Time-dependent diffusion in skeletal muscle with the random permeable barrier model (RPBM): Application to normal controls and chronic exertional compartment syndrome patients

Abstract

Purpose

To collect diffusion tensor imaging (DTI) at multiple diffusion times T_d in skeletal muscle in normal subjects and chronic exertional compartment syndrome (CECS) patients and analyze the data with the random permeable barrier model (RPBM) for biophysical specificity.

Materials and Methods

Using an IRB-approved HIPAA-compliant protocol, seven patients with clinical suspicion of CECS and eight healthy volunteers underwent DTI of the calf muscle in a Siemens MAGNETOM Verio 3-T scanner at rest and after treadmill exertion at 4 different diffusion times. Radial diffusion values λ_rad were computed for each of 7 different muscle compartments and analyzed with RPBM to produce estimates of free diffusivity D₀, fiber diameter a, and permeability κ. Fiber diameter estimates were compared with measurements from literature autopsy reference for several compartments. Response factors (post/pre-exercise ratios) were computed and compared between normal controls and CECS patients using a mixed-model two-way analysis of variance.

Results

All subjects and muscle compartments showed nearly time-independent diffusion along and strongly time-dependent diffusion transverse to the muscle fibers. RPBM estimates of fiber diameter correlated well with corresponding autopsy reference. D₀ showed significant (p<0.05) increases with exercise for volunteers, and a increased significantly (p<0.05) in volunteers. At the group level, response factors of all three parameters showed trends differentiating controls from CECS patients, with patients showing smaller diameter changes (p=0.07), and larger permeability increases (p=0.07) than controls.

Conclusions

Time-dependent diffusion measurements combined with appropriate tissue modeling can provide enhanced microstructural specificity for in vivo tissue characterization. In CECS patients, our results suggest that high-pressure interfiber edema elevates free diffusion and restricts exercise-induced fiber dilation. Such specificity may be useful in differentiating CECS from other disorders or in predicting its response to either physical therapy or fasciotomy.

INTRODUCTION

Diffusion tensor imaging (DTI) provides quantitative markers of tissue microstructure and can be used as a probe in the evaluation of skeletal muscle disease [1–4]. In chronic exertional compartment syndrome (CECS) [5–7], muscle compartments retain fluid following exercise, leading to elevated pressure, hypoperfusion, pain, and ischemia. Since current standard diagnostic and therapeutic tools, including intracompartmental pressure measurements and fasciotomy, are invasive and inconsistently successful, noninvasive MRI markers present an attractive alternative. T₂ signal increase is a known marker of post-exercise muscular edema [8–10], and recently long diffusion time DTI [11] showed that diffusion anisotropy changes occur in CECS.

Restricted or hindered diffusion is the core contrast mechanism for most clinical applications of diffusion-weighted MRI. So far, most implementations employ a single diffusion time, often driven by practical concerns of acquisition time, hardware limitations or signal-to-noise ratio. However, controlled variation of the diffusion time, and corresponding modulation of the degree of restricted diffusion, can provide enhanced specificity and quantification of such properties as pore size, shape, connectivity, and permeability [12].

The initial reduction of measured diffusion below its bulk value at short diffusion time is given by the ratio of the diffusion length to the confining length scale (or more generally, to the inverse surface-to-volume ratio) [13–15]. This short-time limit applies when the diffusion length is much smaller than the curvature radius and/or compartment size, which practically requires very short diffusion times, of the order of milliseconds for <10 μm pores. Correspondingly, the utility of this universal limit has been appreciated in high-gradient and/or large-pore studies in, e.g., water-saturated geological media [16], water-filled trabecular bone [17], fluid-filled bead packs [13], or gas diffusion in bead packs [18]. In clinical MRI systems, unfortunately, the short diffusion time regime is typically less accessible for probing mesoscopic architecture (~1–10 μm) in biological tissue, due to gradient strength limitations.

