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The apparent diffusion coefficient (ADC), as measured by diffusion-weighted MRI has proven useful in the diagnosis and evaluation of ischemic stroke. The ADC of tissue water is reduced by 30-50% following ischemia and provides excellent contrast between normal and affected tissue. Despite its clinical utility, there is no consensus on the biophysical mechanism underlying the reduction in ADC. In this work, a numerical simulation of water diffusion is used to predict the effects of cellular tissue properties on experimentally measured ADC. The model shows that the biophysical mechanisms responsible for changes in ADC post-ischemia depend upon the time over which diffusion is measured. At short diffusion times, the ADC is dependent upon the intrinsic diffusivity, while at longer, clinically relevant diffusion times, the ADC is highly dependent upon the cell volume fraction. The model also predicts that at clinically relevant diffusion times, the 30-50% drop in ADC after ischemia can be accounted for by cell swelling alone when intracellular T2 is allowed to be shorter than extracellular T2.
The idea to use NMR to measure diffusion was introduced by Torrey in 1956 . The clinical utility of diffusion-weighted MRI (DWMRI) was realized in the early 1990s in the evaluation of ischemic stroke. Within minutes of onset, DWMRI exhibits hyper-intensity in regions of the brain affected by acute stroke, while T2-weighted images remain unaffected. The apparent diffusion coefficient (ADC), a quantitative measure of the diffusion of water in tissue, decreases 30-50% in ischemic regions of the brain [2,3]. While these results have had significant clinical utility, there remains no consensus on the biophysical mechanisms causing the drop in ADC. Several mechanisms have been proposed, including increases in the intracellular volume fraction (IVF) , increased tortuosity of extracellular spaces , increased membrane permeability  and decreases in the diffusion of water in the intracellular space [5,6]. Because a large number of tissue parameters have been hypothesized to affect the ADC, mathematical models of water diffusion are useful to assess the role of each parameter on the ADC.
In free diffusion, the signal from DWMRI decays exponentially with increasing b-value, and is characterized by the intrinsic diffusion coefficient, D. However, the diffusion of water in tissue is not free: water interacts with lipid membranes, macromolecules, and other cellular and extracellular contents causing the signal decay to deviate from the monoexponential decay observed in free diffusion [7,8]. The ADC lumps all of these interactions into a single ‘apparent’ diffusion coefficient, which is calculated by fitting the DWMRI signal to an exponential decay over a specific range of b-values, typically between 0 and 1000 s/mm2. Therefore, the ADC provides little indication of the specific biophysical mechanisms contributing the DWMRI signal decay. Other analyses of diffusion fit the non-monoexponential signal decay to bi- or triexponential decays. It was originally hypothesized that each component of the signal decay described a tissue compartment. However, the calculated volume fractions of intracellular and extracellular compartments from multi-component analysis did not match physiologic volume fractions measured by histology .
Several researchers have developed mathematical models to describe diffusive signal decay explicitly in terms of physical tissue properties. Tanner et al., using a short gradient pulse (SGP) approximation, found an analytical expression for signal decay due to diffusion between two reflective parallel plates . Other models of restricted diffusion have been published using the Gaussian phase distribution approximation to find signal decays from diffusion restricted between parallel plates or within a sphere [10,11]. These analytic models have been used to represent the diffusion of water inside cells restricted by the lipid membrane, but do not account for water in an extracellular compartment or exchange between these compartments. Stanisz et al. generalized the Tanner et al. model by solving a set of differential equations to describe exchange between compartmentalized restricted intracellular and tortuous extracellular water . Vestergaard-Poulsen et al. extended the Stanisz et al. model to incorporate compartmental dependent T2 decay and the effect of a finite echo time, TE . In addition to these compartmental models, several computational models of diffusion have been developed. These models utilize a finite element (FE) or finite difference (FD) approximation to solve simplified Block-Torrey equations on an explicitly defined geometry [14,15]. For example, Chin et al. used a FD approximation to investigate the effect of myelin on diffusion restricted in axons . Using the SGP approximation, it is possible to utilize a Fourier transform relationship between the diffusive signal decay and the effective diffusion propagator (EP), a volume averaged diffusion profile . This formalism has been used in both in computational models as well as experiments and constitutes the theoretical basis of q-space-imaging.
