Osmium tetroxide (OsO₄) is one of the most commonly used contrast agents for lipid staining in electron microscopy due to its large atomic number and its ability to integrate into cellular membranes (Porter et al., 1945; Palade, 1952; Watson, 1958). The immersion times for OsO₄ vary between a few minutes to several days, depending on the sample size and tissue type, and are usually determined empirically (Genoud et al., 2018; Hua et al., 2015; Tapia et al., 2012; Mikula and Denk, 2015). However, the kinetics of OsO₄ staining of biological tissue are not well understood and so far had to be determined experimentally by trial and error. We therefore set out to measure the diffusion and accumulation of OsO₄ by placing 4 mm punches of aldehyde-fixed mouse brains in 2% buffered OsO₄ solution. The osmium diffused into the tissue and accumulated in the sample during the staining process (Figure 1b). The intensity of X-ray absorption in the projection images scales linearly with the local concentration/density of osmium (Figure 1c). The density of OsO₄ in the tissue increased beyond the density in the surrounding staining solution, indicating that the density of binding sites for OsO₄ in the tissue is higher than the concentration of the OsO₄ in the staining solution (Figure 1b). The diffusion and spatio-temporal accumulation of OsO₄ results in a staining front that propagates towards the center of the sample (Figure 1b). As expected for diffusive processes (Carnevale et al., 1979), the propagation of the OsO₄ staining front can be approximated by a quadratic model (Figure 1d). As in the case of other fixatives (Medawar, 1941), the osmium staining penetration in aldehyde fixed tissue obeys a quadratic scaling law, or rather a “rule of thumb”, for how the necessary incubation time t depends on the staining depth or sample size x:(1)t ∝ x2

This means, for example, that if one would adapt an established OsO₄ staining protocol for 3 X larger samples, one would have to prolong the incubation time by 9 X in order to produce comparable staining results. Similarly, the time it takes for any point in the sample to reach a given concentration is proportional to the square of its distance to the sample surface (Carnevale et al., 1979).

After 20 hr of incubation the staining density of osmium is homogeneous across the first 1000 μm of cortical depth (Figure 1e). In addition, the ultrastructure is well preserved (Figure 1—figure supplement 2). Reduced osmium is another commonly used staining agent that is known to result in higher contrast for electron microscopy images than non-reduced OsO₄. Typically, a buffered 2% OsO₄ solution is reduced with 2.5% potassium ferrocyanide (K₄[Fe(CN)₆]) (Hua et al., 2015; Willingham and Rutherford, 1984; Mikula and Denk, 2015), which we will call Ferro-redOs throughout this manuscript. Consistent with previous reports on Ferro-redOs staining in large samples (Hua et al., 2015; Mikula and Denk, 2015) we found that for Ferro-redOs the accumulation of heavy metals peaks at a depth between 300 and 800 μm (Figure 1e), whereas the tissue above or below that depth is stained less. Traditionally, this band of more heavily stained tissue has been associated with precipitated osmium that hinders the diffusion and prevents homogenous staining (Genoud et al., 2018; Hua et al., 2015; Mikula and Denk, 2015), in particular deeper in the tissue. No such band is present in tissue immersed in non-reduced OsO₄. (Mikula and Denk, 2015) reported that homogeneous staining of large samples with Ferro-redOs was achieved by adding formamide to the Ferro-redOs solution. However, the underlying mechanisms by which formamide acts are not known (Mikula and Denk, 2015; Genoud et al., 2018). (Mikula and Denk, 2015) hypothesized that formamide might prevent precipitation by generally solubilizing compounds or that it might allow highly charged molecules to cross membranes more easily. Adding formamide to the Ferro-redOs solution indeed resulted in more homogeneous staining (Figure 1e, ), but the heavy metal density was lower than in osmium only. (Mikula and Denk, 2015) reported that the tissue tends to expand at high concentrations of formamide (>50%). But we found that the tissue expands even at lower formamide concentrations, while the amount of expansion depends on both the formamide concentration and the incubation duration (Figure 1f). For a concentration of 11.25% formamide, the sample height increased by about 15% within 20 hr of incubation, whereas for a formamide concentration of 5.56%, the tissue height expanded only by approximately 10% (Figure 1f). Note, in the first 3 hr of incubation in Ferro-redOs with 5.56% formamide the tissue actually shrank, suggesting that there are opposite forces acting on the tissue. Indeed, in Ferro-redOs solution without any formamide, the tissue shrank by about 5% in height (Figure 1f). In contrast, we found that in 2% buffered OsO₄ solution the sample height expands by about 5% (Figure 1f). However, we found no sample expansion if the reduced osmium solution was prepared with 2.5% potassium ferricyanide K₃[Fe(CN)₆]+ 8 a,b), another commonly used reducing agent for OsO₄.