Very long diffusion times, on the other hand, are more widely feasible, especially in the clinic, and can maximize restricted diffusion contrast. However, the tortuosity encoded in this limit is not always microstructurally specific. Relating the asymptotic diffusion coefficient D_∞ to the mesoscopic geometry of restrictions has remained a challenging problem over more than a century.

In this work, we propose that protocols employing intermediate diffusion times spanning and quantifying the transition between these regimes may potentially provide both sensitivity and enhanced quantification of mesoscopic structural complexity in vivo for certain tissues. Some examples exist within biological tissue characterization [19,20] and in clinical scanners. Hyperpolarized gas diffusion studies in the lung make specific use of diffusion time given the range of length scales available to gas phase nuclei [18,21–23]. Time-dependent diffusion results in ex vivo muscle tissue [24,25] showed clear reduction of radial diffusion with time in the time range of 30–800 ms, as compared with largely time-independent axial diffusion along the fiber direction. Time-dependent diffusion of metabolites within in vivo skeletal muscle has shed light on phosphate diffusion in normal muscle metabolism [26]. Several muscle studies in clinical MR scanners have begun to employ stimulated echo sequences at somewhat longer diffusion times (200–400 ms), both for adjusted relaxation weighting and higher structural sensitization [27–29]. A recent study demonstrated stronger differentiation of normal controls from CECS patients at long diffusion times (1000 ms) than at conventionally short ones (30 ms)[11]. The diffusion time range of 500–1000 ms is thus highly sensitive to structural confinement, primarily via the sarcolemma membrane.

Overall, maximizing structural sensitivity is only one part of a successful diagnostic tool; optimal results are expected when such data is combined with an appropriate mesoscopic model for the tissue of interest. The recently proposed random permeable barrier model (RPBM) [30] considers water diffusion hindered by a network of randomly oriented semi-permeable membranes, a pattern that is well-suited to the topology and axial symmetry of skeletal muscle (Figure 1, taken from Ref. [30] and [31]), as shown in its initial validation in ex vivo data [32] and in normal controls [29,33]. RPBM has been validated both analytically and in simulations over a wide range of parameters [30]. In the context of myofiber architecture, it allows analysis of transverse (or radial) diffusion at several diffusion times to estimate three parameters: free diffusion D₀, fiber diameter a calculated from the surface-to-volume ratio S/V of the sarcolemma, and its permeability κ. In the present study, time-dependent diffusion data is collected and analyzed with RPBM in the calf muscles of normal control volunteers and suspected CECS patients, in order to better understand and illustrate the microscopic pathophysiology of CECS and contribute to the development of tools to guide appropriate therapy.

Figure 1.

(a) Cross-sectional histology slide (from Ref. (31)) of skeletal muscle fibers, showing myofibers surrounded by thin endomysium (En). (b) Sketch of permeable barrier pattern (adapted from Ref. (30)) underlying the random permeable membrane model (RPBM), evoking the same topology of skeletal muscle.

METHODS

In this HIPAA-compliant study approved by the local institutional review board (IRB), seven patients (4 F, age 22±6 years, range 15 to 30 years; 3 M, age 24±10 years, range 18 to 36 years) with clinical suspicion of CECS, and eight healthy volunteers (2 F, age 29±1 years, range 28 to 29 years; 6 M, age 27±3 years, range 21 to 29 years) provided written informed consent and underwent diffusion tensor imaging in the lower-leg muscles in a wide-bore MAGNETOM Verio 3-T scanner (Siemens AG, Healthcare Sector, Erlangen, Germany) with a unilateral 15-channel knee coil. Subjects’ legs were not suspended, resting on the table and coil. A portion of these data at fixed diffusion times were previously reported [11]. DTI results were obtained both before and after at least 10 minutes of treadmill exercise challenge. Subjects came off the scanner table and jogged on a commercial exercise treadmill (Reebok S 9.80, Model#RBTL69608.0), at their own pace and resistance level, for 10 minutes or, in the case of patients, until pain onset, up to a maximum of 30 minutes. Subjects were then quickly repositioned on the scanner bed for the post-exercise scan. The average time between the last pre-exercise scan and the first post-exercise scan was 17.7± 5.0 minutes. The first and last DTI scans occurred 3.9 ± 1.9 minutes and 12.4 ± 6.7 minutes, respectively, following repositioning on the scanner table.

