Measurements (mean ± SD) were obtained in normal volunteers using diffusion tensor MRI. (With permission from Pierpaoli et al.^{41})
Standard Article
Methods and Applications of Diffusion MRI
Published in 2007
Published Online: 15 MAR 2007
DOI: 10.1002/9780470034590.emrstm0309
Copyright © 2013 John Wiley & Sons, Ltd
Book Title
eMagRes
Additional Information
How to Cite
Bihan, D. L. 2007. Methods and Applications of Diffusion MRI. eMagRes. .
Publication History
- Published Online: 15 MAR 2007
1 Introduction
- Top of page
- Introduction
- Diffusion MRI
- Diffusion in Biological Systems: Effects of Microdynamics and Microstructure
- Clinical Applications
- Conclusion
- Biographical Sketch
- Related Articles
- References
Diffusion is the process by which matter is transported from one part of a system to another as a result of random molecular motion, also called Brownian motion. This motion, of thermal origin, results in the macroscopic flux of different molecular species, which can be observed in nonuniform systems and is characterized by a diffusion coefficient. Classically, diffusion coefficients may be determined by measuring the concentration of molecular species at different times using either physical or chemical methods based on the classical first Fick Law.^{1} Such approaches rely on the introduction of a tracer in the medium, as similar as possible to the studied molecular species, and then monitoring the concentration of the tracer in the medium by chemical or radiotracer techniques.^{2} Microscopic displacements on the scale of millimeters can be seen with such tracers over diffusion times of minutes. Such tracer methods have been successfully applied to biological systems, such as the brain^{3} but are, of course, extremely invasive. An alternative in studying diffusion is to monitor the random walks of molecules. On a statistical scale, the diffusion reflects the meansquare distance traveled by molecules in a given interval of time (m^{2} s^{−1}). Diffusion NMR, which relies on these principles, is the only method today available to provide noninvasively such information on molecular displacements which occur over diffusion distances that extend largely beyond elementary molecular jumps, justifying the considerable success of diffusion NMR in physics and chemistry, and more recently in biology.^{4} Taking typical values for water diffusion ∼1 × 10^{−3} mm^{2} s^{−1}) and diffusion time (∼20 ms) achievable on conventional MRI equipment free water molecules diffuse over distances on the order of 6 µm, which is about the size of many tissue structures. NMR has, in principle, a displacement sensitivity of around 100 nm.^{5} Because of its noninvasive nature, it is especially suited to probing the molecular dynamics and structural information of biological systems, as well as transport processes. The recent combination of such principles with MRI^{6-8} represents a spectacular and somewhat unpredicted development in the field of medical sciences. Measuring molecular displacements in biological tissues in vivo may have enormous value, from the determination of the molecular organization in tissues to the emergency management of stroke patients or the monitoring of laser surgery. In this chapter, the basic principles of diffusion measurements with NMR will be presented, as well as the various methods that have been proposed to produce images of diffusion. Some technical considerations will be discussed, followed by a description of the effects on diffusion measurements of microdynamic and microstructure in biological tissues. Emphasis will be given to the difficulties encountered when implementing diffusion imaging and the potential of diffusion measurements for clinical applications.
2 Diffusion MRI
- Top of page
- Introduction
- Diffusion MRI
- Diffusion in Biological Systems: Effects of Microdynamics and Microstructure
- Clinical Applications
- Conclusion
- Biographical Sketch
- Related Articles
- References
Although the effects of diffusion on the NMR signal have been described very early,^{9, 10} most diffusion NMR studies started after the seminal work of Stejskal and Tanner, who introduced the bipolar pulse field gradient method (Figure 1).^{11} This approach not only provides much better control over what is actually measured but also simplifies the understanding of the encoding of the diffusion process in the NMR signal. The purpose of these gradient pulses, the duration and the separation of which are classically represented, respectively, as δ and Δ, is to label spins carried by diffusing molecules magnetically. To some extent, these labeled spins are like the endogenous tracers that can be monitored with classical diffusion measurement tracer methods. However, the behavior of these labeled spins in an NMR experiment leads to more peculiar and specific results that are unique to diffusion NMR, and diffusion data obtained with NMR may not always be directly comparable with those obtained by other means.
The first gradient pulse induces a phase shift φ_{1} of the spin transverse magnetization, which depends on the spin position z_{1}:
- (1)
where γ is the gyromagnetic ratio and G is the gradient strength (here, along the z axis). After the 180° rf pulse, φ_{1} is inverted to −φ_{1}. The second pulse will produce a phase shift φ_{2}:
- (2)
where z_{2} is the spin position during the second pulse. The net transverse magnetization Mxy is:
- (3)
where all relaxation effects have been incorporated into M_{0}.
Obviously for ‘static’ spins, z_{1} = z_{2} and the bipolar gradient pair has no effect. For ‘moving’ spins, however, there is a net dephasing that will depend on the spin history during the time interval Δ between the pulses. Indeed, we must now consider a population of spins, which may have different motion histories. This leads to an attenuation of the Mxy and the resulting signal. Assuming δ << Δ (negligible displacement during δ), these equations give a time-dependent part that represents the attenuation owing to diffusion, A:
- (4)
where P(z_{1}) is the probability for a labeled spin to be at initial position z_{1} and P(z_{2}, z_{1}, Δ) is the diffusion propagator, i.e., P(z_{2}, z_{1}, Δ)dz_{2} is the conditional probability to find a labeled spin initially at position z_{1} in position z_{2} after a time interval Δ. For a true, isotropic diffusion process:
- (5)
and P is Gaussian, at t = Δ:
- (6)
where D is the diffusion coefficient.
Combining Equations 1 and 2 gives:
- (7)
where the root mean square diffusion displacement 〈z^{2}〉 that occurred during the diffusion time Δ is easily interchangeable with D according to Einstein's equation, 〈z^{2}〉 = 2DT_{d}. Diffusion coefficients or root mean square diffusion distances are classically obtained by varying either G or δ and by measuring the slope of the semilogarithmic plot of the signal intensity versus (δG)^{2}. However, it is the diffusion path that the NMR experiment is actually sensitive to, not the diffusion coefficient. Therefore, in this simple bipolar gradient experiment, clear information on spin displacements can directly be obtained from the signal attenuation A, assuming P is Gaussian. This model is valid only for diffusion in an infinite and homogeneous medium. If diffusion is restricted or anisotropic (see below), the signal attenuation must be calculated using a more general formalism based on diffusion tensors.^{12}
2.1 Diffusion and the NMR Signal: Different Approaches
Almost any NMR sequence can be designed to measure diffusion.
2.1.1 Constant Field Gradient Spin Echo Method
In the presence of a simple constant linear gradient G_{0}, the echo attenuation is:^{9, 10}
- (8)
where TE is the echo time and n the echo number in a multiple echo experiment. This simple approach is difficult to combine with NMR imaging because the presence of a constant gradient during the rf pulses of the imaging sequence severely impairs the selected slice profile. In addition, the relaxation time T_{2} of the medium must not be too short to allow enough diffusion attenuation to occur during TE.
2.1.2 Bipolar Gradient Pulse Spin Echo Technique
The bipolar gradient pulse spin echo technique is, by far, the sequence most widely used for NMR diffusion measurements. In practice, as δ is usually not negligible compared with Δ, movement during the application of the gradient pulses can no longer be neglected. Taking into account spin diffusion during δ, Equation 7 becomes:^{11}
- (9)
The diffusion time can be formally defined as (Δ − δ/3), but its physical significance is clear only when δ is sufficiently short compared with Δ. Tissues with short T_{2} times require short TE values, which may not allow sufficiently long diffusion times to produce enough diffusion effects.
2.1.3 Stimulated Echo Technique
A stimulated echo is generated from a sequence consisting of three rf pulses separated by time intervals τ_{1} and τ_{2}. Gradient pulses must be inserted within the first and the third periods of the stimulated echo sequence (Figure 2).
The diffusion time now includes τ_{2} and can be much longer than with the spin echo sequence without T_{2}-related signal decay because only longitudinal relaxation occurs during this time interval. The Stejskal–Tanner relation [Equation 9] still applies, provided that the period τ_{2} is included in Δ.^{13} The longer diffusion time is useful for studying very slow diffusion rates or for compensating for the unavailability of large gradients.^{7} Unfortunately, the signal is reduced by 50% compared with the spin echo signal.
2.1.4 Gradient Echo Technique
The effect of diffusion on the amplitude of a gradient echo formed by a bipolar gradient pulse pair of reversed polarity does not differ from that of a spin echo sequence.^{14} However, if the gradient echo is part of a steady-state free precession (SSFP) sequence, where some degree of phase coherence is propagated throughout successive cycles, multiple echo paths with different diffusion times and different diffusion weighting must be considered.^{15} In MRI, the contrast-enhanced (CE-Fast) scheme has been proposed.^{16-18} Unfortunately, this sequence remains very sensitive to motion artifacts.^{19} Moreover, the effects of diffusion and relaxation are mingled so that diffusion measurements are always contaminated to some degree by relaxation effects.^{20}
2.1.5 Diffusion Measurements with B_{1} Field Gradients
Diffusion measurements can also be achieved by means of the rf (B_{1}) field produced by a NMR rf coil oriented perpendicularly to the main transmit/receive NMR coil.^{21-24} With rf gradients, extremely short switching times can be achieved since there are no eddy currents. Furthermore, substantial gradient strength may be produced from the rf transmitter, allowing measurements of very low diffusion coefficients.