Despite the large number of models available, no single model has described all aspects of the diffusion-weighted MR signal and the changes seen in the ADC of tissue water following ischemia. Given the large number of biophysical parameters with the potential to influence the ADC, it is useful to assess the effect of changes in each parameter on the ADC. The goal of this paper is to evaluate cellular and experimental MR parameters that have a significant influence on DWMRI results. A 3D finite difference model of water diffusion is presented, which incorporates T2 decay into the EP formalism. The model calculates diffusion within a geometry of regularly spaced cubic cells taking into account the experimental parameters of diffusion time (Δ), and TE as well as tissue parameters, including intracellular volume fraction (IVF), cell size (Lcell), membrane permeability (Pmem), intracellular and extracellular diffusion coefficients (Dint and Dext), and intracellular and extracellular T2 coefficients (T2int and T2ext).
The diffusion equation describing MR signal, including T2 decay is
where C is the concentration of signal from water, D is the free diffusion coefficient, and T2 is the T2 relaxation rate. In the current model, tissue is represented as an array of cubic cells arranged at regular intervals with an intracellular volume fraction, IVF = (Lcell/Lsep)3 (Fig. 1a). Eqn.  was solved using an explicit forward-time center-spaced finite difference method on a regular mesh of grid points (Fig. 1b). The domain is extended in the diffusion-weighting direction such that an insignificant amount of water reaches the ends of the domain at the longest diffusion time, Δ. The array of cells is assumed to be periodic in the directions perpendicular to diffusion-weighting. By symmetry, zero-flux boundary conditions apply on the sides of the domain. Unless otherwise noted, simulations were run with parameters Dint = 1.0 μm2/ms [12,13], Dext = 3.0 μm2/ms, T2ext = 150 ms, IVF = 80% [18,19], Lcell = 10 μm , Pmem = 0.001 cm/s [12,13,21,22], Δ = 50 ms, and TE = 80 ms. The value of Dext was set to the free diffusion coefficient of water at body temperature . The value of T2ext was set between values measured in gray matter and CSF . However, simulations showed that ADCs were insensitive to T2ext in a range of 100 to 250 ms. The effect of values and changes in the other modeling parameters is the subject of the paper and are described in the Results and Discussion sections below. The intracellular exchange rate is given by ki = Pmem ·SA/V = Pmem ·6/Lcell, where SA/V is the surface area to volume ratio. The model was implemented in C++ with Message Passing Interface (MPI) and run on a Linux cluster.
As shown in Fig. 2, the time domain in the simulation is divided into three regimes based upon the type of signal decay occurring. In all three regimes, the signal from water decays due to T2 relaxation. However, signal decay also takes place because of diffusion between the two gradients. Since the motion and decay of water in each regime is described by Eqn. , concentration distributions are calculated with a finite difference approximation to Eqn.  for each regime, and include the effect of T2 decay. The simulation calculates an EP for the regime between the diffusion gradients, while weighting each diffusion distribution to match initial and final conditions between the three regimes. If T2 is uniform across the model tissue, the weights in the EP are also uniform. However, if there are compartmental differences in T2, the weights vary between compartments, causing a longer lived compartment to be more heavily weighted in the composite signal decay. Exchange of water across the cell membrane causes weights to vary spatially within the two compartments.
To calculate water diffusion within the arena of cells, a series of FD diffusion simulations are carried out, in which the initial condition is a unit concentration at an individual grid point, i.e. C(x x0,t = 0) = δ(x – x0). Given the symmetry of the geometry, only a few grid points in and around the central cell have to be considered. During diffusion weighting, the molecular motion of water in the direction of the diffusion gradient causes a phase shift, Δϕ, proportional to the translational displacement, given by Δϕ = γGδ (x – x0), where γ is the gyromagnetic ratio, G is the gradient pulse magnitude, and δ is the width of the gradient pulse. The signal decay S at a particular diffusion time, Δ, is then calculated by taking the expectation on the phase, eiϕ. In 1D, for simplicity, this is expressed as
The inner integral is the expectation on the phase of the diffusion distribution, while the outer integral averages over all starting locations. Changing the order of integration and introducing a change of variables gives
and the EP is
which is a measure of the average water displacement.
