Diffusion time dependency of extracellular diffusion

Abstract

Purpose:

To quantify the variations of the power-law dependences on diffusion time $t$ or gradient frequency $f$ of extracellular water diffusion measured by diffusion MRI (dMRI).

Methods:

Model cellular systems containing only extracellular water were used to investigate the $t/f$ dependence of $D_{ex}$, the extracellular diffusion coefficient. Computer simulations used a randomly packed tissue model with realistic intracellular volume fractions and cell sizes. DMRI measurements were performed on samples consisting of liposomes containing heavy water (D₂O) dispersed in regular water (H₂O). $D_{ex}$ was obtained over a broad $t$ range (~1 – 1000 ms) and then fit power-law equations $D_{ex}\left(t\right)=D_{\text{const}}+\text{const}\cdot t^{-{\vartheta }_t}$ and $D_{ex}\left(f\right)=D_{\text{const}}+\text{const}\cdot f^{{\vartheta }_f}$.

Results:

Both simulated and experimental results suggest that no single power-law adequately describes the behavior of $D_{ex}$ over the range of diffusion times of most interest in practical dMRI. Previous theoretical predictions are accurate over only limited $t$ ranges e.g. ${\theta }_t={\theta }_f=-\frac{1}{2}$ is valid only for short times, while ${\theta }_t=1$ or ${\theta }_f=\frac{3}{2}$ is valid only for long times but cannot describe other ranges simultaneously. For the specific $t$ range of 5 – 70 ms used in typical human dMRI measurements, ${\theta }_t={\theta }_f=1$ matches the data well empirically.

Conclusion:

The optimal power-law fit of extracellular diffusion varies with diffusion time. The dependency obtained at short or long $t$ limits cannot be applied to typical dMRI measurements in human cancer or liver. It is essential to determine the appropriate diffusion time range when modeling extracellular diffusion in dMRI based quantitative microstructural imaging.

Introduction

Diffusion MRI (dMRI) based quantitative microstructural imaging aims to derive specific histopathological information on biological microstructures non-invasively (1). DMRI may acquire data using multiple $b$ and/or multiple diffusion times ($t_{\mathit{\text{diff}}}$ or $t$) and a variety of diffusion gradient waveforms (2), and then fit biophysical models to dMRI signals to extract quantitative microstructural parameters such as mean cell size (3–5) or cell size distribution (6). Such information has significant clinical potential such as providing a more specific characterization of tumor status (7–9) and monitoring earlier and more accurate tumor response to anti-cancer treatments (10–13).

Biological tissues comprise intracellular and extracellular spaces, so any realistic dMRI-based quantitative model should separate signals arising from these major, separate compartments. A reasonable strategy is based on the remarkable differences in the constraints on free diffusion within the two environments. Intracellular diffusion is usually restricted due to the enclosure of cell membranes. If cells are modeled as simple impermeable geometries such as cubes, cylinders, spheres (14), composite spheres (15), or ellipsoids (16), analytical equations can be derived to predict how dMRI signals depend on model parameters such as cell size and experimental parameters such as diffusion weighting factor $b$ and diffusion time $t$. These equations have been successfully validated using computer simulations and cell experiments for several diffusion sequences, such as PGSE (17), sin- or cosine-modulated OGSE (18), and sin- (19) and cosine-modulated (20) trapezoidal OGSE.

By contrast, modeling extracellular diffusion is more challenging due to the nature of the complex geometry of extracellular space. Early studies typically assumed extracellular diffusion could be considered as highly confined in narrow, irregular, interstitial spaces, so the extracellular diffusion coefficient $D_{ex}$ reaches a long diffusion time limit that is a constant related to tortuosity (21). This approximation was included in several modeling studies despite their use of a broad range of diffusion times (16,18,22). Computer simulations with regularly packed cylinders or spheres were performed to validate this approach and achieved reasonable results (14,16,18,23). However, assuming regular packing is unrealistic and recent studies have shown that this assumption may lead to faster signal decays which then artificially reduce the real diffusion time dependence (24,25). Further experiments have shown pronounced diffusion time dependences at least with short (26) and long (27,28) diffusion times. Therefore, it is more accurate to model extracellular diffusion not as a constant but as a function of diffusion time. This has been further confirmed by studies that included a diffusion time dependence of extracellular diffusion which thereby produced improved estimates of the sizes of axons (29) and cancer cells (4).

