The IMPULSED (imaging microstructural parameters using limited spectrally edited diffusion) model combines multiple low-frequency OGSE measurements (f_OGSE < 150 Hz) and a single PGSE acquisition in the long time regime (Figure 4A) to quantify the characteristic size of restriction and ICS fraction [14, 46].

This approach was shown successful in estimating cancer cell size in vitro in the range (5–10) μm using only a small subset of measurements on murine (MEL) and human leukemia cells (K562) [14]. In vivo, the correlation between histology and IMPULSED-based cellularities were found superior than between histology and conventional PGSE measurements, in three different colorectal cancer xenograft tumor models (DiFi, HCT116, and SW620) [46].

This model assumes that the ECS diffusion varies linearly with frequency f_OGSE in the range 50–150 Hz. This assumption was motivated by (i) the empirical linear behavior of the overall ADC (intra- and extracellular) measured in the healthy mouse brain [34] and (ii) simulations in extra-axonal space derived from histology samples [43]. Unfortunately, this would only be valid of a two-dimensional problem (d = 2 in Equation 3) and the correct formula for the ECS diffusion around spheres at long times is given by Equation (5) instead. However, the linear approximation can be considered as an approximation in a narrow frequency range, with little impact on estimated parameters.

The POMACE (Pulsed and oscillating gradient MRI for assessment of cell size and extracellular space) model combines multiple OGSE and PGSE measurements in different time domains (Figure 4B). Microstructural parameter estimation is performed in two steps. The surface-to-volume ratio and free diffusivity are first evaluated using high-frequency OGSE in the short-time regime [18] using Equation (2). These values are then used as constraints when fitting the low-frequency OGSE and PGSE data (Figure 5, f_OGSE < 88 Hz) to a model of impermeable spheres bathing in ECS [15].

Using a dedicated histology coil [47], the validity of POMACE was tested ex vivo. ICS maps correlated well with optical microscopy performed on the same samples used for MRI [15]. In vivo, ICS estimates were found in agreement with ECS estimates from an effective medium theory [25], while cell sizes matched electron microscopy measurements in mice gliomas (GL261).

The POMACE framework was later applied to the in vivo assessment of treatment response in GL261 gliomas and 4T1 mammary carcinomas [48]. Following tumor treatment with 5FU and bevacizumab, a significant ECS decrease was observed with POMACE, while the absence of impact on S/V or cell radius suggested partial membrane deterioration and/or a decrease of the apparent restrictive surface due to increased cell packing in both cell lines.

VERDICT (vascular, extracellular, and restricted diffusion for cytometry in tumors) is the only model to consider the impact of tumor vasculature on the directionality of diffusion (Figure 4C). Cancer cells are modeled by spheres, the extracellular diffusivity by an isotropic diffusion tensor, and the vascular compartment by an additional highly anisotropic tensor [16], although its precise form can vary depending on the application [49].

This more complex modeling comes at the expense of a large number of parameters to estimate. To ensure fit robustness, the free diffusivities in the ICS and ECS are fixed. Six independent parameters are estimated: intracellular and extracellular volume fractions f_ics and f_ecs, cell size R, the pseudo-diffusion coefficient of water inside blood vessels P, and two angles characterizing the directionality of the vascular compartment. The intravascular fractions is then calculated as f_v = 1 − f_ics − f_ecs.

This model successfully differentiated two human colorectal carcinoma cell lines based on their vascular fraction [16]: SW1222 xenografts exhibited dense perfusion (f_v = 0.22) while LS174T (Figure 5) were properly categorized as densely packed (f_ecs < 0.05) with low perfusion (f_v = 0.12). In addition, significant changes in intravascular and intracellular volume fractions were observed in response to a chemotoxic agent leading to cell apoptosis (gemcitabine), as confirmed by flow cytometry [16].

Focusing on PGSE acquired at several diffusion times (10–40 ms), diffusion contrast is plentiful and VERDICT can be easily adapted to a clinical setting. After preliminary work on model selection, the previous model was modified to (i) consider the vascular compartment as isotropic and (ii) fix the free diffusivities and pseudo-diffusion coefficient to 2 and 8 μm²/ms, respectively. In vivo, the new model (with only three independent parameters) was able to distinguish tumor from benign prostatic areas in eight patients at 3T under acceptable scan times (35 min) [49].