T2 decay between the two diffusion gradients is accounted for by including T2 in Eqn . To incorporate T2 decay outside the diffusion gradients into the model, diffusion distributions are weighted to match final and initial conditions between the three regimes in the simulation. The EP is modified to
where w1 and w2 are weights for the T2-decay regimes before and after the diffusion gradients, respectively:
The variables x1 and x2 represent the starting locations for water distributions in the T2-weighting regimes.
A change of variables, b = q2Δ , allows the signal decay to be fit to an exponential decay, S = S0e −bADC for b values between 0 and 1000 s/mm2, which is consistent with clinical convention. While the simulated decay curves are not monoexponenital over a large range of b-values, they are adequately fit to a single exponential decay over this small range.
When exchange occurs between the intracellular and extracellular compartments, water can experience both compartments over the course of the experiment, blurring the distinction between intracellular and extracellular water. An apparent intracellular EP can be produced by considering only water residing within the intracellular compartment at the time of data acquisition. From the compartment specific EP, an apparent intracellular signal decay can be calculated as described in the previous section.
Simulated signal (total signal and intracellular signal) is plotted vs. b-value in Fig. 3, for values of membrane permeability corresponding to a range from free diffusion (completely permeable) to completely impermeable. The analytical solution to signal decay in free diffusion is a monoexponential decay, which is represented in Fig. 3 as a solid line, assuming a diffusion coefficient of D = 3.0 μm2/ms. The signal decay of the intracellular compartment calculated by the FD model agrees with the Tanner et al. model (dotted line) for the case of impermeable membranes.
ADC values calculated from fitting decay curves between 0-1000 s/m2 are plotted as a function of membrane permeability in Fig. 4 at different values of Dint (3.0, 1.0 μm2/ms), T2int (150, 50, 25 ms) and Δ (10 – 70 ms). The shaded area represents physiologically relevant values of membrane permeability. In general, the ADC decreases as the cell membrane becomes less permeable, and this effect is more pronounced at longer diffusion times. With large differences between T2int and T2ext, the ADC increases as the membrane becomes impermeable. This is a combined effect of T2 filtering and exchange. As the membrane restricts the exchange of water between the two compartments, a shorter T2 in the intracellular compartment decreases the influence of the restricted intracellular compartment on the ADC. Changes in Dint have a significant effect on the ADC when the membrane is permeable or at short diffusion times, but Dint has little effect on the ADC at low Pmem and longer diffusion times.
The physiologic values of membrane permeability of water diffusion have been measured for red blood cells at a value of Pmem = 0.003 cm/s, which are considered more permeable than other cell lines [21,22]. Stanisz et al.  and Vestergaard-Poulsen et al.  both fit to membrane permeability in the range of 0.0006 cm/s to 0.0019 cm/s. In the results presented below, a value of Pmem = 0.001 cm/s was used.
ADC is plotted vs. intracellular diffusivity, Dint, in Fig. 5 at different Δ (10 – 70 ms) and T2int (150, 50 and 25 ms). Increases in Dint at short diffusion times tend to increase the ADC while at long diffusion times Dint has relatively little effect on ADC. At short diffusion times, a significant fraction of the water has not experienced the restrictive membrane and the ADC is sensitive to Dint. However, at longer diffusion times much of the water has already experienced the cell membrane, making the ADC relatively independent of Dint. This effect is independent of T2int, although ADCs are higher with decreased T2int due to T2 filtering.
The effect of intracellular T2 on the ADC is shown in Fig. 6 at different diffusion times and values of Dint. At T2int = 150ms, T2 is homogeneous in the model. In general, a decrease in T2int reduces the influence of restricted intracellular water on the signal decay, and increases the calculated ADC. The three panels represent simulations run at different values of Dint, and shows that the ADC is relatively independent of diffusion time at longer diffusion times or at low Dint.
The ADC has a strong inverse correlation with IVF, as shown in Fig. 7. The IVF is increased by swelling the cells, increasing the volume of the intracellular space at the expense of the extracellular space. Over the range of IVF investigated (0.63 – 0.90), the ADC decreases with increasing IVF, and the decrease is sensitive to differences in T2 between the intracellular and extracellular spaces. As T2int decreases, not only does the ADC increase, but the decrease in ADC with IVF also increases.