To date, a power-law fitting has been widely used to describe how extracellular diffusion depends on diffusion time, i.e., $D_{ex}\left(t\right)=D_{\text{const}}+\text{const}\cdot t^{-{\vartheta }_t}$ for PGSE acquisitions, or $D_{ex}\left(f\right)=D_{\text{const}}+\text{const}\cdot f^{{\vartheta }_f}$ if OGSE is used, where $f$ is the frequency of oscillating gradients. Note that it is usually believed that $f~1/t$ (30–32). Within the short $t$ range, Mitra et al. argued that ${\vartheta }_t=-\frac{1}{2}$ (i.e., $D_{ex}\left(t\right)-D_{t=0}~\sqrt{t}$) (33) or ${\vartheta }_f=-\frac{1}{2}$ (i.e., $D_{ex}\left(f\right)-D_{f\to \infty }~f^{-\frac{1}{2}}$ (31,34). Within the long $t$ range, Novikov et al. showed that PGSE-obtained $D_{ex}\left(t\right)-D_{t\to \infty }~t^{-1}$ (different from the instantaneous $D_{ins}\left(t\right)-D_{t\to \infty }~t^{-\frac{3}{2}}$) (35,36) or OGSE-obtained $D_{ex}\left(f\right)-D_{f\to 0}~f^{\frac{3}{2}}$ for three-dimensional, randomly-packed systems (24). However, diffusion times of practical interest are within a range of ~5 – 70 ms on regular human MRI systems. It remains unclear if this range should be considered short, long, or neither, for extracellular diffusion of water in typical biological systems, and whether either of these theoretical predictions is valid for realistic applications. This is not a trivial question in practical imaging since it affects the accuracy of extracellular diffusion modeling and the derivation of quantitative microstructural information.

The current work aims to elucidate the diffusion time dependence of extracellular diffusion over a broad range of diffusion times ~1 – 1000 ms, and investigates how the power-law fit of extracellular diffusion is dependent on $t$ and $f$ empirically, particularly those available in typical human dMRI measurements. Both computer simulations and MR experiments with a liposome phantom system were performed with signals arising from extracellular space only, which provides an opportunity to exclude any possible contamination from intracellular diffusion and significant intrinsic background gradients. Note that biological tissues are highly heterogeneous with a large variety of different arrangements, shapes, sizes, and orientations. Therefore, this work will focus on mimicking typical isotropic, solid tissues such as extracranial human tumors (8) or livers (37).

Methods

1.1. Computer simulations

1.1.1. Tissue model

Previous computer simulation studies usually used tissue models consisting of regularly-packed (such as face-center cubic for spheres), mono-sized, and hard-surfaced cells with shapes such as cubes (14), spheres (15), or truncated octahedrons (21). These tissue models have three main limitations: (i) real tissues contain a distribution of cell sizes; (ii) regular packing results in unrealistic diffusion time dependences for extracellular diffusion; and (iii) the intracellular volume fraction is usually lower than most tissue values (such as ~75% in the brain (21)). To overcome these limitations, we propose a new approach to generating tissue models for computer simulations. Briefly, it includes three steps

1. Create a distribution of spheres with a range of radii with a reasonable mean cell diameter such as 10 μm as shown in Figure 1A.

2. Perform random-close packing of polydisperse hard-surfaced spheres based on the approach developed in (38).

3. Allow cell “swelling” to further increase intracellular volume fraction. Specifically, cell boundaries are allowed to be compressed when encountering neighboring cells. This can further increase the intracellular volume fraction to 75% or above. A diagram of this process is demonstrated in Supporting Information Figure S1.

Figure 1.

A representative distribution of cell radii (A) and the diagram of the corresponding randomly-packed tissue model used in the simulations (B) with an intracellular volume fraction = 75%.

A representative tissue model of an image voxel is shown in Figure.1B with an intracellular volume fraction of 75%. Note that this tissue voxel will be repeated in three-dimensional space to mimic influences from neighboring voxels. A revised periodic boundary condition can be used in simulations to deal with the linearity of diffusion gradients in periodic structures (39).

1.1.2. Finite difference simulation

A finite difference method was used to simulate diffusion and estimate diffusion coefficients as a function of diffusion time (39). The computation domain was a 128×128×128 matrix with a spatial step of 1 μm and a temporal step of 0.05 ms. Proton concentrations were 0 and 1 in the intra- and extracellular spaces, respectively. T₂ relaxation was homogenous everywhere and the cell membrane permeability was assumed to be zero. A broad range of diffusion times 1 – 1000 ms was used with $b$ = 300 s/mm². Three types of diffusion pulse sequences were simulated (Because PGSE and STEAM use the same effective bipolar diffusion gradients, only PGSE was simulated)

• 1PGSE with $\delta$ = 12 ms and $\Delta$ = 14 – 1004 ms.
• 2PGSE with $\delta$ = 1 ms and $\Delta$ = 2.3 – 1003.3 ms.
• 3Cosine-modulated trapezoidal OGSE with gradient duration 20 – 40 ms and the number of cycles $n$ = 1 – 5, corresponding to diffusion times 1 – 10 ms.