Axial DTI used a SPAIR fat-saturated stimulated-echo diffusion sequence with echo-planar imaging (EPI) readout (TR/TE = (10700−12400)/(42) ms, 64 × 64 × 10 matrix, 3 × 3 × 5 mm³ resolution, 6 diffusion directions, b = 0, 500 s/mm², 3 averages) and adjustable mixing time TM. Four different effective diffusion times T_d = TM + TE/2 of 30, 170, 520, and 1020 ms were acquired by varying TM. One patient and one volunteer had T_d of 70 ms rather than 170 ms. One patient had TE of 31 ms for TM=150 ms and 1000 ms. The b-value of 500 s/mm² was used for consistency with the literature for optimum muscle sensitivity [34–37].

The stimulated-echo diffusion preparation, a prototype vendor sequence, used monopolar gradients in the first and third intervals of the sequence, followed by an echo-planar imaging (EPI) readout [28,38], to enable long diffusion times and correspondingly large diffusion lengths up to (2DT_d)^1/2 ~ 50 μm for amplified microstructural sensitivity. Spoilers were included in the mixing time interval and, for the b=0 acquisition, in the 1^st and 3^rd intervals. For the b-value of 500 s/mm² at the longest diffusion time (1020 ms), the diffusion gradient moment on one axis (59 mT/m ms) was close enough to those of unbalanced spoiler (17 mT/m ms) and slice-selective imaging gradients (8 mT/m ms) that the full diffusion-encoding matrix (“b-matrix”) was required in the diffusion tensor inversion to obtain quantitative accuracy [39–42]. The prototype vendor sequence software on the MAGNETOM Verio platform calculated the full b-matrix for both the nominally unweighted (b₀) and diffusion-weighted images, as previously reported [11]. The sum of the diagonal b-matrix elements for the nominally unweighted (b₀) images ranged from 3 s/mm² at T_d = 30 ms to 145 s/mm² at T_d = 1020 ms, illustrating the necessity for full b-matrix consideration.

Phantom validation

In order to validate the analysis procedure, DTI scans with the same imaging protocol described above were acquired in the Verio 3-T scanner with a range of mixing times TM (30 ms to 1000 ms) in a water phantom having isotropic, time-independent diffusion. The calculated b-matrices were employed in the DTI estimation of the water data. A rectangular region of interest (ROI) was placed on the DTI parametric maps, and mean values and standard deviations of all parameters (λ₁, λ₂, λ₃. MD, FA) were collected for all diffusion times both with and without the b-matrix corrections. Mean values and standard deviations for ROIs in 10 slices were computed.

Human subjects

DTI data were processed offline (Igor Pro, Wavemetrics, Portland, OR, USA), incorporating the full b-matrices generated by the vendor software, to generate maps of MD, FA, and diffusion eigenvalues (λ₁, λ₂, λ₃) for each diffusion time. Regions of interest (ROI) were manually segmented on axial b₀ images, separately for pre- and post-exercise scans, enclosing entire muscle compartments: anterior tibialis (AT), extensor digitorum longus (EDL), peroneus longus (PL), posterior tibialis (PT), soleus (SOL), gastrocnemius medialis (GM), and gastrocnemius lateralis (GL). For each muscle compartment, mean DTI metrics were evaluated in each slice ROI, and the set of values for all slices were averaged as a representative value. The same ROIs were used to sample the same muscle compartments in each diffusion time dataset. Signal-to-noise ratio (SNR) of DWI at b = 0 and b = 500 s/mm² were separately evaluated for each subject, exercise condition, and muscle compartment by dividing the mean signal intensity in each compartment ROI by the standard deviation of signal intensity in a region outside the leg, and averaging results from different slices, directions, and compartments in a given subject. Similarly, standard deviation across averages and diffusion directions, normalized to the mean signal intensity, was also calculated for each muscle compartment separately, and the results averaged for each subject to generate a normalized standard deviation (NSD).

