1. Introduction

X-ray images of biological cells often have a low contrast between the diverse contents and the surroundings. This can result in difficulties for imaging because a high fluence (incident photons per unit area) and dose (energy deposited per unit mass in the specimen) is required to obtain the necessary contrast and resolution. However, radiation damage can limit the dose that can be applied. In this article, estimates are given for the contrast and dose achievable for three categories of X-ray imaging. The categories considered are absorption, coherent scattering and incoherent scattering. Coherent scattering can be categorized as methods that require phase determination, perhaps aided by interference effects such as holography, and methods where images are obtained via an objective lens. For the purposes of this article, incoherent scattering includes techniques such as fluorescence measurements (for which the signals add incoherently) and Compton scattering.

For 3D images, some of these methods require a tomographic approach where 2D projections are obtained first. The dose-fractionation theorem (Hegerl & Hoppe, 1976) states that 3D images can be built up from 2D projection data at a total dose corresponding to that required to see a feature in a 2D projection image.

A 3D reconstruction of a frozen hydrated biological cell was obtained by coherent diffraction imaging (CDI) (Rodriguez et al., 2015), giving an estimated resolution of 74–99 nm rather than in the 10 nm range predicted by Howells et al. (2009) in a calculation that used the dose-fractionation theorem. In a comments article, Robinson (2015) investigated possible reasons for this. It was speculated that the dose fractionation might not apply in a simple manner for this case and, following on from that, the scaling of required dose with a resolution d might scale as 1/d⁵ rather than the 1/d⁴ seen in the theory. In this case, the resolution d is the size of a uniform-density voxel corresponding to a feature to be located.

The dose required to image a feature using CDI was also analysed by Villanueva-Perez et al. (2016) and this was later compared (Villanueva-Perez et al., 2018) with scanning Compton microscopy (SCM). Both were analysed as 2D projections with the assumption that the dose-fractionation theorem could be applied. The theoretical case for Compton microscopy included an analysis showing that the sample size and resolution dependence of the required dose performed as W/d⁴ (given as L/d⁴ in the article), where W is the thickness of the object and d is the resolution. For CDI, the dependence was W²/d⁶, where W is the width of the sample in two dimensions – a much worse situation despite the fact that, for CDI, the phases were assumed to be known. A W²/d⁶ dependence for the required fluence and dose for imaging by CDI would have severe consequences for high-resolution imaging of larger particles by this technique on synchrotrons and perhaps even more on free-electron laser (FEL) sources. For the latter, ptychographic and other scanning techniques will have limited applicability if the aim is to circumvent radiation damage by exploiting the short pulse length of the source. In addition, the flux per pulse is a limitation, particularly for larger specimens where the available photons from an FEL source will be spread out over a larger area when carrying out CDI. Villanueva-Perez et al. (2018) also backed up the theoretical analysis with simulations demonstrating a significant advantage for imaging by SCM compared with CDI for the case of unknown phases. The reasons for the poor apparent performance for CDI (in both theoretical analysis and simulations) are covered in this article.

Both the application of the dose-fractionation theorem and the scaling of required dose with object size and feature size are therefore important issues for the application of imaging methods. The analysis given here is applied to the case of imaging biological cells but some of the principles will apply to imaging other materials at various X-ray energies. A discussion of absorption and phase contrast in X-ray microscopy, together with a comparison with electron microscopy, can be found in the work by Du & Jacobsen (2018).

2. The dose-fractionation theorem

If an extra feature (a voxel of dimension d) replaces a voxel of the same dimension in an object and we calculate the dose required to see this feature in projection, then one should be able to fractionate this dose among an arbitrary number of projections to get a statistically significant 3D image. This is the dose-fractionation theorem (Hegerl & Hoppe, 1976). Because of the reduced dose, each projection would no longer contain a statistically significant signal showing the feature but the feature would be visible in 3D. The following quotations from relevant articles are given.

`A statistically significant 3D image can be computed from statistically insignificant projections, as long as the total dose that is distributed among these projections is high enough that it would have resulted in a statistically significant projection, if applied to only one image', McEwen et al. (1995).