By allowing a compartmental difference in T2, a dependence of the ADC on TE could be predicted. The effect of TE on the ADC is shown in Fig. 8 at a diffusion time, Δ = 60 ms. T2 filtering is observed as an increase in the ADC with TE due to differences in T2. When T2int and T2ext are identical, there is no T2 filtering and, as would be expected, there is no dependence of ADC on TE. At T2int = 50 ms, one third that of T2ext, there is a slight dependence of ADC on TE and at T2int = 25 ms, this dependence is amplified.
The model described in this paper reproduces results of several other previously reported models and adds new insight into the potential influences of intracellular T2 on experimentally measured ADC. The model predicts an exponential signal decay from free diffusion, and accurately reproduces the well-known Tanner model of restricted diffusion in the case of impermeable membranes . With homogenous T2 (T2int = T2ext = 150 ms), diffusion models have predicted a decrease in ADC with restriction and an ADC that decreases with diffusion time , which is also predicted by the present model.
Fig. 4 plots the ADC within a complete range of membrane permeability, from free diffusion to impermeable membranes. Within the physiologic range, the change in ADC with membrane permeability depends primarily upon the T2 difference between intracellular and extracellular compartments. With homogenous T2, an increase in ADC is found with membrane permeability. Decreasing T2int reduces the change with membrane permeability, and eventually inverts the trend. A recently published model also predicted a decrease in the ADC with exchange rate when there are large differences between T2int and T2ext . The initial decrease in ADC with increasing permeability (from impermeable in the right hand column plots of Fig. 4) is a direct result of exchange affecting T2 filtering of the intracellular and extracellular compartments. Further increases in permeability reduce the effect of membrane restriction and raise the ADC to values consistent with the fast exchange limit.
The model predicts that differences in T2 between compartments can have a striking influence on the ADC. A faster decaying intracellular compartment weights the acquired signal toward the tortuous, yet ultimately unrestricted, extracellular compartment, causing the ADC to increase. By decreasing T2int relative to T2ext the predicted ADC is in the range measured clinically in normal tissue, between 0.7 and 1.0 μm2/ms. Such compartmental differences in T2 have been considered in previous diffusion models [13,25]. Szafer et al. investigated compartmental T2 filtering, but neglected T2 differences after their model suggested up to a 2x difference between compartmental T2 values had negligible effect on the ADC. More recent work has suggested larger T2 differences between compartments [13,26,27].
T2 filtering is mediated by the difference in the compartmental T2 values and the intracellular exchange rate relative to the experimental timescale. The dependence upon the exchange rate is evident in Fig. 4 where the ADC increases as the membrane becomes impermeable. Given a low exchange rate, and compartmental difference in T2, T2 filtering can be observed as a change in ADC with TE. Increases in ADC with TE have been previously measured, although the increase did not reach statistical significance . Similarly, this model predicts an increase in the ADC with TE when there are T2 differences between the two compartments. The increase is strongly dependent upon the magnitude of the T2 difference, but also dependent upon the exchange rate of water between the two compartments.
Published estimates on the compartment specific values of T2 vary widely. For instance, Assaf and Cohen hypothesize T2ext to be faster than T2int after measuring an increase in the volume fraction of a slow diffusing component of water when carrying out diffusion-weighted spectroscopic experiments with TEs from 70 to 550 ms in excised bovine optic nerve suspended in a saline solution . However, the tissue was not perfused during the experiments, making the measurements hard to relate to living tissue. Also, their experimental setup (TE range from 70 to 550 ms) would make it impossible to detect T2 populations in the few tens of milliseconds. In contrast to this, Buckley et al. predict a T2int as low as 20 ms after observing a slow diffusing component of water to decrease with TE in perfused hippocampal slices . Vestergaard-Poulsen et al. also predicted a low T2int, approximately 20 ms, by fitting experimental data obtained from gray matter to a model of diffusion in which T2 is allowed to vary within tissue compartments . Silva et al. estimated T2int to be 42 ms, but this measurement was done in the presence of a susceptibility contrast agent in the extracellular space . The only direct experimental measurements of T2int were conducted by Schoeniger et al., where they measured a cytoplasmic T2 = 34 ms in single neurons isolated from Aplysia californica . More experimental work is needed to directly measure values of T2int and T2ext to validate these findings and the results of this model.