From the ROI-sampled DTI data, radial diffusivity was calculated as λ_rad = (λ₂ + λ₃)/2 for each T_d and muscle compartment, and its time dependence λ_rad(t) was fitted to the random permeable membrane model (RPBM) [30,33] to extract free diffusivity D₀, the surface-to-volume ratio S/V from which we determine characteristic fiber diameter a, and myofiber membrane permeability κ. The RPBM analysis was performed with Matlab software (Mathworks, Inc., Natick, MA, USA), whereby the input for the non-linear least-squares fitting procedure is the RPBM time-dependent diffusion coefficient D(t). As D(t) has no closed form, it was derived from the closed-form expression of the RPBM frequency-dependent diffusion coefficient D(ω), Ref. [30].

$$\frac{D_0}{D(\omega )}=1+\zeta +2z_{\omega }(1-z_{\omega })\left[\sqrt{1+\zeta /{(1-z_{\omega })}^2}-1\right]$$ (1)

using the numerical Fourier integration procedure outlined around Eq. (3) of Ref. [30] In Eq. (1) above, the effective volume fraction of membranes ζ = (D₀/2κ d)(S/V) quantifies the membranes’ ability to hinder the diffusion in d=2 dimensions, with S/V the surface-to-volume ratio of the membranes. The parameter $z_{\omega }=i\sqrt{i\omega \tau }$ characterizes a dimensionless frequency, while τ = D₀/(2κ)² represents an effective time scale associated with transport through a single membrane [30]. The numerical integration yielding D(t) was performed in Matlab using the quad routine. Characteristic fiber diameters were obtained based on the fitted S/V value via an approximate relation a = 4V/S and compared with literature values from corresponding muscle groups from quantitative analysis of normal autopsy specimens [43].

Statistical analysis

For each muscle group of each subject a response factor ratio was computed for each parameter as the value post-exercise divided by the value pre-exercise. Unequal variance t tests were used to compare volunteers to the combined group of CECS-affected muscle and normal muscle groups among patients in terms of the mean response factor ratio and the pre- and post-exercise levels of each parameter for each muscle group. Pooling data from all muscle groups into a single overall analysis, mixed-model two-way analysis of variance was used to compare volunteers to patients in terms of the response factor ratio and the pre- and post-exercise levels of each parameter averaged over muscle groups. The model included muscle compartment as a fixed blocking factor and subject group as a fixed classification factor. For the mixed-model analysis of the response factor ratios, the dependent variable was the observed ratio minus unity. In this way, a test of the intercept assessed whether the response factor ratio was different from 1 and a significant group effect still implied that the groups were different in terms of the ratios on their original scale of measurement. To account for statistical dependencies among measures from the same subject, the correlation structure was modeled by assuming responses to be symmetrically correlated when acquired from the same subject and independent when derived from different subjects. The error variance was allowed to differ across comparison groups to remove the unnecessary assumption of variance homogeneity. All tests were conducted at the two-sided 5% comparison-wise significance level using SAS 9.3 (SAS Institute, Cary, NC, USA).

RESULTS

Water phantom

Water phantom DTI metrics as a function of mixing time with and without b-matrix correction are shown in Figure 2. Without correction, the water mean diffusivity shows artificial reduction with diffusion time, and fractional anisotropy artificially increases. With the correct b-matrices, the mean diffusivity is time-independent, and the FA is at a minimal level for all diffusion times.

Figure 2.

Diffusion coefficient of a water phantom measured with STEAM-DTI at variable diffusion time T_d with (closed symbols) and without (open symbols) complete b-matrix correction.

Human subjects

Table 1 shows SNR and NSD results for the subjects in this study as a function of b-value, diffusion time, and exercise condition. SNR drops at higher diffusion time due to T₁-weighting, but remains sufficiently high (~36 for a single DWI) even at b = 500 s/mm², so that noise bias is negligible. NSD values are consistent with other studies [38]. Exercise, diffusion-weighting, and longer diffusion time all tend to increase NSD, but only slightly, suggesting a comparable noise statistics for all diffusion times in this study. Regionally, average NSD values were similar (within 17% of the mean) for all muscle compartments except the posterior tibialis (PT), whose average NSD values were 33% higher than the mean. This additional variance likely originates from pulsation of neurovascular bundles contained in the PT. However, the NSD values are not sufficiently elevated to merit exclusion of this compartment from the study.

