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Dualplane coupled phase retrieval for nonprior holographic imaging
PhotoniX volume 3, Article number: 3 (2022)
Abstract
Accurate depiction of waves in temporal and spatial is essential to the investigation of interactions between physical objects and waves. Digital holography (DH) can perform quantitative analysis of wave–matter interactions. Full detectorbandwidth reconstruction can be realized based on inline DH. But the overlapping of twin images strongly prevents quantitative analysis. For offaxis DH, the object wave and the detector bandwidth need to satisfy certain conditions to perform reconstruction accurately. Here, we present a reliable approach involving a coupled configuration for combining two inline holograms and one offaxis hologram, using a rapidly converging iterative procedure based on twoplane coupled phase retrieval (TwPCPR) method. It realizes a fastconvergence holographic calculation method. Highresolution and fullfield reconstruction by exploiting the full bandwidth are demonstrated for complexamplitude reconstruction. Offaxis optimization phase provides an effective initial guess to avoid stagnation and minimize the required measurements of multiplane phase retrieval. The proposed strategy works well for more extended samples without any prior assumptions of the objects including support, nonnegative, sparse constraints, etc. It helps to enhance and empower applications in wavefront sensing, computational microscopy and biological tissue analysis.
Introduction
Accurate detection of wave propagation in spatial and temporal domains is essential for the investigation of wave–matter interactions. Holographic imaging records the entire complexamplitude wavefront by interference [1, 2], becoming a powerful technique of quantitative phase imaging. Holographic recording and replaying of waves have been widely applied in various fields, such as quantitative phase measurements for stainfree biological cells [3,4,5,6,7], wavefront shaping [8, 9], particle analysis and 3D tracking [10, 11], etc.
Two main holographic configurations exist, i.e., offaxis and inline holography. Offaxis holography can achieve separation of the object wave and conjugate wave by introducing a tilted reference wave. It is widely used in phase microscopic imaging and instruments [12,13,14,15,16]. The maximum bandwidth of the sample should be less than a half of the bandwidth of the detector [17]. Nevertheless, the inline configuration is affected by the fact that the main ray of the object wave and its conjugate wave share the same optical path. The full bandwidth utilization of the detector can be achieved but the superposition of twin images interferes with the quantitative amplitude and phase map. It has been applied in live cell observation [18, 19] and the imaging of whole pathological sections [20,21,22,23]. Several strategies have been reported for extracting the object phase including singleplane phase retrieval [24,25,26,27,28], multiimage phase retrieval [22, 23, 29,30,31,32,33,34,35,36,37,38,39,40,41] and noniterative method [42,43,44,45,46]. The singleplane method carries out an iterative procedure between the measured plane and the estimated plane. It requires prior constrains of the object to escape local convergence in the iterative procedure. The multidistance phase retrieval (MPR) method can achieve highfidelity and stable reconstruction at the expense of measuring many observation planes through axial scanning [30, 31, 34,35,36,37,38,39,40,41]. Objectbased constraints can be transferred to detectionbased constraints without prior assumption. The accuracy of MPR highly depends on the number of measurements. When the paraxial approximation is satisfied, the distribution of the object's diffraction field at different distances is approximately a transversal elongation effect; thus, the hologram at another distance can be obtained by stretching the hologram [47].
Phase retrieval (PR) solves the inverse problem of nonconvex optimization under the known constraints with an iterative procedure [48]. The convergence of nonconvex optimization suffers from phasestagnation problem with a limited initial guess. The support prior, sparsity assumption, nonnegative assumption or random modulation can help to escape local convergence [49,50,51,52,53,54]. The phase solution of transport of intensity equation (TIE) between multiple planes can be considered as the initial phase guess to assist the convergence of iterative reconstruction [22, 55, 56]. Accurately solving the TIE initialphase (TP) estimation requires imposing boundary conditions and certain approximations [57]. Higher convergence speed can be obtained by using offaxis optimization phase in the hologram plane while prior knowledge and nonnegative constraints still need to be imposed on the object [49, 58,59,60,61]. Here, we propose a twoplane coupled phase retrieval (TwPCPR) method to achieve stable and versatile spatial complexamplitude reconstruction. No constraint or prior knowledge of the object is needed in the proposed approach. Two inline holograms at two distances z _{1} and z _{2} and one offaxis hologram at z _{1} (or z _{2}) are recorded with the hybrid optical setup. It realizes a fastconvergence holographic calculation method.
