse sequences are based on the Carr-Purcell-Meiboom-Gill (CPMG) experiment. In one, each 180° pulse is sandwiched between two gradient pulses of identical magnitude to encode and then decode spatial information (Fig. 2A); the pulse sequence is called the Pulsed Field-Gradient Multiple Spin-Echo (PFGMSE) experiment [17]. The delay between the initial 90° pulse and the first 180° is τ and the delay between each 180° is twice this value, 2τ, and each echo corresponds to a different gradient magnitude. The number of echoes that are recorded is the number of points used in the diffusion decay analysis, so a prolonged series of echoes provides a better estimate of D. To extract the value of D, the ratios between two consecutive echoes must be calculated (Eq. 3). It is then a simple matter to determine D from the slope of the graph:

2 2 2 2 2 ( 1) 2 ln ' ( ) S n n g D S n T τ γ δ + = -- Δ (3)
where n denotes to the nth echo that is recorded. The second pulse sequence, that is based on a CPMG RF-pulse train, uses magnetic field gradient pulses that are of constant amplitude (Fig. 2B) [18]. The diffusion coefficient is extracted from the linewidths of the Fourier transform of the echo train. The width of the resonance at half of the peak height, Δν 1/2 , of the adsorption signal in the frequency domain, for a Lorentzian lineshape is:

2 2 2 1 2 2 1 1 2 6 g D T δ υ γ δ π τ τ ⎡ ⎤ Δ ⎛ Δ = + - ⎞ ⎢ ⎥ ⎜ ⎝ ⎠ ⎟ ⎣ ⎦ (4)
To fit the data, the value of the first term in parentheses of Eq. ( 4) is required. Hence an experiment is run with the gradient pulse-amplitudes set to 0 thus eliminating the second term of the equation.

To extend this method to the case of non-Lorentzian lineshapes and/or overlapping peaks in the echo spectrum, the first 90° hard pulse is substituted by a 90° selective one. Fig. 2 emphasizes the similarity between both of the CPMG-based pulse sequences. They differ only in the variation or not of the gradient magnitudes. [17]; and B, the method of Chandrakumar et al. [18]. For A the gradient amplitude is increased systematically, while for B it is constant. The notation is identical to that in Fig. 1 except that τ is the delay between the 90° and 180° RF pulses. The subscripted square-brackets around the pulse sequence indicates that it is repeated N times.

The third method in the present class involves recording a series of echoes that are created by alternation of the diffusion gradient [19]; these are called gradient echoes. Each echo is attenuated further by an additional diffusion-gradient pulse. The expression for the signal amplitude differs between even (Eq. 5) and odd (Eq. 6) echoes: )

3 γ τδ δ - - -= i ( 6
)
where A n is the amplitude without diffusion attenuation, and the vector dot product g 0 •g accounts for the interaction between the gradient pulses and a static magnetic field. Fitting the signal amplitude using both previous equations is used to estimate D. Finally, a recent approach called Difftrain has been described [20]. This is like the previous one because it exploits attenuation of an echo by an additional gradient pulse. The position of a nucleus is encoded and then it is decoded multiple times. This yields all the information necessary to fit the Stejskal-Tannner equation (Eq. 1). To conserve chemical shift information, the pulse sequence uses the inversion recovery method that is routinely used for measuring longitudinal relaxation times [21]. Positions of the spins are encoded in the transverse plane and then promoted to the longitudinal direction with an RF pulse; then a small proportion of the spin population is returned to the transverse plane for the detection via a spin echo. After detection, the residual magnetization is destroyed by a spoil gradient, before applying another gradient and RF pulse to detect the next echo.

Difftrain can be used to measure the distribution of droplet sizes in an oil-water emulsion that is undergoing evolution during phase separation [22]. A fast-diffusion experiment (taking less than 4 s) was needed to characterize the system because the emulsion was thermodynamically unstable. In contrast, the acquisition time for a classical PGSE NMR experiment is too long to capture the evolving droplet sizes in the system. The authors also describe the possibility of measuring the apparent velocity of water flow through an ion-exchange (desalting) column.

All the abovementioned experiments take significantly less time than the classical PGSE NMR diffusion experiments to yield their data. They are still based on the evolution of a signal whose variation is described by an expression that contains D. Comparing the methods, the one described by Morris et al. [16] appears to encompass the best compromise between the difficulty of setting up the experiment on the NMR spectrometer and processing the resulting data. Other approaches have been developed to measure D in a single-scan, but the major difference is that they do not require a change in gradient magnitude during the experiment.

Pulse sequences with a small number of gradients

Burst pulse sequence

The Burst pulse sequence [23], was initially developed to rapidly acquire magnetic resonance images (MRI) under conditions of high spectral resolution [24]. A train of echoes is created from a series of low flip-angle RF pulses. To enable image formation, the pulses are applied in the presence of a persistent magnetic field gradient; and the echoes are detected by using a magnetic 'read' gradient (Fig. 3). To analyze the effect of this pulse sequence on a spin system we note that the first series of RF pulses is simply a DANTE pulse train [25]. This realization suggests the use of the 'small-angle approximation' [26] that is used in predicting the excitation envelope of the pulse train. The analysis is then extended to include the effects of diffusion. Accordingly the signal intensity for each echo is described by the equation:

2 2 2 3 n n n g D n S e δ γ δ ⎛ ⎞ - Δ- ⎜ ⎝ = ' ⎟ ⎠ (7)
where Δn = nτ+d 2 and δn = nτ+d 1 . D is estimated by fitting Eq. ( 7) to a graph of the intensity of the NMR signal versus the grouped parameters in the exponent of the expression.