2.2 Combining Diffusion NMR with MRI
2.2.1 Principles
Diffusion-weighted (DW) images are obtained by inserting gradient pulses into any NMR imaging sequence. For quantification, it is necessary to determine the degree of diffusion weighting of the sequence. When inserting gradient pulses for diffusion, however, the combination of the imaging and the diffusion gradient pulses produces cross-terms. These cross-terms, which depend on the sequence design, may also lead to significant diffusion-related attenuation effects,^{14, 25} which must be taken into account; consequently the Stejskal–Tanner relation [Equation 9] is incorrect in most cases. A more general formalism must then be developed to solve the Bloch–Torrey equation.^{26} Solutions may become analytically complex,^{27} and it has been suggested that all gradient effects should be combined in a term generally known as the ‘b factor’:^{14, 28}
- (10)
The signal attenuation for, unrestricted, isotropic diffusion is then reduced to a simple, convenient expression:
- (11)
This b factor characterizes the diffusion sensitivity of the sequence, as TE characterizes the degree of T_{2} weighting of a spin echo sequence. Without additional gradients, typical spin echo imaging sequences have intrinsically very low b values, typically less than 1 s mm^{−2}; consequently diffusion effects are negligible (for pure water at room temperature the attenuation is less than 1%).^{14, 29}
As the raw signal depends not only on diffusion but also on other MR parameters, such as T_{1} or T_{2}, a quantitative determination of the diffusion coefficient requires at least two images acquired with different diffusion sensitivities (i.e., b factor values), but better accuracy is obtained when more than two images are used (see below). With the b factor values associated with each image, it is possible to compute diffusion images, i.e., images where the diffusion coefficient is determined for each pixel. The computation of such diffusion images is obtained by fitting the signal intensity of each pixel obtained for different b factor values with Equation 11^{4, 6, 7, 14, 28} using a regression analysis (Figure 3).
On diffusion images, diffusion values are displayed using a gray scale where brightness corresponds to high, fast diffusion and darkness to low, slow diffusion (Figure 4). To save on acquisition and processing time, some investigators have proposed to limit the diffusion studies to the analysis of the raw images obtained with some degree of diffusion sensitivity or weighting. For this reason, these images are called DW images. Contrast in these images is opposite to that found in true diffusion images: regions with high diffusion have more pronounced MRI signal attenuation and appear dark, while regions with low diffusion appear bright. The DW image may be convenient to use, but its content is affected by many parameters other than diffusion, as it is also usually strongly T_{1} and T_{2} weighted. As these parameters may not have the same behavior as diffusion, variations in image intensity may be difficult to interpret. For instance, acute stroke lesions have decreased diffusion (see below) and appear bright on DW images, while subacute infarcted areas, which are associated with increased diffusion, may also appear bright on DW images because T_{2} has increased (T_{2} ‘shine-through’ effect). DW images may be convenient to use in clinical practice, but, whenever possible, absolute diffusion images should be preferred.
2.2.2 Diffusion MRI Sequences
Using the spin echo sequence, it is possible to vary the strength or the duration of the diffusion-sensitizing gradients or their direction in order to enhance anisotropic diffusion effects^{14} (see below). The spin echo two-dimensional Fourier (2DFT) method is certainly the simplest to implement.^{6, 8, 28, 29} Other sequences^{4} that have been investigated are stimulated echo sequences,^{7} line-integral projection reconstruction,^{30-32} variants of the SSFP technique; and ‘turbo’ sequences such as HASTE,^{33} TurboFlash^{34} and fast spin echo.^{35} These schemes have been suggested to overcome, at least partially, some of the problems encountered with the spin echo 2DFT method. However, echo planar imaging (EPI) remains generally the imaging method of choice.^{36-39} With EPI, the entire set of echoes needed to form an image is collected within a single acquisition period (single shot) of 25–100 ms. This is obtained by switching the echo signal formation in a train of gradient echoes by means of a large gradient in which polarity is very rapidly inverted as many times as is required to achieve the desired image resolution.^{40} EPI may easily be sensitized to diffusion (Figure 5).^{36-39}
Sensitization consists of providing a pair of large compensated gradients for a period of time before rapid gradient switching and data acquisition. The refocusing may be achieved either by simply reversing the polarity of the gradient halfway through the period over which it is applied or by inserting a 180° rf refocusing pulse at the midpoint without reversing the gradient polarity.
With EPI, motion artifacts are virtually eliminated. The accuracy of assessments of diffusion achieved with EPI is generally extremely good, as many images differently sensitized to diffusion can be generated because of the very short acquisition time (typically less than 100 ms). The EPI technique is the method of choice for in vivo diffusion imaging, although it is very vulnerable to susceptibility artifacts, which are responsible for image distortion or signal dropout and to chemical shift artifacts, which require efficient fat suppression. Despite this, diffusion-EPI has become widely available on many clinical scanners and has been successfully used for measuring diffusion of water in the human brain in volunteers (Figure 4)^{37, 41} and patients (see below).
Recently, alternative fast acquisition schemes, such as BURST^{42, 43} and spiral MRI,^{44} have been proposed to overcome some EPI artifacts, such as ghosting or susceptibility artifacts.
2.2.3 Sequence Optimization
Although the b factor can be exactly determined for any pulse sequence,^{27, 45} the diffusion time is impossible to define accurately for a MR sequence where gradient pulses of finite duration are scattered all over the pulse sequence. A mixture of diffusion-related events with differently weighted contributions are obtained depending on their sequence in time with respect to the many pulse gradients. That is why it has been suggested that the parameter derived from the NMR signal attenuation using Equation 7 should be called an ‘apparent diffusion coefficient’ (ADC).^{28} MR sequences should be designed to minimize these problems, i.e., by placing gradient pulses to get as close as possible to the simple Stejskal–Tanner bipolar pulse sequence. However, the use of the ADC term remains justified, as biological tissues usually differ significantly from the ideal ‘free, unlimited, isotropic’ medium.
Another issue is to optimize the values and the number of b factors to be used. A simple calculation^{4} shows that the best accuracy for the diffusion coefficient is obtained from a set of two acquisitions if the two b factors, b_{1} and b_{2} differ by about 1/D.^{46} In the brain, this translates to (b_{2} − b_{1}) ∼ 1000 to 1500 s mm^{−2}. If more than two acquisitions are to be used, error propagation theory^{47} shows that it is better to accumulate n_{1} and n_{2} measurements at each of the small and large b factors, b_{1} and b_{2}, than to use a range of b factors. The accuracy dD/D can be obtained from the raw image signal-to-noise ratio (SNR):
- (12)
Acquisition performed with linear sets of b value ranges remains, however, extremely useful to validate the sequence/hardware quality and to assess the nature of the diffusion process or the presence of multiple diffusion compartments as it allows the exponential diffusion attenuation to be visualized according to Equation 11.
2.3 Experimental Considerations in Diffusion MRI
2.3.1 Gradient Hardware
Since the minimum length of molecular diffusion paths detectable with gradient-pulsed NMR is primarily determined by the intensity of the gradient pulses, hardware must be capable of providing stable gradients of the utmost intensity.^{48} This requirement may be extremely challenging when considering whole-body instruments designed for clinical studies. The lack of gradient power is usually compensated by using somewhat long gradient pulse widths; as a result, the classical condition δ << Δ is not satisfied. The case of such finite pulse widths is significantly more difficult to treat and to interpret compared with the situation with δ-sharp pulses. If the duration of the pulses is ignored, significant underestimation of the diffusion distances may result.^{49} For instance, to achieve at least a 20% signal attenuation and limit signal loss from transverse relaxation, the minimum diffusion coefficient D_{min} measurable is 37 × 10^{−3} mm^{2} s^{−1} using δ = Δ/10, Δ = T_{2} = 100 ms, and G_{max} = 10 mT m^{−1}, which is more than 16 times the diffusion coefficient of free water!^{14} Maximum sensitivity is achieved when δ = Δ (equivalent to a constant gradient) and Δ = T_{2}/2. Then D_{min} is 0.4 × 10^{−3} mm^{2}s^{−1}. There are partial solutions to this problem, such as using stimulated echoes to increase the effective diffusion time without penalizing the signal by T_{2} relaxation effects,^{13} or, better, by building high-performance gradient coils (e.g., up to 50, 100 or even 200 mT m^{−1}). Safety should not be too much a concern, considering that slew rates will remain low, as there is no need to switch such gradient coils very rapidly (which also limits eddy currents). Hence, diffusion effects during the application of the gradient pulses can no longer be neglected, although their contribution to the overall signal attenuation does not scale the same way with time as for diffusion occurring between pulses. The diffusion time then becomes more difficult to define, although (Δ − δ/3) is often taken instead of Δ as the efficient diffusion time.^{11}
Another concern is that any mismatch between the diffusion-sensitizing gradient pulses may cause artifactual signal losses through an improper spin rephasing. Two major sources of problem are commonly seen. One is gradient instability, which may arise when gradient amplifiers are driven hard for fast switching of large gradient intensities. Variations from shot to shot of the bipolar gradient balance result in widely distributed ghost artifacts. While such artifacts do not occur when a single-shot technique such as EPI is used, it is still essential to have high-quality gradient amplifiers with some reserve capacity for accurate diffusion imaging. The second problem results from eddy currents generated mainly in the cryostat when switching large gradient pulses rapidly. Eddy currents may also be a major cause of image distortion and misregistration between images obtained with different b values, leading to ADC miscalculation. The best solution to this problem is undoubtedly to remove eddy currents at the source by using actively shielded gradient coils, which have no fringe fields and, therefore, do not generate eddy currents. The use of a gradient coil of small dimensions, which remains at a fair distance from the magnet core and with which it is easier to generate large gradient amplitudes, is certainly an attracting alternative.^{21} If residual eddy current effects persist, other approaches can be combined.^{50-52}
In summary, there is clearly a need for very-high-performance gradient coils. With the hardware that is now becoming available,^{53, 54} shorter diffusion times or larger b values can be reached while maintaining high SNR, giving access to slow diffusion components (e.g., intracellular water, metabolites).