Hua et al., 2015 suggested an alternative approach to achieve homogeneous staining in 1 mm brain punches with Ferro-redOs without formamide, in which osmium and the reducing agent are applied separately. First, the samples are immersed in buffered 2% OsO₄ solution for 90 min. Subsequently, the samples are placed for 90 min in buffered 2.5% K₄[Fe(CN)₆] without any washing step in-between. We repeated this procedure for 4 mm brain punches: First, the samples were incubated in 2% buffered OsO₄ for 22 hours resulting in homogeneous staining throughout the sample. Next, we placed the sample directly without any washing step in 2.5% buffered K₄[Fe(CN)₆] solution. (Hua et al., 2015) hypothesized that the main effect of reducing agents such as K₄[Fe(CN)₆] is to convert VIII-oxidized water-soluble osmium into an VI-oxidized water-soluble form which is thought to generate additional, non-polar OsO₂ to be deposited in the membrane increasing the heavy metal content and the contrast in electron microscopy. However, we found that the heavy metal density in the K₄[Fe(CN)₆] immersed samples did not increase, but rather decrease from the sample surface towards the center as time progressed (Figure 2a). This suggests that K₄[Fe(CN)₆] removes or “washes out” osmium from the sample (Litman and Barrnett, 1972) resulting in an inverse staining gradient. This “washing effect” is stronger for longer incubation times as well as close to the sample surface and around blood vessels (anecdotal observation in electron microscopy images, data not shown). For incubation times longer than 12 hr, K₄[Fe(CN)₆] readily dissolves and disintegrates the tissue (Figure 2—figure supplement 1). Interestingly, no ‘washing effect’ was observed if the same procedure was repeated with 2.5% potassium ferricyanide K₃[Fe(CN)₆], but the sample height shrank by about 2% (Figure 1—figure supplement 3,f, ). Similarly, only a small reduction in heavy metal density could be observed when the sample was washed with double-distilled H₂O for 20 hr (Figure 2—figure supplements 2 and 3).

The chemistry and diffusion-reaction-advection kinetics of OsO₄ staining in aldehyde-fixed tissue are not well understood. In brain tissue, OsO₄ is thought to mainly react with carbon-carbon double bonds, in particular of unsaturated lipids, but it has also been reported to react with some amino acids and nucleotides (Khan et al., 1961; Korn, 1967; White et al., 1976; Schroeder, 1980). At the beginning of the OsO₄ incubation, the density of OsO₄ steeply increases and peaks close to the surface of the sample. As the incubation time progresses, the OsO₄ continues to diffuse and accumulates deeper in the tissue (Figure 2b, d and e). Due to the curved surface of the cortical tissue sample, the projected tissue thickness is lower (Figure 2f) and there are fewer binding sites for osmium, which results in a lower density of osmium closer to the sample surface (Figure 2b). At any given time, the accumulation seems to plateau from the surface towards the center of the sample, indicating that the binding of OsO₄ saturates (Figure 2d and e). However, we found that the saturated density in the samples slowly increased over time from about 129 nmol/mm³ after 1 hr of incubation to 160 nmol/mm³ after 10 hr of incubation (Figure 2b, d and e), while the density of the solution remained stable at about 82 nmol/mm³. This suggests that additional binding sites for OsO₄ become available slowly over time. The reactions underlying the ‘unmasking’ of binding sites are not known, but we hypothesize that it depends at least partially on the availability and the local concentration of unbound, freely diffusing osmium. Hence, at the macroscale level, the accumulation of osmium in biological tissue can be described by four coupled processes (Figure 2g): As the osmium diffuses passively into the tissue, it binds to the available binding sites in the tissue and additional binding sites are slowly being unmasked, while the tissue expands by around 5% within 20 hr of incubation (Figure 1f). The kinetics of these four processes can be described with the following coupled (non-linear) partial differential equations: (2)∂f(x,t)∂t=De⋅∂2f(x,t)∂x2−kon⋅s(x,t)⋅f(x,t)−∂∂x[f(x,t)⋅v(x,t)] (3)∂b(x,t)∂t=kon⋅s(x,t)⋅f(x,t)−∂∂x[b(x,t)⋅v(x,t)] (4)∂s(x,t)∂t=−kon⋅s(x,t)⋅f(x,t)+kunmask⋅m(x,t)⋅f(x,t)−∂∂x[s(x,t)⋅v(x,t)] (5)∂m(x,t)∂t=−kunmask⋅m(x,t)⋅f(x,t)−∂∂x[m(x,t)⋅v(x,t)]