Table 1.

Signal-to-noise ratio (SNR) and normalized standard deviation (NSD) for all muscle compartments evaluated in this study before/after exercise at two different diffusion times.

	b=0 s/mm2 (pre)	b=0 s/mm2 (post)	b=500 s/mm2 (pre)	b=500 s/mm2 (post)
SNR (Td = 30 ms)	84.7±18.5	85.9±14.1	51.3±10.8	49.3±8.1
SNR (Td = 1.02 s)	43.9±8.4	45.2±9.2	35.6±7.4	36.1±7.5
NSD (Td = 30 ms)	0.136±0.025	0.139±0.015	0.156±0.023	0.161±0.019
NSD (Td = 1.02 s)	0.148±0.028	0.154±0.019	0.166±0.026	0.173±0.019

Figure 3 shows exemplary b₀ images and axial and radial diffusion maps at 4 different diffusion times in a healthy volunteer. The constant axial diffusion and the decrease in radial diffusion with diffusion time are evident. Figure 4 shows average axial and radial eigenvalues as a function of diffusion time for 4 different muscle compartments from the healthy volunteers in this study, along with reported values in normal controls in the same compartments from several literature studies. Diffusion times were defined by (a) one-half of the echo time TE for studies [1,36,37,44] employing a standard twice-refocused spin echo sequence, or (b) one-half TE plus the mixing time TM for studies ([11,27,28] and this study) employing a stimulated echo sequence. In each compartment, the global trend confirms the significant time dependence of the radial diffusion and the near time independence of the axial diffusion. For example, in the anterior tibialis (AT), axial diffusion (λ₁) values from this study decrease by only 12% with diffusion time, while transverse diffusion (λ₂, λ₃) values and mean diffusivity (MD) decrease markedly with diffusion time by more than 30% of their short-time value. This enhanced structural sensitivity is particularly emphasized by the studies with longer diffusion times. While some variability is to be expected given the range of acquisition parameters (TR/TE, b-values) in the cited studies, the global trend remains evident in the combined dataset.

Figure 3.

Example unweighted (b₀) images, axial diffusion (λ₁) maps, and radial diffusion (λ_rad) maps for a healthy volunteer as a function of diffusion time T_d. Axial diffusion remains relatively constant with diffusion time while radial diffusion sharply decreases due to myofiber restrictions.

Figure 4.

Volunteer calf DTI measurements in the anterior tibialis (AT), soleus (SOL), gastrocnemius medialis (GM), gastrocnemius lateralis (GL) muscle from a range of literature studies as a function of experimental diffusion time T_d.

Time-dependent radial diffusion values and RPBM fits are shown in Figures 5a and 5b before and after exercise in a volunteer GM compartment and CECS patient GM compartment, respectively. These examples show the typical good fit quality found in this study (pre/post exercise R² values of 0.9953/1.0000 for this volunteer and 0.9997/0.9998 for this patient). The volunteer case showed an increase in both D₀ (5.8%) and a (19.7%), while the patient case featured an increase in D₀ (7.7 %) and decrease in a (9.2%); these features are reflected in the qualitative shape of the time-dependent curve fits. Globally, model fitting was successful (i.e. R² values ranging from 0.9808 to 1.0000) in all 15*7 = 105 muscle compartment fits except 3 GL cases which produced unphysical order of magnitude estimates and were excluded from the statistical analysis.

Figure 5.

Time-dependence and model fits of radial diffusion in (a) volunteer GM compartment and (b) CECS patient GM compartment.