1.2. MR experiments of liposomes

1.2.1. Design of an experimental model system with extracellular diffusion only

Because packed cells contain signals arising from both intra- and extracellular compartments, previous studies have proposed packed glass or polystyrene beads (40) to investigate extracellular diffusion only. However, there are a few limitations on packed beads: (i) there are limited choices of discretized bead sizes. This leads to a relatively low intracellular volume fraction with random packing of close to monodispersed spheres (38). (ii) beads may cause strong background gradients that cause additional dephasing of MR signals, particularly for STEAM with long diffusion times. To overcomes these limitations, we here introduce a novel liposome model system to mimic cellular tissues without intracellular diffusion signals.

Liposomes are vesicles most often composed of phospholipids. Although most types of liposomes are very small (< 500 nm), MVL (multivesicular liposomes) can be 1 – 50 μm, covering typical cell sizes. The liposome enclosed space may be occupied by heavy water (D₂O, deuterium oxide) that does not contribute signals to proton dMRI measurements, so a phantom may be made that contains extracellular diffusion signals only. Also importantly, these liposomes can be randomly packed with high packing density and without significant background gradients, all of which mimic realistic tissues. Figure 2 shows a diagram of this model system. Cholesterols are added not only to strengthen bilayer membranes (41) but also to reduce any possible membrane permeability (42,43). Note that micelles are also produced during liposome preparation. They are hydrophobic and are small (< 1 μm) so they serve as obstacles to extracellular diffusion.

Figure 2.

Diagram of the liposome model system. All liposome enclosed space is occupied by heavy water (D₂O) that does not contribute to proton dMRI signals. Hence, measurements are determined by interstitial extracellular regular water (H₂O) only.

1.2.2. Sample preparation

Multivesicular liposomes (MVL) were prepared with a double-emulsion procedure developed by Kim et al. (41). Briefly, phospholipids, cholesterol, and oil mixtures dissolved in an organic solvent were emulsified with D₂O twice. Various sized MVLs with D₂O inside all compartments were formed after the removal of organic solvent. The interstitial D₂O was then replaced with regular H₂O by concentration, filtration, and washing. More details about the procedure are provided in the Supporting Information. Six MVL samples were obtained by varying packing density, size range, and degree of micelle removal. This provides a variety of liposome properties that may affect diffusion measurements.

1.2.3. MR measurements

All MR experiments were performed on a 4.7T Varian/Agilent small animal scanner (Agilent Technologies, Santa Clara, CA) with a 10 mm, homemade volume coil for both transmit and receive. A 4-mm-thick slice crossing the center of the liposome sample was acquired. Single-shot EPI was used for acquisitions with OGSE, PGSE, and STEAM to cover short, intermediate, and long diffusion time ranges. TR/TE = 3500/102 ms. For OGSE acquisitions, $\delta /\Delta$ = 40/45 ms, number of cycles 1 – 10, corresponding to diffusion times ~ 1 – 10 ms. For PGSE, $\delta$ = 3 ms and $\Delta$ = 11 – 101 ms, corresponding to diffusion times ~ 10 – 100 ms. For STEAM, $\delta$ = 3 ms and $\Delta$ = 201 – 1001 ms, corresponding to diffusion times ~ 200 – 1000 ms. All diffusion measurements used $b$ = 300 s/mm² and were repeated twice with the gradient direction reversed, and the geometric means of the two signals were used to calculate ADC to remove the cross-terms of background gradients. A doped water phantom with 0.1 mM MnCl₂ was used to ensure there is no artificial diffusion time dependence caused by hardware or pulse sequence imperfection as shown in Figure S2 in Supporting Information.

After dMRI measurements, liposome samples were filtered to remove interstitial H₂O, repacked with D₂O, and then scanned with another quick MR experiment to check if any proton signals from liposome enclosed intracellular space.

