1. Introduction

X-ray Coherent Diffractive Imaging (CDI) is rapidly developing as a routine method for biological imaging because the short wavelength and specific interaction of X-rays with matter can be exploited to image intracellular features [1–8] with spatial resolution beyond the limits imposed by image-forming optics [9]. Phase-diverse Fresnel CDI (FCDI) is an extension of ptychography [10] in which an object is translated in three dimensions through illumination with phase curvature introduced by a focusing optic [11]. It has been shown to provide increased contrast with reduced X-ray dose, while imaging extended areas virtually free from artefacts [10–12]. Recently, phase-diverse FCDI has been shown to provide high-contrast, high-resolution images of biological samples with significantly lower X-ray dose compared to other CDI methods [7,13].

CDI involves recording a coherent diffraction pattern of an object in the far-field and reconstructing the complex exit surface wave from the object using iterative phase retrieval algorithms. These algorithms depend on a set of constraints defined by the measured intensities and known properties of the object and illumination [14,15]. The reconstructed exit surface wave provides a quantitative measure of the complex transmission function of the object. A priori knowledge of the object composition can therefore be used to define a constraint that is applied in the iterative algorithms. For example, the magnitude, T, and phase, ϕ, of the complex transmission function can be independently constrained to fall within physically reasonable limits (see, for examples [16,17]). For homogeneous areas of an object of known composition, the ratio between the logarithm of the magnitude of the transmitted wave, $\ln \left(\left|T\right|\right)$, to the phase change, ϕ, in the illuminating wave as it passes through the object can be constrained to a physically appropriate value. Known as a complex constraint, this is found to improve the convergence of the phase retrieval algorithm and the quality of the reconstructed object image [18]. However, for heterogeneous objects such as biological specimens the ratio $\ln \left(\left|T\right|\right)/\varphi$ cannot be constrained to a single value. It was previously asserted that constraining this ratio to a range that encompasses the values for all known components of a heterogeneous object will improve convergence of the phase retrieval algorithm [18]. Here we test this assertion by applying a complex constraint to whole red blood cells infected with the malaria parasite Plasmodium falciparum using phase-diverse Fresnel coherent diffractive imaging at X-ray energies of 520 eV and 2535 eV. We show that by constraining the ratio $\ln \left(\left|T\right|\right)/\varphi$ to a range encompassing the values appropriate for the cell and parasite components the quality of the reconstruction of both the phase and magnitude components of the complex transmission function are improved, achieving consistency between their values that is not otherwise possible.

2. Method

The application of the complex constraint employed here follows the description given by Clark et al. [18]. The X-ray transmission and phase retardation in the complex wavefield in each finite area (or imaging pixel) of the object are, respectively:

The ratio C at each pixel can be constrained to this range at each iteration of the phase retrieval algorithm, in a way that is analogous to the application of a positivity constraint [14]. In this case, however both the phase and magnitude are constrained. Equations (3) and (4) together set upper and lower limits of the permissible range of the values of the phase and magnitude. As with other constraints, values that fall outside of this range are set to the nearest limit of the range, defined as;

Phase-diverse FCDI [11] data at X-ray energies of 520 eV and 2535 eV was used to illustrate the effectiveness of the method, with the experimental and data processing details explained in [20] (520 eV) and [7] (2535 eV). In each case, the sample was a red blood cell infected with the malaria parasite P. falciparum. As the parasite develops, it consumes the hemoglobin contained in the host cell, reducing the hemoglobin volume by 75% compared to an uninfected cell [21]. The digested hemoglobin is stored as hemozoin crystals in the parasites digestive vacuole [22], and membrane protein structures such as Maurer’s clefts are observed in the host cell [23,24]. The samples were fixed and dried prior to imaging, with full details described elsewhere [7,20]. Consequently, the samples contain very little water and the level of undigested hemoglobin would likely have been small due to perforations in the host cell membrane caused during fixing and drying. As a result, we expect the samples to be heterogeneous and largely made up of proteins in cell membranes, exhibiting similar properties to many biological samples. In this case, hemozoin is likely to concentrated in moderated abundance in the digestive vacuole of the parasite, and absent from other areas. Values for the ratio C (Eq. (3)) for materials likely to be present in the sample are listed in Table 1. As any single pixel contains information from a selection of materials at relative abundances listed in Table 1, the complex constraint, Eq. (4), was chosen to limit $\ln \left(\left|T\right|\right)/\varphi$ to within a range of $-0.25\le C\le -0.19$ for the data at 520 eV, and $-0.033\le C\le -0.025$ for the data at 2.5keV. In addition to the complex constraint, constraints limiting the phase and magnitude $-\pi <\varphi <0$ and $0<T<1$respectively were applied. We also applied the modulus and support constraints, which replace the diffraction intensity at each iteration with the measured diffraction intensity, and limit the extent of the object in the sample respectively [15,18].

Table 1. Relative abundance of materials likely to be found in the samples and approximate values for C (Eq. (3)) obtained from [26].