2.3.2 Motion Artifacts
As the sequences used are deliberately sensitized to motion by the addition of large gradients, a major problem occurring with in vivo imaging of diffusion arises from motion of the object. Artifacts result from discontinuities that occur between successive cycles of an imaging sequence. Results of such temporal incoherence are commonly visible as ‘ghosts’ along the phase-encoding direction. These ghosts are particularly intense in the presence of the diffusion gradients and render the diffusion measurements meaningless. Cardiac gating has been used to mitigate this problem, but even this motion is not strictly cyclic. It is difficult to compensate diffusion imaging sequences for motion, since the use of successive bipolar gradient pulses considerably reduces the value of the b factor.^{4, 14} Chenevert et al. have suggested eliminating any phase encoding from the acquisition by sacrificing one dimension of the image, which is then reduced to a single line.^{55, 56} Ultimately, however, the best way to avoid motion artifacts is to use a single-shot technique, such as EPI, and to secure patients comfortably within the magnet using cushions or inflating devices. In spite of this, motion artifacts may persist, especially when multiple shots are used, as with diffusion spectroscopy (see below) or segmented EPI. Motion effects can be estimated by monitoring the phase of the signal and corrected.^{57, 58} Navigator echoes have been used with some success by several investigators.^{59-62}
2.3.3 Background Inhomogeneities
In addition to the magnetic field gradient pulses, residual field inhomogeneities arising from the imperfect shimming of the magnet and from inhomogeneities in the sample must be considered. These inhomogeneities result from susceptibility variations, especially at high field, and may not be negligible.^{63} By adding a constant, linear gradient G_{0}, it is clear that two contributions of the residual gradients coexist.^{11} One is a unique contribution (term in ); the other results from a cross-term between the residual and the applied gradients (GG_{0} term).^{14} While the first contribution can easily be eliminated from a series study where only G is varied, the second term is more difficult to handle. There are however, diffusion pulse sequences that have been designed to reduce significantly or to eliminate effects of background gradients.^{64-70} Most of them are based on multiple rf pulses. For instance, cross-terms can be completely eliminated by alternating the gradient pulses between the successive 180° rf pulses of a quadrupole spin echo sequence.^{71}
If cross-terms with the background gradients are not eliminated, the logarithm of A is no longer linear with the b factor and the ADC overestimates the true diffusion coefficient.^{25} The ADC will, furthermore, appear as a function of the diffusion time. The examination of the linearity of the relationship between log A and b is therefore, the first step in interpreting any NMR diffusion study in biological tissues, although there are several possible causes that would explain a deviation from linearity. Notably, similar nonlinear behavior will be seen if the b factor is miscalculated, i.e., cross-terms coming from all applied (diffusion and imaging) gradient pulses are not appropriately considered in the calculation.^{27} The b factor must be carefully calculated before the presence of background gradients can be assessed.
However, finding a linear plot for log A versus b does not exclude background gradients. First, the effect may be small. Second, the interpretation of the data may become very complicated when gradients are nonuniform or nonlinear and slowly varying in space, as in the case of capillaries filled with materials with a high susceptibility constant (e.g. contrast agent). Simulations and experiments have shown, in this case, that the ADC is decreased and that the parameter to consider is the variance of the internal gradients.^{72, 73}
The ultimate test to evaluate background gradients, however, is to vary TE while δ and Δ are kept constant. The measured ADC should normally be independent of TE. Otherwise, the contribution of background gradients must be suspected. Furthermore, as self-induced gradients may depend on the orientation of the sample with respect to the direction of B_{0}, they may cause isotropic diffusion to appear anisotropic.^{65} In this case also, the measured diffusion coefficient depends on the diffusion time.
Once it is established that background gradients are present, it may be desirable to estimate them from the attenuation behavior in order to obtain information on the structure of the medium. For instance, in the ideal case where the internal gradients result from inhomogeneities caused by the presence of closely packed particles, a rough estimation of the mean size R, of these particles can be obtained from:^{74}
- (13)
where χ_{0} is the magnetic susceptibility constant. Furthermore, these susceptibility effects may be of biological interest,^{75} as variations in the oxygenation level in brain capillaries and small vessels following brain activity have been shown to induce susceptibility effects measurable by T_{2}*.^{76, 77} A significant drop of the measured ADC has also been observed during status epilepticus,^{78} cortical electroshocks^{79} and spreading depression.^{80-82} In all cases, large increases in oxygenated blood flow could result in important reductions in local internal gradients and apparent variations to diffusion, although cytotoxic edema and cell swelling associated with energy failure has been mainly evoked as the reason for this diffusion drop.
2.4 Other Approaches to Diffusion
2.4.1 Diffusion Tensor Imaging
Diffusion is a three-dimensional process. However, the molecular mobility may not be the same in all directions. This anisotropy may be a result of the physical arrangement of the medium (liquid crystal) or the presence of obstacles that limit diffusion in some directions. The result is that the ADC appears different when pulse gradients are imposed in different directions; this has been observed in muscle^{83} and, more recently, in brain white matter.^{37, 56, 84}
The proper way to study anisotropic diffusion is to consider the diffusion tensor.^{1, 11} Diffusion is no longer characterized by a single scalar coefficient but by a tensor D, which fully describes molecular mobility along each axis and the correlation between these axes:
- (14)
In the reference frame [x′, y′, z′] that coincides with the principal or main directions of diffusivity, the off-diagonal terms do not exist and the tensor is reduced only to its diagonal terms D_{x′x′}, D_{y′y′}, D_{z′z′}, which represent molecular mobility along axes x′, y′, and z′, respectively. The echo attenuation then becomes:
- (15)
where b_{ii} are the elements of the b matrix (which now replaces the b factor) expressed in the coordinates of this reference frame.
In practice, measurements are made in the reference frame [x, y, z] of the gradients, which usually does not coincide with that of the tissue. Therefore, one must also consider the coupling of the nondiagonal elements b_{ij} of the b matrix with the nondiagonal terms D_{ij}(i ≠ j), of the diffusion tensor (now expressed in the gradient frame), which reflect correlation between molecular displacements in perpendicular directions:^{85}
- (16)
Calculation of the value of b may quickly become very complicated when many gradient pulses are used,^{45} however, the full determination of the diffusion tensor is necessary to assess anisotropic diffusion properly and fully.
To determine the diffusion tensor, DW images are collected along several gradient directions. As the diffusion tensor must be symmetric, measurements along only six directions are mandatory (instead of nine), along with an image acquired without diffusion sensitivity (b = 0). A typical set of gradient combinations that preserves uniform space sampling and similar b values along each direction is as follows (coefficients for gradient pulses along the x, y, z axes, normalized to a given amplitude, G):
This minimal set of images may be repeated for averaging, as the SNR may be low^{12, 85} In the case of ‘axial symmetry’, only four directions are necessary (tetrahedral encoding),^{86} as suggested for the spine.^{87} The acquisition time and the number of images to process is then reduced, but efforts should be made to collect data along as many directions in space as possible to avoid sampling direction biases for applications such as the determination of fiber orientation and gain in SNR (see below).^{88} A second step is to estimate the D_{ij} values from the set of diffusion/orientation-weighted images, generally by multiple linear regression, using Equation 23 below. The final step is to determine the main direction of diffusivities in each voxel and the diffusion values associated with these directions. This is equivalent to determine the reference frame [x′, y′, z′] where the off-diagonal terms of D are null. This ‘diagonalization’ of the diffusion tensor provides ‘eigen-vectors’ and ‘eigen-values’, λ, which correspond, respectively, to the main diffusion directions and associated diffusivities,^{12, 85} (the eigen-diffusivities λ are not to be confused with the tortuosity factors; see below).
As it is difficult to display tensor data with images (multiple images would be necessary), the use of ‘diffusion ellipsoids’ has been proposed.^{12, 85} Ellipsoids are a three-dimensional representation of the diffusion distance covered in space by molecules in a given diffusion time (T_{d}). These ellipsoids, which can be displayed for each voxel of the image, are easily calculated from the eigen-diffusivities λ_{1}, λ_{2}, and λ_{3} (corresponding to D_{x′x′}, D_{y′y′}, and D_{z′z′}):
- (17)
where x′, y′, and z′ refer to the frame of the main diffusion direction of the tensor. These eigen-diffusivities represent the unidimensional diffusion coefficients in the main directions of diffusivity of the medium. The main axis of the ellipsoid gives the main diffusion direction in the voxel (coinciding with the direction of the fibers), while the eccentricity of the ellipsoid provides information about the degree of anisotropy and its symmetry (isotropic diffusion would be seen as a sphere). The length of the ellipsoids in any direction in space is given by the diffusion distance covered in this direction. In other words, the ellipsoid can also be seen as the three-dimensional surface of constant mean squared displacement of the diffusing molecules.