Essentially, all the constraints of the TwPCPR method are derived from measurements without any prior constraints of the object, making it expand the ability to become a wavefront sensor. Offaxis optimization phase provides an effective initial guess to avoid stagnation and minimize the required measurements of the MPR method. Two imaging sensors capture holograms simultaneously to avoid the mechanical shift. We demonstrate the superiority of the proposed method by reporting imaging reconstruction of a variety of objects, i.e., an amplitude resolution target, phase resolution targets and biological samples, using both numerical simulations and experiments.
Materials and methods
The optical setup coupled DH system
The geometrical configuration of the lensless hybrid holographic system is shown in Fig. 1(a), and it also includes microscopic mode, respectively. The light source is a HeNe laser with a wavelength of 632.8 nm. The illumination plane wave is divided into two beams by using beam splitter (BS) 1. The imaging sensor chip has the resolution of 1200×1920 with a pixel size of 5.86 μm (QHY174, QHYCCD Co., Ltd.). The total field of view (FOV) of this lensless system is 7.032×11.251 mm^{2}. Two sensors are used to detect holograms simultaneously without mechanical shift. The system includes an inline device and an offaxis device. They are combined by BS2 and switched by a shutter. In the lensless system, the sample is placed in front of BS2, and the reference wave illuminates from its adjacent surface. The distance between the object and the sensor can be adjusted by changing the size of the BS. In the microscopic system, the parameters of the microscope objective (MO) are 10× Mitutoyo Plan Apo infinity corrected objective with 0.28 NA (MY10X803, Thorlabs). The MO is combined with a tube lens (TL) to form a microscopic imaging system. The focused positions magnified object can be adjusted by axially moving the sample. The sample is placed before the MO and the focused image can be directly obtained by the detectors.
The TwPCPR principle
The distribution of the hologram is I =  P(O)+ R^{2}. O and R are the distributions of the object and reference waves, respectively, and P is the propagation vector from the object plane to the hologram plane. The MPR method receives N holograms as input, and the phase field is initially unknown. The complexamplitude field is solved to make the current estimate closely fit the input actual captured holograms, and the quantification is given by the objective function as follows [48]:
where ‖. ‖ is the Euclidean norm, I_{k} is the intensity of the k_{th} captured hologram, and O is the current estimated function corresponding to the object. Operator P_{k} is the transmission process from the object plane to the k_{th} hologram plane. N is the number of detections. The process provides a relatively correct solution in early iterations but then overshoots, which can often be attributed to the nonconvex characteristic of phase retrieval [48]. The convergence depends on more known detectionbased constraints. The lateral displacement error accumulates between each hologram as the number of measured holograms increases. Fewer constraints are desired to reduce the accumulation of errors, but having a weak convergence of iteration. Efficient subpixel registration can be achieved in the lateral displacement between different holograms [62,63,64].
