Abib Alimi
email: abib.alimi@inria.fr

Samuel Deslauriers-Gauthier

Felix Matuschke

Andreas Müller

Sacha Ea Muenzing

Markus Axer

Rachid Deriche

Sascha E A Muenzing

Analytical and fast Fiber Orientation Distribution reconstruction in 3D-Polarized Light Imaging

Keywords: fiber orientation distribution, 3D-PLI, polarized light imaging, diffusion MRI

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

To improve our understanding of the human brain structure and function as well as its normal development and complex disorders, it is compulsory to develop and combine different imaging techniques at multiple spatial scales. With a millimeter resolution, diffusion Magnetic Resonance Imaging (MRI) [START_REF] Le Bihan | Mr imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders[END_REF][START_REF] Basser | Mr diffusion tensor spectroscopy and imaging[END_REF][START_REF] Tuch | High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity[END_REF][START_REF] Jones | Diffusion mri[END_REF][START_REF] Hagmann | Understanding diffusion mr imaging techniques: from scalar diffusion-weighted imaging to diffusion tensor imaging and beyond[END_REF][START_REF] Tournier | Diffusion mri in the brain-theory and concepts[END_REF][START_REF] Leemans | Diffusion mri of the brain: The naked truth[END_REF] is the only in vivo and non invasive imaging technique able to assess the structural architecture of human fibrous organs such as the brain and the heart and to infer their respective connectivity [START_REF] Mori | Fiber tracking: principles and strategies-a technical review[END_REF][START_REF] Descoteaux | Deterministic and probabilistic tractography based on complex fibre orientation distributions[END_REF][START_REF] Jeurissen | Diffusion mri fiber tractography of the brain[END_REF][START_REF] Henssen | Ex vivo visualization of the trigeminal pathways in the human brainstem using 11.7 t diffusion mri combined with microscopy polarized light imaging[END_REF]. However, in vivo diffusion MRI suffers from a lack of ground truth [START_REF] Maier-Hein | The challenge of mapping the human connectome based on diffusion tractography[END_REF] and a relatively low spatial resolution. Therefore, 3D-Polarized Light Imaging (3D-PLI) (Axer et al., 2011a;[START_REF] Jouk | Three-dimensional cartography of the pattern of the myofibres in the second trimester fetal human heart[END_REF][START_REF] Larsen | Polarized light imaging of white matter architecture[END_REF][START_REF] Reckfort | A multiscale approach for the reconstruction of the fiber architecture of the human brain based on 3d-PLI[END_REF]Alimi et al., 2017a) has been investigated and, thanks to its high spatial resolution, it is presented as a complementary and potential technique for the validation and guidance of diffusion MRI fiber orientation estimates and tractography. 3D-PLI is an optical imaging technique that utilizes the birefringence property of brain tissue to map its spatial organization at micrometer resolution (Axer et al., 2011a;[START_REF] Jouk | Three-dimensional cartography of the pattern of the myofibres in the second trimester fetal human heart[END_REF]. The term '3D' in this context is first of all related to the within-section 3D fiber orientation estimation, and only secondly in the sense of 3D volume reconstruction of serial sections. The technique, thus, provides high-resolution 3D microscopic fiber orientation measurements recovered from each unstained histological tissue section.

Complementing diffusion MRI with 3D-PLI measurements, therefore, requires to bridge the spatial scale information from micro to millimeter dimensions. Thus, the fiber orientation distribution (FOD), which is widely used in diffusion community [START_REF] Tournier | Direct estimation of the fiber orientation density function from diffusion-weighted mri data using spherical deconvolution[END_REF][START_REF] Alexander | Multiple-fiber reconstruction algorithms for diffusion mri[END_REF][START_REF] Tournier | Robust determination of the fibre orientation distribution in diffusion mri: non-negativity constrained super-resolved spherical deconvolution[END_REF][START_REF] Daducci | Quantitative comparison of reconstruction methods for intra-voxel fiber recovery from diffusion mri[END_REF][START_REF] Dell'acqua | Modelling white matter with spherical deconvolution: How and why?[END_REF], is a suitable tool to compare diffusion MRI and 3D-PLI fiber orientation estimates in 3D. For validation purpose, 3D-PLI has been used to reconstruct brain white matter (WM) fiber pathways and presented as an adequate means to validate diffusion MRI fiber tractography (Alimi et al., 2019a). Moreover, [START_REF] Caspers | Target sites for transcallosal fibers in human visual cortex-a combined diffusion and polarized light imaging study[END_REF]; [START_REF] Zeineh | Direct visualization and mapping of the spatial course of fiber tracts at microscopic resolution in the human hippocampus[END_REF]; [START_REF] Henssen | Ex vivo visualization of the trigeminal pathways in the human brainstem using 11.7 t diffusion mri combined with microscopy polarized light imaging[END_REF] used high-resolution 3D-PLI, or more specifically the generated Fiber Orientation Map (FOM) which reflects the determined 3D fiber courses best, in order to identify and to follow the tiny fiber tracts to complement diffusion MRI results from the same region, rather than addressing long-distance tractography in both modalities. Concerning the brain's fiber architecture, histology-based studies compared FODs reconstructed from diffusion MRI to those estimated either in 2D from stained tissue sections [START_REF] Leergaard | Quantitative histological validation of diffusion mri fiber orientation distributions in the rat brain[END_REF][START_REF] Budde | Examining brain microstructure using structure tensor analysis of histological sections[END_REF], or more recently, in 3D via optical coherence tomography [START_REF] Wang | Structure tensor analysis of serial optical coherence scanner images for mapping fiber orientations and tractography in the brain[END_REF] and confocal microscopy [START_REF] Khan | 3d structure tensor analysis of light microscopy data for validating diffusion mri[END_REF][START_REF] Schilling | Comparison of 3d orientation distribution functions measured with confocal microscopy and diffusion mri[END_REF]. However, to our knowledge, 3D-PLI has not yet been used to evaluate 3D fiber orientation distributions obtained from diffusion MRI, and building PLI-based FODs paves the way for it.