A prospective study—INNOVATE [50]—recently started combining the VERDICT framework with novel blood and urine sampling based potential biomarkers in an attempt to affine patient screening and promote the use of multi-parametric MRI before biopsy for the diagnosis of prostate cancer. Patient follow-up is needed before assessing the potential improvement in patient care by diagnosing early aggressive prostate cancer.

A linear increase of ADC vs. OGSE frequency was reported in the ex vivo mouse brain in the range 0–150 Hz [34]. Regions of large ADC changes (Δ_f ADC) colocalized well with Nissl staining and densely packed neuronal regions, suggesting a link between ADC dispersion and ICS volume and/or cell size.

A theoretical justification for this effect can be found in Equation (7) and Figure 3. At low frequency OGSE, the intracellular diffusion in small structures (R ≤ 5 μm) does not approach the asymptotic short-time limit and can be considered linear with f_OGSE, as a first approximation in a narrow frequency range. Fixing D₀, the slope of this linear relationship increases with cell size (Figure 3A) and ICS volume, assuming slower diffusion time-dependence in the ECS.

The ADC dispersion rate averaged in the range 50–250 Hz was shown sensitive to treatment of colorectal tumor SW620 with barasertib (AZD1152) [51], known to induce the formation of new chromosomic structures at subcellular level, increased cell size and eventually apoptosis [52].

A closer look on Figure 2C highlights that the instantaneous dispersion rate ∂D/∂f is non-monotonous with OGSE frequency. A maximum is reached around 0.4×D0/R2, suggesting potential for characterizing the tissue characteristic restriction scale. This was demonstrated in vitro using two cell lines with different radius (R = 5/10 μm for MEL/K562) with significantly different instantaneous dispersion rate around 60 Hz [53].

Although ex vivo experiments performed on kidney and liver tissue highlighted very little contrast with dispersion rate compared to conventional ADC [53], these result are dependent on sample preparation and fixation, and should be reproduced in vivo. Larger diffusivities might shift the oscillation frequency range of interest.

As discussed in the first section, the universal behavior of short-time diffusion is described for PGSE and OGSE by Equations (1) and (2). In this regime, any system can be considered made of two spin populations. Regardless of the particular geometry, some random walkers will never experience the cell walls (and freely diffuse with D₀) while the displacement of the population within diffusion length of the wall (with volume fraction: height×surface/volume=D0t×S/V) will be restricted. At such short times, neither the curvature nor the permeability of the cell walls impact diffusion [10, 19].

The validity of the short diffusion-time regime was demonstrated in vivo and ex vivo in mice gliomas (GL261, R~5 μm) in the range 88 Hz ≤ f_OGSE ≤ 225 Hz [18]. The decoupling of diffusive and geometric properties was assessed ex vivo by varying the sample temperature, only impacting the term D₀ in Equation (2). Parametric maps of S/V and D₀ were easily accessible in vivo, paving the way for robust—thanks to linear fitting—and unambiguous interpretation of TDD in tumors.

Potential applications in a clinical setting include characterization of breast cancer. Recently, this regime was demonstrated in vivo at 3T in healthy breast tissue using STEAM and diffusion times in the range 80–900 ms [54], following up on muscular studies [55, 56] with similar restriction scale (hundreds of microns).

On the other side of the spectrum, the effective medium theory (EMT) only focuses on the macroscopic properties of tissue. At long times, molecules have diffused around and inside each structure, so that microscopic information such as cell size is lost. Using an EMT analogous to that of conductivity in porous media, the long time limit of diffusion in a biological system {permeable spheres + ECS} was derived [17]. In the impermeable case, Equation (2) from Latour et al. [17] becomes:

where f is the ICS volume fraction and D₀ the free extracellular diffusivity. Since microstructural information is lost, changing cell shapes should not affect Equation (8).