The inline and offaxis recording modes can be switched by the presence or absence of a reference wave by using the shutter, where the reference wave is \(R\left(x,y\right)=A\exp \left\{i\frac{2\pi }{\lambda}\left[x\sin \left({\theta}_x\right)+y\sin \left({\theta}_y\right)\right]\right\}\). Here, θ_{x} is the angle between the projection of the reference wave illumination direction on the x–z plane and the z axis, and θ_{y} is the angle between the projection of the reference wave illumination direction on the y–z plane and the z axis. The process of the TwPCPR method is shown in Fig. 1b1. Two inline holograms at z_{1} and z_{2} and one offaxis hologram at z_{1} (or z_{2}) are obtained for the reconstruction. Half of irradiation energy is lost before the camera by using BS. The bandwidth limitations of different holographic methods are shown at the bottom of Fig. 1b2. The numerical aperture (NA) of the system determines the optical bandwidth of the object wave. The reconstruction is also limited by the sensor pixel size based on the Nyquist theorem. The highest frequency in the horizontal or vertical direction of the wave collection in detector can be described as min{1.22NA/(Mλ), 1/(2Δ)} under the coherent diffraction limit [3], where M is the magnification, and Δ is the pixel size of the camera. Onedimensional is considered for analysis here. If \(\frac{2.44 NA}{M\lambda}\le \frac{1}{4\varDelta }\), the object wave can be completely separated from its conjugate wave and autocorrelation (AC) term in the spatial frequency domain (SFD). If \(\frac{1}{4\varDelta}\le \frac{2.44 NA}{M\lambda}\le \frac{1}{2\varDelta }\), the object wave can be separated from its conjugate wave, but it overlaps with its AC term to form a slight offaxis hologram [65]. For lensless imaging systems, the offaxis bandwidth requirement is usually not satisfied (See supplementary). However, an estimated lowfrequency complexamplitude envelope of the sample can be extracted from it. In Fig. 1b3 and c, an object is assumed with a spectrum of full camera’s bandwidth. 2Δ is the smallest detection period of the camera based on the Nyquist theorem. The inline reconstruction is obtained by using backpropagation method. Full bandwidth reconstruction can be achieved but twin image is superimposed on the reconstructed object. The slight offaxis reconstruction is the inverse Fourier transform after filtering the 1^{st} order spectrum under the circular pupil function with the radius of a quarter of camera’s bandwidth. Twin image is almost removed, but highfrequency details are missing and some artifacts remain in the object due to incompletebandwidth reconstruction. A faithful reconstruction of the TwPCPR method is shown as a result of combining the advantages of the two holographic configurations. It works for more extended samples without any prior assumptions in the object plane.
The specific optimization process is described by the following steps:

Step 0: Initialization. The size of each hologram is M _{1} × M _{1} pixels. The inline holograms I _{1} and I _{2} are located at objectdetector distances z _{1} and z _{2}, respectively. The offaxis hologram is located at z _{1} (or z _{2}). In the MPR method, the initial phase φ _{0} are set to constant or random. In the TwPCPR method, the initial phase φ _{0} comes from offaxis hologram. The diffraction wave is filtered in the SFD of the offaxis hologram. First, the offaxis hologram is performed by Fourier transform. Second, the 1^{st} order spectrum is filtered by using a pupil function. The radius of the pupil function is onesixth of the bandwidth for the camera (See supplementary). Thirdly, the inverse Fourier transform is performed in the filtering spectrum and the phase is initial phase φ _{0}. Then, the initial wave field at distance z _{1} is
Step 1: The complexamplitude field U _{2} = t _{2} exp(iφ _{2}) at the distance z _{2} is obtained from the complexamplitude field U _{1} by using the angular spectrum propagation method [2].

Step 2: The amplitude of U _{2} is replaced with the square root of the intensity of the hologram I _{2}. The phase value is maintained. Operator \({P}_{z_1{z}_2}\) represents the function of propagation from distances z _{1} and z _{2}. The complexamplitude field is obtained:

Step 3: The complexamplitude field U _{1} = t _{1} exp (iφ _{1}) is obtained from the complexamplitude field U _{2} by using the angular spectrum propagation method.

Step 4: The amplitude of U _{1} is replaced with the square root of the intensity of the hologram I _{1}. The phase value is maintained. The complexamplitude field U _{1} is obtained:
The next iteration is calculated through steps 1–4. Then, backpropagation to the object plane is performed to achieve the reconstruction from the complex amplitude U _{1} or U _{2}. The distances z _{1} and z _{2} are calculated by using the autofocusing algorithm [10, 66]. The interval between two hologram planes can be determined in this system.
Samples for testing
Three types of samples are considered: an amplitude USAF1951 resolution target, two phase resolution targets and two biological tissues, respectively. The amplitude USAF target is fabricated by etching lines onto a glass plate using nano processing methods. The phase object is the Quantitative Phase Microscopy Target (QPMT, Benchmark Technologies). It is a completely transparent measurable artifact which provides a wellcharacterized optical path difference for quantitative phase imaging. The optical path difference is achieved by increasing the thickness of pattern on the glass plate. It is made of an acrylate polymer (The refractive index is 1.52) on Corning Eagle XG Glass. The phase USAF1951 resolution target includes group 610 with the feature height of 250 nm. The other QPMT is an optically focus star on simulate circular structures with the feature height of 250 nm. The third type of sample includes two biological sections, one of which is a muscle cell section on the glass slide. It includes the transverse and longitudinal morphology of the skeletal muscle cells mainly composed of muscle fibers. The other is the mitotic section of horse roundworm fertilized egg section. It is drawn from the crosssectional of the uterus of horse roundworm. The uterus contains cells at various stages of division.