f(x,t) is the density of freely diffusing OsO₄, b(x,t) is the density of osmium bound to the tissue, s(x,t) is the density of available binding sites and m(x,t) corresponds to the density of masked binding sites at time t and a radial depth x from the cortical surface. D^e is the effective diffusion coefficient for OsO₄, k^on is the reaction rate constant for the binding of osmium to the tissue and k^unmask is the reaction rate constant at which masked binding sites get unmasked. In addition, the local change of the tissue geometry, e.g. the expansion as observed for OsO₄, results in a velocity field v(x,t) that describes the local spatio-temporal flow of the density of each component c(x,t):(6)∂∂xc(x,t)⋅v(x,t)=∂c(x,t)∂x⋅v(x,t)+c(x,t)⋅∂v(x,t)∂x

The flow consists of an advection term ∂c(x,t)∂x⋅v(x,t), corresponding to elemental volumes moving with the local flow, and a dilution term c(x,t)⋅∂v(x,t)∂x that describes the local volume change. While we assume that the tissue density is homogeneous across the cortical thickness, we consider differences in the sample thickness due to the curvature of the cortical surface. Additional information on the model assumptions can be found in the methods section.

We fitted the parameters of the system of reaction-diffusion-advection equations for a tissue sample with the projection geometry shown in Figure 2f to the average measured spatio-temporal accumulation of OsO₄ (Figure 2b–e) as well as to the individual samples (n=13, ). The fitted parameters of the diffusion-reaction-advection model (Figure 2—figure supplement 4) capture the measured diffusion-reaction-advection kinetics of osmium staining well (Figure 2c; residual standard error SE_res = 0.026 ± 0.005, mean ±s.d.).

Animal use procedures were approved by the Princeton University Institutional Animal Care and Use Committee (protocol number 2000) and carried out in accordance with National Institutes of Health standards (AAALAC International Institutional Number: Unit #1001, PHS assurance ID D16-00273). The experiments were carried out with C57BL/6 J mice (Jackson Laboratories, Bar Harbor, Maine, USA) of both sexes, age ranging from 13 to 58 weeks (mean ± sd: 27±9 weeks). Prior to tissue sample extraction, the mice were anesthetized by isoflurane inhalation (4%) and euthanized with an intraperitoneal injection of a ketamine (100 mg/kg) and xylazine (10 mg/kg) overdose. Next, the animals were perfused transcardially with ≥50 ml fixative solution (1.3–1.5% glutaraldehyde (GA) and 2.5–2.6% paraformaldehyde (PFA) in 0.14–0.15 M cacodylate buffer with 2.0–2.1 mM CaCl₂ at pH 7.4; (GA, PFA) and cacodylate buffer Electron Microscopy Sciences, Hatfield, PA; CaCl₂, Sigma Aldrich, St. Louis, MO). After perfusion, the animal was left for 15–60 min on ice. Subsequently, the brain was removed in order to extract tissue samples at the center of the cortical surface of each hemisphere using biopsy punches of 4 mm diameter. The liver tissue was extracted with a 4 mm biopsy punch from the center of the liver after perfusion. The samples were kept in fixative solution at 4 °C for 36 hr (4 mm biopsy punches) or for 12 hr (whole mouse brain). Next, the samples were washed 6–7 times for 30 min in 0.15 M cacodylate buffer with 2 mM CaCl₂.

Glass vials were prepared prior to the experiments with a 1–2 mm layer of Sylgard 184 (Electron Microscopy Sciences, Hatfield, PA) at the bottom. The whole mouse brain was glued with a small drop of Vetbond tissue adhesive (3 M, Hanna Pharmaceutical Supply Co., Inc, Wilmington, DE) to the bottom of the vial. For the 4 mm biopsy punch samples, a hole with a diameter of 4 mm was cut out of the Sylgard layer. Next, the samples were placed into the sylgard hole and were glued to the bottom of the glass vial with a small drop of Vetbond tissue adhesive. Subsequently, the samples were immersed in fresh cacodylate buffer solution for the transfer to the X-ray microscope. At the microscope, presets (see X-ray microscopy section below) were loaded for image acquisition. After that the washing solution was replaced with the staining solution and the sample was placed into the X-ray microscope. The acquisition of X-ray images was typically started within 5.66±2.35 min. (mean ±s.d.) after adding the staining solution. The following staining solutions have been assessed at room temperature for this study (all buffered with 0.15 M cacodylate buffer and 2 mM CaCl₂):