On this account, [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF] proposed an estimate of the FOD derived from high-resolution 3D-PLI data, called pliODF. This technique enables the bridging of spatial scales from micro to macroscopic dimensions, for instance, to the millimeter resolution of diffusion MRI. pliODF benefits from the important concept of super-voxel (SV) which allows downsampling of the high-resolution data, and then estimating the FOD on the unit sphere by expanding a normalized directional histogram with a series of spherical harmonics (SH) functions. By changing the SV size, the FOD can be computed at different spatial scales [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF][START_REF] Alimi | An analytical fiber odf reconstruction in 3d polarized light imaging[END_REF][START_REF] Alimi | Analytical fiber odf reconstruction in 3d polarized light imaging: Performance assessment[END_REF]. [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF] refer to a tissue voxel containing a single high-resolution fiber orientation as 'native voxel' and the super-voxel regroups K = n x × n y × n z native voxels. The pliODF, though, is derived from an empirically parameterized directional histogram, which makes it a discrete method. It relies on a histogram binning procedure which can introduce angular errors in the recovered FODs [START_REF] Alimi | An analytical fiber odf reconstruction in 3d polarized light imaging[END_REF], and since it is represented on a SH basis, it can also limit the order of its SH expansion (Alimi et al., 2017b). In order to overcome these issues, Alimi et al. (2017b) proposed to regularize the pliODF using the Laplace-Beltrami operator and therefore improved its results.

In this work, we present an analytical procedure to determine the FOD from 3D-PLI data in order to assess the spatial distribution of fiber orientations within a super-voxel. Our method is based on defining each fiber orientation in a native voxel as a 2D Dirac delta function on the unit sphere and the FOD as a sum of these Diracs within each super-voxel. The FOD is continuously described on a spherical harmonics basis and analytically computed via the spherical Fourier transform and by means of the Diracs. We remind that the FOD is the spatial distribution of the fiber orientations within an image voxel as known in diffusion MRI [START_REF] Tournier | Direct estimation of the fiber orientation density function from diffusion-weighted mri data using spherical deconvolution[END_REF][START_REF] Dell'acqua | Modelling white matter with spherical deconvolution: How and why?[END_REF]. In the context of 3D-PLI, [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF] introduced a discrete approximate of the FOD called the 'pliODF' and, here, we propose an analytical FOD denoted 'aFOD'. This paper reviews and extends our previous works in which the analytical method was introduced [START_REF] Alimi | An analytical fiber odf reconstruction in 3d polarized light imaging[END_REF] and its performance evaluated [START_REF] Alimi | Analytical fiber odf reconstruction in 3d polarized light imaging: Performance assessment[END_REF]. Here, we give (1) a full description of our method to analytically compute the aFOD from high-resolution 3D-PLI vector data, then (2) the used datasets and the different evaluation criteria are presented for a further/extended comparison with pliODF.

Results demonstrate (3) the high angular resolution and precision of the aFOD and the dependence of pliODF on its empirical histogram binning to perform similarly, and (4) that the aFOD is computationally very efficient and very fast. We finally (5) compute aFODs on human brain data at different spatial scales while preserving the original high-resolution organization of WM fiber bundles and (6) present it as a means for diffusion MRI to assess potential resolution limits to track specific fascicles in the human brain.

Theory

The fiber orientation distribution provides a comprehensive statistical description of integrated 3D-PLI vector data [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF][START_REF] Alimi | An analytical fiber odf reconstruction in 3d polarized light imaging[END_REF][START_REF] Alimi | Analytical fiber odf reconstruction in 3d polarized light imaging: Performance assessment[END_REF] and positions itself as a suitable tool for multimodal analysis and comparisons [START_REF] Schilling | Comparison of 3d orientation distribution functions measured with confocal microscopy and diffusion mri[END_REF]. As a function defined on the unit sphere S 2 , the FOD can be expanded as a linear combination of spherical harmonics. In this section, we give a brief background information on spherical harmonics before presenting the pliODF estimate by [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF] and the analytical approach -aFOD-we propose.

Spherical harmonics

The spherical harmonics of order l and phase factor m are defined as

Y m l (θ, φ) = N m l P m l (cos θ)e jmφ (1)
where N m l is a normalization coefficient, P m l are the associated Legendre polynomials, the angles θ ∈ [0, π] and φ ∈ [0, 2π) are the colatitude and azimuth, respectively. The spherical harmonics {Y m l : -l m l, l = 0, 1, ...} form an orthonormal basis over L 2 (S 2 ) and any square integrable function f (θ, φ) ∈ L 2 (S 2 ) can be expressed as

f (θ, φ) = ∞ l=0 l m=-l c lm Y m l (θ, φ) (2)
where c lm are the SH coefficients of f . These coefficients are obtained by computing the spherical Fourier transform [START_REF] Healy | Spherical deconvolution[END_REF] of f defined as

c lm = S 2 f (w)Y m l (w)dw (3)
for w ∈ S 2 and the over-bar denotes the conjugation. The unit vector w is related to (θ, φ) by

w(θ, φ) = [sin θ cos φ sin θ sin φ cos θ] T (4)
where • T indicates transposition. In this paper, both functions f (θ, φ) and f (w) are assumed to be equal and written interchangeably to simplify notations.

pliODF, a discrete solution

Axer et al. (2016) proposed an estimate of the FOD from high-resolution 3D-PLI vector data, which they called pliODF. For this, the collection of all native voxels, which represents the fiber orientation map (FOM) (Axer et al., 2011a;[START_REF] Jouk | Three-dimensional cartography of the pattern of the myofibres in the second trimester fetal human heart[END_REF], is downsampled into cubic compartments or super-voxels (SV) containing n x × n y × n z native voxels each. This allows to determine the spatial distribution of 3D fiber orientations in a given SV. Thus, starting from high-resolution FOMs, as illustrated in Fig.

(3) in [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF], their approach is implemented as follows:

1. Define super-voxels of the FOMs measured from unstained histological tissue sections.

2. Create a normalized directional histogram on the unit sphere from discretized distribution of fiber orientation vectors in each super-voxel.

3. Approximate the directional histogram using spherical harmonics, see Eq. ( 2).

In practice, the SH approximation is truncated at a maximum order or bandlimit L max . Moreover, in the original implementation of the pliODF technique, the SH coefficients are determined with a linear least square method by minimizing ||h -Bc|| 2 , yielding

c = (B T B) -1 B T h (5)
where, in matrix format, B is the basis matrix of the truncated SH, h the vector entries of the directional histogram, and c the coefficients vector of the expansion up to order L max .