1.3. Data analysis

Power-law equations $D_{ex}\left(t\right)=D_{\text{const}}+\text{const}\cdot t^{-{\vartheta }_t}$ and $D_{ex}\left(f\right)=D_{\text{const}}+\text{const}\cdot f^{{\vartheta }_f}$ were fit to both simulation and experimentally derived extracellular diffusion coefficients. Three specific diffusion time ranges were used in the data fittings:

• 4Short diffusion time range: $t$ < 5 ms and $f$ > 50 Hz
• 5Intermediate diffusion time range: 5 ≤ $t$ ≤ 70 ms and $f$ < 50 Hz. Note that this is the diffusion time range available on typical human MRI systems (6,20).
• 6Long diffusion time range: $t$ > 200 ms.

Mitra’s (${\vartheta }_t$ or ${\vartheta }_f=-\frac{1}{2}$) and Novikov’s (${\vartheta }_t=1$ or ${\vartheta }_f=\frac{3}{2}$) equations were fit only within the short and long diffusion time ranges, respectively. The fitted equations were then extrapolated across the whole diffusion time range to evaluate to what extent these equations can explain the data. No constraints were enforced during the fittings of ${\theta }_t$ and ${\theta }_f$ with the practical intermediate diffusion time range. Moreover, to investigate the optimal exponents (${\vartheta }_t$ and ${\vartheta }_f$) within the practical diffusion time range used in MR cell size imaging on regular human MRI, $D_{ex}$ with diffusion times 5, 10, and 70 ms (20) were fit with fixed ${\theta }_t$ and ${\theta }_f$ values, sweeping the ranges of $-3\le {\vartheta }_t\le 3$ and $0\le {\vartheta }_f\le 3$, respectively, with an interval of 0.1. Note that because PGSE $f$ is usually assumed to be 0, $f^{{\vartheta }_f}(f=0)\to \infty$ if ${\vartheta }_f<0$. Hence, ${\vartheta }_f$ can be only zero or positive by theory. All fittings were performed in MATLAB with the fit function with random initial values.

Results

Figure 3 shows the simulated extracellular diffusion coefficient $D_{ex}$ dependence on diffusion time $t$ and gradient frequency $f$ and the corresponding fitted curves with typical power-law exponents in different $t/f$ ranges. As expected, Mitra’s and Novikov’s equations explain the data well only within selected short and long $t$ ranges, respectively. However, neither can explain the data within the intermediate $t$ range that is used in regular human dMRI measurements, shown as a shaded area in Figure 3. By contrast, ${\vartheta }_t=1$ or ${\vartheta }_f=1$ explains the intermediate $t$ range well with the data shown in this work. Note that the $D_{ex}$ intercept fitted using ${\vartheta }_t=\frac{3}{2}$ is much larger than any $D_{ex}$ acquired with practical $\delta /\Delta$, suggesting it fails to describe both PGSE and OGSE data simultaneously.

Figure 3.

Simulated extracellular diffusion coefficient $D_{ex}$ dependent on diffusion time $t$ and gradient frequency $f$ (markers) and the corresponding fitted curves with typical power-law exponents in different $t/f$ ranges. The shaded area indicates the typical $t/f$ ranges available on clinical MRI systems. A base-10 logarithmic scale is used on the $t$ axis.

PGSE $D_{ex}$ with short ($\delta$ = 1 ms) and long ($\delta$ = 12 ms) gradient durations show different values. This is a well-known effect caused by the limited validity of the short pulse approximation. Moreover, PGSE $D_{ex}$ shows a significant dependence on $t$ within the practical intermediate $t$ range, and values are widely spread out at $f$ = 0 Hz. Both these results suggest that PGSE $D_{ex}$ values are affected by both gradient duration $\delta$ and separation $\Delta$. Because PGSE $D_{ex}$ is usually simplified as $D_{ex}(f=0)$ when analyzing the $f$ dependence of PGSE+OGSE data (30,44–47), ignoring the $\delta$ and $\Delta$ influences may cause additional variations.