The distribution of mean values of each pre-exercise RPBM parameter (D₀, a, κ) in volunteers and patients across muscle compartments is shown in Figure 6a–c. Anterior muscle compartments (AT, EDL, PT, PL) tend to have larger free diffusivity D₀ and smaller fiber diameter a than posterior compartments (SOL, GM, GL). Permeability κ is largely equivalent across all muscle groups, with the exception of larger values in GL. No significant control/patient differences between absolute RPBM parameters at the muscle compartment level were found. We found correlation of pre-exercise fiber diameters in several muscle compartments with estimates from autopsy literature [43], showing good qualitative agreement that posterior compartments (SOL, GM) have generally larger fiber diameter than anterior compartments (AT, PL) (Figure 6d). Quantitatively, the fiber diameter estimates from diffusion MR are generally less than those from autopsy analysis. This bias may originate from the conversion of surface-to-volume ratio (the native RPBM parameter) to fiber diameter via a = 4V/S, which may not be accurate for the two-dimensional geometry of random barriers (Figure 1). Also, while an intuitive estimate of size, the fiber diameter is somewhat ill-defined since myofiber cross-sections are not circular. In the future, comparison to a histological measurement of the S/V ratio may provide a more unbiased validation for the RPBM.

Figure 6.

Mean pre-exercise values of RPBM parameters (a) D₀, (b) a, and (c) κ in separate muscle compartments in volunteers and CECS patients. Error bars reflect standard deviation within the subject group. (d) Correlation of fiber diameters in several muscle compartments derived from time-dependent diffusion MRI in this study to those obtained from autopsy reference (43). Error bars reflect standard deviation within the subject and compartment group.

Figure 7 and Table 2 show the subject group comparison of response factors of each RPBM parameter, as well as axial diffusion λ₁ at T_d = 1.02 s, including response factors from all muscle compartments; as normalized quantities, this collective consideration is dimensionally consistent albeit regionally nonspecific. Long-time λ₁ increased significantly for both volunteers (3%, p<0.05) and patients (5%, p<0.05), and the λ₁ response factors significantly differentiated volunteers from patients (p<0.05). D₀ showed significant (p<0.05) change from baseline for controls, and significant fiber diameter increases (p<0.05) for volunteers. Quantitatively, free diffusion D₀ increased by 5% in volunteers and 10% in CECS subjects. Fiber diameter a increased by 21% on average in volunteers, but changes negligibly (−3%) in patients. Permeability κ decreased slightly (−7%) in volunteers but increased by 54% in patients. The volunteer-vs.-patient group comparisons of fiber diameter a and permeability κ trended towards significant differences (p=0.07 in both cases).

Figure 7.

Comparison of response factors of (a) long time axial diffusion λ₁ and model parameters (b) free diffusivity D₀, (c) fiber diameter a, and (d) permeability κ between volunteers and CECS patients. Error bars reflect 95% confidence intervals.

Table 2.

Response factors and t-test comparisons for RPBM parameters and long-time axial diffusion. For each subject group, the p-value is given from a two-sided t-test within the mixed model to assess whether the response factor ratio for each parameter was significantly different from 1. The p-value testing for significant differences between volunteer and patient groups is also given for each parameter.

Parameter	Volunteers		Patients		p (Vol vs. Pts)
λ1(1.02s)	1.03 ± 0.03	<0.001	1.05 ± 0.04	<0.001	0.014
D0	1.05 ± 0.09	0.0259	1.10 ± 0.12	0.0689	0.294
a	1.21 ± 0.55	0.0263	0.97 ± 0.29	0.7082	0.070
κ	0.93 ± 0.42	0.3368	1.54 ± 1.02	0.1799	0.074

Significant (p<0.05) results are shown in bold.

DISCUSSION

This study combines the collection of diffusion tensor metrics as a function of diffusion time with a biophysical model to produce structural metrics relevant to the target anatomy of skeletal muscle. The diffusion times required to interrogate the spatial scale of myofibers (30–50 μm) are several hundred milliseconds, which is accessible via stimulated-echo diffusion in clinical scanners. The stimulated-echo sequence is thus a useful module for providing DTI parametric maps with sufficient SNR for both image quality and quantification (see SNR values in Table 1) as well as the opportunity for more specific quantitative metrics. Likewise, fiber diameter estimates for individual compartments are in fairly good agreement with estimates from autopsy studies, as found previously [33], up to an overall scaling factor possibly related to transition from the surface-to-volume ratio to the fiber diameter, as discussed above. These results are an important demonstration of the quantitative potential associated with combining appropriately sensitized data with a biophysical model parametrized with the characteristics of the measured regime.