This technique benefits not only from the 3D nature of 3D-PLI dataset but also from the super-voxel which allows the estimation of the FODs at different spatial scales. Nevertheless, the reconstructed pliODF is an empirical and discrete approximation of the FOD. Indeed, it is a directional histogram defined by its bin centers and the number of 3D-PLI fiber orientation vectors falling in each bin is dependent on the bin size or corresponding dihedral angle. Therefore, the L max of the SH expansion of the pliODF can be limited as shown in (Alimi et al., 2017b).

Analytical FOD in 3D-PLI

We demonstrate, in this section, that without any discretization which leads to loss of orientation information, the aFOD can be analytically reconstructed from high-resolution 3D-PLI orientation measurements.

We show how we model our aFOD before we obtain the exact SH coefficients that uniquely characterize it.

aFOD as sum of k Diracs

In each voxel of the fiber orientation map (visualizing the dominant fiber orientation), we propose to describe the unit fiber orientation vector as a two dimensional Dirac delta function δ on the sphere.

Therefore, in a super-voxel containing K orientations (θ k , φ k ), the fiber orientation distribution function f can be modeled as the sum of K Diracs [START_REF] Deslauriers-Gauthier | The application of a new sampling theorem for non-bandlimited signals on the sphere: Improving the recovery of crossing fibers for low b-value acquisitions[END_REF][START_REF] Deslauriers-Gauthier | Sampling signals with a finite rate of innovation on the sphere[END_REF], that is

f (θ, φ) = 1 K K k=1 δ(cos θ -cos θ k )δ(φ -φ k ) (6)
Note that f (θ, φ) is completely defined by parameters (θ k , φ k ) of the K orientations located in the SV. In the next step, the coefficients of the SH expansion of f are determined.

Computation of the SH coefficients

Unlike [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF] who used a linear least square method to get the coefficients of the SH expansion, we directly recover the coefficients that uniquely define our aFOD f by computing the spherical Fourier transform through Eq. ( 3) since we know the parameters (θ k , φ k ) of all fiber orientation vectors determined from 3D-PLI analysis (Axer et al., 2011a;[START_REF] Jouk | Three-dimensional cartography of the pattern of the myofibres in the second trimester fetal human heart[END_REF]Alimi et al., 2017a) , in a given super-voxel. That is,

c lm = S 2 f (w)Y m l (w)dw = 2π 0 π 0 f (θ, φ)Y m l (θ, φ) sin θdθdφ
then f is replaced by the collection of Diracs in Eq. ( 6) and Y m l by its expression in Eq. ( 1) to get

c lm = N m l K K k=1 1 -1 δ(cos θ -cos θ k )P m l (cos θ)d cos θ × 2π 0 δ(φ -φ k )e -jmφ dφ
and finally applying the sifting property of the Diracs gives

c lm = 1 K K k=1 N m l P m l (cos θ k )e -jmφ k = 1 K K k=1 Y m l (θ k , φ k ) (7)
This solution for the c lm is the analytical and continuous equivalent of the discrete one approximated for pliODF. Eq. ( 7) yields, with respect to L max , the exact solution which requires no directional histogram and, therefore, all the information from the high-resolution fiber orientation vectors are taken into account in the FOD computation. Furthermore, it can be evaluated at any L max .

Using Eq. ( 2), consequently, the analytical FOD f is simply computed through matrix-vector multiplication:

f = Bc (8)
with B the SH basis matrix and c the vector of the analytically recovered SH coefficients.

Note that although our reconstruction approach uses the SH to represent the aFOD, it is independent of the SH basis used and here the real and symmetric basis defined in [START_REF] Descoteaux | Regularized, fast, and robust analytical q-ball imaging[END_REF] is chosen. In this case, the number of terms of the SH basis of order l is L = (1/2)(l + 1)(l + 2), where only the even orders are considered.

Methods

Synthetic data generation

The simulated 3D-PLI dataset consists of two 3D fiber bundles intersecting at an angle X in the XYplane. A 3D collision solving algorithm, described in [START_REF] Matuschke | Dense fiber modeling for 3d-polarized light imaging simulations[END_REF], is performed to ensure that the individual fibers do not overlap. For this purpose, the fibers are divided into short segments so that two segments can be checked for a collision. In the event of a collision, the two segments are slowly pushed apart. This happens simultaneously for all colliding objects and is repeated until there is no more collisions. The 3D models generated in this way are then used for the simulation using SimPLI tool [START_REF] Dohmen | Understanding fiber mixture by simulation in 3d polarized light imaging[END_REF][START_REF] Menzel | A jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue[END_REF] to produce synthetic 3D-PLI datasets. The noise of the CCD camera is modelled as in [START_REF] Schmitz | Derivation of fiber orientations from oblique views through human brain sections in 3d-polarized light imaging[END_REF] i.e. with a negative binomial distribution in which the variance σ depends on the expected light intensity µ by σ = 3µ. From the generated 3D-PLI signal, high-resolution orientations are reconstructed based on the approaches in (Axer et al., 2011a;[START_REF] Schmitz | Derivation of fiber orientations from oblique views through human brain sections in 3d-polarized light imaging[END_REF]. The generated data consists of 10 slices of 60 µm thick, with a pixel resolution of 64 × 64 µm 2 .

Experimental brain data

The post mortem human brain tissue used for this study was acquired in accordance with the local ethic committee of our partner university at the Heinrich Heine University Düsseldorf. As confirmed by the ethic committee, postmortem human brain studies do not need any additional approval, if a written informed consent of the subject is available. For the research carried out here, this consent is available.