Recent work has demonstrated the feasibility of diffusion tensor imaging (DTI) in animal models, as well as in the human brain.^{41, 86, 89, 90}
2.4.2 The Diffusion Propagator and q-Space Imaging
In the case where diffusion deviates from the Gaussian model, as with restricted diffusion (see below), it becomes extremely difficult to get any meaningful information on molecular displacements from the measured NMR signals. The distribution of the propagator is no longer Gaussian; consequently the value obtained for the diffusion coefficient using Equation 11 may not necessarily properly reflect tissue structure of properties. Diffusion coefficients may become, in these conditions, meaningless. The main difficulty is that the medium structure is generally not known in detail and modeling is required. In particular, one must be cautious in the use of Fick's Law and derived relations. The probability distribution driven by the diffusion process (Equation 6] may now significantly deviate from a Gaussian distribution; as a result the relationship between A and b is no longer exponential as would be expected from Equation 7. A successful analysis should, therefore, start from a reformulation of the probability distribution, taking into account the medium structure and particular boundary or limit conditions. Hence, more useful information may be obtained by directly inferring molecular displacements from the MR signal attenuation without going through a diffusion coefficient formulation. Molecular displacement profiles, or more exactly their probability density distribution, can be obtained using q-space imaging.^{91, 92} This does not differ, in principle, from diffusion imaging. Assuming P(z_{1}) is homogeneous throughout the medium, Equation 4 can be rewritten by substituting the relative diffusion displacement z for (z_{2}−z_{1}) and by calling q the quantity (γδG) as:
- (18)
It now appears that P(z, Δ) is just the Fourier transform of the attenuation A(q, Δ). By inverting Equation 15, direct and unequivocal data on molecular displacements can be obtained. In practice, data are collected for a series of different gradient strengths and an inverse Fourier transform is applied to the corresponding set of signal attenuation values. The experiment can be repeated with different values of Δ in order to characterize the probability distribution for different diffusion times.^{93} With q-space imaging, a more comprehensive picture of the system under study is obtained than when using a unique ADC.^{92} It then becomes straightforward to verify whether P(z, Δ) is a Gaussian distribution and to determine the value of the diffusion time when the value of P(z, Δ) may start to deviate from a Gaussian behavior, suggesting that molecules have reached another structural order in the medium, or to obtain the molecular displacement distribution in different conditions. Localized q-space imaging has been used, for instance, to obtain water-displacement profiles from normal and ischemic mouse brain, showing an increase of the fraction of molecules undergoing displacements of less than 10 µm.^{94} Such findings are very useful for examining the microstructural changes occuring during acute brain ischemia (see below).
2.4.3 Diffusion Spectroscopy
Water may not be the most suitable molecule for diffusion measurements in biological tissues because of its ubiquity and the permeability of most tissue interfaces, such as membranes, to water molecules. Consequently, diffusion measurements of larger molecules that are more specific to a tissue or compartment appear to be more promising for tissue characterization. However, such measurements are technically more challenging, because of the low concentration of such molecules and their relatively low diffusion coefficients. Further, this requirement for signal averaging, and, therefore, long acquisition times, makes in vivo diffusion spectroscopy particularly sensitive to motion.^{57, 95}
Recent progress made in in vivo Fourier NMR spectroscopy allows the extension of diffusion measurements to molecules other than water. For a given species, the chemical shift can be used to determine independent diffusion coefficients of compounds in complex mixtures.^{96} Diffusion of phosphocreatine, for instance can be studied by ^{31}P spectroscopy.^{97, 98} Phosphocreatine is a true intracellular space probe, in contrast to water, which diffuses across cell membranes; as a result it can be used to observe true restricted diffusion. Phosphocreatine may be used to provide unique information on the intracellular medium, such as viscosity or geometry. Similarly, diffusion of N-acetylaspartate (NAR) and myo-inositol may provide useful information on neuronal and glial cell populations, respectively. Monitoring of exchanges of metabolites or drugs through cell membranes could also benefit from diffusion filter techniques designed to separate intracellular from extracellular compounds.^{99} Although diffusion coefficients of metabolites are low because of their size, for nuclear species with spin >1/2 or for coupled-spin systems, multiple quantum experiments are less demanding on gradient hardware because the effective gradient amplitudes are increased by the power of the coherence order (n). With n = 2, a fourfold increase in the diffusion effect would occur. Such spectral editing techniques have been used for selective measurement of diffusion properties of J-coupled compounds such as lactate.^{100, 101}
3 Diffusion in Biological Systems: Effects of Microdynamics and Microstructure
- Top of page
- Introduction
- Diffusion MRI
- Diffusion in Biological Systems: Effects of Microdynamics and Microstructure
- Clinical Applications
- Conclusion
- Biographical Sketch
- Related Articles
- References
Significant technical progress has been achieved in biological systems to ensure that meaningful data can be acquired with minimal artifacts. However, even in the ideal case with perfect data, the interpretation of diffusion NMR experiments performed in biological tissues remains challenging. The diffusion coefficient of water in tissues has been found to be two to ten times less than that of pure water (Table 1).^{14} This can be largely understood by considering that water molecules are obliged to divert tortuously around obstructions presented by fibers, intracellular organelles, or macromolecules. Water molecules may be confined in bounded compartments or retained by attractive centers or surfaces. Biological systems, therefore, differ greatly from an ‘infinitely large medium’. They are very heterogeneous and made of multiple subcompartments (microstructure). Depending on the permeability of the barriers that limit these compartments, exchanges and transport between them (microdynamics) may occur. A classic treatment of the NMR signal may not then reflect properly tissue structure or properties. Water molecules will ‘sense’ all these obstacles only if the time over which diffusion is measured is made sufficiently long that significant interaction of the water molecules with cellular compartments may occur. The concept of ‘diffusion time’ is, therefore, central to any diffusion study in biological tissues. While it may appear that using long diffusion times may reveal more tissue interactions, it is perhaps even more interesting to make the diffusion time as short as possible, up to the point where the diffusion coefficient will reach its value in pure water. By doing so, and plotting measured diffusion coefficients versus diffusion times, the many mechanisms contributing to in vivo diffusion may be revealed and identified. This approach, which will benefit from the newly developed gradient hardware allowing very large gradient strength, may be compared in some ways to that used in NMR dispersion studies, where multiple elementary relaxation mechanisms can be identified and separated by studying dispersion curves.^{102} Diffusion coefficients may become, in these conditions, meaningless if the measurement time scale or the measurement direction are not provided.
Mean diffusivity (10^{−3} mm s^{−1}) | Anisotropy (1–volume ratio) | |
---|---|---|
| ||
Cerebrospinal fluid | 3.19 ± 0.10 | 0.02 ± 0.01 |
Gray matter (frontal cortex) | 0.83 ± 0.05 | 0.08 ± 0.05 |
Caudate nucleus | 0.67 ± 0.02 | 0.08 ± 0.03 |
White matter | ||
Pyramidal tract | 0.71 ± 0.04 | 0.93 ± 0.04 |
Corpus callosum (splenium) | 0.69 ± 0.05 | 0.86 ± 0.05 |
Internal capsule | 0.64 ± 0.03 | 0.70 ± 0.08 |
Centrum semiovale | 0.65 ± 0.02 | 0.27 ± 0.03 |
The main difficulty is that the medium structure is generally unknown in detail. The issue is, therefore, to infer meaningful information on tissue from the measured NMR signals. Initially, this problem could be reversed to consider how known tissue features relevant to tissue microstructure and dynamics affect diffusion NMR signals. Diffusion NMR has been used to study fluid-filled porous media and derive information on microstructure (shape of the pore space) and fluid permeability (porosity).^{103-105} It is clear that many of the questions addressed in those studies are relevant to diffusion measurements in biological media.^{5} Differences in diffusion coefficient and available diffusion space can be used to distinguish compartments and exchanges between them.^{99}
3.1 Effect of Temperature
The first obvious effect on diffusion is that of temperature, as diffusion directly results from the molecular thermal motion. The diffusion sensitivity to temperature is high, about 2.4% for 1°C change (Figure 6),^{106, 107} consequently temperature must be carefully controlled in diffusion MRI experiments. Based on the strong and unique relationship that exists between temperature and molecular diffusion, diffusion imaging has been proposed for the real-time and noninvasive monitoring of temperature. Noninvasive and nondestructive temperature imaging in biological systems may be particularly useful to monitor hyperthermia treatments in real-time, whether using rf electrical fields^{108} or focused ultrasound.^{109, 110} It could also be used to study and control tissue interactions in surgical and medical laser procedures. However, in the context of this chapter, the diffusion-temperature relationship also provides some indication about diffusion mechanisms in tissues.
This relationship has been established semi-empirically in liquids^{1} and verified experimentally.^{106, 111}
- (19)
where k is the Boltzmann constant, T is the temperature, D_{0} is the diffusion that would be obtained at an infinite temperature, and E_{a} is introduced as the translational diffusion activation energy, which is approximately equal to the energy required to break hydrogen bonds, both in pure water and in vivo (E_{a} = 0.2 eV).^{107, 111}
This important result can be understood by considering that the mechanism at the molecular scale for water to move, and thus diffuse, involves continuous breaking and reforming of hydrogen bonds. The fact that E_{a} is identical for water molecules diffusing in vitro and in vivo is not really surprising as the laws of physics at this level should not be different.
