Diffusion weighted imaging produces images in which the contrast reflects differences in the distances (measured as mean squared displacements) moved by water molecules during a time that is selected experimentally. The contrast also depends on the strengths of the gradients used to induce the motion-dependent dephasing of the water signal from tissues. Random Brownian motion causes reductions in the MRI signal to a degree that depends on the self-diffusion coefficient of the water, D, which may be measured quantitatively by acquiring images at different values of the gradient strength;
S= Soexp(-bD)
where S is the signal, b is an experimental factor whose value depends on the gradient strength and duration, and So the signal when b=0. Tissue is a complex medium in which water may reside in different compartments (e.g. intracellular and extracellular) that undergo exchange to different degrees, so within a typical imaging voxel of ≈ 1mm3 the value of D may vary. Then, only an averaged value is obtainable from a single measurement of the ratio of S/So. Moreover, whereas in free solution the mean displacement in three dimensions of water molecules (L) in a time t is given by the Einstein relationship;
L2=6Dt
in tissues L increases less rapidly with time t because of the effects of hindrances and restrictions to free movement. Tissue contains macromolecules that hinder motion, while some structures such as membranes may be relatively impermeable and thus restrict the maximal displacement possible. At very short times, before there are possibilities of colliding with obstructions, the Einstein relation may hold and D have a relatively high value, the intrinsic value for the fluid. At longer times, the probability of interacting with interfaces increases, L no longer increases as rapidly with time, but still using the Einstein relation, D appears to decrease. In principle, the measured variation of D with t can provide information on the scale of microstructures within the medium, such as cell sizes. Taken together, the effects of multiple compartments and time-dependent diffusion are summarized as the measured apparent diffusion coefficient or ADC.
Numerous studies have shown that MRI diffusion measurements are sensitive to the presence and density of membranes and other structures that hinder or restrict free diffusion. Measurements of ADC have therefore been used to assess changes in tumor cellularity, which describes the density of cells within a tumor, or region of tumor, and is often thought to indicate the aggressiveness of a tumor’s growth. Regions with high cellularity are usually indicative of rapid cell division and tumor growth, while necrotic regions of tumors, which may form in the wake of rapid growth or in response to treatment, exhibit low cellularity. Quantitative maps of ADC, acquired in vivo, have proven to be valuable tools for both treatment planning and monitoring. However, the interpretations of ADC changes are not unambiguous because ADC itself may depend on several different factors. Therefore, more elaborate mathematical models have been developed that, when applied to multiple measurements of ADC obtained with different experimental parameters, can be used to derive explicit estimates of intrinsic properties of tissues that can be displayed as parametric maps. These models vary in their complexity and practicality as dictated by the number of parameters that characterize the model and their underlying assumptions and approximations. A common approach is to model tissue as comprising two major water compartments, the intracellular and extracellular spaces, so typical parameters of potential importance include the intracellular and extracellular volume fractions, their intrinsic diffusion rates, the average cell size, the cellularity, the rate of exchange between the compartments (determined by the membrane permeability), the extracellular tortuosity, and contributions from other compartments such as the vasculature, all of which may affect the ADC. In some tissues such as brain and prostate, further consideration of the possible anisotropic nature of diffusion may need to be given, by incorporating, for example, diffusion tensor measurements.
Most current studies of ADC use pulsed gradient spin echo (PGSE) methods, in which signal changes caused by diffusion are measured over a specific interval, the diffusion time Δ. Because of technical constraints on the design of gradients and the overall sensitivity of MRI, nearly all current and published measurements of ADC correspond to the regime of “long” diffusion intervals. In practice Δ is many (typically, 20 - 80) milliseconds, during which time tissue water may move ≈ 9 – 18 μm, long enough for most water to encounter restricting barriers when cell sizes are of that order or smaller. Measured ADC values then reflect the influence of cell density, and differences between tissues detected by current PGSE methods are sensitive mainly to alterations in the permeability and density of boundaries on the whole cell scale. However, they are unable to distinguish changes at smaller scale (such as intracellular changes or changes in cell sizes) unless measurements are made in a regime before all nuclei encounter the restricting barriers in the time Δ. This is not practical using PGSE methods because extremely strong gradients are needed to produce measurable diffusion attenuation in short times. For gradient strengths and diffusion times available clinically, ADC varies little with Δ and so, for example, estimates of cell sizes are challenging.
A modification of conventional methods uses field gradients oscillating cosinusoidally at well defined frequencies. The effective diffusion time is related to the oscillation frequency f;
Δeff=1/4f
For f varying between 25 and 100 Hz (the range accessible using clinical scanners, which is smaller than that available for animal studies on smaller bore systems) this approach probes ADC over the range 10 - 2.5 ms, or displacements of order 3 - 6 μm, smaller than a typical cell. In this regime, not all water is restricted and ADC varies significantly over this range of Δ , thus providing more information on microstructure. Measurements of the dependence of the diffusion coefficient of tissue water on the frequency f provide temporal diffusion spectra which represent the manner in which various spectral components of molecular velocity correlations vary in a particular medium, which reflect properties of the geometrical structures that restrict or hinder free movements. Measurements at different gradient frequencies reveal information on the scale of restrictions or hindrances to free diffusion, and the shape of a spectrum reveals the relative contributions of spatial restrictions at different distance scales. Experimentally, oscillating gradients at moderate frequency are more feasible for exploring restrictions at very short distances, which in tissues correspond to structures smaller than cells. Computer simulations and experiments in animal models suggest that diffusion spectral measurements may be sensitive to intracellular structures such as nuclear size, and that changes in tissue diffusion properties may be measured before there are changes in cell density.
Analytical expressions for the behavior of the diffusion coefficient as a function of gradient frequency or diffusion time in simple geometries with different dimensions have been derived. These can be incorporated into simple mathematical models. One approach models cells as spheres in which the explicit dependence of ADC on gradient frequency or diffusion time and cell size can be calculated. By modeling the theoretical dependence of the intracellular and extracellular diffusion behaviours, and by combining measurements of ADC at low and higher frequencies or diffusion times, the cellular density, cell size, compartmental volume fractions along with their intrinsic diffusion rates can be extracted. The assumptions behind such models e.g. ignoring exchange between compartments, have been justified both by computer simulations and by the success of these methods in experiments in cells and tissues with independent verification of their predictions.
The value of deriving quantitative parameters remains to be proven but there are strong reasons for believing these approaches will prove valuable. For example, one major target of treatments of tumors is to activate apoptosis, which is a programmed cell death associated with distinct cellular morphologic changes that have been well characterized. For example, Taxol is used to treat breast cancer and targets tubulin, producing defects in mitotic spindle assembly, chromosome segregation, and cell division. This blocks the progression of mitosis beyond the G2/M phase, and prolonged activation of the mitotic checkpoint triggers apoptosis. There are morphologic changes including cell shrinkage, cytoplasm condensation and apoptotic body formation. These changes are likely detectable by changes in parameters derived from appropriate ADC measurements and modelling. For example, the cytoplasm condensation at an early stage of apoptosis yields an enhanced cytoplasm viscosity which can lead to a decreased intracellular diffusion coefficient, while cell shrinkage causes decreases in cell size, intracellular volume fraction, and extracellular tortuosity, which then leads to an increase of extracellular diffusion coefficient. Conversely, radiation effects can lead to increases in cell size accompanied by increases in intracellular structure and polyploidy, which result in a distinctly different set of changes in diffusion-based metrics. Hopefully, the creation of maps of derived diffusion parameters will reveal such underlying pathological changes more sensitively, earlier, or with less ambiguity than current approaches.