The brain was removed 24 h after the donor's death (male, age: 54 years, suffered from multiple system atrophy). The right hemisphere was fixed in 4% buffered formaldehyde solution for 4 years. After immersion in a 20% solution of glycerin with Dimethyl sulfoxide (DMSO) for cryoprotection, the brain was deep frozen and cryo-sectioned at 70 µm thickness (with Polycut CM 3500, Leica, Germany) along the coronal plane, orthogonal to the line connecting the anterior and posterior commissures determined from a post mortem turbo spin-echo MRI scan pre-sectioning. Before each sectioning step, en face images (referred to as blockface images) were taken from the frozen surface of the brain block to generate reference images used for serial section image alignment. Each section was finally mounted unstained on a glass slide, immersed in a 20% solution of glycerin to avoid dehydration, and cover-slipped. (Axer et al., 2011b). Pixel size of all images was set to 64 × 64 µm 2 . The experimental protocol provided one planar measurement and four tilting position measurements (in north, west, east, and south direction with 8 • tilting angle), and each measurement was composed of a set of 18 images taken at 10 • equidistant polarization filter rotation angles. Image analysis included calibration, segmentation, and tilting analysis to determine the local fiber structure orientations, utilizing the analysis method described in [START_REF] Schmitz | Derivation of fiber orientations from oblique views through human brain sections in 3d-polarized light imaging[END_REF].

Three-dimensional reconstruction of the brain slab volume consisting of the 50 sections was achieved by a multi-step registration approach: Firstly, the 3D-PLI images were aligned to their corresponding blockface images via affine transformations utilizing in-house developed software based on the software packages ITK [START_REF] Yoo | Engineering and algorithm design for an image processing api: a technical report on itk-the insight toolkit[END_REF] and elastix [START_REF] Klein | Elastix: a toolbox for intensity-based medical image registration[END_REF]. Secondly, non-linear registration using the ANTs toolkit [START_REF] Avants | A reproducible evaluation of ants similarity metric performance in brain image registration[END_REF] was performed. Finally, all 3D-PLI modalities were spatially transformed using the obtained deformation fields and orientations were corrected accordingly.

One tissue section was measured with the Polarizing Microscope (PM) setup without tilting stage, as described in [START_REF] Reckfort | A multiscale approach for the reconstruction of the fiber architecture of the human brain based on 3d-PLI[END_REF]. Pixel size of was set to 1.3 × 1.3 µm 2 . The experimental protocol provided just the planar measurement with each measurement being composed of a set of 18 images taken at 10 • equidistant polarization filter rotation angles. Image analysis included calibration, segmentation, and flat image analysis to determine the local fiber structure orientations, utilizing the analysis method described in [START_REF] Reckfort | A multiscale approach for the reconstruction of the fiber architecture of the human brain based on 3d-PLI[END_REF].

Evaluations

To assess the performance of aFOD and pliODF reconstruction approaches and see the relationship between them, we do tests on synthetic and on experimental human brain 3D-PLI data.

Angular resolution

In diffusion MRI, it is generally assumed that the local maxima or peaks of the FOD coincide with the fiber directions [START_REF] Tournier | Direct estimation of the fiber orientation density function from diffusion-weighted mri data using spherical deconvolution[END_REF][START_REF] Tuch | Q-ball imaging[END_REF][START_REF] Descoteaux | Regularized, fast, and robust analytical q-ball imaging[END_REF][START_REF] Daducci | Quantitative comparison of reconstruction methods for intra-voxel fiber recovery from diffusion mri[END_REF]. Here, the angular resolution corresponds to the critical angle τ under which the peaks of the two crossing fiber bundles are confounded, that is, they can not be detected separately. To find the peaks of the FOD, the sh2peaks1 algorithm implemented in Mrtrix3 [START_REF] Tournier | Mrtrix3: A fast, flexible and open software framework for medical image processing and visualisation[END_REF]) is used, with the minimum peak amplitude set to 0.5 and the other parameters to default.

Angular precision

Let 'single fiber' denote a bundle of coherently organized/oriented fibers. The angular precision is the deviation or angular error of the peak of the reconstructed FOD from the simulated ground-truth. Here we investigate the of both aFOD and pliODF, as well as the effect of the directional histogram bin width on . The angular precision is evaluated in degrees as follows,

= 180 πM M m=1 arccos(v T m w m ) (9)
which is the average distance between v m the m th ground truth orientation and w m the closest of the recovered peak orientations of the FOD and M is the number of fibers in the super-voxel.

Computational runtimes

In order to ensure a fair comparison and due to the large size of 3D-PLI human brain data, both reconstruction algorithms are performed on the JURECA (Jülich Supercomputing Centre, 2016) supercomputer at Forschungszentrum Jülich, in Germany. The supercomputer consists of 1872 compute nodes and each node is equipped with two Intel Xeon E5-2680 v3 Haswell 24-core processors operating at 2.5 GHz. The considered compute nodes contain 128 GB of main memory each, and their number is set to 5, for a total of 120 cores in order to keep the compute time in an acceptable range. The super-voxel size and the spherical harmonics bandlimit L max are taken as parameters, plus the histogram bin size for the pliODF generation.

Results

Single Fiber Orientation

We recall that single fiber refers to a single bundle of fibers. in the rest of the experiments, in order to get results closer to our analytical approach, bin widths of 3.6 • (consistent with [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF]) and 1.8 • will the considered for the estimation of the pliDOFs.

The previous observations -and choices of bin width-are corroborated in Table 2 which quantitatively compares the of recovered aFOD and pliODF for the single fiber population with varying SH L max . The angular precision of the aFOD is always < 1 • regardless of L max . This applies to the of pliODF, except for L max = 4 when bin size is 1.8 • , and L max ≤ 6 for a bin width of 3.6 • . Note, though, that the pliODF produces more erroneous results (e.g. up to > 30 • ) when larger bin width are used (Fig. 2).

Crossing Fiber Orientations

Angular Resolution

Fig. 3 displays the angular resolution of both aFOD and pliODF relative to L max . It can be observed that 90 • -crossings can be separated at any SH L max ≥ 4, and at least L max = 6 is necessary to discriminate 60 •crossings and L max = 10 for 40 • -crossings. These observations are valid for the aFOD as well as the pliODF with the selected bin widths, though only results for bin size 3.6 • are shown for illustration. Moreover, one can also notice shape differences between the aFOD and the pliODF at 40-and 60 • -crossings.

Angular Precision

Fig. 4 displays the angular precision of the FODs reconstructed from crossing fiber bundles using both methods at different spherical harmonics L max . The green curve represents the of aFOD and the red and black dotted curves represent the of pliODF estimated with bin sizes of 3.6 • and 1.8 • , respectively. The following observations can be made. First, globally for all algorithms, decreases as the crossing angle X increases: it goes from around 7 • at X = 40 • and L max = 10 down to less than 1.5 • at X = 90 • and L max = 6.