Figure 4 shows MR measured values of the extracellular diffusion coefficient $D_{ex}$ of six liposome samples and how they depend on $t$ and $f$, respectively. Medium packing densities (MVL#1, #3, and #4) result in higher $D_{ex}$ values than higher packing densities in MVL#3 and #4. Moreover, $D_{ex}$ with very long diffusion times approaches to zero in MVL#1, #3, and #4, indicating a significant restriction in highly packed liposomes. Note that micelles and liposome debris were not removed from MVL#2 so micelles and debris may accumulate in interstitial space and effectively reduce liposome packing density, leading to higher $D_{ex}$ values. Despite these different $D_{ex}$ values cause by different liposome preparation procedures, the diffusion time dependence in each sample is similar to the simulated results shown in Figure 3. Mitra’s and Novikov’s equations still explain the data well only within the short and long $t$ ranges, respectively. Although MR measured $D_{ex}$ show large variations across six samples because of different preparations, the simple linear frequency dependence $D_{ex}\left(f\right)-D_{\text{const}}~f$ or $D_{ex}\left(t\right)-D_{\text{const}}~1/t$ fit the data well in the practical intermediate $t$ range for all six samples. This suggests the simple linear frequency dependence $D_{ex}~f$ or $~1/t$ may be appropriate to describe $D_{ex}$ used in practical human dMRI measurements.

Figure 4.

MR measured extracellular diffusion coefficient $D_{ex}$ of six liposome samples dependent on diffusion time $t$ (A) and gradient frequency $f$ (B). The shaded area indicates the typical $t$ range 5 – 70 ms and $f$ range $f$ < 50 Hz used on clinical MRI systems. A base-10 logarithmic scale is used on all $t$ axes.

To further investigate the optimal power-law exponents within the intermediate $t$ range, the fitting NRMSE (normalized root mean square error) is plotted against a range of power-law exponents and the results are shown in Figure 5. Except for sample#1, NRMSEs reach minima when ${\vartheta }_f$ is close to 1 for all simulated and experimental results. This is consistent with previous studies that $D_{ex}\left(f\right)-D_{\text{const}}~f$ provides accurate fits of the behavior of ADC for cultured cells in vitro and animal tissues in vivo (4,8). By contrast, NRMSEs reach minima within a range of ${\vartheta }_t$ between ~0.5 to 1.5. This is slightly different from widely-used predictions ${\vartheta }_t={\vartheta }_f$. Except that experimental variations such as noise may cause this discrepancy, another possible reason is that there might exist multiple diffusion times in each OGSE measurement (48,49). When ${\vartheta }_t$ or ${\vartheta }_f$ = 0, i.e., no diffusion time dependence of $D_{ex}$ assumed in the model, NRMSEs always reach a maximum. This suggests that the exclusion of the diffusion time dependence in any extracellular diffusion models decreases model accuracy. Interestingly, NRMSEs < 5% when $-0.5<{\vartheta }_t<1.5$ or $0<{\vartheta }_f<1.5$ for simulations and all six heterogeneous liposome samples. This suggests that, even if the minimum NRMSE may not be achieved, any power-law exponents within a reasonable range may provide good fittings with small errors.

Figure 5.

NRMSE (normalized root mean square error) dependent on power-law exponents in simulations (A) and liposome experiments (B). Note that ${\vartheta }_f$ can only be zero or positive because PGSE $f$ is assumed to be 0.

Discussion

It is a long-known fact that in many media including biological tissues, measurements of diffusion coefficients depend on diffusion time and different diffusion sequences such as OGSE, PGSE, and STEAM provide capabilities to probe different diffusion time ranges (50). What often remains unclear is the precise diffusion time regime relevant to a specific study of interest and selected experimental parameters. This is particularly true for diffusion in irregularly shaped extracellular spaces. There have been some significant improvements in our understanding of the physical mechanisms that affect time-dependent diffusion recently (24,27,28). With a long diffusion time limit, intra- and extracellular diffusion time dependences can be separated (25), and the power-law exponent ${\vartheta }_t=(p+d)/2$ where $d$ is the order of spatial dimension and $p$ is a structural exponent (24). For random packing, $p=0$ and hence ${\vartheta }_t/{\vartheta }_f=\frac{3}{2}$ for three-dimensional tissues. Note that for ${\vartheta }_t>1$, the PGSE tail of $D\left(t\right)$ at long diffusion time still obeys ${\vartheta }_t=1$, while instantaneous $D_{\mathit{\text{inst}}}\left(t\right)$ has $\stackrel{-}{{\vartheta }_t}=\frac{3}{2}$ (35,36). However, this conclusion is different from experimental findings that ${\vartheta }_f=1$ (i.e., $D_{ex}\left(f\right)-D_{\text{const}}~f$) provides good fitting results cells and animal data in vivo (4,8). Because those experiments contained signals from both intra- and extracellular spaces, cross compartment comtamination may be responsible. The current work provides an opportunity to elucidate the time dependence of extracellular diffusion without contamination from intracellular diffusion. The results suggest a power-law with ${\vartheta }_f=1$ fits $D_{ex}$ with the smallest errors with the practical intermediate diffusion time range, consistent with experimental studies. One possible explanation is that the typical diffusion time range (5 – 70 ms) is still not long enough for the extracellular diffusion to reach the long $t$ limit.