Multiple Modulation Multiple Echoes (MMME)

A further method to rapidly estimate D of a molecular species is to record the spinecho from a series of different coherence pathways. This approach increases the information content over that of a single-transient [27]. For spin-1/2 nuclei such as 1 H, we can define three states of magnetization: M 0 , M -and M + . For a train of RF pulses, a coherence pathway is characterized by a series of N + 1 numbers from q 0 to q N where q N is the magnetization after the N th pulse and q 0 is the magnetization-state before the first pulse. Each coherence pathway can be written as a product of three terms:

M Q = A Q BB Q C Q (8)
where Q represents one coherence pathway between 0 and N; A Q is the frequency spectrum; BB Q is a term that describes diffusion; and C Q is the relaxation-attenuation factor. To write an expression for B Q B , it is necessary to introduce the instantaneous wavevector k(t) [START_REF] Callaghan | Principles of Nuclear Magnetic Microscopy[END_REF][START_REF] Sodickson | [END_REF], the magnitude of the constant field-gradient g and the instantaneous value of q, q(t'):

(9) 0 ( ) ( ') ' t k t g q t dt γ = ∫
In the case of diffusion in an isotropic unbounded medium the attenuation of the signal is given by,

B t D k t dt Q e -∫ = ( 10
)
where t = 0 specifies the beginning of the pulse sequence and T is the time at the maximum of the spin echo. Now consider a persistent magnetic field gradient and a train of four RF pulses with tipping angles α 1 ,…,α 4 and a time spacing between them of τ 1 to τ 3 , respectively. Fig. 4 shows this sequence using α 1 = α 2 = α 3 = 90° and α 4 = 180°. During the τ periods in the pulse sequence of Fig. 4, transverse magnetization (q = ±1) acquires a phase of qgτ ι (i = 1, 2 or 3) and the spin echo appears when the total phase is zero:

1 1 2 2 3 3 4 0 q q q τ τ τ τ + + -= (11)
where τ 4 is the delay after the last pulse and prior to acquisition of the free induction decay. For a ratio τ 1 = τ 2 /3 = τ 3 /9, Eq. 11 is expressed by the ratio τ 4 /τ 1 which represents the phase index:

τ 4 /τ 1 = q 1 + 3q 2 + 9q 3 (12)
Therefore, fourteen different phase indices are obtained, and by using the convention, q 0 = 0 and q 4 = -1 to detect the signal it is seen that an echo is detected every τ 4 = nτ 1

where n is the phase index. Thus we detect a new spin-echo signal every τ 1 seconds as indicated in Table 1.

Table 1: Different coherence pathways in a MMME4 pulse sequence. One free induction decay (coherence pathway 0,0,0,0,-1) with 13 echoes are recorded.

Phase index

q 0 q 1 q 2 q 3 q 4 0 0 0 0 0 -1 1 0 1 0 0 -1 2 0 -1 1 0 -1 3 0 0 1 0 -1 4 0 1 1 0 -1 5 0 -1 -1 1 -1 6 0 0 -1 1 -1 7 0 1 -1 1 -1 8 0 -1 0 1 -1 9 0 0 0 1 -1 10 0 1 0 1 -1 11 0 -1 1 1 -1 12 0 0 1 1 -1 13 0 1 1 1 -1
For each coherence pathway, molecular diffusion is responsible for signal attenuation which is described by the expression:

2 3 Q b D g Q e γ τ - = B ( 13
)
where b Q is calculated for each coherence pathway. The C Q term contains the longitudinal and transverse relaxation times, T 1 and T 2 , respectively. The previous experiment can be extended to a sequence of N pulses whereupon it is called MMMEN. For N pulses, the number of coherence pathways that are generated is given by [30]: 2 shows the number of coherence pathways that are recorded as a function of the number of RF pulses. In order to record spin echoes at particular times we set the period τ i to be a power of 3:

[3 (N-1) -1]/2 + 1 (14) Table
τ i = 3 i-1 τ 1 ( 15
)
Because the experiment is performed with a field gradient that is on all the time, and the sample extends beyond the region of the RF coils, each pulse must be slice-selective so that off-resonance signals are avoided. Using these conditions, the spin-echo signals and line shapes are able to be rigorously defined. Thus a single-transient experiment is sufficient to measure D. If the sample size is not sufficiently large spin-echoes may not have uniquely determined shapes and/or areas. In this case, two transients must be acquired using two different τ values (τ and τ'); then the diffusion coefficient is obtained from the ratio of the intensities of both echoes, as follows:

2 2 3 3 ( ' ( ) ( ') Q b D g Q Q S e S γ τ τ τ τ - - = ) (16)
This method has been successfully implemented in the SMART imaging pulse sequence [31,32]. The pulse sequence [31] measures both D and T 2 , and for unspecified reasons it requires a phase cycle of four transients. MMME has also been applied to measuring the velocity of flowing fluid in a single transient [33]. By adding magnetic field gradient pulses in multiple directions, it is possible to record a diffusion tensor that characterizes diffusion in all three directions of a Cartesian coordinate system [34]. Thus the values of D xx , D yy , and D zz can be deduced for each dimension in anisotropic media using a single-transient.

Fast-CRAZED

In this experiment RF pulses are used to excite spin systems in the sample so that a dipolar magnetic field is transiently established. This so called 'distant dipolar field' (DDF) is then used to measure simultaneously both D and T 2. Thus the CRAZED pulse sequence led to fast-CRAZED [35] due to its ability to rapidly measure the NMR parameters [36] (Fig. 5).

In this pulse sequence only one gradient pulse is used. The DDF created by the modulated magnetization performs the role of field gradient pulses by refocussing dephased magnetization. To select intermolecular zero-quantum transitions mediated by the DDF, a two-step phase cycle is used with a transient recorded with a β RF pulse (Fig. 5) of 45° subtracted from one recorded with β = 135°. A train of 180° pulses is used as a sandwich of 90°-180°-90° pulses [37] to minimize the loss of magnitude of the DDF. The first 90° RF pulse can be an adiabatic one thus selecting particular coherence pathways. The total time for an experiment using this sequence is ~10 seconds; the slightly longer time than perhaps expected is due to the long train of 90°-180°-90° pulses.

Fig. 5: Fast-CRAZED pulse sequence [38]. β is a 45 o or 135° RF pulse. t 1 is the period between the end of the adiabatic pulse and the middle of the β pulse and τ is the delay between the midpoints of the β pulse and the sandwiched 180° pulse. The sandwiched train is repeated N times. For other items of notation see Fig. 1.

The theoretical underpinning of this experiment relies on the non-linear term that is added to the Bloch-Torrey equations to account for the presence of the DDF [39][40][41]; the parameter D is incorporated into this term. An analytical solution of this equation is available only if diffusion-mediated signal-attenuation is much stronger than the rephasing of the magnetization due to the DDF. Thus diffusion attenuation is encapsulated in the relationship 1/(D(γgδ) 2 ) while the dipolar characteristic time is given by: 0 0

1 d M τ μ γ = ( 17
)
where μ 0 is the magnetic permeability of a vacuum, and M 0 is the equilibrium magnetization. The signal from the pulse sequence represented in Fig. 5 is given by Eq. ( 18) that includes the DDF in its local form [42] assuming that (see Fig. 5) t 1 << T 1 .