Second, in all combinations of X and L max , the aFODs have a smaller angular error than pliODFs have, except for crossing X = 40 • and L max = 10 as can be seen in the plot at the right of Fig. 4. Third, as for pliODF, the smaller the bin size, the lower the angular error. Finally, as seen qualitatively in Fig. 3 for both methods, the fiber crossing at X = 90 • can be correctly reconstructed at any SH order with a maximum error = 1.3 • , the X = 60 • is resolved when L max ≥ 6 with maximum < 6 • and, the X = 40 • only when L max ≥ 10 with maximum error ≈ 7 • .

Computational runtime

The FODs' computational runtime measurements on the system JURECA (Jülich Supercomputing

Centre, 2016) are presented in [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF][START_REF] Alimi | Analytical fiber odf reconstruction in 3d polarized light imaging: Performance assessment[END_REF]. Second, at high resolution, the analytical method is faster than the pliODF technique by a factor between 24 and 73 at SV size of 5 × 5 × 5 corresponding to 320 × 320 × 350 µm 3 resolution (in Table 1). Third, when the SV size is > 25 isotropic native voxels, it is 3 to 6 times quicker to generate pliODFs than aFODs, and both reconstruction approaches are completed within seconds. In between these SV sizes, the analytical solution runs faster. This could be accounted for the continuous setting of the analytical approach, which makes it compute the SH coefficients from every native voxel located in the super-voxel, regardless of its size. However, the discrete pliODF approach estimates the SH coefficients from the defined directional histogram in the considered super-voxel [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF][START_REF] Hänel | Interactive level-of-detail visualization of 3d-polarized light imaging data using spherical harmonics[END_REF].

Therefore, pliODF runs moderately faster when the SV size is large, at least 50 isotropic native voxels since the number of super-voxels in the whole dataset decreases i.e. there is a few directional histograms to estimate the SH coefficients from. Conversely, with a small SV size, the number of super-voxels as well as histograms is large, thus the analytical technique is faster. S V = 25 × 25 × 25 (see Table 1), while preserving their original high-resolution structure. From these same maps, one can also appreciate the fibers fanning into the gyri.

Stratum sagittale

In Fig. 6 is shown a FOD map of the stratum sagittale (SS), yellow RIO in (A). The SS consists of longitudinal fibers (going out of the plane), lateral to the ventricles and located in the deep WM region of the occipital lobe. It can be reported that: 1) Going from smaller S V = 5×5×5 (B) to larger S V = 50×50×50 (C,D), the spatial resolution gets (10 times) coarser (3.2 × 3.2 × 3.5 mm 3 , see Table 1), however, the global WM tissue structure of the SS is conserved by both the pliODF (C) and the aFOD (D) with crossings.

2) Overall, the shapes of the aFOD and pliODF are very similar, although the latter sometimes presents some spurious lobes at the center of the crossings.

3) The aFOD is more sharply defined as the spherical harmonics L max increases as shown in (E). This observation is also valid for pliODF (but not displayed, with eventual small spurious lobes). A zoom in on the white RIO is displayed in Fig. 7.

Diffusion MRI resolution limit

Fig. 7 displays a zoom in on the aFOD map of the stratum sagittale. At high resolution (A), the two layers of the SS are distinguishable as shown by [START_REF] Vergani | Intralobar fibres of the occipital lobe: a post mortem dissection study[END_REF] in their postmortem dissection of the occipital lobe, even though they mentioned the difficulty to appreciate them on coronal sections. From our FOD maps build from high-resolution 3D-PLI vector data, one can separately identify them up to a coarser resolution of 640 × 640 × 700 µm 3 in (C) where the thinner layer intersect another vertical fiber bundle to form crossing FODs. We can see loss of WM fibers at this scale, suggesting, thus, that higher resolutions of 128 × 128 × 140 µm 3 and 320 × 320 × 350 µm 3 could be considered as suitable spatial scales for diffusion MRI to reach and apply bundle-specific tractography [START_REF] Rheault | Bundle-specific tractography with incorporated anatomical and orientational priors[END_REF] of the SS. Indeed, using a SH L max = 6 at these particular scales, the specific WM tracts of the SS can be tracked, maybe more preferably at SV size of 2 × 2 × 2 equivalent to 128 × 128 × 140 µm 3 resolution as observed in our study.

Discussion

In this paper, we propose an analytical and fast approach to resolve the FOD reconstruction problem in 3D-Polarized Light Imaging. We quantitatively compare the performances of our approach and the discrete pliODF method on both highly rich synthetic and human brain 3D-PLI datasets.

On Angular Precision, Angular Resolution. As pointed out by [START_REF] Parker | A pitfall in the reconstruction of fibre odfs using spherical deconvolution of diffusion mri data[END_REF], "characterisation of any error in the resolution of the single fiber orientation will provide an easy way of identifying any underlying systematic problems inherent in certain (FOD reconstruction) techniques...". Thus, both reconstruction techniques are performed on a single fiber synthetic model. Results show that, unlike our aFOD, pliODF is dependent on the number bins of its discretized directional histogram. This demonstrates that pliODF requires high number of bins to reach results similar to the analytical solution. Indeed, in this experiment (see Sect. 4.1), the angular precision of aFOD is always smaller than 1 • while the of pliODF converges to the aFOD's by decreasing its bin width.

Our aFOD is analytically defined, therefore, the produced angular errors in the single and crossing fiber experiments could be explained by the truncation of the spherical harmonics basis as reported in the previous chapter (and by potential biases from the FOD peaks detection method). In the multiple fiber context, except for X = 60 • -crossing where increases when L max goes from 6 to 8, the angular precision improves when the SH truncation order increases, independent of the crossing configuration. In the particular cases of crossing at X = 0 • and X = 90 • , the deviation is always very low, i.e. less than 1 • . Moreover, using high SH orders helps sharpen the aFOD (and pliODF with proper bin width) and gives more defined peaks very useful in fiber-tracking applications [START_REF] Tournier | Diffusion mri in the brain-theory and concepts[END_REF].