This EMT establishes the well-known relationship between PGSE measurements at long times and cellularity for a simple system [3, 4]. Provided the cell size is of little interest, estimating the tortuosity limit with multiple PGSE in the long time regime (Figure 1C) is indeed an alternative way of estimating the size of the ICS. A priori knowledge on D₀ is however required to quantify f using Equation (8). Additional information on D₀ can be gathered in the short-time regime using Equations (1) and (2).

An EMT approach was successfully demonstrated in mice gliomas using only four diffusion times (6–31 ms) [15]. ICS estimates were found in excellent agreement with that of POMACE, fewer acquisitions were required, and fit estimates found very robust. Unfortunately, cell sizes could not be estimated using this technique.

Accuracy represents the closeness of fit estimates compared to the ground truth. In the absence of a ground truth, a commonly used approach is to generate synthetic data based on the model, add noise, and compare the “noisy” fit outputs to the initial “clean” input. For preclinical brain studies, typical in vivo SNR values range were reported between 100 and 150 [15].

We consider here the case of impermeable spheres within the ECS. As in POMACE, the ECS is modeled in the tortuosity limit for low-frequency measurements, and in the short time regime for high-frequency OGSE acquisitions. The problem is further simplified by initiating the fitting algorithm from the ground truth in order to minimize the influence of local minima when estimating parameters. Synthetic data is generated from the parameters best describing TDD in murine glioblastoma [15].

Without PGSE, a large range of solutions emerge from noise propagation (Figure 6A) despite high SNR (SNR = 120) and multiple OGSE time-points (10 f_OGSE steps from 60 to 225 Hz). This model is obviously not well suited to characterize tumor microstructure accurately, its solutions are not centered on the ground truth.

This issue can be resolved here by increasing the SNR or incorporating additional data points (b-values and/or diffusion times), for instance PGSE measurements. Incorporating a priori knowledge on the system—by fixing one parameter—will also help by reducing the degree of freedom of the problem. A combination of the last two approaches was chosen to improve the robustness of the POMACE framework [15], as can be seen in Figure 6B. Another “angle” can be to use the directionality of diffusion, only useful when properly accounted for in tissue modeling [16].

Similarly, synthetic diffusion data was generated using the IMPULSED framework [46] in order to mimic TDD in colorectal tumors (see Table 1). Multiple instances of gaussian noise (typical in vivo SNR = 120, n = 2,500) were added to the signal before fitting. Although the distribution of fit estimates were not found normal (Figures 6C,D), average fit estimates matched the ground truth (SNR = ∞) with good accuracy for most parameters (Table 1, relative bias below 3/13% for f, R, D_ics for DiFi/HCT116 cell lines). The matlab code used to generate synthetic tumor data and plot parameter accuracy with POMACE and IMPULSED is readily available for download at https://github.com/oreynaud/FIT_TDD.

The primary objective behind the development of TDD was to attribute the changes in the diffusion signal to specific microstructural metrics, without ambiguity. Even if the fit is accurate, microstructural variations—in space or time—can only be reliably estimated if they exceed the fit precision, defined by the reproducibility of parameter estimation.

Small changes in volume fraction are likely to be picked up by POMACE (Figure 6B and standard deviations in Table 1), due to the large amount of data acquired in the long time regime. On the other hand, the IMPULSED framework is well suited to detect small variations in cell size (Table 1) and would benefit from a reduced scan time. Results might depend on the particular microstructure, as illustrated by the different precision available on diffusivities estimates between the two colorectal cell lines.

In general, it appears unreasonable to attempt to detect variations below the following thresholds: Δf _min = 3% and ΔR_min = 0.5 μm. Since low image SNR is extremely detrimental to the fit precision, smoothing and/or averaging the signal within regions of interest might be preferred to raw single-voxel parametric mapping in order to enhance the robustness and specificity of the analysis. Special care should however be taken in very heterogeneous tumors.

Cell permeability does not impact diffusion measurements at very short times: Equations (1) and (2) are always valid regardless of cell permeability κ. However, as time increases, diffusion departs from the previous equation and can be expressed as in Sen [19] and Sen [58]: (9)D1(t)=D1(1−S1V1[4D1t9π−D2(D2+D1)6D1κt                 − ρt6+D1t12〈1R1+1R2〉R]·) + O(D0t3/2) Here ρ is the surface relaxivity, and D_i, S_i/V_i, and R_i the free diffusivity, surface-to-volume ratio, and radius of curvature of compartment i = {1,2}. A similar expression describes the diffusion in the second compartment, by interchanging the subscripts {1,2} and the sign of the last term 〈1R1+1R2〉R, representing the average inverse curvature radius over the interior surface.