3.2 Restricted Diffusion
Diffusion is restricted when boundaries in the medium prevent molecules from moving freely.^{112, 113} Restriction must be related to the experimental parameters. When measurement times are very short, most molecules do not have enough time to reach boundaries and so they behave as if diffusing freely. Once the diffusion time increases, an increasing fraction of molecules will strike the boundaries, and diffusion will deviate from the free, Gaussian behavior. Hence, the usual way to check for restricted diffusion is to plot the diffusion distance, calculated as for free diffusion using Einstein's equation, as a function of the square root of the diffusion time. In the case of restricted diffusion, this plot shows a curvature and finally a leveling off when the diffusion distance reaches the size of the restricting compartment. The effects of restriction will, therefore, appear in the NMR signal for diffusion times such that molecular displacements are in the order of the size of the restricting volumes. These effects will depend on the type of restriction (impermeable or permeable barriers, attractive centers, etc.), the shape of the restricting volumes (spherical, cylindrical, parallel walls, etc.), and the type of NMR experiment (constant or pulsed gradients). As a result, there is not a unique analytical expression that could describe any configuration. A simple example is represented by molecules diffusing between two impermeable parallel walls separated by a distance a.^{112} If the theoretical, free diffusion distance greatly exceeds a, the echo attenuation A in the case of the bipolar gradient pulse experiment significantly deviates from an exponential decay and becomes independent of the diffusion time,^{112} implying that molecules are trapped in the direction of the applied gradient:
- (20)
Another interesting, but somewhat more complicated, case is represented by diffusion restricted in a spherical cavity of radius R_{0}. In the limit where the theoretical, free diffusion distance largely exceeds R_{0}, the attenuation is again independent of the diffusion time^{112, 114} and the measured ADC decreases when the diffusion time is increased:
- (21)
corresponding to an asymptotic ADC value, D_{asymp} of . The factor 5 would be replaced by 3 if diffusion was confined to the surface of the sphere.
Whatever the geometry of the restrictive medium, the deviation from linearity in the semi-log plot of the signal attenuation versus b is crucial to determine whether diffusion is restricted, although other causes may be responsible, such as diffusion in inhomogeneous systems or anisotropic diffusion. The ultimate test is to show that the measured diffusion coefficient or the signal attenuation varies when the diffusion time is changed. Such studies can theoretically lead to the determination of the geometry and size of the restricting boundaries, as with the q-space concept. To avoid restricted diffusion effects, the diffusion time must be decreased to ensure that the diffusion distance during that period remains less than the size R of the restricted region. Unfortunately, the diffusion effect in these conditions becomes small unless considerable gradient intensities are used. A further complication is that in biological tissues walls may not be reflecting boundaries but rather occur as partially absorbing borders.^{115}
3.3 Permeable Barriers
When the restrictive barriers become permeable to diffusing molecules, the restricted diffusion pattern changes. The mathematical treatment of diffusion in systems partitioned by permeable barriers is far from simple. An example was given by Tanner for equally spaced, plane barriers having a permeability constant k.^{116} For short diffusion times, the ADC is D_{0}. When the diffusion time increases, the ADC decreases, as expected for restricted diffusion, but saturates at D_{asymp}, which depends on the permeability constant:
- (22)
where a is the barrier spacing. This spacing can be estimated by the equivalent free diffusion distance that would be obtained from Einstein's equation with D = (D_{0} + D_{asymp})/2 as the diffusion coefficient and , the corresponding diffusion time.^{117} The plot of D versus T_{d} then shows a typical sigmoid pattern and it becomes possible to estimate the barrier permeability k by fitting the data with Equation 22.^{118} This approach is, however, very optimistic, as the geometrical arrangement of the medium is generally not known. In particular, this formalism does not apply to the case where the system consists of spherical cavities separated by permeable barriers.
3.4 Hindered Diffusion
True ‘restricted’ diffusion, which occurs in bounded media and leads to a decrease of diffusion with the diffusion time should be distinguished from ‘hindered’ diffusion. With hindered diffusion, diffusion is decreased by the presence of obstacles but there is no limit to the diffusion distance; consequently diffusion does not change with the diffusion time (once the hindered diffusion regime has been reached). Perhaps the most powerful concept associated with hindered diffusion is that of ‘tortuosity’, a concept that has been widely used in solid porous media studies and more recently in brain diffusion experiments using external tracers.^{119-121} The idea is that because of the presence of obstacles such as fibers, macromolecules, and organelles water molecules must travel longer paths to cover any given distance. In other words, molecules can no longer go straight from A to B, but must diffuse around structures that are impermeable to them (Figure 7). This situation results in a longer diffusion time to diffuse from A to B, or to an apparent decrease in the diffusion distance covered in a given diffusion time and in the measured ADC. This ‘hindered’ diffusion effect is classically expressed quantitatively using a ‘tortuosity’ coefficient, λ, such that:
- (23)
where D would be the diffusion coefficient observed in the absence of obstacles.
Furthermore, as there is no real barrier, molecules can, in principle, diffuse over very large distances compared with those seen in restricted diffusion. Therefore, no curvature would be seen in the plot of the diffusion distance versus the square root of the diffusion time. Similarly, the measured ADC would not depend on the diffusion time, unless the diffusion time is very short, and hindered diffusion paths would not differ significantly from free diffusion paths (Figure 8). It would then be difficult to distinguish between hindered diffusion and diffusion restricted by permeable barriers.
3.5 Anisotropic Diffusion
Diffusion in tissues may be different for different directions of motion; this anisotropy will give rise to variations in the measured diffusion coefficient with the direction of measurement. Diffusion anisotropy has been observed in muscle,^{83} in brain white matter,^{37, 56, 84} (Figure 9) and, recently, in gray matter.^{122, 123} It may result from restriction of diffusion inside fibers (intracellular water) or from an increased tortuosity when diffusion occurs around fibers (extracellular water). Diffusion can be both anisotropic and unrestricted. This behavior is well known in nematic liquid crystals^{124} and can be found in the water lamellar phase of amphiphilic lyotropic systems.^{125}
Using DTI, it appears that diffusion data can be analyzed in three ways to provide information on tissue microstructure and architecture for each voxel or region of interest:^{126} the mean diffusivity, which characterizes the overall mean-squared displacement of molecules (average ellipsoid size) and the overall presence of obstacles to diffusion; the degree of anisotropy, which describes how much molecular displacements vary in space (ellipsoid eccentricity) and is related to the presence of oriented structures; and the main direction of diffusivities (main ellipsoid axes), which is linked to the orientation in space of the structures. These three DTI ‘meta-parameters’ can all be derived from complete knowledge of the diffusion tensor. However because of the complexity of data acquisition and processing for full DTI, and the sensitivity to noise of the determination of the diffusion tensor eigen values, simplified approaches have been proposed.
3.5.1 Mean Diffusivity
To obtain an overall evaluation of the diffusion in a voxel or region, anisotropic diffusion effects must be avoided and the result limited to an ‘invariant’, i.e., a quantity that is independent of the orientation of the reference frame.^{12, 86} Among several combinations of the tensor elements, the trace of the diffusion tensor,
- (24)
is such an invariant. The mean diffusivity is then given by Tr(D)/3. A slightly different definition of the trace has proved useful in assessing the diffusion drop in brain ischemia^{127} (see below).
Unfortunately, the correct estimation of Tr(D) still requires the complete determination of the diffusion tensor. Diffusion coefficients obtained by separately acquiring data with gradient pulses added along the x, y, and z axes cannot be used as these measured coefficients usually do not coincide with D_{xx}, D_{yy}, and D_{zz}, respectively. The reason is that the diffusion attenuation that results, for instance from inserting gradients on the x axis, is A = exp[−b_{xx}D_{xx} + 2b_{xy}D_{xy} + 2b_{xz}D_{xz}] and not simply exp[−b_{xx}D_{xx}] unless diffusion is isotropic (no nondiagonal terms) or there are no gradient pulses at all on the other axes (y and z, here) (no localization) during the diffusion measurement time.