As for pliODF, the presence of angular error can be attributed to the SH basis truncation as well, however, it could be mainly due to its discrete nature. pliODF by definition uses an arbitrary discretized directional histogram and only provides an approximate of the FOD [START_REF] Axer | Estimating fiber orientation distribution functions in 3d-polarized light imaging[END_REF], which is the main limitation of the approach. Indeed, the directional histogram is defined by its bin centers and the number of native orientation vectors falling in each bin is dependent on the bin size or corresponding dihedral angle.

This could explain the fact that, in the single fiber experiment when the histogram bin size is < 4 • , pliODF has lower than aFOD does. In fact, the closer this bin center to the orientation of single fiber ground truth, the smaller the angular error of pliODF. Conversely, the further the bin center, the larger the as shown in Fig. 2 and Table 2. In the crossing fiber experiment, results in Fig. 4 show again that the aFOD presents less angular deviation than the pliODF.

The use of the discretized directional histogram has also a significant influence on the angular resolution of the pliODF in the crossing fiber experiment, but the bin sizes we have selected (3.6 • and 1.8 • ) are 'empirically' chosen so as to get pliODF results close to the aFOD solution. As said earlier, when the bin width gets smaller, i.e. the number of histogram bins gets larger, pliODF is expected to behave as well as our the aFOD (Fig. 3, Fig. 4). Nevertheless, one should be cautious about possible over-binning of the directional histogram which could eventually cause some undesirable effects such as the appearance of spurious lobes (Fig. 3, Fig. 6) or eventual extended computation runtimes.

As aforementioned, the value of the spherical harmonics order L max determines the maximum achievable FOD angular resolution. By increasing L max , more acute crossing fiber populations could be resolved as shown in Sect. 4.2. Here again we set the directional histogram bin size and both approaches give very similar angular resolutions τ but aFOD in general demonstrates better angular precision (Fig. 4).

As for imaging noise influence, in a previous study [START_REF] Alimi | Analytical fiber odf reconstruction in 3d polarized light imaging: Performance assessment[END_REF], we tested our analytical method on a synthetic dataset and it was revealed very little influence of noise on both the angular resolution and angular precision of the aFOD. We expect pliODF to perform as well, if the good number of radial histogram bins is used. Indeed, the noise in 3D-PLI data we deal with comes mainly from the CCD camera and its removal is performed during the estimation of the high-resolution fiber orientation vector (Axer et al., 2011a,b;[START_REF] Kleiner | Classification of ambiguous nerve fiber orientations in 3d polarized light imaging[END_REF][START_REF] Wiese | Polarized light imaging of the human brain: a new approach to the data analysis of tilted sections[END_REF]Alimi et al., 2017a;[START_REF] Schmitz | Derivation of fiber orientations from oblique views through human brain sections in 3d-polarized light imaging[END_REF].

On computational runtimes. The analytical approach is very fast and computationally efficient. This could be due to the choice of defining each fiber orientation as a 2D Dirac function on the unit sphere, which simplify and accelerate the computation of the SH coefficients, analytically. Note that in our study, only a subset of the human brain was analysed. An entire human brain of about 1200 cm 3 has, however, about 100 times more voxels when scanned at 64 × 64 × 70 µm 3 resolution and it would theoretically last about 100 times longer to calculate the FODs. Moreover, at a (ultra) higher resolution of 1.3 × 1.3 × 70 µm 3 the dataset gets about 2 × 10 5 times larger and the computations will require more time. We recall that to reach millimeter resolutions, the SV size should be very large, therefore, pliODF runs faster as seen in Sect. 4.3 and might still be applicable, given a suitable bin size is used. However, since the calculation time at this scale is already quite fast (seconds for the current dataset), the use of the aFOD is still advocated, since it gives the exact solution and has no bias of the histogram binning.

Limitation. One limitation of representing the fiber orientation distribution in a truncated SH basis is the presence of Gibbs ringing effects as discussed in a recent review by Dell'Acqua and Tournier (2019).

However, this issue can be resolved or drastically mitigated for the aFOD by introducing an apodized delta function in its definition as recently proposed by [START_REF] Raffelt | Reorientation of fiber orientation distributions using apodized point spread functions[END_REF] and [START_REF] Dhollander | Track orientation density imaging (todi) and track orientation distribution (tod) based tractography[END_REF] in diffusion MRI.

On human brain data. Finally, our analytical algorithm preserves the global integrity of the WM fiber architecture in the human brain. Indeed, experiments in deep WM region (Fig. 6) as well as in the cortex (Fig. 5) where the fiber structures are well known [START_REF] Sachs | Das Hemisphärenmark des menschlichen Grosshirns: Der Hinterhauptlappen[END_REF][START_REF] Vergani | Intralobar fibres of the occipital lobe: a post mortem dissection study[END_REF], show the conservation of the general anatomical organization of the stratum sagittale (SS) and the U-fibers at different spatial resolutions. It is noticed though that pliODF sometimes presents some spurious lobes, which could be due to the empirical (over-or under-) binning of the directional histogram. Conversely, high SH orders improve the sharpness of the FOD as well as its angular resolution and precision as demonstrated in the simulations (Fig. 3, Fig. 4), and commonly known in diffusion MRI [START_REF] Tournier | Robust determination of the fibre orientation distribution in diffusion mri: non-negativity constrained super-resolved spherical deconvolution[END_REF][START_REF] Descoteaux | High angular resolution diffusion MRI: from local estimation to segmentation and tractography[END_REF].

Further analysis of the SS illustrates the loss of its thinner layer or simply its intersection with a differently oriented fiber population to form more complex fiber configurations. This is visible at a coarse resolution of about 640 × 640 × 700 µm 3 in Fig. 7C. Whereas at higher resolutions of around 128 × 128 × 140 µm 3 , both layers can be clearly appreciated. This observation would suggest a potential spatial resolution limit, for diffusion MRI to reach, at which the WM fibers of the SS could be reconstructed and possibly improve their tracking.

Note, that the state-of-the-art 3D-PLI signal analysis only provides a single orientation estimate, i.e. one orientation vector per native voxel. Consequently, an initial image resolution of 64 × 64 µm 2 leads to fiber orientation descriptions (including their fiber tract crossings) at larger spatial scale. This does not mean, that there is no crossing present in human brain data shown in Figs. (5,6,7). As demonstrated in Fig. 8, the studied region of interest in the sagittal stratum, for example, is full of fiber crossings which are revealed by 3D-PLI at axonal resolution, i.e. at 1.3 × 1.3 µm 2 pixel size. Future research will certainly address aFOD determination in serial 3D-PLI measurements at the one micrometer scale.