Beyond the scale estimations at baseline, the second element of this study addresses exercise response. As an interpretive summary, Figure 8 shows a hypothesized diagram of myofiber structural changes consistent with the RPBM parameter responses. In normal volunteers, myofibers dilate with exertion with little change in the extracellular (endomysium) space. In CECS patients, trapped interfascicular edema accumulates in the interfiber space, elevating pressure and prohibiting dilation, consistent with the negligible changes in fiber diameter a. This fluid accumulation also contributes to a higher free diffusivity D₀ and possibly to higher apparent permeability κ due to the enlargement of a less confined compartment than the myofiber interior.

Figure 8.

Sketch of hypothesized muscle fiber configurations with pre- and post-exercise in volunteers or CECS patients.

While the changes following exertion may conceivably have multiple contributions [44–46], the RPBM model narrows the possibilities. Morphological dilation of myofibers (i.e. hypertrophy) is a well-known response to repeated mechanical loading, as has been observed in numerous studies of short- or long-term resistance training with quantitative histological analysis [47–50], particularly of type II fast-twitch fibers [51,52]. The biological processes driving this adaptation are complex, but typically involve production of more intrafiber contractile protein, requiring an enlargement of the sarcolemma. The current study evaluates acute response immediately after a single session of exercise, which may potentially involve various processes (temperature, edema, vascular changes, fiber structure). Elevated temperature likely plays a significant role. Water diffusion in muscle is known to increase by 2 %/°C [45]. Given the average increase of 3–5% in axial diffusion λ₁ with exercise at long diffusion times (Table 2), it is likely that part (though not all) of the elevation in the transverse free diffusion D₀ is due to temperature changes, especially in volunteers. Additionally, the RPBM modeling process and the group statistics of its parameters’ responses indicate a role of myofiber hypertrophy at the acute exercise response phase, mirroring and consistent with longer-term training changes. This observation is an example of the clarification brought about by appropriate biophysical modeling of time-dependent diffusion, which may not be as clear using single-time-point DTI [2,11,36,37,53,54]. Finally, the finding that CECS muscle groups do not dilate as pronouncedly as normal muscle, possibly due to the influence of high-pressure interfascicular edema [55], suggests diffusion MRI may differentiate CECS from other pathologies presenting with enhanced T₂ signal intensity on conventional imaging.

As a complement to the interpretations above, discussion is warranted on the approximations inherent to the current form of RBPM and its application to the data in this study. Firstly, the single-compartment picture adopted in RPBM does not include an extracellular space (i.e. endomysium in skeletal muscle), which may have biased model estimates, since neither multi-compartment diffusion nor multi-compartment relaxation is considered, though both are known to be present in biological tissue, and may change independently with exercise. Given a fairly tight packing of myofibers in a bundle, we can view the “cells” in RPBM (Figure 1) as reflecting both intracellular and extracellular properties. This is an obvious oversimplification, because true extra-cellular space is topologically connected (one can travel across it without entering cells). In the limit of highly impermeable membranes, the RPBM would then strongly underestimate the overall tortuosity limit D_∞. However, given that the effective “hindrance strength” of the membranes is not very large (ζ~1, i.e. permeability is not small), it is as probable for a water molecule to traverse cells as to travel around them within the extra-cellular space. Qualitatively, this means that whether the voids between cells are completely enclosed by membranes (as in RPBM) or are connected (as in reality), should not drastically affect model parameters. Similarly, within the single-compartment approximation, changes in the extracellular space may indirectly induce changes in apparent permeability κ, whose increase may be due to this change in the geometry rather than a change in the sarcolemma water transmission rate. In a related sense, a numerical study has not yet been performed to optimize the number and choice of measurements times or to analyze potential correlated errors between model parameters in the presence of random noise. Vascular effects are also not explicitly modeled. Given these issues, the exercise response phenomena represented in Figure 8 are at this point promising qualitative trends that would benefit from further quantitative verification. Finally, intracellular ultrastructure such as organelles and the sarcoplasmic reticulum are represented in RBPM within the ‘bulk’ hindered diffusivity D₀, primarily because by the shortest diffusion time employed in this study (30 ms), water diffusion has fully explored their tortuosity and thus cannot fully quantify their scale. Thus, continued validation of the RBPM in physiological systems and clinically available sampling intervals is warranted to support its adoption as a clinical tool. Despite these important caveats, we postulate that the significant volume fraction of the fiber interior, the temporal regime accessible in clinical scanners, and the normalization of T₂-relaxation effects from diffusion weighting, provide cogent justification for the use of RPBM for probing first-order microstructural values and alterations with exercise.