(

) 1 2 2 2 2 2 1 1 1 2 2 ( 2 ) 2 ( ) 0 3 1 2 2 1 ( 2
- + - - + - = × × + - ( 18
)
where θ is the angle between G and BB 0 , and k is the wave vector of spatial modulation of the magnetization caused by the magnetic field gradient; it is equal to γgG. A log-normal plot of the signal amplitude versus t 2 allows the extraction of estimates of both D and T 2 as follows:

(i) if 2 2 1 1 1 2 t Dk T +
, T 2 is estimated from the slope of the function that is fitted to the data.

(ii) if 2 1 1 2Dk
T , D is obtained from the data by considering the value of T 1 to be effectively infinite and fitting the resulting expression.

One-dimensional DOSY

As noted above, a well-known data representation from PGSE NMR is called DOSY [7,9]; the experiment and subsequent data analysis generate a 2-dimensional contour map with the estimates of D along one axis and chemical shift along the other. In the realm of experiments used to rapidly measure D the name of a new method is derived from the fact that information on both diffusion and chemical shift are recorded using a singletransient; hence it is called one-dimensional DOSY [43,44].

The principle of the method is that information on diffusion is encoded in the lineshape of each resonance. To achieve this outcome there must be a spatial dependence of the chemical shift, and diffusion will broaden peaks during signal acquisition. There are two reported methods based on this principle that record data in a single transient, and from which the data are then analyzed to yield an estimate of D. The two methods differ in the way they create the spatial dependence of chemical shift. In the first method [43], this dependence is created by using a non-uniform magnetic field gradient that is generated by an additional current in the z 2 shim coils, after a hardware modification that enables its switching on and off during the pulse sequence.

In the second method [44], spatial encoding is generated using an adiabatic frequency-swept 180° RF pulse that is applied while the diffusion-detecting magnetic field gradients are applied.

For both pulse sequences, the peak broadening that is invoked during signal acquisition is brought about with a weak linear 'read' gradient. But because the second method does not require hardware modification it is easier to implement than the first one. Therefore henceforth we focus on this pulse sequence (Fig. 6). The diffusion gradients G d are chosen to bring about different chemical shifts (off-set frequencies) as a function of spatial position. During the first of the magnetic field gradient pulses an adiabatic RF pulse of duration t p , is swept through a range of off-set frequencies. The spins are nutated through the x',y'-plane by this pulse at different times during the application of the magnetic field gradient; thus they are exposed to diffusion sensing over different times depending on their spatial location in the sample. A schematic representation of this effect is shown in Fig. 7. For spins located at one end of the sample, the adiabatic RF 180° pulse exerts its effect immediately and their magnetization is dephased by the gradient field for a time t p . On the other hand, magnetization in the middle of the sample is dephased during t p /2, then the adiabatic swept-frequency RF 180° pulse is applied; finally the magnetization is dephased by the gradient pulse during the last t p /2 period. This last period leads to refocusing of the spin magnetization vectors. The combination of gradient and sweep pulse allows the acquisition of a net signal that depends on the spatial position of the spins. To record the signal attenuation along the sample, signal acquisition occurs in the presence of a readgradient G r . Therefore spectral peaks are obtained which are diffusion-weighted images of the sample, but the weighting varies along the sample. To analyze the data, the net phase acquired by spin isochromats at the end of the adiabatic pulse needs to be expressed as a function of the experimental parameters and D.

Hence, if a spin at position z experiences RF-induced 180 o flipping at time α(z)t p , where 1 ≥ α(z) ≥ 0, the following relationship holds:

[ ] [ ] ( ) 1 ( ) 1 2 ( ) d p d p d G z z t G z z t z G zt γ α γ α α γ - - =-- p ( 19
)
During the frequency-sweep pulse, the spin isochromats precess as if they were under an effective gradient of strength

G eff : [ ] ( ) 1 2 ( ) eff d G z z G α = - (20)
In this simplified analysis it is assumed that the spin isochromats experience a 180° nutation at the instant the sweep is on-resonance. A more exact approach is described in the Appendix of the original paper [44].

After calculating the effective magnetic field gradient associated with each off-set frequency (chemical shift position, hence z) a predicted lineshape is calculated for a value of D fit using a slightly modified form of Eq. ( 1):

2 2 ( ) 0 ( ) eff fit A z D S z S e γ - Δ = (21)
where S 0 is a scaling factor, and A eff corresponds to G eff (z)t p .

Limitations of fast-recording methods

General

None of the methods described above have been developed to estimate two different values of D from the one spectral peak. The systems that have been studied have been well defined ones, and applications to more complex heterogeneous samples are awaited. In the particular case of biological samples where the temperature is relatively high convection can introduce a systematic overestimate of the value of D, and yet none of the fast-diffusion methods are convection compensated [3,45]. In addition, the pulse sequences have not yet been used with systems that have restricted diffusion, such as water in red blood cells [46].

Another general limitation is the available signal-to-noise (S/N) ratio in the NMR spectra. The gain in time efficiency is off-set by a lower S/N ratio; only in systems in which the detection is facile, i.e., a high solute concentration, can the experiments be used.

The particular fast-diffusion methods that are based on the recording of spin-echoes [17][18][19][20]24,27,36] also have the same drawbacks related to S/N. To avoid the overlap of spin echoes, the intensity of each subsequent echo must be significantly less than the previous one. The second drawback is the loss of chemical shift information in these latter methods, making them inappropriate for studies on mixtures of solutes [18][19][20].

Limitations of experiments with trains of gradients

A common observation with respect to the estimates of Ds obtained with fastdiffusion experiments is that they are high relative to those obtained with standard PGSE methods. The coefficient of variation on the estimates of D made from a peak that is broad in the diffusion dimension of a 2-dimensional (DOSY) map, is routinely higher from a fast-diffusion pulse sequence. In the method of Morris et al. [16] errors arise from the presence of a small extra amount of signal that survives the pulse train due to an insufficient absolute difference in gradient area between the two last gradient pulses. This undesired residual signal increases the signal intensities above what is expected at the higher magnitudes of the field gradient pulses. This results in a smaller estimate of the apparent D. On the other hand, for the pulse sequences that use a CPMG RF-pulse train [17,18], an imperfect 180° refocusing pulse gives an error in the signal intensity that manifests itself as a faster decay and hence an artifactually high estimate of D.

For both of the CPMG-based methods, obviously the total acquisition time must be shorter than the T 2 of the nuclei of interest. Unfortunately, in most cases, this condition can not be satisfied. The method of Van Gelderen et al. [19] allows the capture of chemical shift information when a large number of points are sampled in each gradient echo but the subsequent data processing is long and complex: both the chemical shift (off-set frequency) and gradient-induced evolution evolve in the opposite way. And, the data set from each echo must be processed separately.