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3. Results

Figure 1 shows the results of applying the complex constraint as described in Section 2 to a red blood cell infected with the malaria parasite P. falciparum at an X-ray energy of 520 eV. Figures 1(a) and 1(c) shows the effect of applying a constraint whereby the phase [Fig. 1(a)] and magnitude [Fig. 1(c)] are limited to $-\pi <\varphi <0$ and $0<T<1$respectively, while Figs. 1(b) and 1(d) show the effect of applying, in addition, the complex constraint, $-0.25\le C\le -0.19$. It is evident in Figs. 1(a) and 1(c) that inconsistencies between the magnitude and the phase in the recovered complex transmission function (Fig. 1(a), arrow), are avoided in Figs. 1(b) and 1(d).

 

Fig. 1 The phase (top) and magnitude (bottom) of the complex transmission function of a red blood cell infected with the malaria parasite P. falciparum at the trophozoite life-cycle stage. Results in A and C were obtained by independently constraining the phase and magnitude within $-\pi <\varphi <0$ and $0<T<1$_, respectively. In B and D, the ratio, C, is applied as discussed in the text.

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A similar approach was applied using a higher X-ray energy of 2.5 keV applied to a red blood cell infected with the malaria parasite P. falciparum at the sexual gametocyte life-cycle stage, with results shown in Fig. 2. At this energy the ratio C is dominated by the δ component of the complex refractive index, yielding a magnitude image with little information and artifacts seen in the magnitude image that do not relate to features in the corresponding phase image (Fig. 2(c), arrow). Accordingly, in this case, we apply the complex constraint to update the magnitude information based on the phase using Eq. (5b) only, rather than updating both the phase and magnitude as in the previous case. Figures 2(a) and 2(c) used the same constraint as in Figs. 1(a) and 1(c), limiting the phase and magnitude to within $-\pi <\varphi <0$ and $0<T<1$respectively, while Figs. 2(b) and 2(d) were obtained by also applying the complex constraint, C as outlined in Section 2. This application resulted in the elimination of the previously mentioned artifacts from the magnitude image, and results in a magnitude image that is, as expected, more consistent with the phase image.

 

Fig. 2 The phase (top) and magnitude (bottom) of the complex transmission function of a red blood cell infected with the malaria parasite P. falciparum at the sexual gametocyte life-cycle stage. Results in A and C were obtained by independently constraining the phase and magnitude to within $-\pi <\varphi <0$ and $0<T<1$ respectively. In B and D the complex constraint, C, was applied as discussed in the text.

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We found that in these samples the χ² error between the measured intensity and the intensity calculated from the reconstructed transmission function was not useful discriminator between the different reconstructions, consistent with earlier findings [25]. However, as the reconstructed complex transmission function provides quantitative results of the phase and magnitude, measurements of the thickness and composition of the sample can be made using Eqs. (2a) and (2b). The relative sample thickness obtained from the phase and magnitude provides an easily interpretable comparison to judge consistency, and was calculated for a line profile (dashed line indicated in Fig. 1(c) and 2(c)). Figure 3 shows relative thickness for the images in Figs. 1(a)–1(d) [Fig. 3(a)] and Figs. 2(a)–2(d) [Fig. 3(b)], for the central value of C (lines) and allowing for the range in C described in Section 2 (corresponding shaded area). It is clear that the application of the complex constraint (black and grey) provides a much more self-consistent result, with the thickness determined from the phase (dashed lines) and magnitude (solid lines) in much better agreement. It is also evident from this analysis that without the application of the complex constraint (red) at lower energies, neither the phase nor magnitude independently give a reliable reconstruction. However, at higher energies, the phase reconstruction is similar with or without the application of the constraint.

 

Fig. 3 Line profiles through the reconstructions of the two samples at the positions indicated in Fig. 1(c) [Fig. 3(a)] and Fig. 2(c) [Fig. 3(b)], plotted as the thickness using the central value of the ratio C (solid and dashed lines) and the range of C (corresponding shaded area) for the phase (dashed lines) and magnitude (solid lines). Red and grey refers to the case when the complex constraint was not applied, and the case where the complex constraint was applied respectively. The label for each line refers to the corresponding panel, (A)-(D), in Figs. 1 and 2.

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4. Discussion and conclusion

We have shown that although a complex constraint based on a single material cannot be applied to CDI for the biological sample used here, a complex constraint based on the expected average composition can greatly improve the self-consistency of the reconstructed complex transmission function. The measured thickness derived from measurements of the real and imaginary components of the refractive index show a much higher agreement with each other compared to the case when only the real and imaginary parts of the transmission function are independently constrained. This demonstrates an enhancement to the quantitative nature of CDI. By adjusting the range (Eq. (4)) for specific samples using limited a priori knowledge, the method can readily be applied to a variety of biological and other heterogeneous specimens. The method described here can readily be applied other two and three-dimensional CDI techniques.