To avoid this problem and to simplify the approach, several groups have designed sequences based on multiple echoes or acquisitions with tetrahedral gradient configurations to cancel nondiagonal term contributions to the MRI signal directly.^{86, 128-130}
3.5.2 Diffusion Anisotropy Indices
Several scalar indices have been proposed to characterize diffusion anisotropy. Initially, simple indices calculated from DW images^{84} or ADC values obtained in perpendicular directions were used, such as ADC_{x}/ADC_{y} and displayed using a color,^{131} scale. Other groups have devised indices mixing measurements along the x, y, and z directions, such as maximum [ADC_{x}, ADC_{y}, ADC_{z}]/minimum [ADC_{x}, ADC_{y}, ADC_{z}] or the standard deviation of ADC_{x}, ADC_{y}, and ADC_{z} divided by their mean value.^{127} Unfortunately, none of these indices is really objective as they do not correspond to a single meaningful physical parameter; more importantly, they are clearly dependent on the choice of directions made for the measurements. The degree of anisotropy would then vary upon the respective orientation of the gradient hardware and the tissue frames of reference and would generally be underestimated. Here again invariant indices must be found to avoid such biases and provide objective, intrinsic structural information.^{132}
Invariant indices are made of combinations of the terms of the diagonalized diffusion tensor, i.e., the eigen-values λ_{1}, λ_{2}, and λ_{3}. The most commonly used invariant indices are the relative anisotropy, RA, the fractional anisotropy, FA, and the volume ratio, VR:
- (25)
where 〈λ 〉 = (λ_{1} + λ_{2} + λ_{3})/3
- (26)
- (27)
RA, a normalized standard deviation, also represents the ratio of the anisotropic part of D to its isotropic part. FA measures the fraction of the ‘magnitude’ of D that can be ascribed to anisotropic diffusion. FA and RA vary between 0 (isotropic diffusion) and 1 ( for RA) (infinite anisotropy). As for VR, which represents the ratio of the ellipsoid volume to the volume of a sphere of radius 〈λ〉, its range is from 1 (isotropic diffusion) to 0; as a result some authors prefer to use (1–VR).^{133}
Once these indices have been defined, it is possible to evaluate them directly from DW images, i.e., without the need to calculate the diffusion tensor.^{134} For instance, A_{σ} which is very similar to RA, has been proposed as:^{135}
- (28)
Also, images directly sensitive to anisotropy indices, or anisotropically weighted images, can be obtained.^{136} Finally, the concept of these intravoxel anisotropy indices can be extended to a family of intervoxel or ‘lattice’ measures of diffusion anisotropy, which allows neighboring voxels to be considered together, in a region of interest, without losing anisotropy effects resulting from different fiber orientations across voxels.^{133} Clinically relevant images of anisotropy indices have been obtained in the human brain (Table 1).^{41, 137}
3.5.3 Fiber Orientation Mapping
The last family of parameters that can be extracted from the DTI concept relates to the mapping of the orientation in space of tissue structure. The assumption is that the direction of the fibers is colinear with the direction of the eigen-vector associated with the largest eigen-diffusivity. This approach opens a completely new way to obtain directly in vivo information on the organization in space of tissues such as muscle, myocardium, and brain or spine white matter, which is of considerable interest, clinically and functionally. Direction orientation can be derived from DTI directly from diffusion/orientation-weighted images or through the calculation of the diffusion tensor. A first issue is to display fiber orientation on a voxel-by-voxel basis. The use of color maps was first suggested,^{131} followed by representation by ellipsoids,^{12, 133} octahedra^{138} or vectors pointing in the fiber direction.^{139, 140} A second issue is to assess connectivity from DTI data in order to visualize anatomical connections between different parts of the brain on an individual basis. Studies of neuronal connectivity are tremendously important to the interpretation of functional MRI data and for establishing how activated foci are linked together through networks.^{141, 142} This issue is difficult, as continuity of fiber orientation from voxel to voxel has to be inferred. Fiber orientation may appear to be varying because of the occurrence of noise in the data. In a given voxel fibers may be merging, branching or dividing. In addition, several fascicles may cross in a given voxel, which cannot be detected with the diffusion-tensor approach in its present form. Structures that exhibit anisotropic diffusion at the molecular level can be isotropically oriented at the microscopic level, resulting in a ‘powder average’ effect that is difficult to resolve.^{143} The semilog plot of the signal attenuation versus b may not be linear in this case.^{125} This deviation from linearity can be ascribed to anisotropy and not to restricted diffusion because the diffusion measurements are independent of the diffusion time. Several groups have recently approached the difficult problem of inferring connectivity from DTI data in the rat brain^{144} in vitro and in the living human brain^{145} (Figure 10).
3.6 Diffusion in Multiple Compartment Systems
Most diffusion measurements in biological tissues refer to an ADC and yet it is generally considered that diffusion in the measurement volume (voxel) has a unique diffusion coefficient. This simplification may not always be legitimate, since partial volume effects may occur and most tissues are made of multiple subcompartments (with at least intracellular and extracellular components). Assuming measurement times are short and diffusion is unrestricted in each subcompartment i, and that there is no exchange, the signal attenuation is:
- (29)
where ρ_{i} is the density of molecules diffusing in compartment i, and D_{i} the associated diffusion coefficient. In this case, the ADC that would be measured would depend on the range used for the b values and would not reflect properly the diffusion in the voxel. Measurements with low b values would then be more sensitive to fast diffusion components. The ideal approach would be to separate all subcompartments by fitting the data with a multiexponential decay. Unfortunately, the values for D_{i} are often low and not very different from each other; consequently very large b values and very high SNR would be required.
Relaxation effects must also be considered if compartments have different relaxation rates.^{146} If all spins do not have the same transverse relaxation during the pulse sequence, Equation 5 must be complemented with a decay term:^{115}
- (30)
For two compartments, assuming measurement times are short and, therefore, exchanges are small, the diffusion signal is given by:
- (31)
where b is the sequence diffusion factor, D_{1}, D_{2} are the diffusion coefficients and F_{1}, F_{2} are relative weights to the signal of the two compartments. Estimates of F_{1}, F_{2} and D_{1}, D_{2} are obtained by fitting a series of signals acquired with multiple b values. For a spin echo sequence with TR ≫ T_{1}, F depends on the volume fraction f and the relaxation time T_{2} of each compartment:
- (32)
Two diffusion compartments have been found in the rat brain.^{147} Assignment of these compartments to extra- and intracellular water is not, however, straightforward, as the values found for F_{1} and F_{2} are apparently opposite to the known volume fractions for the intra- and extracellular spaces (82.5% and 17.5%, respectively). A first explanation for this discrepancy is that T_{2} effects must be taken into account when converting F values into f values. A second factor is that the exchange rate may not be negligible. It has been shown that if the residence rate in each compartment is considered, the contribution of the fast diffusing component is overestimated.^{148} It is, therefore, not yet certain that the two compartments that have been observed do correspond to the intra- and extracellular compartments.
The situation is even more complex when measurement times are longer. First, restricted diffusion may be seen in the smallest subcompartments. Second, molecular exchanges may occur between communicating compartments. As a result, the analytic treatment becomes difficult. Applying the central limit theorem for statistically distributed compartments in the case of long diffusion times, the use of a single apparent diffusion constant D_{a} can be justified:
- (33)
This result is also consistent with NMR dispersion studies,^{102} which consider that cell membranes can be ignored on the NMR time scale. In intermediate situations, the geometrical arrangement and the diffusion coefficients of each compartment must be considered as well as their rates of exchange.^{149} The comprehensive analysis of the diffusion attenuation curves obtained with different diffusion times may lead to an accurate description of the medium microstructure.
It is clear that the ADC depends on the range used for the b values and would not reflect properly diffusion in the tissue. Measurements with low b values (less than 1000 s mm^{−2}) would be more sensitive to fast diffusion components, such as those occurring in the extracellular compartment. In clinical studies and most animal experiments, especially when data are fitted to a single exponential, diffusion patterns observed in tissues have to be explained by features of the extracellular space, although this compartment is physically small. Changes in the ADC calculated in these conditions should be interpreted in terms of changes in the diffusion coefficient of the extracellular space (tortuosity) and in its fractional volume relative to the intracellular volume. This explains why diffusion has appeared as a sensitive marker of the changes in the extracellular/intracellular volume ratio, as observed in brain ischemia, spreading depression,^{80-82} status epilepticus,^{78} or extraphysiological manipulations of the cell size through osmotic agents.^{150} Using large b values, it becomes increasingly possible to assess separately the intra- and extracellular compartments and their relative volumes in vitro^{151} and in vivo, as attempted in the rat brain^{147} and the human brain.^{152}
3.7 Metabolites
Data on magnitudes and even directional anisotropy of diffusion coefficients of molecules such as choline, creatine/phosphocreatine and NAA in animals,^{97, 153, 154} and human brain^{95} have been made available. Although diffusion rates of metabolites in pure water are lower than that of water because of the difference in molecular weight and hydration layers, the ADC values of these metabolites in brain are considerably smaller. Typical values from a series of 10 normal volunteers (18 cm^{3} voxels located in white matter, diffusion time 240 ms) are 0.13 × 10^{−3} mm^{2} s^{−1} for choline and creatine and 0.18 × 10^{−3} mm^{2} s^{−1} for NAA compared with 1.24 × 10^{−3} mm^{2} s^{−1} for choline and 0.85 × 10^{−3} mm^{2} s^{−1} for NAA at 35°C in vitro.^{155} The fact that metabolites, as well as molecules such as fluorodeoxyglucose 6-phosphate,^{156} with different molecular weights had very similar ADC values in vivo is striking and remains to be explained. The very low values of the ADC obtained with long diffusion times are, of course, compatible with restriction of these metabolites in compartments that were about the same size. For instance, recent experiments varying the diffusion times have shown that diffusion of NAA is considerably restricted in a manner compatible with two compartments, one of 7–8 µm and one of approximately 1 µm, possibly representing the cell bodies and the intra-axonal space.^{157} However, intracellular obstacles may also play an important role, as water molecules could diffuse easily in small spaces between small obstacles, such as mitochondria, organelles or macromolecules, while larger metabolites would ‘feel’ a more obstructed medium. Tortuosity factors may, therefore, be larger for metabolites than for water and this would explain, at least partially, the diffusion decrease for intracellular metabolites that has been observed during ischemia.^{158, 159}
3.8 Summary: Application to Brain White Matter
The concepts of restriction, hindrance, tortuosity, and multiple compartments are particularly useful to understand diffusion findings in brain white matter. Water diffusion is highly anisotropic in white matter^{37, 56, 84} and this anisotropy is observed even before fibers are myelinated, though at a lesser degree.^{137, 160-166} ADC values obtained by measurements made parallel and perpendicularly to the fibers do not seem to depend on the diffusion time^{167, 168} at least for diffusion times longer than 20 ms (see Figure 8).