Beyond the approximations inherent to RPBM, this study had a number of limitations. The total number of subjects was small and the volunteer cohort was not number-, gender-, or age-matched, which may have contributed to the lack of significant differentiation at the individual compartment level. No eddy-current corrections were performed on the raw DW images prior to DTI processing. Our estimates of SNR were approximate but adequate for purposes of ensuring data fidelity. The DTI acquisitions did not use cardiac gating [38]. Regions of interest were prescribed directly onto the moderate resolution DW images, rather than transferred from those prescribed on co-registered higher resolution anatomical imaging. Parametric maps of DTI parameters were sampled at the ROI level for RPBM analysis, excluding interrogation of RPBM parameter intracompartmental heterogeneity. The RPBM does not include an extracellular space and could be generalized to incorporate one for a more accurate representation, albeit at a cost of acquiring more diffusion times to reliably determine more parameters. Finally, some variability existed in the timing of the exercise procedure, as previously noted [11] and indicated in the Methods section.

In conclusion, we have performed time-dependent DTI at 3T in the leg muscles of healthy volunteers and suspected CECS patients before and after exercise, and analyzed the results with a random permeable barrier model (RPBM). Model fitting detects the expected myofiber dilation in normal volunteer muscles, while this dilation is notably absent in CECS patients. Free diffusivity and apparent permeability metrics also show significantly different exercise response between the control and patient groups; increased interfiber fluid accumulation (muscular edema) may be a driving component of these observations. This study shows an example of quantitative specificity that is provided by the collection and analysis of time-dependent diffusion. In the case of CECS, this differentiation may help guide treatments that either relieve pressure directly (fasciotomy) or increase fiber integrity to modulate exchange. It is expected that analogous approaches may also provide useful biomarkers for other muscle diseases.

ABBREVIATIONS

DWI

diffusion weighted image

DTI

diffusion tensor imaging

CECS

chronic exertional compartment syndrome

MRI

magnetic resonance imaging

RPBM

random permeable barrier model

S/V

surface-to-volume ratio

EPI

echo-planar imaging

ROI

region of interest

AT

anterior tibialis

EDL

extensor digitorum longus

PL

peroneus longus

PT

posterior tibialis

SOL

soleus

GM

gastrocnemius medialis

GL

gastrocnemius lateralis

TR

repetition time

TE

echo time

TM

mixing time

T_d

diffusion time

SNR

signal-to-noise ratio

NSD

normalized standard deviation

MD

mean diffusivity

FA

fractional anisotropy

T₂

transverse relaxation time

T₁

longitudinal relaxation time

b₀

unweighted diffusion image (b=0)

λ₁

primary diffusion eigenvalue, axial diffusivity, longitudinal diffusivity

λ₂

secondary diffusion eigenvalue

λ₃

tertiary diffusion eigenvalue

λ_rad

radial diffusion

D₀

free diffusivity

a

fiber diameter

κ

permeability

ζ

membrane volume fraction

z_ω

dimensionless frequency

τ

membrane transport time