Hoppe & Hegerl (1981) include `The signal-to-noise ratio of this difference is, by the way, independent of the thickness of the specimen' and `The excellent economy of 3D reconstruction is by no means trivial. The important point seems to be that the pieces of information from different sections of the specimen are not only contained in the projections without loss of information, but also that they are coherently added'.

The dose-fractionation theorem was developed to demonstrate that atoms which had a statistically significant contribution in a 2D projection would also be visible in a 3D image at the same accumulated dose. For this case, the contribution in the 2D projection had a defined depth of approximately the size of a single atom. The mathematical basis for the dose-fractionation theorem is not in doubt but there is confusion in the literature about its applicability. For example, it is an incorrect application of the dose-fractionation theorem to claim that a high contrast and resolution in a projection means that the same contrast and resolution with the same overall dose can be obtained in a 3D reconstruction. This would only be the case if the contrast in the 2D image resulted from a feature of depth d. Line scans across a projection can give optimistic measurements of the contrast and resolution if, for example, they cross membrane boundaries with the plane of the membrane at an orientation normal to the projection.

The dose-fractionation theorem was originally applied to imaging via electron microscopy. This article analyses the conditions for X-ray imaging under which the required dose-fractionation conditions apply and when the signal given by a feature within the projection is dependent on sample thickness. It is assumed that the aim is to identify a feature of size d within an object of size W, with both W and d applying in all three dimensions. Some imaging methods work in projection. In this case, it is assumed for the calculations that the object has a dimension W × W in the projection and the path length through the object is also W. Similarly a depth d of the feature contributes to an area of d × d in the projection.

3. Additional considerations for all X-ray imaging techniques

4. Types of imaging

The three overall types of imaging are analysed below in terms of the dependence of the required fluence and dose on object size and resolution to obtain sufficient contrast to identify a feature. The following analysis assumes that a feature (e.g. a protein molecule) has to be imaged against the surrounding density of an object (e.g. the cytosol in a biological cell). Parameters of the feature and object are given the subscripts f and o (e.g. T_f and T_o for the transmission) while the contrast between them is given the subscript c.

5. Comparison of dose and resolution estimates

Many of the methods involve imaging by more than one technique. Coherent-based methods such as CDI, holography and ptychography can be used to obtain the complex refractive index. Zernike methods also depend on both phase and absorption contrast while SCM can obtain a signal from both coherent and incoherent scattering. The most appropriate technique will depend on factors such as the thickness of the specimen as well as the contrast mechanism.

Table 1 gives a comparison of fluence and dose estimates obtained from this article and the literature for identifying features within biological cells. Different assumptions are present for the various analyses. For example, in the works by Howells et al. (2009) and Nave (2018) the assumption was that, for coherent imaging, a value of K = 5 applied to the intensity is required in each relevant reciprocal volume. Starodub et al. (2008) suggested that a value of 2 × 5^1/2 (corresponding to a distribution of the number of photons with an RMS value of 6.25 rather than 25) was sufficient as the image is derived from the modulus of the scattered amplitude rather than the intensity. If this is applied, the dose and fluence figures given by Nave (2018) and Howells et al. (2009) for phase contrast would be a factor of four lower, or improvements in resolution would be obtained for the same dose. These values are from analytical calculations with known phases. Values of dose and fluence from Starodub et al. (2008) using numerical simulations with unknown phases in 2D are given for comparison. In addition, Starodub et al. (2008) also discussed the dependence of required fluence on the type of feature to be examined. Most of the examples in Table 1 use uniform-density voxels but 2D Gaussian features were used for the numerical simulations by Villanueva-Perez et al. (2018). Where the same approach is used, the table provides guidance for the dependence on parameters such as sample contrast and energy dependence. With these parameters fixed, the table provides guidance on the different approaches (e.g. with known phases or unknown phases).