In the CPMG methods the magnetic field gradient pulses must be switched on and off very rapidly, thus promoting eddy currents in conducting parts of the NMR probe. These give rise to distortion of spectral line shapes and lead to artifacts in the estimates of D.

The Difftrain pulse sequence has two drawbacks: The first is the requirement that the sample has long relaxation times, and there must be plenty of samples with a high S/N. The second, is the need to acquire two transients to estimate D: the first transient is recorded without the magnetic field gradient on to record the signal decay due to T 1 ; while the second one is recorded with gradients to measure both diffusion and relaxation effects.

Limited gradient pulse methods

The estimates of D obtained with the one-dimensional DOSY methods are often lower than those estimated with the more conventional PGSE methods. In performing the experiments the spectroscopist should have this consideration in mind. In order to have a coefficient of variation in the estimate of D of ~1.5%, the S/N ratio should be greater than ~100. Because the CPMG-diffusion methods are based on analyzing diffusion-broadened spectral peaks, the likelihood of peak-overlap is increased; therefore fitting of the requisite lineshape function to the data can be problematical. To apply the methods to estimate the D of solutes in mixtures requires well-resolved peaks as a precondition.

For both the MMME and Burst pulse sequences, a long period is used for the evolution of the magnetization of the system, and this occurs in the presence of magnetic field gradients. This is especially true of MMME (Fig. 4) in which the gradients are on for the full duration of the pulse sequence. Even if the gradient magnitudes are only a few gauss per centimeter, the total experimental time is critical. The manufacturers recommend using gradient pulses that are on for less than 10 ms at the maximum current. If we consider a linear relationship between current and g, the maximum experimental time obtained by using only 1% of the maximum current is 1 second. Hence, this duration determines the maximum duration of the MMME pulse sequence.

The final inconvenience lies in the restricted domain of parameter values in which the equation that is used for estimating D from the data remains valid. For the Fast-CRAZED method, Eq. ( 18) is valid only in regimes of highly diffusion-attenuated signals. This is generally found in systems that contain small molecules in solution or in the gas phase. Furthermore, the number of transients that are needed to retrieve the parameters to fit with Eq. ( 18) increases with the number of components. Thus it appears that fastdiffusion experiments have only been applied, at most, with binary mixtures with each component being 'non dilute'. And with the Burst pulse sequence the total DANTE angle is made less than 30° to improve the veracity of the estimate of D. With the latter, and indeed all the pulse sequences discussed here, a significant amount of effort must be expended on trial experiments that are used to optimize performance on each particular NMR spectrometer.

Conclusions

A knowledge of the diffusion coefficients of solvent and solutes in mixtures of various origins is valuable for predicting the physical and chemically-reactive properties of the system. If systems of interest evolve rapidly on the time scale of conventional PGSE NMR diffusion experiments then the pulse sequences described here might be of value.

There are principally two different classes of NMR fast-diffusion experiments. The first records the signal as a function of the magnitude of the magnetic field gradients that are used in the experiment. To speed up signal acquisition various pulse sequences have been designed to minimize the extent of phase cycling; or to record a series of spin-echo signals from a range of magnitudes of the magnetic field gradients.

The second class of experiments uses only one or two magnetic field gradient magnitudes. To estimate D, spatial information is encoded by using a different approach from that commonly used in, say, the Stejskal-Tanner experiment [6]. Each method is based on a different physical phenomenon. The first uses a DANTE RF-pulse train, while the second pulse sequence achieves in one transient the sampling of many different magnetization-coherence pathways. The third method uses a distant dipolar field (DDF) and both adiabatic and gradient pulses are combined to give spectra from which D is estimated.

Each of the methods described presents its own limitations. None of the pulse sequences have been used to study restricted diffusion for which long diffusion times are usually needed. Systems in which there is convection pose another problem, as none of the methods are convection compensated. Some of the methods record a train of spin-echoes, and chemical shift information is lost or is barely accessible, thus limiting the application of the methods to only simple mixtures of solutes.

Overall the one-dimensional DOSY method appears to be the method of choice for the (biological) systems which we study [3]. Currently, this method is being developed in our laboratory to study rapidly evolving systems such as the morphological changes in the human red blood cell, its membrane flickering, and the variation of D of guest molecules inside an extended support such as gelatin that undergoes changes in length [47,48].

Fig. 1 :

 1 Fig. 1: PGSE pulse sequences that use composite pulses. A, classical BPPLED sequence B, the pulse sequence designed by Morris et al. [16] that uses unequal gradient pulses in each bipolar pair. Additional gradient pulses can be used in B to refocus the lock signal and to dephase unwanted coherences. Spoil gradients can also be added to the BPPLED pulse sequence. RF denotes the

Fig. 2 :

 2 Fig.2: CPMG-based PGSE methods for measuring D in a single multiple-echo transient. A, the PGMSE method[17]; and B, the method of Chandrakumar et al.[18]. For A the gradient amplitude is increased systematically, while for B it is constant. The notation is identical to that in Fig.1

Fig. 3 :

 3 Fig.3: The Burst pulse sequence modified to measure D[24]. On the upper right of the diagram each arrow represents an echo; d 1 is the delay between the end of the last pulse and switching off the gradient; d 2 is the period between the last RF pulse and switching the read gradient on; τ is the period between the midpoints of two consecutive RF pulses. For other notation, see Fig.1.

Fig. 4 :

 4 Fig. 4: MMME4 pulse sequence [27] used to measure D in a single transient. RF, Gz and g are defined in Fig. 1. τ 1 is chosen and then τ 2 and τ 3 are calculated using Eq. (15).

Fig. 6 :

 6 Fig.6: One-dimensional DOSY pulse sequence[44]. Two negative gradient pulses are added to reduce eddy currents. t p is the adiabatic pulse duration, G d and G r are, respectively, the intensity of the diffusion and read gradients, and Δ is the diffusion time.

Fig. 7 :

 7 Fig. 7: One-dimensional DOSY. Effect of the first 90 o hard pulse and then the adiabatic pulse and diffusion gradient, as a function of the position of spins (spin isochromats) in the sample. The time scale is that of the adiabatic RF pulse of duration t p .

Table 2 :

 2 Number of coherence pathways recorded for a pulse sequence incorporating from 2 to 7 RF pulses[26].

	Number of RF pulses	2	3	4	5	6	7
	Number of coherence pathways	2	5	14	41	122	365

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors) Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007 © 2007, G. Pages

Acknowledgements

The work was supported by a Discovery Grant from the Australian Research Council to PWK.