Results and discussion
The TwPCPR method
Figure 2 shows the reconstruction of complex objects by using the proposed method and the MPR method. For the MPR method, constant initial phase and TIE initial phase (TP) are both considered. The simulation parameters are as follows: the wavelength is 500 nm, the pixel pitch is 6 μm, and the propagation distances of the three planes are 4.00, 4.05, and 4.10 cm, respectively. Figures 2a1 and a2 show the amplitude and phase of the object 1. Figure 2(b) is one of the three inline holograms of the original scene. The pixels number of diffraction field is 1000×1000 and the central area with 400×400 pixels is shown. Figures 2c1c4 are the amplitudereconstructions by using the 2plane MPR (2MPR), 3plane MPR (3MPR), 3plane MPR with TP (3MPRTP) and TwPCPR method with 20 iterations, respectively. Figures 2d1d4 are the phasereconstructions by using the 2MPR, 3MPR, 3MPRTP and TwPCPR method with 20 iterations, respectively. The constant initial phase is used in 2MPR and 3MPR method. The reconstructions by using random initial phase are presented in supplementary. The normalized mean square errors (MSEs) \({E}_s=\left[\sum_{ii}\sum_{jj}{\leftr\left( ii, jj\right){r}_0\Big( ii, jj\Big)\right}^2\right]/\left[\sum_{ii}\sum_{jj}{\left{r}_0\left( ii, jj\right)\right}^2\right]\)of the reconstruction are calculated in Fig. 2, where r(ii, jj) is the estimated distribution and r _{0}(ii, jj) is the initial distribution. Figure 2e1 and e2 show the logarithm MSE of the reconstructed amplitude and phase with different numbers of iterations, respectively. The TP in the plane 2 are calculated by using the axial derivative of plane 1 and plane 3. The object 1 is centrally isolated in the image FOV. The hologram can be approximately regarded as the flat boundary, which is recovered regardless of the selected boundary conditions [57]. The universal solution is used to obtain the TP here [68]. The 2MPR and 3MPR method remain the crosstalk between the amplitude and phase. A higher convergence speed and better reconstructed quality can be achieved by increasing the sampling plane in the MPR method [31, 67]. The usual expectation is a reduction in the complexity with fewer detection planes but higher reconstructed quality. The TP can speed up convergence of MPR method and it effectively avoids the stagnation in the MPR method [22, 56, 57]. The TP and the offaxis initial guess perform initial input to minimize the required measurements. The crosssections of the reconstructed amplitude and phase are shown below Fig. 2c1d4, respectively. The phase errors of TP originate from the nonlinear components related to the finite difference approximation [55, 56]. The estimation using TP provides limited convergence assistance in this parameter of the system. Figure 2f1 and f2 show the amplitude and phase of the object 2 without flat boundary. Figure 2(g) is one of the three inline holograms of the original scene. The whole diffraction field with 750×750 pixels is shown. Figure 2h1h4 and i1i4 are the reconstructed amplitude and phase by using the 2MPR, 3MPR, 3MPRTP and TwPCPR method with 20 iterations, respectively. The TwPCPR method shows the advantage of parameterfree of the imaging with insensitivity to the interval between different planes, boundary conditions, object assumptions and paraxial approximation. It is reflected in the reconstructed MSE of the amplitude and phase with different numbers of iterations in Fig. 2e1, e2, j1 and j2, respectively. The proposed TwPCPR method can achieve a lower MSE than those of the MPR methods after a few iterations.