Table 1 A comparison of the required fluences and doses for X-ray imaging NA = not applicable. Feature acronyms: heterochromatin (HC), inner mitochondrial membrane (IMM), lipid droplet neutral core (LDNC), starch granule (SG) and heterochromatin potassium [HC(K)]. Data that appear in bold are used for the plots in Fig. 1. Where a resolution other than 10 nm is given for the theory and simulations, additional estimates are provided assuming 10 nm resolutions, using the theoretical model. Reference Contrast Feature Object Energy (keV) d  (nm) W (µm) Fluence (photons µm−2) Dose (Gy) Theoretical studies (assume known phases for coherence-based methods) Nave (2019)† Absorption Protein Water 0.52 10 NA 3.29 × 109 1.01 × 108 Nave (2019)† Absorption Protein Water 4 10 NA 1.14 × 1015 6.78 × 1012 Schneider (1998)‡ Absorption Protein Water 0.52 30 10 4.00 × 107 2.00 × 106 Schneider (1998)‡ Absorption Protein Water 0.52 10 10 3.24 × 109 1.62 × 108 Nave (2018)† Phase Protein Water 0.52 10 NA 1.31 × 109 4.01 × 107 Nave (2018)† Phase Protein Water 4 10 NA 3.30 × 1011 1.96 × 109 Howells et al. (2009) Phase Protein Water 10 10 NA 2.00 × 1012 2.00 × 109 Schneider (1998)‡ Phase (Zernike) Protein Water 0.52 30 10 1.00 × 107 5.00 × 105 Schneider (1998)‡ Phase (Zernike) Protein Water 0.52 10 10 8.10 × 108 4.05 × 107 This article§ Phase Protein Water 0.52 10 NA 3.27 × 108 1.00 × 107 This article Fluorescence HC (K) Nucleosol 3.61 30 5 1.09 × 1012 7.60 × 109 This article Fluorescence HC (K) Nucleosol 3.61 10 5 8.82 × 1013 6.15 × 1011 Nave (2019)† Absorption HC Nucleosol 0.52 10 NA 5.77 × 1010 2.09 × 109 Nave (2019)† Absorption IMM Cytosol 0.52 10 NA 1.71 × 1010 5.23 × 108 Nave (2019)† Absorption LDNC Cytosol 0.52 10 NA 1.03 × 1010 3.15 × 108 Nave (2019)† Absorption SG Cytosol 0.52 10 NA 3.83 × 1010 1.17 × 109 Nave (2018)† Phase HC Nucleosol 0.52 10 NA 2.24 × 1010 6.85 × 108 Nave (2018)† Phase IMM Cytosol 0.52 10 NA 7.13 × 109 2.18 × 108 Nave (2018)† Phase LDNC Cytosol 0.52 10 NA 5.50 × 109 1.68 × 108 Nave (2018)† Phase SG Cytosol 0.52 10 NA 6.30 × 109 1.93 × 108 Nave (2018)† Phase HC Nucleosol 4 10 NA 3.81 × 1012 2.27 × 1010 Nave (2018)† Phase IMM Cytosol 4 10 NA 3.18 × 1012 1.89 × 1010 Nave (2018)† Phase LDNC Cytosol 4 10 NA 2.40 × 1012 1.43 × 1010 Nave (2018)† Phase SG Cytosol 4 10 NA 4.32 × 1011 2.56 × 109   Simulations (unknown starting phases for coherence-based methods) Starodub et al. (2008)¶ Phase Protein Water 10 10 NA 1.20 × 1014 1.20 × 1011 Villanueva-Perez et al. (2018)†† Phase Protein Water 10 34 5 1.43 × 1015 1.16 × 1012 Villanueva-Perez et al. (2018)†† Phase Protein Water 10 10 5 2.21 × 1018 1.79 × 1015 Villanueva-Perez et al. (2018)†† Compton Protein Water 64 34 5 1.15 × 1014 3.63 × 109 Villanueva-Perez et al. (2018)†† Compton Protein Water 64 10 5 1.53 × 1016 4.84 × 1011   Experimental results from biological cells Rodriguez et al. (2015) CDI Various Cytosol 8 75 ∼1–5 — 4.55 × 108 Shahmoradian et al. (2017) Ptychography Axons 1.8 M sucrose 6.2 115 ∼40 — 2.0 × 107 Diaz et al. (2015) Ptychography Various Cytosol 6.2 180 — — 6.7 × 105 Kosior et al. (2012, 2013)‡‡ Fluorescence K, Zn, etc. Various Cytosol 17 150 — — 9 × 108 Chichón et al. (2012)§§ Absorption, full field Various Cytosol 0.51 55 — — 5.0 × 109 Maser et al. (2000)¶¶ Absorption, STXM Various Cytosol 0.52 100 — — 5.0 × 1011 †The calculation of dose assumes the energy of the absorbed photons is spread throughout the object rather than just within the feature. ‡Estimated from Figs. 5 and 7. Assumes attenuation through an ice-layer thickness of 10 µm. §As for row 5 but with the Rose criterion applied to the amplitude rather than the intensity giving a factor of four reduction in fluence and dose. ¶Dark line, Fig. 7. ††Tables 1 and 2. Note that there appears to be a mistake for the density of the biomolecule in Table 1 which should presumably be 1.35 g cm−3. The 34 nm value for W corresponds to 2σ for a Gaussian feature. ‡‡Scanning fluorescence X-ray microscopy in 2D. The probe size is 150 nm. §§Soft X-ray tomography with zone-plate objective (the efficiency is ∼10%). ¶¶Scanning transmission X-ray microscopy (STXM) observation. The dose figure is that at which observable mass loss occurs.