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Jacques Wéry

Closing the yield gap under uncertain weather: risk analysis of cropping system intensification in family agriculture of the tropics

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Closing the yield gap under uncertain weather: risk analysis of cropping system intensification in family agriculture of the tropics.

Affholder F. 1 , Madeira Netto J. 2 , Scopel E. 1 , Wery J. 3

1 : CIRAD-URSCA, 2 : EMBRAPA, 3 : Montpellier Supagro -UMR System. affholder@cirad.fr How far do risks related to rainfall inter-annual variability limit the shift from extensive to intensive farming systems in family agriculture of the tropics? We explored this question in the Brazilian cerrado, where high average rainfall (up to 1500mm y -1 ) but large inter-annual variability of weather translates into strongly varying annual crop performance. While many farms maintain extensive farming and cropping systems (CS), many others have shifted to intensive dairy systems within a decade, with associated intensification of maize production as an essential step [START_REF] Bainville | Les transformations de l'agriculture familiale de la commune de Silvânia : une petite révolution agricole dans les Cerrados brésiliens[END_REF]. The study reported here aimed at assessing maize yield inter-annual variability under the range of environments and CS observed in the region to better identify biophysical constraints and management tactics to help close the yield gap. The latter is defined here as the gap between yields obtained by farmers and the potential yield that would be obtained in this region with the same cultivars in the absence of limiting factors other than radiation and temperature.

Materials and Methods

A previously performed regional agronomic diagnosis (RAD, [START_REF] Dore | Methodological progress in on-farm regional diagnosis[END_REF] provided a typology of maize CS across environments (E), corresponding to levels of yield, and with the key interactions between E and CS that explain yield variations. A version of the Stics crop model [START_REF] Brisson | An overview of the crop model STICS[END_REF] was elaborated specifically for Family Agriculture in Central Brazil (Stics_facb). It accounted for these key interactions and reproduced the variations of yield and yield gap across CS and E as observed in the network of farmers' fields used for the RAD [START_REF] Affholder | Diagnosis of the productivity gap using a crop model. Methodology and case study of small-scale maize production in central Brazil[END_REF]. Such validation based on data collected in farmers' fields showed that the model was able to account for interactions between crop establishment operations, soil N and water status, germination and emergence dynamics, stand density, and weeds competing with maize for light, water and nitrogen.

A simulation experiment was designed in which each class of the CS x E typology was simulated over a series of 21 years of weather data. We present here the most contrasted cases, corresponding to two CS types crossed with two E types consisting in a deep Latosol on plateau (DL) and a superficial Cambisol on sloping land with stony topsoil (SC). Both soils had high aluminum saturation rates. The CS included an extensive one (Ecs), practiced by subsistence-oriented farms, and an intensive CS (Ics), typical of the new dairy systems. In both CS, soil preparation was performed using tractors owned and managed by associations of 10 to 50 farmers, and weeding was horse drawn. In Ics, liming was practiced to maintain aluminum saturation rate under the threshold of toxicity to root growth. Sowing was horse drawn at 3 plants m -2 and tractor drawn with 5 plants m - 2 , and total N fertilization was 15 and 70kg ha -1 , respectively for Ecs and Ics. Two variants of Ics were considered, corresponding to two types of management of tractors under farmers' associations. In IcsR ("responsive"), soil tillage and sowing were performed at dates calculated as a function of soil water status, and with adequate seed depth and density, reflecting the possibility, offered by some associations, for each farmer to adjust the dates of operations and the settings of the machinery specifically for his fields. In IcsI ("imposed"), a random "noise" was used to modify the parameters relative to crop establishment operations, independently for each simulated year, reflecting the rules set by most associations, under which farmers were not permitted to change the settings of the drill or to operate the tractor by themselves, and they had no control over the tractor schedule.

Results and discussions

Simulations showed that yield gains from intensification were highly dependant on soil type and quality of crop establishment operations. The more constraining the soil or lack of flexibility in access to machinery, the smaller the yield response to increasing N fertilizer and sowing density (fig 1). On SC, there was a probability of circa 0.2 that yield is not increased when shifting from Ecs to IcsR. Details (not shown) of the simulation data indicated that this was due to a high frequency of water deficits on these soils with low water storage capacity. This probability increased to 0.5 when considering IcsI, i.e. poorly managed farm machinery due to delayed sowing (relative to last tillage operation) and inadequate seed depth jointly contributed to low stand density and consequently high weed pressure. There was a general awareness in the region that crop establishment was not adequately managed, and the RAD study per se had already pointed it out. Our modeling analysis brought the advantage of providing quantitative data on the yield increase that could be expected from improvements of crop management, such as better adapting the sequences of tillage and sowing operation to weather events and to the specific soil characteristics of each field. This advantage was critical to improve awareness of farmers' associations about the value of improving rules for use of collectively-owned machinery.

Conclusion

The study confirmed the value of extrapolating to series of climatic data the results of a RAD carried out over a limited number of years, as a way to better identify strategies for yield gap reduction in a small region, and to tailor such strategies to the specific conditions of farmers' fields. The study also suggested that biophysical constraints were severe enough to make maize intensification too risky to be viable at the field level in some cases. This could explain why certain farms did not adopt intensified dairy in the region. Such hypothesis would deserve to be further explored, which would require accounting for the way yield variations translate into variations in farm income.


Timothy J Larkin

Guilhem Pages

Allan M Torres

Philip W Kuchel
email: p.kuchel@mmb.usyd.edu.au

Rapid Acquisition of NMR Diffusion-Diffraction q-Space Plots from Erythrocytes with Varying Gradient Orientation

Keywords: erythrocyte, diffusion-diffraction, Monte-Carlo, q-space, rapid-measurement of diffusion, red blood cell. Abreviations: BSA, bovine serum albumin, NMR, nuclear magnetic resonance, PBS, phosphate buffered saline, PGSTE, pulsed field-gradient stimulated echo, RBC, red blood cell, SGP, short gradient pulse

The rapid-acquisition of q-space data from 1 H 2 O undergoing restricted diffusion in suspensions of red blood cells (RBCs) is made possible using a recently implemented pulse sequence, where the phase cycling of the radio-frequency pulses is reduced by using unbalanced pairs of bipolar magnetic field-gradient pulses. The q-space plots obtained with this pulse sequence show a shift in the position of the first diffraction minimum when compared to data from classical pulsed field gradient stimulated echo experiments. Diffusion simulations were used to investigate the effect of the additional delay introduced by the bipolar gradient pulses on the form of the q-space plots. RBCs of normal discocyte shape align with an external magnetic field, and the angular dependence of q-space spectra from suspensions of RBCs was examined using a linear combination of gradients applied along the y-and z-axes. The resulting q-space plots showed a gradual disappearance of the first diffraction minimum as the angle at which the gradients were applied was changed from 0° (along the z-axis) to ~40°, beyond which the q-space plots showed no diffraction features. These experimental results were also evident in Monte-Carlo random walk simulations of diffusion in RBCs with field-gradients applied at varying angles with respect to cells aligned with B 0 .