Reconstruction on pure amplitude objects
Figure 3(a) shows the lensless holographic system. The red lines represent the route of the illuminating wave. Figure 3b1 is one of the raw inline holograms. The distance between the two diffraction planes of two sensors is 2.5 mm. For the MPR method, the distance interval between different hologram planes is 0.5 mm. Figure 3b2 is the raw offaxis hologram. Figure 3c1 is the reconstruction from the inline hologram. It is the result from Fig. 3b1 by using backpropagation to the object plane. Figure 3c2 is the reconstruction from the offaxis hologram. The radius of the pupil function is onesixth of the camera’s bandwidth (See supplementary). The advantages and disadvantages of the two types of holographic imaging are clearly shown. The inline hologram provides a highresolution reconstruction but fails to give a correct estimate of the object, while the offaxis hologram has the opposite function. The inline reconstruction in Fig. 3c1 shows the best resolvable feature with all of group 5 (line width from 8.77 to 15.6 μm) and even in group 6, element 3 (line width 6.2 μm, close to the pixel size). The offaxis reconstruction in Fig. 3c2 shows the best resolvable feature with group 4, element 4 (line width 22.1 μm). The inline reconstruction makes full use of the bandwidth of the entire camera to achieve the extreme resolution but with twin image. Figure 3(d) is the reconstruction achieved by using the TwPCPR method with only 5 iterations. Figure 3e1e4 are enlargements of Fig. 3(d) by using the 2MPR, 3MPR, 3MPRTP and TwPCPR method, respectively. The TwPCPR method outputs contrastenhanced images in which the artifacts are remarkably removed. The resolution of this system is close to the pixel size of the detector. The best resolvable feature corresponds to group 6 element 4 of USAF 1951 (6.2 μm line width). The minimum distance between the object and the sensor is the width of the BS.
Reconstruction for biological section
To further demonstrate the capability of complexamplitude reconstruction, an experiment on skeletal muscle cells is performed in Fig. 4. The transverse and longitudinal morphology of the skeletal muscle cells are prepared in one specimen. The whole specimen can be detected in the area of sensor. Figure 4(a) shows the muscle cells on a glass slide. Figure 4b1 is the raw measured inline hologram. Figure 4b2 is the raw offaxis hologram. Figure 4(c) shows the amplitudereconstruction achieved by using the TwPCPR method with 5 iterations. Figures 4d1 and d2 are the reconstructed amplitudes by using the 3MPR and 3MPRTP with 50 iterations, respectively. Figures 4(e)(f) are the corresponding phase, respectively. The reconstructed phase is calculated by the phase unwrapping algorithm [69]. Observation field by using TwPCPR method includes the transverse and longitudinal morphology of the muscle cells. They can be clearly distinguished from the reconstructed amplitude and phase. Twin images are covered in the detailed texture in Fig. 4f1f2. The reconstructions achieved by using the TwPCPR method show that it eliminates the noise artifacts and accomplishes contrastenhanced reconstruction with fewer iterations.
Reconstruction for microscopic imaging on quantitative phase target
At the aim to verify the universality of the TwPCPR method, the geometrical configuration of the hybrid DH microscopy imaging system is shown in Fig. 5(a). The principle of configuration is similar to Fig. 1(a), excepts that a microscopic system is considered between the sample and BS2. The distance difference between the two the two diffraction planes of two sensors is 3.3 mm. For the MPR method, the interval between different hologram planes is 1 mm. The bandwidth limit of this system belongs to the scope of inline holography in Fig. 1b2 according to the bandwidth calculation (See supplementary). Figure 5(c) shows the reconstruction of QPMT with the feature height of 250 nm by using TwPCPR method with 5 iterations. Figure 5(d) shows the red area of USAF 1951 in Fig. 5(c) by using 3MPR, 3MPRTP and TwPCPR method, respectively. The TwPCPR method outputs the detected height of the pattern with only 5 iterations. According to the refractive index of the QPMT and the wavelength of source, the pattern height can be calculated as 249.58 nm. The best resolvable feature corresponds to group 8 element 5 of USAF 1951 (1.23 μm line width). Figure 5(e) shows the red area of optically focus star in Fig. 5(c) by using 3MPR, 3MPRTP and TwPCPR method, respectively. The TwPCPR method reflects the height of 249.06 nm with 5 iterations. The convergence of the MPR method is limited with a small number of captured planes in the score of inline bandwidth limitation. The initial estimation of the offaxis phase can reduce the required number of captured planes for convergence.