After staining, the samples were washed with double-distilled H₂O 3 times for 30 min. Next, the tissue was dehydrated with iteratively increasing concentrations of ethanol (EtOH, 200 proof, anhydrous,≥99.5%, Sigma Aldrich, St. Louis, MO) in double-distilled H₂O for 30 min per step: 25% EtOH, 50% EtOH, 75% EtOH, 90% EtOH, 95% EtOH, 100% EtOH, 100% EtOH. Subsequently, the samples were transferred to 100% acetonitrile (ACS grade, Electron Microscopy Sciences, Hatfield, PA) for two 30-min washes. The tissue was embedded in hard Epon in an iterative concentration series: 1:2 Epon: acetonitrile for 12 hr, 1:1 Epon: acetonitrile for 24 hr, 2:1 Epon: acetonitrile for 2 days with fresh resin every 8–12 hr. Finally, the sample was left on the bench at room temperature for 6 days, with fresh resin every 8–12 hr, before it was placed in a 60 °C oven for curing for 5–7 days.

Serial sections of ~40 nm thickness were collected using an UC7 ultramicrotome from Leica. To assess the ultrastructural preservation of the tissue, the sections were imaged with a Zeiss Sigma SEM (Zeiss Microscopy, Oberkochen, Germany) with a backscattered electron detector (BSD) and the following acquisition parameters: 38 μs dwell time, 4 nm pixel size, probe current 200 nA, EHT voltage 6 kV, BSD Gain = High, all sectors of the detector ON.

All X-ray microscopy experiments were performed on a Zeiss Xradia 520 Versa 3D X-ray microscope (Zeiss, Thornwood, NY). The samples were immersed in heavy metal solution in glass containers tightly sealed with Parafilm (Bemis Company, Inc, Neenah, WI) just before being placed in the recording chamber of the Xradia. Test images were acquired to check imaging parameters. For the biopsy punch samples, projections at a fixed view angle (rotation angle step size = 0°) were taken every 22–60 s resulting in about 1300–1350 images per sample over 22 hr of continuous acquisition. For the whole mouse brain, projections were taken every 22–60 s and after each projection the sample was rotated by 1°. All images were acquired using the image acquisition software Scout-and-Scan Control System (Zeiss, Thornwood, NY) with the following parameters:

First, the individual images and metadata of each projection of each sample were extracted from the proprietary *.txrm and *.xrm files as saved by the Xradia image acquisition software. In addition, a time stamp of the acquisition time point was extracted from the image metadata for each projection. A custom cross-correlation based image alignment procedure (Wanner et al., 2016) was used to correct for translational offsets between images of subsequent time points. The projection images were flat-field corrected to account for radial intensity variations due to the cone-shaped X-ray beam as follows:

The pixel intensities of the detected absorbed X-rays were used as a proxy for the density of heavy metal accumulation in the tissue. In each sample, we quantified the pixel intensities in each projection taken at time t along a cross-section that pointed from the cortical surface radially towards the white matter (Figure 1b). The cross-sections were manually selected such that they started at a well detectable surface boundary and they were located at least 1 mm away from the vertical sample borders. For each sample i, the intensity wi(x,t) of the transmitted X-rays was quantified by averaging the pixel intensity of the cross-section at a depth x at time t within a width of ±25 pixels perpendicular to the axis of the cross-section. x=0 corresponds to the cortical surface of the brain samples or the surface of the liver for the liver samples. The intensity of X-ray absorption was defined as ui(x,t)=10,000-wi(x,t). For each time point t, the baseline intensity bi(t) was calculated by averaging the pixels outside of the sample within –50 μm and –150 μm from the sample surface along the cross-section within a width of ±25 pixels perpendicular to the axis of the cross-section. To account for baseline fluctuation in the staining solution and the X-ray illumination the intensity of X-ray absorption was baseline-corrected by uib(x,t)=ui(x,t)-bi(t)-median(bi) .

To standardize uib(x,t), the following quantities where calculated for each sample:

Finally, uib(x,t) was standardized to u^i(x,t)=uib(x,t)-uiref/uinorm.

This procedure ensures that the origin is at 0 for the linear relationship between the X-ray absorption intensities in the projection images and the different OsO₄ concentrations as shown in Figure 1c and that a pixel intensity of 0.9 roughly corresponds to the saturated staining in 2% OsO₄. For this, the factors a=0.61749 and m=6420 were determined empirically.