Conclusion

We have presented an analytical and fast method to define and reconstruct the fiber orientation distribution in 3D-PLI. To achieve it, we model each high-resolution fiber orientation vector contained in the supervoxel as 2D Diracs delta function on the unit sphere and the aFOD as their sum. The aFOD is analytically described on a spherical harmonics basis and efficiently computed via the spherical Fourier transform and by means of the Diracs. This makes the aFOD not only computationally very fast but also with high angular resolution and precision and the ability to close the resolution gap with diffusion MRI. This work serves as a important step towards a major goal which is the validation of diffusion MRI fiber orientation estimates and histology guided tractography via 3D-Polarized Light Imaging technique , more specifically, it leads to multimodal brain imaging analysis across the scales, through the fiber orientation distribution. [START_REF] Menzel | A jones matrix formalism for simulating three-dimensional polarized light imaging of brain tissue[END_REF]. Each fiber is modelled as a filled cylindrical object with an absorption coefficient µ = 1/mm 3 and a birefringence ∆n = -0.001. The deriving microscopic images have a in-plane resolution of 60 × 60 µm 2 . The signal noise was calculated with a gain factor of g = 3 (see [START_REF] Schmitz | Derivation of fiber orientations from oblique views through human brain sections in 3d-polarized light imaging[END_REF]) and an initial light intensity of 26000 a.u. The signal from the 3D-PLI pipeline is then analyzed to calculate the FOM.

Figure 1 :

 1 Figure 1: (A) 3D visualization of the simulated fiber bundles crossing at angle X = 90 • by the FAConstructor (Reuter et al., 2019). (B)-(D) The resulting Fiber Orientation Maps of 3D-PLI simulations through 60 µm sections with a crossing at X = 30 • , 60 • and 90 • , respectively. The 3D orientations of entire fiber structures (A) and determined local fiber orientations (B)-(D) are color coded as indicated by the color sphere at the bottom right.

Fig

  Fig 1Ashows an example of the resulting fiber configuration for the X = 90 • crossing case. Note that X is varied from 0 • to 90 • with 10 • increments to give 10 different crossing situations. More details on the fiber generation can be found in Appendix A.

Figure 2 :

 2 Figure 2: Angular precision of 'single fiber' FODs as a function of the directional histogram bin size, in degrees. (A)-(C) For decreasing bin size, the of pliODF (red dotted line) converges to the constant < 1 • of aFOD, in green horizontal line. At L max = 12, when bin size is very small, for both FODs < 1 • , this is illustrated in (D) where the blue square in (C) is zoomed in.

Figure 3 :

 3 Figure 3: Effects of SH L max on the angular resolution of aFOD and pliODF (with bin size 3.6 • ) estimated at different crossing configurations.

Figure 4 :

 4 Figure 4: Angular precision of the FODs reconstructed at varying SH L max from different crossing fiber configurations X.

Figure 5 :Figure 6 :

 56 Figure 5: FOD map of U-shaped fibers connecting adjacent gyri at varying resolutions from (B) to (D) with a high-resolution retardation map displayed in (A). The global fiber organization is preserved through the scales. FODs overlay a recomputed retardation map and are colour-coded according to their local orientation (red: left-right; green: anterior-posterior; blue: inferiorsuperior).

Figure 7 :

 7 Figure 7: Zoom in on the white rectangle of Fig. 6: details about the layers of the stratum sagittale. The spatial resolution is 128 × 128 × 140 µm 3 in (A), 320 × 320 × 350 µm 3 in (B) and 640 × 640 × 700 µm 3 in (C). The blue and the red arrows point the thinner and the more pronounced layers of the SS, respectively. By reducing the scale, the thinner layer form crossing aFODs with a vertical fiber bundle.

Figure 8 :

 8 Figure 8: Fiber orientation map of a coronal brain section in the region of the stratum sagittale (A) observed with 3D-PLI at 1.3 × 1.3 × µm 2 resolution. Details of crossings (radial and horizontal fibers) in a transition zone between white and gray matter is shown in (B). (C) highlights the crossing of three pathway crossings at the bottom right (dark spot).

Table 1 :

 1 Imaging spatial resolutions resulting from super-voxels.

	Super-voxel size Resolution [mm 3 ]
	1 × 1 × 1	.064 × .064 × .07
	2 × 2 × 2	.128 × .128 × .14
	5 × 5 × 5	.32 × .32 × .35
	10 × 10 × 10	.64 × .64 × .7
	25 × 25 × 25	1.6 × 1.6 × 1.75
	50 × 50 × 50	3.2 × 3.2 × 3.5
	50 consecutive sections were selected for the present study and measured with the Large-area Polarimeter
	(LAP) setup with tilting stage, as described in

Table 2 :

 2 Single fiber angular precision in degrees [ • ] as a function of L max . For aFODs is always < 1 • , and the difference with of pliODFs is not very significant with regard to the considered bin sizes.

	Angular Error [ • ]

https://www.mrtrix.org

The dataset is actually masked and only

38% of the voxels are considered. These voxels contain the brain tissue.

Acknowledgements

This work was partly supported by ANR "MOSIFAH" under ANR-13-MONU-0009-01, the ERC under the European Union's Horizon 2020 Research and Innovation Program (ERC Advanced Grant agreement No. 694665:CoBCoM). M.S, F.M and S.M were partly funded by the European Union's Horizon 2020 Research and Innovation Program under Grant Agreement No. 785907 (HBP SGA2). The authors gratefully acknowledge the computing time granted through JARA-HPC on the supercomputer JURECA (Jülich Supercomputing Centre, 2016) at Forschungszentrum Jülich.

Appendix A. Synthetic Data Generation

The nerve fiber model imitates two fiber bundles in the XY-plan crossing at an angle X which varies from 0 • to 90 • , with 10 • increments. To build this configuration, the following steps were performed:

1. Two inital fibers bundles with a length of 3840 µm and a radius of 640 µm were placed at an crossing angle X.

2. These initial fiber bundles were then filled with smaller fiber bundles with a radius of 64 µm, on which the collision algorithm [START_REF] Matuschke | Dense fiber modeling for 3d-polarized light imaging simulations[END_REF] was performed for 25 steps, to allow a 'macroscopic' weave pattern.