It is often assumed that slow exchange in combination with compartmental differences in T2 should indicate a multi-component T2 decay in tissue. Such a multi-component decay has not been observed in experiments carried out in gray matter . However, the ability to detect multi-compartment T2 is dependent upon the absolute difference between compartmental T2 values, exchange between the compartments, and SNR . For example, at the biophysical values used in this work, Pmem = 0.001 cm/ms, T2int = 25 ms, and T2ext = 150 ms, the signal decay with TE (sampled at TE = 12.5 by 12.5 to 200 ms) is statistically bi-exponential (F-test rejected at the 0.05 significance threshold) above SNR ~ 200. By increasing Pmem to 0.002 cm/ms, the signal decay with TE only becomes bi-exponential above SNR ~ 600. Even with this higher membrane permeability, the trends in Fig. 7 are still observed. ADC decreases with increasing IVF and this decrease is greater at the short T2int (right panel in Fig. 7) Therefore, it is possible that compartmental differences in T2 could play an important role in filtering compartments in diffusion data, while not exhibiting multi-exponential T2 decay, even at high SNR.
Many models of restricted diffusion have predicted a decrease in the ADC of water with increasing diffusion time [22,25,31], but clinical and experimental results have shown a relatively constant ADC with diffusion time . Figs. Figs.55 and and66 show conditions at low Dint and low T2int for which the ADC is effectively independent of diffusion time. While low Dint has been previously hypothesized, allowing a low T2int further decreases the diffusion time dependence of the ADC, consistent with experimental results.
The strong decrease in ADC with IVF predicted by the present model agrees with previously published models [13,25]. Furthermore, the present model shows that the percentage decrease is strongly dependent upon the difference in T2 between the two compartments. For instance, at Δ = 50 ms, an increase in the IVF from 0.8 to 0.9 reflects only a 24% decrease in the ADC when the T2 is matched between the intracellular and extracellular compartments. However, at the shorter T2int = 25 ms, the same increase in IVF produces a 42% decrease in ADC. This dependence upon T2int is due to the decreased dependence of the ADC on unrestricted diffusion in the extracellular space as the cell swells to a higher volume fraction. Similarly, Vestergaard-Poulsen et al. predicted a large decrease in ADC with low T2int compared to T2ext, resulting in a 45% decrease while increasing the cell volume fraction from 0.81 to 0.96.
The primary observation of DWMRI after ischemia is a 30-50% decrease in the ADC of tissue water. Several changes in biophysical properties have been hypothesized to explain this observation. Table 1 presents a few possible scenarios of cellular changes in response to ischemia, and the predicted drop in ADC calculated from the FD model. In tissue, the IVF has been measured to increase from approximately 0.8 to between 0.9 and 0.95 during ischemia [18,19]. When cells swell at the expense of the extracellular space, the increase in IVF alone is responsible for an ADC decrease of 32.3% when T2int = 50 ms and 42.4% when T2int = 25 ms. A change in IVF would also tend to change the T2 from the global tissue measured by MRI. Because such a change in T2 is not observed experimentally, we have raised the T2int such that the apparent T2 is the same before and after swelling. Such an increase could result from the dilution of the intracellular space because of cell swelling. This only accounts for an additional 1.5-3.1% decrease in the ADC. As discussed earlier, inducing changes in Dint minimally affects the ADC, causing only an additional +/−2% change in the ADC with a decrease in Dint from 1.0 to 0.7 μm2/ms or an increase from 1.0 to 1.2 μm2/ms. The model also predicts that changes in the ADC with membrane permeability depend upon the difference in compartmental T2. An increase in membrane permeability, Pmem, to 0.02 cm/s could account for an additional 3.3% decrease in ADC when T2int = 25 ms, while at T2int = 50 ms the change in ADC decreases by 5.4%.
The present model predicts that, although many factors may play a role in the decreases in ADC observed following ischemia, a change in the IVF due to cell swelling is a dominant factor. When compartmental differences in T2 are present, cell swelling is sufficient to account for the 30-50% decrease in ADC observed after ischemia. Other modeling results have been interpreted to limit the effects of IVF on the ADC. For instance, Ford et al. predicted little decrease in the ADC with cell swelling when modeling diffusion perpendicular to infinitely long axonal fiber bundles . However, Ford et al. neglected compartmental T2 differences, which is required in the present model to reach the 30-50% observed in acute stroke. Also, the ADC perpendicular to axon fiber bundles has been shown to be less sensitive to ischemia than the mean diffusivity in isotropic tissue .