The X-ray absorption/pixel intensity of cacodylate-buffered solutions with 1%, 2% and 3% OsO₄ was measured the same way as for the X-ray absorption in stained tissue.

Intensity based heuristics were used in a semi-automated procedure to detect the cortical surface in each image of each sample. The glass vial bottom was used as a reference point to measure the distance to the cortical surface H(t) at time t. Right after placing the samples in the X-ray the height of the samples was H0=H(t=0) and the change in sample height (expansion or shrinkage) was quantified as H(t)-H0H0⋅100%.

The expansion of the tissue in buffered OsO₄ solution was modeled by the monomolecular, saturating growth function g(t)=a1-a2⋅exp(-tτ). The built-in function lsqcurvefit of MATLAB was used for parameter fitting. The fitted parameters for the average expansion in Figure 1f were a1=5.8697%, a2=5.6924%, τ=588.23 min. Under the assumption that the expansion of the tissue domain in buffered osmium solution is isotropic, the corresponding velocity field v(x,t) of the flow in Equation 6 can be characterized by:v(x,t)=−x⋅∂(1+g(t)/100)∂t=−x⋅a2100⋅τ⋅exp(−tτ).

While we assumed that the tissue density is homogeneous across the brain punch, we took into account the variation in the tissue thickness of the 4 mm brain punch. Figure 2f shows a sketch of the modeled tissue geometry. The geometry of the 4 mm brain punches is modeled as a cylinder with a curved surface. The diameter of the cylinder is R and the curvature of the sample surface is approximated by a circle with radius 12H⋅H2+14R2 . The thickness of the sample as a function of the depth x from the sample surface is then given by d(x):

d(x)=2⋅xH⋅H2+14R2-x2 , for 0≤x≤H and d(x)=R, for x>H.

The diffusion-reaction-advection model described by Equations 2–5 makes the following assumptions:

Because the projected intensity also includes any X-ray absorbing components surrounding the sample, we have to consider the change in free osmium density outside of the sample o(x,t) for fitting the model to the experimental data:(7)∂o(x,t)∂t=-∂∂xo(x,t)⋅v(x,t)

The initial conditions (t=0) for the system of coupled partial differential Equations 2–5; 7 were chosen as follows:

The boundary conditions of the system of coupled partial differential Equations 2–5; 7 for x∈0,L with maximal depth L=3mm were chosen as follows:

The MATLAB function pdepe was used to solve/simulate the initial-boundary value problem described above.

C0 was calculated by averaging uib(x,t) over x∈[-150,-50] μm from the sample surface (i.e. outside of the sample) and t∈[0,1200] min and dividing it by the vial diameter D=25mm and by the intensity-to-concentration scaling factor FI2C=214.6308mm3mmol⋅mm . FI2C was calculated from the linear fit in Figure 1c. In each fitting iteration, simulated data was produced according to the diffusion-reaction-advection model above. Within the fitting procedure, FI2C⋅(o(x,t)+f(x,t)+b(x,t)) was fitted to uib(x,t) in a least square displacement sense within the domain x∈[100,700] μm from the sample surface and t∈[0,1200] min by using the trust-region-reflective algorithm of the nonlinear least-squares solver lsqnonlin in MATLAB to optimize the N=6 model parameters De , kon , kunmask , S0 , M0 and H. These parameters were fitted to the average experimentally measured osmium accumulation from n=13 samples within the domains x∈[100,700] μm and t∈[0,1200] min. To evaluate the quality of the model fits, the residual standard error was calculated as SEres=∑(x,t)uib(x,t)-(FI2C⋅(o(x,t)+f(x,t)+b(x,t)))2df , were df corresponds to the degree of freedom.

The average phospholipid density in the mouse brain is 56,241±4000 nmol/g (Barceló-Coblijn et al., 2007) and for an average mouse brain volume of 508.91±23.42 mm³ (Badea et al., 2007) with a weight of 0.427±0.024 g (The Jackson Laboratory, 2021), a cubic millimeter brain tissue contains about 47.19 nmol phospholipids. Lipids in the brain carry on average 2.5 double bonds (Fitzner et al., 2020) which results in 117.975 nmol/mm³ double bonds to which osmium tetroxide could bind through osmium mono- or di-ester formation. Our proposed model predicts 2.999 times more binding sites (353.774±27.460 nmol/mm³), even without unmasking (Figure 2—figure supplement 4).