The models simplifies under two assumptions: ρ ≪ κ (true for most biological systems) and D₁ = D₂. The curvature terms cancel each other in the overall diffusion D = fD₁ + (1 − f)D₂. From there follow that the linear dependence of the diffusion directly represents the influence of permeability κ. Estimates are weighted by the surface-to-volume ratio counted twice, as random walkers explore the walls both from inside and outside the cells.

Using the diffusivity and permeability calculated for packed erythrocytes (red blood cells) in Latour et al. [17], Sen [19] estimated that permeability only becomes a relevant model parameter when diffusion times approach or exceed 60 ms.

Since in vivo diffusion deviates from the short-time limit regime around f_OGSE = 88 Hz for cancer cells [18], one could wonder whether permeability might already impact low-frequency diffusion measurements. However, the mismatch between experimental data and Equation (2) could not be fitted by a linear relationship with f_OGSE.

The impact of permeability on diffusion at long times can be derived using the EMT proposed by Latour et al. [17]: (10)(D∞−D*Decs−D*)×(DecsD∞)1/3=(1−f),with                                         D*=DicsκR/(κR+Dics) Equation (10) was successfully used to estimate membrane permeability in bovine red blood cell samples around 6.3 × 10⁻³ cm/s [17]. Diffusion results were found well in agreement with extensive literature in red blood cell permeability [59, 60].

This equation simplifies for f = 1 (i.e., no ECS) to the well-known equation derived by Tanner [61] for a stack of flat layers with characteristic length R: D-1=Dics-1+(κR)-1. This approximation was later used to accurately measure cells permeability in yeast suspensions [62].

The lesser tortuosity expected from permeable cells according to Equation (10) was observed experimentally on human leukemia K562 cells treated with saponin [63], for the multiple diffusion times and oscillation frequency available on preclinical scanners.

The impact of non-zero permeability on parameter estimation was simulated using a finite difference method within the IMPULSED framework [64]. The robustness of most fit estimates (f, R, D_ics) was demonstrated under two conditions: the image SNR must remain large (≥50) and the water exchange time τ–related to permeability via κ-1=3τ/R-R/5Dics – must exceed 100 ms.

Such results would in all likelihood hold for other frameworks, provided tissue exchange times exceed the longest diffusion times used to probe diffusion. In cancer, water residency times were estimated around hundreds of milliseconds [65–67], suggesting that permeability could bias but not severely impact parameter estimation performed using PGSE and OGSE. However, changes of cell permeability due to treatment during longitudinal studies could impair the specificity of the model via the apparent variation of other microstructural estimates, such as cell size and ICS diffusivity.

The analytical solutions for the diffusion inside impermeable spheres D_spheres(R, D₀) or between planes D_planes(R, D₀) predict D = 0 at infinite times. Rather than evaluating the contribution of a second—extracellular—compartment, a handful of studies attributed the disparity between these models and the observed diffusion (D_∞ ≠ 0) to membrane permeability, and modeled the TDD of a biological system using empirical formulas such as: D = D_∞ + D_{planes/spheres}(R, D₀ − D_∞) [28, 35]. It should be noted that permeability cannot be estimated using those models.

Such models can be used to sensitize MRI to small tissue changes thanks to a restricted number of fit parameters, and were shown useful in assessing tumor treatment efficacy in two ovarian human cell lines (OVCAR-8 and NCI/ADR-RES) [35].

However, interpretation of the results is limited as only a mere qualitative insight into tissue structural changes is possible. Without ECS, the pseudo-intracellular diffusivity D₀ − D_∞ is void of physical meaning. At best, R can represent a “restriction index”, based on D₀ − D_∞ and the characteristic oscillation frequency (Figure 2C).