Introduction

The rapid-acquisition of diffusion-diffraction spectra from water diffusing in a suspension of red blood cells (RBCs) has been recently demonstrated [1], enabling the kinetic analysis of cell shape changes on the several-minute time scale. The method uses a pulse sequence developed by Pelta et al. [2], with a pair of unbalanced diffusion encoding bipolar gradient pulses (Fig. 1) to reduce the number of transients required per field gradient strength used. However, it was observed that by using this rapid-acquisition pulse sequence a shift occurred in the position of the diffraction minima to higher q values when compared with data from the classical pulsed field-gradient stimulated echo (PGSTE) diffusion experiment (Fig. 2).

Fig. 1: Rapid-acquisition diffusion pulse sequence of Pelta et al. [2] which uses unbalanced pairs of bipolar gradient pulses. RF refers to the time train of the radiofrequency pulses, while G refers to the field gradient pulse time train. Δ is the diffusion time, δ is the gradient duration, τ is the time between the midpoints of the antiphase field gradient pulses, g is the gradient amplitude, and α is the unbalancing factor. Diffusion-measuring nuclear magnetic resonance (NMR) spectroscopy encodes spatial information in the phase of nuclear spins through the application of magnetic field gradient pulses. Translational displacement between a pair of gradient pulses can then be measured, and hence diffusion coefficients calculated [3]. A geometry that restricts diffusion can result in spatial coherences of the magnetisation phases in the sample, giving rise to interference and diffraction-like effects in the data. These effects termed 'diffusion-coherence' [4,5] are readily observed by plotting normalised signal intensity versus the spatial wave-number vector q (units, m -1 ):

( )

1 2π γδ - = q g ( 1
)
where γ is the magnetogyric ratio, δ is the duration of the magnetic field-gradient pulses applied during the experiment, and g is the magnetic field-gradient vector. PGSTE experiments on the diffusion of water in suspensions of RBCs produce q-space plots with several well-defined features (Fig. 2); these arise because the cells align in the uniform magnetic field [5,6]. The first feature is a shoulder due to water 'hopping' between the pore-like spaces between the cells as it diffuses in the extracellular medium, followed by several diffraction minima arising from water undergoing restricted diffusion inside the cells [7]. The position of diffraction minima in q-space plots reports on the amplitude of the average propagator, and hence on the size of the restricting geometry; this average propagator is calculated by taking the Fourier transform of the q-space data [8,9].The features of the calculated average propagator can be enhanced by the use of a digital filter, followed by application of the second derivative to suppress both the strong initial signal, and the noise at high q-values [8]. Under conditions where the short gradient-pulse (SGP) approximation is not valid, the position of diffraction minima are dependent on the duration of the gradient pulses relative to the time interval over which diffusion is measured [10,11]. Therefore, we hypothesised that the shift in the position of the minima is introduced by using bipolar gradient pulses. To test this idea we conducted Monte-Carlo random walk simulations of diffusion in RBCs with varying separation of the bipolar pulses. We also aimed to explore a new application of the rapid-acquisition PGSTE sequence to generate q-space plots at varying angles across a sample of RBCs, and correlate these real data with simulations of diffusion in RBCs with variable orientation.

Experimental

Fresh blood was obtained from a healthy donor (TJL); the RBCs were isolated by centrifugation at 3000 × g for 10 min at 4°C, followed by aspiration of the plasma and buffy coat. The RBCs were then washed three times in isotonic phosphate buffered saline (PBS), by centrifugation as just described, and bubbled for 15 min with CO to render the haemoglobin diamagnetic. The cells were suspended in isotonic PBS containing 0.5% (w/v) bovine serum albumin (BSA) and 10 mM glucose, to give a final haematocrit (H t ) of 70%.

The NMR experiments were carried out on a Bruker Avance 500 WB spectrometer equipped with a micro-imaging probe capable of producing gradients of 3 T m -1 along the x-, y-and z-axes. The sample of RBCs was placed in a 5-mm Shigemi tube and height restricted to 1 cm to ensure that the sample was in the constant-gradient region of the probe. The experiments were performed at 25°C using the optimised fast diffusion pulse sequence [1]. The diffusion time Δ was 20 ms, δ was 4 ms, the gradient recovery delay (τδ/2) was 1 ms, and the gradient unbalancing factor α was set to 0.3. Gradients were applied at 0° (along the z-axis), 2°, 5°, 8°, 10°, 15°, 20°, 30°, 40°, 45°, 50°, 60°, 70°, 80°, and 90° (along the y-axis), using a combination of z-and y-gradients, with each q-space plot using 64 linearly-spaced gradient values.

Simulations of restricted Diffusion in RBCs

The Monte-Carlo random walk simulation of the diffusion of water in a suspension of RBCs was performed using programs written in a combination of Mathematica 6.0 (Wolfram, Champaign, IL) and C. Mathematica has a library of C functions for communication with an external C program, which allow for more efficient execution of the simulations. During each time step (t) of the simulation, the diffusing particle was moved one jump length (s), in the x-, y-and z-directions, where s was calculated from the diffusion coefficient (D) and the Einstein equation [12]:

2 s Dt =
(2) The direction of the jump along each axis, either ±s, was determined using a binary digit randomly generated with the function Random in Mathematica. The unit cell for the simulations was based on that described previously [13][14][15][16], consisting of a discocyte-shaped RBC inside a hexagonal prism (Fig. 3a), which allows the transmembrane exchange of water to be included by applying periodic boundary conditions at the prism surface. The RBC surface was described by the equation [17],

( ) ( ) ( ) 2 2 2 2 2 2 2 X Y Z Y Z X 0 P Q R + + + + + + = (3)
P, Q and R are constants related to the dimensions of the RBC, a the maximum thickness, b the minimum thickness, and d the main diameter (Fig 3b). Thus,