Initial reports suggested that the anisotropic water diffusion could be explained by restriction of the water molecules to the axons (anisotropically restricted diffusion) by the myelin sheath.^{169, 170} However, although restricted diffusion has been seen for intra-axonal metabolites, such as NAA, or for truly intra-axonal water,^{171} it now appears that most studies were performed with relatively low b values and are mostly sensitive to the extracellular, interaxonal space. In this condition, diffusion anisotropy in white matter should be linked to the anisotropic tortuosity of the interstitial space between the fibers: diffusion would be more impeded perpendicular to the fibers because of their geometric arrangement (Figure 7).^{167}
Considering a bundle where fibers are organized in the most compact way, molecules would have to actually travel over a distance of πd/2, where d is the fiber diameter, for an apparent diffusion distance equal to the fiber diameter. Therefore, on average, the ADC measured perpendicularly to the fibers would be reduced, whatever the diffusion time, to:
- (34)
where the reference value, D_{0}, is the diffusion value measured parallel to the fibers. This ratio of about 0.4 fits reasonably very well with literature data,^{167} although much larger ratios have been reported.^{133} This rough model also implies that the tortuosity factor would be anisotropic in white matter with a λ_{perpendicular}/λ_{parallel} ratio of π/2(≈1.15). Unfortunately no systematic measurements of λ have yet been made in white matter using ionic extracellular tracers,^{2, 119} although anisotropy has been seen.^{172} Another interesting point about this model is that it is compatible with the fact that no true restricted effects are observed when the diffusion time is increased, as there are no actual boundaries to diffusing molecules. Also, the parallel organization of the fibers may be sufficient to explain the presence of anisotropy before myelination. However, as the axonal membranes should be more permeable to water than the myelin sheaths, the degree of anisotropy should be less pronounced in the absence of myelin, as significant exchanges with the axonal spaces should occur. Oriented filaments within the axoplasm do not seem to play an important role.^{173} Fiber orientation mapping and connectivity studies derived from anisotropic diffusion in white matter will clearly benefit from a better understanding of the respective contributions of intra-axonal and extra-axonal compartments to anisotropy mechanisms.^{174}
4 Clinical Applications
- Top of page
- Introduction
- Diffusion MRI
- Diffusion in Biological Systems: Effects of Microdynamics and Microstructure
- Clinical Applications
- Conclusion
- Biographical Sketch
- Related Articles
- References
Diffusion imaging is a truly quantitative method. The diffusion coefficient is a physical parameter that directly reflects the physical properties of the tissues in terms of the random translational movement of the molecules under study (most often water molecules, but sometimes metabolites. The diffusion coefficient does not depend on the field strength of the magnet or the pulse sequence used, which is not the case for the other classical MRI parameters such as T_{1} or T_{2}. Diffusion coefficients obtained at different times in a given patient, or in different patients, or in different hospitals can be compared without any need for normalization.
4.1 Central Nervous System
The most clinically relevant field of application of diffusion MRI is in the nervous system. Diffusion MRI has already appeared as a breakthrough in two areas: early brain ischemia and white matter diseases.
During the acute stage of brain ischemia, water diffusion is decreased in the ischemic territory by as much as 50%, as shown in cat brain models.^{175} This diffusion slowdown is linked to the cytotoxic edema that results from the energetic failure of the cellular membrane Na^{+}/K^{+} pumping system. The exact mechanism by which diffusion is reduced is still unclear (increase of the slow-diffusion intracellular volume fraction, changes in membrane permeability,^{176} or shrinkage of the extracellular space resulting in increased tortuosity for water molecules.^{177, 178} have been suggested). Diffusion MRI has been extensively used in animal models to establish and test new therapeutic approaches. These results have been confirmed in patients with stroke, offering the potential to highlight ischemic regions within the first hours of the ischemic event, when brain tissue might still be salvageable,^{179, 180} well before conventional MRI becomes abnormal (vasogenic edema). Combined with perfusion MRI, diffusion MRI is under clinical evaluation as a tool to help clinicians to optimize their therapeutic approach to individual patients,^{181} to monitor patient progress on an objective basis, and to predict clinical outcome.^{182-185}
In white matter, DTI has already shown its potential in diseases such as multiple sclerosis^{186-188} leukoencephalopathy,^{189} Wallerian degeneration, Alzheimer disease,^{190, 191} and Creutzfeld–Jacob disease.^{192-194} Mean diffusivity indices, such as the trace of the diffusion tensor, reflect overall water content, while anisotropy indices indicate myelin fiber integrity. It has been shown that the degree of diffusion anisotropy in white matter increases during the myelination process,^{137, 162, 195} and diffusion MRI could be used to assess brain maturation in children,^{196} newborns, or premature babies.^{137, 197} Abnormal connectivity in frontal white matter based on DTI data has also been reported in schizophrenic patients.^{198} The potential of diffusion MRI has also been studied in brain tumor grading,^{199-201} trauma,^{202} hypertensive hydrocephalus,^{203} AIDS,^{204} eclampsia,^{205} leukoaraiosis,^{206, 207} and the spinal cord.^{87, 185, 208, 209}
4.2 Body
The use of diffusion imaging has been less successful in areas of the body apart from the brain because of the occurrence of strong respiratory motion artifacts and of the short T_{2} values of body tissues, which require shorter TE than in the brain and, therefore, leaves less room for the diffusion gradient pulses. These obstacles, however, can sometimes be overcome with ad hoc MR sequences and hardware. Potential for tissue characterization has been shown in the extremity muscles,^{210} the spine,^{211, 212} the breast,^{54, 213, 214} the kidney, and the liver.^{215-218} Muscle fiber orientation can be approached using DTI in organs such as the tongue^{219} or the heart. Myocardium DTI^{220, 221} has tremendous potential for providing data on heart contractility, a very important parameter, but remains technically very challenging to perform in vivo because of heart motion. Other applications include temperature imaging through the sensitivity of diffusion coefficients to temperature.^{107, 108, 210, 222}
5 Conclusion
- Top of page
- Introduction
- Diffusion MRI
- Diffusion in Biological Systems: Effects of Microdynamics and Microstructure
- Clinical Applications
- Conclusion
- Biographical Sketch
- Related Articles
- References
Many tissue features at the microscopic level may influence NMR diffusion measurements. So far, many theoretical analyses of the effect of restriction, membrane permeability, hindrance, anisotropy, or tissue inhomogeneity have been published. These analyses underline how much care is necessary to conduct diffusion NMR studies properly and to interpret the results. Although the results of these analyses have been applied to characterize nonliving systems, such as porous media, much work remains to be done to produce accurate information on microstructure and microdynamics in vivo in biological systems. Powerful tools, such as diffusion spectroscopy of metabolites, DTI or q-space imaging, that are still under development are expected to provide such information.
6 Biographical Sketch
- Top of page
- Introduction
- Diffusion MRI
- Diffusion in Biological Systems: Effects of Microdynamics and Microstructure
- Clinical Applications
- Conclusion
- Biographical Sketch
- Related Articles
- References
Denis Le Bihan. b 1957. Ph.D., 1987, Physics, M.D., 1984, Radiology, University of Paris, France. Former Chief of the Radiology Research Section, Clinical Center, NIH, Bethesda, USA. Presently Research Director, French Atomic Energy Commission, Director, Anatomical and Functional Neuroimaging Laboratory, Orsay, France, and Clinical Professor of Radiology, Harvard University, Cambridge, USA. Corresponding Member, French Academy of Sciences. Approx. 400 articles, book chapters, abstracts and patents. Research interests: diffusion, perfusion and functional MRI.
References
- Top of page
- Introduction
- Diffusion MRI
- Diffusion in Biological Systems: Effects of Microdynamics and Microstructure
- Clinical Applications
- Conclusion
- Biographical Sketch
- Related Articles
- References
- 1Diffusion in Solids, Liquids and Gases, Academic Press, San Diego, 1960.,
- 2Diffusion and Perfusion Magnetic Resonance Imaging. Applications to Functional MRI, ed. D. Le Bihan, Raven Press, New York, 1995, p. 127.,
- 3 , , , and ,
- 4D. Le Bihan (ed.), Diffusion and Perfusion Magnetic Resonance Imaging. Applications to Functional MRI, Raven Press, New York, 1995.