It is seen that much lower doses are required for imaging low-density high-carbon-content components in the water window compared with imaging at higher energy. The combination of absorption and phase contrast has been analysed in terms of `optimum phase contrast' (Schneider, 1998) in the context of imaging protein against water with a Zernike phase plate. Imaging within the water window also requires much lower doses for the cellular components heterochromatin, mitochondrial inner membrane, lipid and, to a lesser extent, high-density starch granules. X-ray imaging at higher energies has been shown to be reasonably successful in 2D imaging of high-density cellular components such as chloro­plasts, pyrenoids and polyphosphate bodies (Deng et al., 2017).

At energies above the water window there are two reasons why the required dose is significantly higher than at energies within the water window. Firstly, the contrast difference between carbon-containing components and water decreases above the water window. Some biological membranes, if lightly loaded with protein, could actually match the cytosol density, effectively making them invisible at higher energies, whereas their high carbon content would make them very visible at 520 eV. Secondly, the water (typically corresponding to 70 to 85% of the cell) absorbs much more strongly at energies above the oxygen absorption edge, therefore increasing the dose for the same fluence.

Fig. 1 indicates the positions of some of the examples given in Table 1 on a dose versus resolution plot. In addition, the Howells et al. (2009) model of the dose threshold as a function of resolution is shown together with a more recent one (Atakisi et al., 2019) obtained from radiation-damage studies of protein crystals. Similar diagrams occur in the works by Shen et al. (2004), Howells et al. (2009) and Villanueva-Perez et al. (2018). The Atakisi radiation-damage model assumes that local disordering reactions occur. If these applied during a tomographic data collection for X-ray imaging the result would be a loss of resolution as well as possible artefacts. The highest attainable resolution shown in Fig. 1 is that for coherent imaging at 0.52 keV for protein contrasted against water, assuming the Rose criterion (K = 5) applied to the amplitude in a Shannon voxel and the Atakisi et al. (2019) model of tolerable dose applied. All these conditions, plus the assumption of efficient phase determination, would have to be met to realize the 3 nm resolution implied in Fig. 1. A 3 nm focal spot is beyond the capabilities of present X-ray optics and this implies that ptychography in the far field is likely to be necessary in order to approach such resolutions. The dose-limited resolution would in any case degrade to ∼5 nm for a 10 µm thick sample owing to the attenuation of the signal at 0.52 keV.