Figure 5 shows the exclusion of diffusion time dependence significantly decreases the accuracy of power-law models. However, it should be pointed out that at least one more fitting parameter needs to be added to the extracellular diffusion models if diffusion time dependence is considered. This in turn will decrease the precision of other fitting parameters, particularly a challenge for human imaging with lower signal-to-noise ratios. Eventually, a trade-off is necessary to either increase the accuracy of the extracellular diffusion model or increase the fitting precision of other biophysical parameters. For example, several animal studies using preclinical scanners found a linear dependence of $D_{ex}\left(f\right)-D_{\text{const}}~f$ that provides reliable fits of quantitative microstructural parameters such as cell size with a frequency range of up to 200 Hz (2,8,10,13). By contrast, a human study using a clinical scanner assumed no diffusion time dependence of $D_{ex}$ with a frequency range up to 50 Hz, which resulted in higher precisions of fitted cell size and intracellular volume fraction (20). Therefore, it is expected modeling time dependence of extracellular diffusion would be case-by-case in specific studies.

There is increasing interest in characterizing tissue with time-dependent diffusion (30,44–47,50). To broaden the diffusion time range, several studies use both PGSE and OGSE acquisitions and usually consider the PGSE diffusion coefficient as $D(f=0)$. Our results in Figure 1 suggest that both $\delta$ and $\Delta$ affect extracellular diffusivities so this induces additional variabilities across different studies. The power-law exponents fitted from both PGSE and OGSE data may be dependent on sequence parameters as well. A possible solution is to investigate diffusion time dependence with OGSE data only. However, this increases the hardware requirement of human MRI scanners, such as gradient strength (49).

One concern with our phantom studies is that bilayer membranes might still have a small permeability. This could cause some H₂O to diffuse into the liposome-enclosed space during experiments and result in some intracellular diffusion signals contaminating measurements. To clarify this, interstitial H₂O was removed from our samples after MR experiments via filters and then the liposomes were re-suspended and repacked in D₂O. After that, another MR experiment was performed to ensure there were no detectable MR signals from the samples. This suggests that our measurements were not affected by any intra-vesicular signals.

Conclusion

We introduced novel simulation and experimental model systems mimicking human tumors with extracellular diffusion signals only to investigate the diffusion time dependence of extracellular diffusion. The results suggest simple power-law fits at short and long diffusion time ranges cannot explain the data within the intermediate time range that is typically used in diffusion MRI measurements. By contrast, $D_{ex}\left(t\right)-D_{\text{const}}~1/t$ or $D_{ex}\left(f\right)-D_{\text{const}}~f$ is a reasonable model to adopt in practical imaging of tumors. It is essential to determine the diffusion time range when modeling extracellular diffusion in any diffusion MRI based quantitative microstructural imaging.

Supplementary Material

Supplementary_Material

Figure S1 Diagram of cell “swelling” to increase intracellular volume fraction in the finite-difference computer simulations. Random-close packing of hard-surfaced spherical cells cannot reach a high intracellular volume fraction. This current simulation allows cells to “swell” while keeping the locations of all cells. This not only maintains random packing but also can increase intracellular volume fraction to 75% or higher.

Figure S2 Diffusion time dependence of a doped water phantom with 0.1 mM MnCl₂. At a bore temperature of ~ 20 °C.

Table 1.

Summary of all six MVL samples with different packing density, size range, and the degree of micelle removal.

Sample ID	Micelle removal	Pack Density	Size Range	Note
MVL#1	Partial	High	0.1 to 20 μm	Spin & 100K Centricon
MVL#2	None	High	0.1 to 20 μm	100K Centricon
MVL#3	Partial	High	1 to 20 μm	Spin & 0.45 μm mini filter
MVL#4	Partial	High	1 to 20 μm	Spin & 0.45 μm mini filter
MVL#5	Partial	Medium	1 to 20 μm	Spin & 0.45 μm mini filter
MVL#6	Partial	Medium	1 to 20 μm	Spin & 0.45 μm mini filter

Data Availability Statement

The core code for the random-close packing of polydisperse hard-surfaced spheres is at https://github.com/VasiliBaranov/packing-generation. The code for increasing intracellular volume fraction is at https://github.com/jzxu0622/mati/tree/main/Examples/Eg06_random_packing.