- 5Principles of Nuclear Magnetic Resonance Microscopy, Oxford University Press, Oxford, 1991.,
- 6C. R. Acad. Sci. Paris, Series II, 1985, T.301, 1109.and ,
- 7 , , and ,
- 8 and ,
- 9 ,
- 10 and ,
- 11 and ,
- 12 , , and ,
- 13 ,
- 14 ,
- 15 , , and ,
- 16Magn. Reson. Med., 1988, 7, 346.,
- 17Magn. Reson. Med., 1989, 10, 324., , and ,
- 18 , , , , and ,
- 19Magn. Reson. Med., 1989, 9, 423., , , , , and ,
- 20 and ,
- 21 , , , , , and ,
- 22Magn. Reson. Med., 1988, 7, 111., , , and ,
- 23 , , , and ,
- 24 , , and ,
- 25 , , and ,
- 26 ,
- 27 , , and ,
- 28 , , , , and ,
- 29 ,
- 30 and ,
- 31Magn. Reson. Med., 1997, 38, 101., , and ,
- 32 , , , , , , , and ,
- 33 , , , , , , and ,
- 34Magn. Reson. Med., 1998, 39, 950., , , , and ,
- 35Magn. Reson. Med., 1997, 38, 638.,
- 36 and ,
- 37 , , , , , and ,
- 38Proc. IXth Annu. Mtg. Soc. Magn. Reson. Med., New York, 1990, p. 5., , , , , , , and ,
- 39Proc. VIIth Annu. Mtg. Soc. Magn. Reson. Med., San Francisco, 1988, p. 80.and ,
- 40 ,
- 41 , , , , and ,
- 42Magn. Reson. Med., 1996, 35, 547., , and ,
- 43 and ,
- 44Magn. Reson. Med., 1999, 41, 143.,
- 45Diffusion and Perfusion Magnetic Resonance Imaging. Applications to Functional MRI, ed. D. Le Bihan, Raven Press, New York, 1995, p. 77., , and , in
- 46 , , and ,
- 47 , , , , , and ,
- 48Diffusion and Perfusion Magnetic Resonance Imaging. Applications to Functional MRI, ed. D. Le Bihan, Raven Press, New York, 1995, p. 67.and , in
- 49 and ,
- 50Magn. Reson. Med., 1996, 36, 960.and ,
- 51Magn. Reson. Med., 1998, 39, 801., , and ,
- 52Magn. Reson. Med., 1999, 41, 95.,
- 53 , , , , , , and ,
- 54Magn. Reson. Med., 1998, 39, 392., , , , and ,
- 55Magn. Reson. Med., 1991, 19, 261.and ,
- 56 , , and ,
- 57 , , and ,
- 58Magn. Reson. Med., 1995, 34, 476., , and ,
- 59 , , , , and ,
- 60Magn. Reson. Med., 1994, 32, 379.and ,
- 61Magn. Reson. Med., 1995, 30, 720., , , and ,
- 62 , , , , , and ,
- 63 , , and ,
- 64 , , and ,
- 65J. Magn. Reson., 1995, B108, 22., , and ,
- 66 and ,
- 67 , , and ,
- 68 and ,
- 69 , , , and ,
- 70 , , and ,
- 71J. Magn. Reson., 1989, 83, 308.and ,
- 72Magn. Reson. Med., 1994, 31, 9., and ,
- 73Diffusion and Perfusion Magnetic Resonance Imaging. Applications to Functional MRI, ed. D. Le Bihan, Raven Press, New York, 1995, p. 110., , and , in
- 74 and ,
- 75 and ,
- 76Magn. Reson. Med., 1994, 31, 601., , , and ,
- 77Magn. Reson. Med., 1995, 34, 555., , , and ,
- 78Magn. Reson. Med., 1993, 30, 241., , , and ,
- 79Magn. Reson. Med., 1997, 37, 1., , , , and ,
- 80Magn. Reson. Med., 1994, 32, 189., , , , and ,
- 81 , , , , and ,
- 82 , , , , and ,
- 83 , , and ,
- 84 , , and ,
- 85 , , and ,
- 86Magn. Reson. Med., 1995, 35, 399., , , and ,
- 87Magn. Reson. Med., 1997, 38, 868., , , , , and ,
- 88 , , , , and ,
- 89Proc. XIIth Annu. Mtg. Soc. Magn. Reson. Med., New York, 1993, p. 584., , , and ,
- 90Proc. XIIth Annu. Mtg. Soc. Magn. Reson. Med., New York, 1993, p. 289., , , and ,
- 91 ,
- 92Magn. Reson. Med., 1990, 14, 435.and ,
- 93Magn. Reson. Med., 1994, 32, 707., , , , , and ,
- 94Magn. Reson. Med., 1997, 38, 930., , , and ,
- 95 , , and ,
- 96 and ,
- 97Magn. Reson. Med., 1990, 13, 467., , , and ,
- 98 , , , and ,
- 99 , , , , , and ,
- 100 and ,
- 101 ,
- 102Magn. Reson. Med., 1990, 14, 482., , , and ,
- 103 , , , and ,
- 104 , , , , , and ,
- 105 and ,
- 106 ,
- 107 , , and ,
- 108 , , , and ,
- 109Med. Phys., 1993, 20, 7., , and ,
- 110 , , , , , , , , and ,
- 111 and ,
- 112 ,
- 113 ,
- 114 ,
- 115 , , , and ,
- 116 ,
- 117 , , and ,
- 118 and ,
- 119 and ,
- 120 and ,
- 121Proc. Natl. Acad. Sci., USA, 1998, 95, 8975.and ,
- 122Magn. Reson. Med., 1997, 38, 662., , , , , , , and ,
- 123 , , and ,
- 124 ,
- 125 and ,
- 126Imaging Brain Structure and Function, New York Academy of Sciences, New York, 1997, p. 123.,
- 127Magn. Reson. Med., 1994, 31, 154., , , , , and ,
- 128Magn. Reson. Med., 1995, 33, 41.and ,
- 129Magn. Reson. Med., 1998, 40, 622., , and ,
- 130Magn. Reson. Med., 1995, 34, 194.and ,
- 131 , , , , and ,
- 132 and ,
- 133Magn. Reson. Med., 1996, 36, 893.and ,
- 134Magn. Reson. Med., 1998, 39, 928.and ,
- 135Magn. Reson. Med., 1996, 35, 399., , , and ,
- 136Magn. Reson. Med., 1998, 40, 160.and ,
- 137 , , , et al.,
- 138Am. J. Physiol., 1998, 38, G363., , , , , and ,
- 139Ann. Neurol., 1997, 42, 951., , , et al.,
- 140J. Neurol. Neurosurg. Psychiatr., 1998, 65, 863., , , , , , , , and ,
- 141 , , , , , and ,
- 142 , , , , and ,
- 143 and ,
- 144Ann. Neurol., 1999, 45, 265., , , and ,
- 145Regularization of MR Diffusion Tensor Maps for Tracking Brain White Matter Bundles, Springer-Verlag, 1998, p. 489., , , , , , , and ,
- 146J. Magn. Reson., 1982, 50, 409.,
- 147Magn. Reson. Med., 1996, 36, 847., , , , and ,
- 148Magn. Reson. Med., 1998, 40, 79., , and ,
- 149 and ,
- 150 , , and ,
- 151Magn. Reson. Med., 1997, 37, 825., , , , , and ,
- 152 , , , et al.,
- 153Magn. Reson. Med., 1993, 29, 125., , , , and ,
- 154Diffusion and Perfusion Magnetic Resonance Imaging. Applications to Functional MRI, ed. D. Le Bihan, Raven Press, New York, 1995, p. 57., , and , in
- 155 , , and ,
- 156Magn. Reson. Med., 1998, 40, 1., , , and ,
- 157 and ,
- 158 , , , and ,
- 159Magn. Reson. Med., 1996, 36, 914., , , and ,
- 160 , , and ,
- 161 , , , , and ,
- 162 , , and .
- 163 and ,
- 164 , , , , , , and ,
- 165 , , , , , and ,
- 166 , , and ,
- 167 , , and ,
- 168Magn. Reson. Med., 1991, 19, 327., , , , , and ,
- 169 , , and ,
- 170 , , and ,
- 171 and ,
- 172 and ,
- 173Magn. Reson. Med., 1994, 32, 579.and ,
- 174Magn. Reson. Med., 1997, 37, 103., , , and ,
- 175 , , , , , , , and ,
- 176Proc. Natl. Acad. Sci., USA, 1994, 91, 1229., , , and ,
- 177 , , and ,
- 178 , , and ,
- 179 , , , et al.,
- 180 , , , , and ,
- 181 , , and ,
- 182Ann. Neurol., 1997, 42, 164., , , et al.,
- 183 , , , , , , , , and ,
- 184 , , and ,
- 185Magn. Reson. Med., 1998, 39, 878., , , , , , and ,
- 186 , , , , and ,
- 187Magn. Reson. Med., 1997, 38, 484., , , , , , , and ,
- 188 , , , and ,
- 189 , , , , , , and ,
- 190 , , , , , and ,
- 191 , , , , , and ,
- 192 , , , and ,
- 193 , , , , , , and ,
- 194 , , , , and ,
- 195 , , and ,
- 196 , , , , , , and ,
- 197 , , , , , , and ,
- 198 , , , et al.,
- 199 , , , , , and ,
- 200 , , , , , , and ,
- 201 , , , , , and ,
- 202 , , , , and ,
- 203 , , , and ,
- 204 and ,
- 205 , , , and ,
- 206 , , and ,
- 207 , , , , , and ,
- 208 , , , , and ,
- 209 , , , , and ,
- 210 ,
- 211 , , , , , , and ,
- 212Am. J. Neuroradiol., 1997, 318, 443., , , ,
- 213 , , , , and ,
- 214Magn. Reson. Med., 1997, 37, 576., , , , and ,
- 215 , , , , and ,
- 216 , , , , and ,
- 217 , , and ,
- 218 , , , , and ,
- 219 , , , , , and ,
- 220Am. J. Physiol., 1998, 44, H2308., , , and ,
- 221 , , , , and ,
- 222 , , , , , and ,