Figure 1 Values for required dose against resolution given in the literature, including this article. For details see Table 1. The plots are for imaging protein against water/amorphous ice and based on theoretical calculations for CDI unless otherwise indicated. The CDI plots assume a 1/d4 resolution dependence of the required dose, except for the Villaneuva-Perez plot which assumes a W2/d6 dependence. The SCM plot assumes a W/d4 dependence. Note that the values of W for the experiments do not necessarily correspond to the simulations in Villaneuva-Perez et al. (2018). Plots for the resolution dependence for the tolerable dose at which the intensity decreases by half are shown with a 100 MGy nm−1 dependence (Howells et al., 2009) and a q1.86 dependence (Atakisi et al., 2019), where q is the wavevector (2π/d).

The example used by Villanueva-Perez et al. (2018) to compare SCM and CDI consisted of a 34 nm feature embedded in a 5 µm sized object and assumed that the dose dependence performed as W/d⁴ for SCM (given as L/d⁴ in the article) and as W²/d⁶ for CDI. The theoretical analysis then showed that the required dose for CDI would be ∼10³ higher than for SCM. If a 1/d⁴ dependence for CDI applied, the dose required for CDI would instead be ∼20 times less than for SCM. The reason for the strong particle size and resolution dependence for CDI in the work by Villanueva-Perez et al. (2016) comes from an assumption in the theoretical section. Equation (9) in this section gives the required SNR (using the nomenclature in the present article) as

where N is the total number of scattered photons from a pixel of size d × d. The analysis was carried out in two dimensions with the assumption that the dose-fractionation theorem applied. The d/W term corresponds to the Shannon sampling requirement in each dimension. The expression for the SNR is equivalent to requiring that each pixel (size d × d) in the object contributes the same number of scattered photons to each Shannon pixel in the scattering pattern and this is independent of the object size. No justification is provided for this and the W²/d⁶ dose dependence is a consequence of this assumption. In coherent imaging, the complex amplitudes from the scattering of the pixels or voxels are added before squaring for the intensity. This preserves the scattering statistics from each pixel or voxel in the presence of the surrounding ones and leads to the 1/d⁴ dependence.

The simulations (Villanueva-Perez et al., 2018) appeared to show an advantage for SCM compared with CDI. These simulations were carried out for Gaussian features in 2D. Other 2D simulations show that up to two orders of magnitude increase in dose can be required for CDI compared with the situation where the phases are known (e.g. Starodub et al., 2008). The advantages of Compton microscopy over CDI therefore depend on whether a larger dependence (e.g. W²/d⁶ as in the work by Villanueva-Perez et al., 2018) for coherent-scattering techniques is valid for the case of unknown phases. There is no theoretical basis for this and the performance of CDI in the simulations might have been because of the less-robust phasing procedures with 2D data and/or the use of Gaussian features. In general, coherent-scattering methods should be more powerful than incoherent-scattering methods. For coherent scattering, the phase relationships between waves scattered by each component are retained as opposed to a simple summing of the intensities.

Near-field holography appears to offer advantages for phase retrieval over CDI (Hagemann & Salditt, 2017; Du et al., 2019). Ptychography can also provide some phase information because of the overlap between probe positions (Bunk et al., 2008). This was confirmed in the work by Du et al. (2019), where an advantage was found for ptychography over holography. These simulations were carried out in 2D and assumed that only phase contrast was present and that the illumination function was known. For the general case, with no assumptions about the specimen, there are four unknowns corresponding to the phase and amplitude for both the sample scattering and the incident beam. Near-field ptychography (Stockmar, 2013; Clare et al., 2015) aims to address these four unknowns. Far-field ptychography could also determine these four unknowns by exploiting the overlap between the probe positions. Use of structured or diffuse illumination might offer a further advantage (Liu et al., 2008). Further development of reconstruction algorithms together with simulations and experiments on realistic samples should hopefully resolve the most appropriate technique for coherent-based imaging. For synchrotron-radiation sources, techniques such as ptychography and near-field holography appear to be replacing CDI. However, if the aim is to outrun radiation damage by exploiting the short pulse length, ptychography has limitations on FELs. This is because radiation-damage effects can cover an area larger than the initial illumination.