Reconstruction for microscopy imaging on biological sections
We demonstrate the biological imaging experiments of the TwPCPR method based on the hybrid DH microscopic system in Fig. 5(a). Figure 6(a) shows the reconstructed amplitude of the mitotic section of horse roundworm fertilized egg on the glass slide by different method. Figure 6(b) shows the corresponding phase. Some cells are distributed along the edge of original FOV without flat boundary condition. The red area (a1)(a2) and (b1)(b2) show the reconstructed amplitude of anaphase mitosis and telophase mitosis of the horse roundworm fertilized egg, respectively. The cell and nucleus of horse roundworm fertilized egg have not degenerated in the normal style. The phase reflects the thickness and refractive index of the cells. The chromosomes in the nucleus gradually begin to split from the equatorial plate to the poles in the anaphase in (a1) and (b1). The thickness of the nucleus increases significantly. A new nucleus is gradually formed in the telophase and the two daughter cells begin to separate, which is clearly shown in red area (a2) and (b2). The reconstructed phase by using MPR method only represent edge contours of the cells. The results of the quantitative phase microscopy and biological experimental validation show that the TwPCPR method can be combined with modern microscopy systems to achieve robust reconstruction of complexamplitude with fast convergence.
The advantage of nonprior TwPCPR reconstruction without constrains on the object is that the spatial complexamplitude field can be reconstructed even if the exact location of the object cannot be obtained, such as atmospheric turbulence and air disturbance. The TwPCPR method can also achieve precise wavefront detection of phase disturbance. But the TwPCPR method still suffers from a few limitations that need to be addressed in the future. Currently, this method captures three images to accurately recover a field. Time division method are required to obtain the offaxis hologram though two cameras are used without realtime calibration and registration of multiimage. The distance between the object and the sensor can be adjusted by changing the size of the BS. But it limits the numerical aperture of the lensless system due to cube shape of BS. With the development of integrated optics, BS by specially designing with large area and small thickness may be help to perform the TwPCPR method in pixel superresolution with fast convergence, and then it can apply in lensless onchip microscopy. Moreover, this method can only handle static aberrations because we calibrate aberrations prior to the measurement, and we assume that they remain unchanged during the acquisition process. An optimization framework to automatically separate the object and aberrations without calibration is great interest to its applications in highprecision and realtime measurement.
Conclusions
We demonstrate that the proposed TwPCPR method performs a reliable, rapidly converging iterative procedure by involving only two inline holograms and one offaxis hologram between only two planes. Fullbandwidth reconstruction can be realized based on inline holography. It realizes a fastconvergence holographic calculation method. The reported results show robust reconstructions of the complexamplitude fields of different types of objects by combining the advantages of inline and offaxis holography. Offaxis optimization phase provides an effective initial guess to avoid stagnation and minimize the required measurements of multiplane phase retrieval. This universal complexamplitude reconstruction method works well for more extended samples. The TwPCPR method does not require any prior assumptions of objects including support, nonnegative, sparse constraints, etc. It realizes both amplitude and phase solutions for complexamplitude wave field analysis and imaging. Experimentally, the high phase reconstructed accuracy enables this method to have important applications in highresolution wavefront sensing. It can be easily combined with modern microscopy system to realize complexamplitude microscopy imaging. The versatility of TwPCPR method can significantly improve the performance of existing technologies such as adaptive optics. This technique helps to perform improvements in DH and quantitative phase imaging, thus enlarging and empowering applications in wavefront sensing, computational microscopy and biological tissue analysis.
Availability of data and materials
The datasets used and analysed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 DH:

Digital holography
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Acknowledgments
We thank Yunhui Gao for imaging the sample.
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National Natural Science Foundation of China (NSFC) (61827825).
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ZH and LC proposed the framework of this research. ZH and LC conducted the experiments. ZH, PM, PF carried out the computations and data processing, ZH, PM, PF, LC analysed and discussed the results. All the authors contributed to the writing of manuscript.
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Huang, Z., Memmolo, P., Ferraro, P. et al. Dualplane coupled phase retrieval for nonprior holographic imaging. PhotoniX 3, 3 (2022). https://doi.org/10.1186/s4307402100046w
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DOI: https://doi.org/10.1186/s4307402100046w