In cancer, randomly oriented flat membranes represent a more realistic model than a stack of flat layers, for which a solution accounting for permeability can be derived [20]. Using the EMT formalism for the diffusion signal proposed in Novikov and Kiselev [31], D(t) is related to the dispersive diffusivity D(ω) via: (11)D(t)=1t∫dω2πe−iωtD(ω)(ω+i0)2 and D(ω), for random permeable barriers, is described by Novikov et al. [20]: (12)D0D(ω)=1+ξ+2zω(1−zω)[1+ξ/(1−zω)2−1] where ξ represents the effective volume fraction of membranes via ξ = S/V × D₀/2κd in d dimensions, and zω=iiD0/2κ is a dimensionless frequency.

Although never applied in tumors, the RBPM geometry is well suited for muscle studies [56], and tissue permeability and cell size were recently estimated in vivo and on clinical scanners [68]. This approach could provide an interesting approach to characterizing sarcomas using TDD in the near future.

A promising alternative to TDD for characterizing cell permeability might lie in apparent exchange rate/filter exchange imaging (FEXI) [69, 70]. In FEXI, a stimulated-echo double diffusion encoding sequence uses two PGSE diffusion blocks separated by a mixing time t_d, during which exchange between intra- and extracellular compartments (where diffusion is assumed to be approximately Gaussian) occur. The water exchange rate is estimated by measuring a mono-exponential decay of diffusion with mixing time t_d [69].

The clinical potential of FEXI was first assessed in the brain, in both healthy and brain cancer patients, where viable and necrotic parts of the tumor could be clearly differentiated based on exchange rate [70]. More recently, FEXI was shown capable of differentiating two brain cancer types (astrocytomas vs. meningiomas) in vivo based on exchange rate using only a small sample size (5–10 subjects) [71]. In breast cancer, FEXI could differentiate between multiple cell lines in vitro, while its potential for in vivo imaging was also demonstrated [72].

For a given gradient strength and duration, bOGSE∝fOGSE-3 for cosine OGSE [27]. As a result, reasonable contrast at large oscillation frequencies can only be achieved by compensating the lesser temporal window allow for molecular diffusion by stronger dephasing, i.e., stronger gradient strength.

This sets an upper bound limit for the frequency of OGSE measurements around 300–350 Hz (using b_OGSE = 0.4 ms/μm² and typical echo times) on preclinical scanners equipped with diffusion-friendly gradients (1 T/m). This in turns sets a lower limit for the range of restriction scales that can be probed using TDD around R~D0/fOGSE ~ 2 μm.

As a result, most preclinical applications of TDD so far have focused on relatively large structures, such as brain glioma or colorectal cells (R ~ 4–20 μm). An obvious downside is that—in the brain—the comparison of diffusion-based tumor microstructural metrics with healthy tissue remains out of reach, since TDD is not adapted to characterize normal brain tissue structure (white/gray matter).

Although sinusoidal OGSE exhibit larger b-values than their cosine counterpart, a DC component is introduced into the frequency spectrum, effectively mixing conventional PGSE and OGSE measurements [27]. Compared to cosine OGSE, the gain in diffusion contrast does not originates from the frequencies of interest.

The situation worsens for clinical scanners, where f_OGSE = 100 Hz can only be achieved with b < 120 s/mm² (3T, gradient strength 80 mT/m). While intra-voxel incoherent motion effects [73] do not affect cosine modulated OGSE or other sequences with no sensitivity to the zero frequency of the diffusion spectrum relating to translation, this results in poor diffusion contrast. In addition, diffusion is in that range already highly restricted in small structures, and microstructural information cannot be retrieved using diffusion time-dependence [74]. The development and availability of high gradients systems is crucial to the eventual translation of the full TDD potential to the clinic.

On the contrary, clinical diffusion—using STEAM and PGSE—is already well adapted to characterizing breast and muscle tissues, where the restriction scale approaches hundreds of microns. TDD applications in sarcomas and breast cancer are well within reach of the current hardware systems, and are expected to flourish over the next few years.

In addition, the potential success of the INNOVATE study [50] on a large cohort could represent a tremendous springboard for prostate cancer characterization using TDD, as well as a major billboard for promoting TDD applied to various forms of cancer.