For coherent-based methods, the Rose criterion should be applied to the modulus of the amplitude (as in the work by Starodub et al., 2008) of the scattered wave rather than to the intensity. However, at any radiation-damage limit, this would only lead to a small improvement in resolution (e.g. from 10 to 7.5 nm) because of the strong dependence of required dose with resolution and the dependence of the radiation-damage limit on resolution.

Methods which utilize an objective lens or involve scanning a small focal spot at the specimen position can suffer from depth-of-focus limitations. As an example, Villanueva-Perez et al. (2018) estimate the depth of focus of their proposed SCM as 10 µm at 64 keV with a focal spot of 10 nm. Methods are being developed to handle depth-of-focus (or depth-of-field) effects (e.g. Tsai et al., 2016; Gilles et al., 2018) but depth-of-focus effects are likely to lead to a decrease in the information content of the data for incoherent-based methods. The other disadvantage of incoherent-scattering methods is the path-length dependence of the signal, meaning that thicker specimens will require a higher dose to achieve the same SNR as thinner specimens. The two effects together will limit the application of fluorescence microscopy as the required dose will increase for thicker specimens if the Wρ_o term is significant.

It is of interest to compare the dose required for imaging a typical cellular feature by fluorescence with that required for imaging via ptychography at the same resolution. With the potassium concentrations used in the work by Nave (2018), the fluence and dose required to image the potassium in heterochromatin (potassium concentration of 0.018 g cm⁻³) at 30 nm resolution against 5 µm of the surrounding nucleosol (0.007 g cm⁻³) can be calculated. The required fluence is 1.09 × 10¹² photons µm⁻² and the required dose is 7.60 × 10⁹ Gy. These values assume that all the fluorescent photons can be collected. The estimate would increase to 6 × 10¹¹ Gy at a resolution of 10 nm. This is much greater than the required dose to image the heterochromatin itself at either 4 keV or 0.52 keV by coherent-imaging methods (see Table 1). This is consistent with the significant damage seen in phase-contrast images after a fluorescence scan (Kosior et al., 2012, 2013). For situations where there is a high concentration of metal atoms in the feature, where the specimen is thinner and where the surrounding material contains a much lower concentration, the required dose will of course be much lower. A discussion of radiation damage in scanning-probe fluorescence microscopy can be found in the work by Jonge et al. (2014).

6. Conclusions

Assuming known phases for coherent scattering, the required dose and fluence for both coherent scattering and absorption follows a 1/d⁴ dependence, where d is the resolution, and a 1/C² dependence, where C is the contrast. For incoherent scattering, the required dose and fluence follows a W/d⁴ dependence if there is a large signal from the bulk of the object and a 1/d³ dependence if the signal from the bulk is negligible. The tolerable dose is much less well determined with possibilities ranging from a 1/d dependence (Howells et al., 2009) to a higher power dependence, e.g. 1/d^1.86 as suggested by Atakisi et al. (2019).

The present article is not in agreement with the theoretical dose dependence (W²/d⁶) for CDI with known phases given in the works by Villanueva-Perez et al. (2018, 2016). The simulations in these articles are carried out in 2D, use Gaussian features and do not cover a range of values of W and d, so only provide a partial test of the theoretical model. It is possible that the lower resolution obtained in the work by Rodriguez et al. (2015) was caused by a combination of factors. Firstly, the cellular components could have had a much lower contrast than that given by a protein molecule in pure water (or amorphous ice). Secondly, the phase-determination step is unfavourable for CDI, compared for example with ptychography and near-field holography. Thirdly, the specimen exposure (fluence) was far below that required for higher-resolution imaging and also below that for any radiation-damage threshold at the resolution obtained. If the dose was increased by a factor of 10, a resolution of 42 nm might have been obtained. The dose would then have been near the radiation-damage limit according to the Howells et al. (2009) model but less than that for the Atakisi et al. (2019) model (see Fig. 1). The radiation-damage models for X-ray imaging do not appear to have been tested in detail over a wide range of resolutions. Mass loss was observed at 100 nm resolution with a dose of 5 × 10¹¹ Gy (Maser et al., 2000) but this does not correspond to the half-dose metric shown in Fig. 1. In addition, the relationship between radiation damage for crystalline specimens and that for non-crystalline specimens would also merit further investigation. It is possible that the damage would be similar if only local disorder occurred rather than longer-range disorder (Atakisi et al., 2019).