1 2 2 2 2 2 2 2 2 2 4 2 4 2 4 1 1 2 2 1 4 4 16 / d a d b P b a d b d Q P b b d d R P ⎛ ⎞⎛ ⎞ = - + - - ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎛ ⎞ = + - ⎜ ⎟ ⎝ ⎠ = - - (4)
In Eq. 3 X, Y and Z represent the modified Cartesian coordinates, , respectively, can be readily included in the simulations. Rotation of the cell about the y-axis allows the orientation of the cell with respect to B 0 to be varied, however, in all the simulations described here the cells were aligned with B 0 (by definition along the z-axis), therefore θ was fixed at 0°. While RBCs align with an external magnetic field, they will adopt random orientations in the x/y-plane, and rotating the cell about the z-axis allows this to be included in the simulations. The dimensions of the hexagonal prism, which was also rotatable about the y-and z-axes, were determined by fixing the volume of the prism (V prism ) with respect to the volume of the cell (V cell ),

prism cell t V V H = ( 6
)
where H t is the hematocrit content and V prism is given by, ( )

2 prism sin π / 3 V w h = (7)
As the RBC membrane is semi-permeable to water, there are two possible outcomes if a jump would take a particle across the cell boundary; the particle could transit the membrane or be reflected at the boundary. The probability (t p ) of a particle transitioning the membrane is calculated using the relationship [13],

/ / / p d in out in out t P s D = (8)
where P d is the membrane permeability to water, and s in/out and D in/out are the jump length and diffusion coefficient of the particle either inside or outside the RBC, respectively. Reflection of a particle at the boundary requires the solution of the intersection of a line with the surface describing the RBC. The coefficients of the resulting quartic equation were calculated using Mathematica (See Appendix). Each spin acquires an additional phase angle (φ) during the gradient pulses of the PGSTE experiment, dependent on its spatial position, and the magnitude of the magnetic field gradient pulse (g). This acquired phase angle is implemented in a Monte-Carlo random walk simulation by updating the value of φ after each jump using the equation, ( )

g gt ϕ γ = r ( 9
)
where t is the duration of a jump, and |r| is the magnitude of the position vector of the particle with respect to the applied gradient. During the first gradient pulse, phase is accumulated (positive g), while during the second gradient pulse g has the opposite sign.

The effect of the 180° RF pulses in the bipolar gradient pulses was ignored, with the first pair of pulses having positive g, and the second pair negative g. The accumulated phase angle at the end of the simulation is converted into a signal intensity (E) by taking the projection of the magnetisation vector onto the -y-axis, using the expression:

( ) ( ) cos E g g ϕ = (10)
The first set of simulations were based on the rapid-acquisition pulse sequence shown in Fig. 1, using the experimental parameters used to acquire real q-space plots from suspensions of RBCs [1]. Specifically, these were Δ = 20 ms, δ = 3 ms, α = 0.3, and gradient recovery delays of 0, 1 and 2 ms, to investigate the effect of this delay on the position of the first diffraction minimum. With a gradient-recovery delay of 0 ms the pulse sequence is equivalent to the classical PGSTE pulse sequence; but a simulation using unbalanced pairs of gradient pulses with δ = 3 ms was run to determine the effect on the shape of q-space plots. The last simulation in this set used the rapid-acquisition pulse sequence with the parameters that were used to conduct the real experiment in which the gradient angles were varied, namely Δ = 20 ms, δ = 4 ms, α = 0.3, and a gradient recovery delay of 1 ms. The second set of simulations involved changing the orientation of the magnetic fieldgradients across aligned RBCs, using a combination of gradients along the z-and y-axes. These simulations used the classical PGSTE pulse sequence with Δ = 20 ms, δ = 2 ms, and gradient orientations as described in the experimental section. For both sets of simulations the dimensions of the RBCs were a = 2.12 μm, b = 1 μm, and d = 8 μm, which gives a model RBC with the mean volume of human RBCs of 86 fL [START_REF] Dacie | Practical Haematology[END_REF]. The membrane permeability was set to the literature value for human RBCs at 25°C of 6.1 × 10 -5 ms -1 [START_REF] Benga | [END_REF]. The magnetogyric ratio of 1 H, 2.62752 × 10 8 rad s -1 T -1 was used, and the magnetic field gradient (g) was varied along the z-axis in a 96 step arithmetic series between 0 and 10 T m -1 . One million particle trajectories were used for each simulation, being divided into packets of 1000 trajectories. For each packet a rotation about the z-axis by the angle ψ was applied in steps of 0.18°, giving a total rotation of the cell of 180°, to generate the varying orientations expected in a real RBC sample. The signal attenuation data obtained from the simulations were interpolated in Mathematica to give 1000 points, using a shifting cubic spline prior to plotting the logarithmic ordinate to enhance visualisation of the coherence features.

The mean length of a chord along the z-axis for cells oriented at differing angles with respect to B 0 was determined by randomly generating 10 5 points inside the cell and calculating the length of the chord parallel to the z-axis though each point. For comparison purposes the mean length of a chord along the z-axis was also calculated for an oblate spheroid that was oriented at various angles to B 0 . The radii of the spheroid were 4 μm and 1.28 μm, giving a volume of ~86 fL.

Results and Discussion

The implementation of the rapid-measurement diffusion pulse sequence simulation was tested by performing simulations of free diffusion using the diffusion coefficient of water at 25°C of 2.30 × 10 -9 m 2 s -1 , Δ = 20 ms and several values of δ, α and τ. The fitting of the resultant signal decays using the equation of Pelta et al. [2] consistently yielded a diffusion coefficient of 2.39 × 10 -9 m 2 s -1 . Therefore the equation describing the signal decay was derived using the Mathematica program described by Momot and Kuchel [20], and found to be different from the one published in [2],

( ) ( ) { } 2 2 2 2 2 0 exp 1 12 1 2 g I D g I γ δ δ α τ α ⎡ ⎤ = - Δ- + - - ⎣ ⎦ (11)
When this new equation was fitted to the signal decays the expected diffusion coefficient of 2.30 × 10 -9 m 2 s -1 was obtained. q-Space plots for simulations of water diffusing in RBCs using the rapid-acquisition pulse sequence with gradient recovery times of 0, 1 and 2 ms are shown in Fig. 4. The curves show a systematic shift in the position of the first diffraction minimum towards higher q values as the delay is increased; this correlates well with the observed shift seen in real experimental data (Fig. 2). The position of the first diffraction minimum in the real data obtained using PGSTE and the rapid-acquisition pulse sequences were 1.9 × 10 5 m -1 and 2.0 × 10 5 m -1 (gradient recovery between bipolar pulses = 2 ms), respectively, while from the Monte-Carlo simulations with the same parameters, the minima appeared at 1.6 × 10 5 m -1 and 1.8 × 10 5 m -1 , respectively. This represents a difference in the apparent mean-displacement of water between the simulated and real data, with a larger apparent displacement in the simulated data. We conclude that the source of the difference was the assumption of an infinitely thin membrane used in the simulations. The position of the first minimum was identical in the simulated data obtained using the rapid-acquisition with no gradient-recovery delay, and the classical PGSTE with δ = 3ms, and therefore we concluded that the source of the shift in the experimental data was from the non-negligible diffusion that occurs during the delay introduced by the bipolar gradient pulses, and was not due to the unbalancing of the gradient pulses, per se.