The effect of cell swelling on the measured ADC has also been studied experimentally. Van Pul et al. reported a 20% drop in ADC due to osmotic swelling (80% of normal osmolarity) of perfused hippocampal slices, and a 40% drop in ADC when similar slices were oxygen and glucose deprived . Similarly, Buckley et al. found only a 19% decrease in ADC after treating hippocampal slices with ouabain, which induces cell swelling by shutting down the Na+/K+ ATPase without altering metabolic activity . At first glance, these results seem to diminish the impact of cell swelling on measured ADC. However, an important detail of these studies is that they are performed at very short diffusion times, tdiff = 7 ms (Δ = 8 ms, δ = 3 ms) and tdiff = 10 ms (Δ = 11.3 ms, δ = 4 ms) respectively. At such short diffusion times, the present model predicts a small percentage decrease in ADC with increasing IVF, as shown in Fig. 7. Also, a decrease in Dint, due to reductions of energy dependent intracellular motion, is predicted to have a much larger influence on the ADC at short diffusion times, as shown in Fig. 5. Therefore, this model predicts that reductions in the ADC following ischemia, measured at very short diffusion times, would be heavily weighted towards changes in Dint and less sensitive to changes in IVF, consistent with these experimental observations.
The intracellular diffusion coefficient is an important parameter, as many groups hypothesize that it plays a role in the decrease in ADC observed following ischemia [5,6]. Many indirect measurements on intracellular metabolites and ions have led to the hypothesis that the intrinsic diffusivity of intracellular water decreases after ischemia, and is the driving force behind the decrease in the ADC of water [5,36]. Given the experimental evidence that IVF and Dint are both affected in ischemia, this model agrees with the partial and complete decrease in ADC with cell swelling and ischemia, respectively, as observed by van Pul et al. and Buckley et al., as well as the ADC drop after ischemia at short diffusion times measured by Does et al. . However, recent work in cell cultures has shown an increase in the ADC of intracellular water after ischemia , which may indicate that trends in the ADC of water are not be coupled to the ADC of intracellular metabolites and ions at clinical diffusion times. This agrees with our model which predicts that, at more clinically relevant diffusion times (Δ > 30 ms), reductions in ADC would be insensitive to changes in Dint and more sensitive to changes in IVF.
While providing insight into the biological properties affecting water diffusion in tissue, and their influence on measured ADC, the model has limitations. While physiologically appropriate values for cell volume fraction and cell size were used, precise values for Dint, Dext, T2int, T2ext, and Pmem for cells in gray matter are not known. Because the results of modeling are sensitive to these parameters, particularly T2int and T2ext, the predictive value of this work may be limited by the accuracy of these parameters. Furthermore, the inherent complexity of gray matter, such as variations in cell shape, size, and density are ignored, and the model neglects the presence of any dendrites, unmyelinated axons, and axon terminals. Further work is needed to explore the effects of these factors on the measured ADC in tissue.
In this work, clinically observed DWMRI results have been predicted using physiologically realistic parameters by a mathematical model that incorporates the effects of membrane permeability as well as differential diffusion coefficients and T2 relaxation times between intracellular and extracellular spaces. The model predicts that the biophysical parameters influencing ADC measurements are highly dependent on the diffusion times at which experiments are carried out. At longer, clinically relevant, diffusion times the ADC is very sensitive to changes in the IVF and insensitive to Dint. At short diffusion times, however, the ADC is less sensitive to IVF and more sensitive to Dint. Using the model, the decrease in ADC following ischemia measured clinically can be fully accounted for by cell swelling alone when the T2 relaxation time is longer in the extracellular spaces than within the cells.
This work supported by NIH grants GM57270 and CA88285 and Arizona Biomedical Research Commission grant 8-078. The authors are grateful to the University of Arizona High Performance Computing Facility for sponsored computing resources, and to Mr. John Totenhagen for helpful discussions.