Measurements in the water window offer a considerable advantage for imaging low-density components in frozen hydrated biological cells because of the increased phase and absorption contrast. If phase retrieval is reasonably efficient there will be an advantage in lensless imaging methods over those based on objectives lenses because of inefficiency and depth-of-focus issues for objective lenses. For thicker specimens, again, provided phase retrieval is reasonably efficient, coherent-imaging methods will offer a significant advantage compared with incoherent methods such as SCM. Given these assumptions, there is no dependence of the required dose with object size until significant X-ray attenuation occurs. Further simulations and tests on realistic samples are obviously required to validate these conclusions. Significant developments are occurring in electron microscopy for thicker samples including techniques such as tomographic imaging, serial sectioning and focused ion-beam milling, for both imaging cells and brain tissue. For a discussion of these electron-microscopy techniques see, for example, Xu et al. (2017) and Hoffman et al. (2020). A detailed comparison of the relative abilities of imaging by electrons and X-rays can be found in the work by Du & Jacobsen (2018).

The above discussion applies to frozen hydrated specimens and assumes that this is the nearest practical approach to a native state when using ionizing radiation to obtain images. This is undoubtedly the case for imaging at near-atomic resolution for protein structure determination but there are a large number of electron-imaging studies using other forms of specimen treatment such as freeze substitution, fixation, resin embedding and staining. If these specimen-preparation techniques are considered to be acceptable at resolutions achievable by X-ray imaging and they offer enhanced contrast, this would improve the capabilities of X-ray imaging. This issue needs to be considered along with the usefulness of imaging biological cells at the relatively low resolution obtainable by X-ray imaging. Many of the processes within biological cells take place at a local level where the interest is in examining the interactions involving individual macromolecules and there is a significant advantage with the higher resolution obtained with electron imaging. X-ray imaging at higher energies is likely to have a significant role for thicker samples such as biological tissue and organoids. For imaging the neurons and synapses in the brain, the interactions of interest could take place over large distances so meaningful data for large objects would be required and potentially available via X-ray imaging. A demonstration of the capabilities is the imaging of axons in mouse-brain tissue (Shahmoradian et al., 2017) examined at cryotemperatures, prepared using chemical fixation and a 1.18 M sucrose cryoprotectant (density is ∼1.23 g cm⁻³), and examined by ptychograpy. The mass densities ranged from ∼0.99 to a maximum of 1.42 g cm⁻³ consistent with there being adequate contrast to image the features. The contrast was attributed to the high density of the cryoprotectant compared with the density of the myelin. Some details of the data collection are given in Table 1, and the dose and resolution values are shown in Fig. 1.

Whether X-ray imaging of intrinsically low-contrast bio­logical specimens can reach the most optimistic estimates of achievable resolution (e.g. 3 nm) depends on a variety of factors. These include which radiation-damage model is correct, the reliability of any phase-retrieval algorithm, the (energy-dependent) contrast for the features of interest, the statistical precision required, the acceptability of contrast-enhancing specimen-preparation procedures and the overall efficiency of the instrument. A systematic study of many of these factors is necessary in order to give a more reliable estimate of the potential of X-ray imaging for these type of materials. At present, there are few observations which can be used to populate diagrams such as Fig. 1 in this article. In order to compare theory or simulations with experiment, standard methods of reporting results covering parameters such as the feature examined (e.g. which cellular organelle), resolution, contrast, depth of the material (for 2D), estimate of noise and dose should be encouraged. Much effort has gone into implementing this for fields such as macromolecular crystallography but it has to be admitted that this is still an ongoing process.