Fig. 4: Simulated q-space plots of water diffusing in RBCs comparing results from the classical PGSTE pulse sequence with δ = 3 ms (blue) with results from the rapid-acquisition pulse sequence with δ = 3 ms, and gradient recovery delays (δ-τ) of 2 ms (black), 1 ms (red), and 0 ms (green). Shown in orange is a q-space plot from a simulation with δ = 4 ms and a gradient recovery delay of 1 ms (experimental conditions used for the gradient orientation study).

Fig. 5: q-Space plots from a real suspension of RBCs where magnetic field-gradients were applied at various angles across the cells using combinations of y-and z-axis gradients. Data were acquired for gradients at several angles between 0° and 30°, however, for clarity only the data for 15° are shown in this domain.

q-Space plots generated from real experiments with the rapid-acquisition sequence, where the angle of the gradients across a suspension of RBCs was varied, are shown in Fig. 5. As the angle of the gradients was increased there was a gradual broadening and shift in the position of the first diffraction minimum towards higher values of q, with the feature being lost in the region of 40-45°. The data obtained for gradients between 0° and 30° were overlapping, and for clarity only the data for 15° are shown in the graph. The gradual shift in the diffraction minimum in samples of RBCs contrasts with experimental data obtained from diffusion in orientated cylinders with large length-to-radius ratio [21,22], where the coherence features were lost after a rotation of only a few degrees. The corresponding data from simulations of diffusion performed on cells orientated with the magnetic field where the gradients were applied at various angles (Fig. 6), showed the loss of the first diffraction minimum for gradient orientations greater than ~45°. The shift in the position of the diffraction minimum was also more pronounced in the simulated plots, with the minima in the real experimental data being broader and shallower than those observed for simulated data. The broadening seen in the real experimental data is, in part, a consequence of the distribution of cell sizes that is normal for real blood samples [23], giving a range of mean displacements along the gradient axis. Such a variation of cell size was not considered in the simulations where all the cells had the same dimensions. Simulated q-space plots for RBCs oriented at 0°, 45° and 90° to the magnetic field have been reported previously [24], but the cells were approximated by a flat cylinder. Nevertheless there is good agreement with the position of the previous diffraction minima [24] with our simulated data presented here, our diffraction minima are significantly broader however. The additional rotation of the cell about the z-axis creates a range of cell profiles along the y-axis, broadening the minima in q-space plots from our simulated data.

The gradual shift and then disappearance of the first diffraction minimum observed in the experimental data can be explained by the curved profile of an RBC that is aligned with B 0 , as shown in Fig. 3b. Small changes in the orientation of the gradients (or of the cells) can be expected to result in only a small reduction in the profile of the cell that is projected along the gradient axis. This is demonstrated in Fig. 7, where the mean chordlength projected along the z-axis of an orientated cell, or oblate spheroid, is seen to decline slowly until an orientation of ~10°; this is followed by a rapid change to ~45°, after which the mean chord-length reaches a plateau. The plateau in mean chord-length for orientations greater than 45° corresponds well with the loss of the first diffraction minimum that was seen in the experimental and simulated data.

Conclusions

We investigated the source of the shift in the position of the diffraction minima in qspace plots from water undergoing restricted diffusion in RBC suspensions when using a rapid-acquisition PGSTE pulse sequence. The delay introduced into the pulse sequence by the use of bipolar gradient pulses is the cause. We also used this pulse sequence to acquire q-space plots from real experiments on RBC suspensions where the field gradients had various angles relative to B 0 , and compared these with simulated q-space plots from orientated RBCs. The demonstrated dependence of the form of a q-space plot on the choice of NMR pulse sequence is an important factor to be considered in deriving estimates of the main diameter of RBCs from q-space data.

Acknowledgements

The work was funded by a Discovery Project Grant from the Australian Research Council to PWK. TJL received a University of Sydney Postgraduate Award. Dr Bob Chapman is thanked for valuable NMR-software developments and discussions, and Professor Bill Price for generously providing access to the 500 MHz-wb NMR spectrometer at the University of Western Sydney.

Fig. 2 :

 2 Fig.2: Comparison of q-space plots obtained using the classical (blue) and rapid-acquisition PGSTE (red) pulse sequences from a suspension of human RBCs. The positions of the important coherence features are shown. We can note a shift in the minimum positions between both pulse sequences. The experimental details are described in the text.

  rotation of the body about the y-and z-axes by θ and ψ (Fig 3c)

Fig. 3 :

 3 Fig. 3: a Schematic of the unit cell used in the diffusion simulations, consisting of an RBC inside a hexagonal prism. The dimensions of the prism are shown with w being the width, and h the depth of the prism; b shows a cross section through the model RBC used for the simulations, a is the maximum thickness of the cell, b is the minimum thickness, and d is the main diameter; and c illustrates the definition of ψ, the rotation angle about the z-axis.

Fig. 6 :

 6 Fig.6: Simulated q-space plots for RBCs with variable gradient orientation angles relative to the aligned cells. The gradient angles and respective colours are identical to those in Fig 5.All curves were generated using the pulse-timing of the classical PGSTE pulse sequence and combinations of gradients along the y-and z-axes.

Fig. 7 .

 7 Fig. 7. Variation of the mean length of a chord in the z-direction of a model RBC (Fig. 3, upper points), and an oblate spheroid (lower points), with the angle of orientation (θ) of the cell with respect to B 0

Appendix

The following is taken from a Mathematica program used to derive the coefficients of the quartic equation describing the intersection of a line with the surface describing the RBC.

The intersection was determined by substitution of the coordinates of the points inside and outside the RBC surface, and then using the Mathematica function Solve to calculate the roots of the quartic. Substitution of the real root in the interval (0,1) into the parametric equation of the line then yielded the coordinates of the intersection point.


Denis Breuillé

Sabine Mercier

Isabelle Papet

Philippe Patureau Mirand

Christiane Obled

Nutritional method

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