3. The resulting fiber bundles were then filled one last time with fibers with a radius of 10 µm. In this state, each fiber is spatially divided into 20 µm segments. To limit the curvature of each fiber, the minimum radius of a bending fiber is limited to 40 µm. This configuration runs with the collision solving algorithm until no collision are detected.

The resulting fiber configuration, for instance as pictured in Fig. 1A where X = 90 • , is then used for the 3D-PLI simulation to generate the resulting FOM. For the simulation, the model is discretized on a volume of 3840 × 3840 × 600 µm 3 to allow the calculations performed by simPLI [START_REF] Dohmen | Understanding fiber mixture by simulation in 3d polarized light imaging[END_REF]


domain: spi.nano sdv sde

text:
Yu-Sheng Lin

Bruno Le Pioufle

Hsiang-Yu Wang

MONITORING THE GROWTH PHASES OF MICROALGAE USING STRAIGHTFORWARD DIELECTROPHORESIS MEASUREMENTS

Keywords: Microalgae, Growth Phases, Dielectrophoresis Measurement

This study demonstrates a straightforward method for differentiating single microalgal cells in different growth phases based on dielectrophoresis (DEP). The balance of DEP force and drag force leads to an easy-to-use linear relation between DEP velocity and the real-part of the Clausius Mossotti factor, which determines the velocity of DEP. The DEP velocity of microalgae in the stationary phase is slower than microalgae in the exponential phase, owing to the cellular compositions changed during growth. Microalgal cells in the stationary phase also showed a higher transition frequency than cells in the exponential phase.

INTRODUCTION

Photosynthetic microalgae convert carbon dioxide into high-value metabolic products such as pigments, proteins, and lipids. Microalgae produce metabolic products such as pigments, proteins, lipids in different growth phases and the changes of cellular compositions can be considerable in a short period of time [START_REF] Gifuni | Current bottlenecks and challenges of the microalgal biorefinery[END_REF]. However, conventional analyses either are time-consuming or require fluorescent labels. Our previous study showed that microalgae cells have apparent differences in dielectrophoresis and electrorotation between generations and growth phases [START_REF] Lin | Using dielectrophoresis and electrorotation techniques to assess the dielectric properties of Scenedesmus Abundans at different growth states[END_REF]. In this study, a straightforward method for identifying cellular status based on dielectrophoretic movements is developed and applied to the monitor of microalgae in different growth phases.

THEORY

Dielectrophoresis (DEP) force, as shown in equation ( 1), drives a cell to the highest (positive DEP) or lowest (negative DEP) electric field regions, depending on the polarizability of the cell compared to its medium. The direction of DEP force (𝐹 ⃗ 𝐷𝐸𝑃 ) is determined by the real part of the Clausius Mossotti factor (𝑅𝑒[𝑓 𝐶𝑀 (𝜔)]).

𝐹 ⃗ 𝐷𝐸𝑃 = 2𝜋𝑅 𝑐𝑒𝑙𝑙 3 𝜀 0 𝜀 𝑚 𝑅𝑒[𝑓 𝐶𝑀 (𝜔)]𝛻 ⃗⃗ 𝐸 𝑅𝑀𝑆 2 (1)
where 𝜀 0 and 𝜀 𝑚 respectively stand for the permittivities of the vacuum and the medium, 𝑅 𝑐𝑒𝑙𝑙 is the radius of the cell and 𝜔is the angular frequency of applied electric field 𝐸.

When the movement of a cell is at steady state, dielectrophoresis force is balanced with the viscous drag as shown in equation ( 2) [START_REF] García-Sánchez | Alternating current electrokinetic properties of gold-coated microspheres[END_REF]. In equation ( 2), v is the velocity of the cell and 𝛿 is the distance between the wall and the microalgae cell. The velocity v, shown in equation ( 3), thus can be obtained by balancing equation ( 1) and [START_REF] Lin | Using dielectrophoresis and electrorotation techniques to assess the dielectric properties of Scenedesmus Abundans at different growth states[END_REF].

𝐹 𝑑𝑟𝑎𝑔 = 6𝜋𝜂𝑅 𝑐𝑒𝑙𝑙 𝑣(-(8/15) 𝑙𝑛 (𝛿/𝑅 𝑐𝑒𝑙𝑙 ) + 0.9588) (2)
𝑣 = 𝑅 𝑐𝑒𝑙𝑙 2 𝜀 0 𝜀 𝑚 𝑅𝑒[𝑓 𝐶𝑀 (𝜔)]𝛻 ⃗⃗ 𝐸 𝑅𝑀𝑆 2 2𝜂((8/15) 𝑙𝑛 (𝑅 𝑐𝑒𝑙𝑙 /𝛿) + 0.9588) (3)

EXPERIMENTAL

The microalgae Scenedesmus abundans was cultured in BG-11 medium. The growth was monitored by UV/Visible spectrometry and dielectrophoretic movements. The growth of S. abundans was measured by a spectrophotometer at wavelengths of 682 nm (OD682) and 220 nm (OD220), which represented the total amount of chlorophylls and nitrate concentration, respectively. The dielectrophoretic movements were measured between 4 planar parabolic electrodes (Figure 1a) with a 75 µm gap on the quartz wafer. The electrodes were fabricated by conventional photolithography technology and the fluidic chamber was constructed by 3M tapes. The conductivity of the medium was adjusted to 0.02 S/m for DEP measurement. The electrodes were then connected to two-channel function generators (AFG3102, Tektronix, USA) with electric signals of 3 Vpp, 180 o phase shift, and 5 frequencies ranging from 37 kHz to 38.6 MHz. At each frequency, at least 10 movements of microalgal cells induced by DEP forces were recorded during 5 seconds by a high-speed camera with a frame rate of 100 fps.

RESULT AND DISCUSSION

The amount of chlorophylls and nitrate consumption of S. abundans were monitored daily to assess the growth conditions of microalgae as shown in Figure 1b. The OD682 (total chlorophylls) doubled and OD220 (nitrate) decreased by 68 % in the first fifth days. The rapid rates of chlorophylls increase and nitrate consumption are a characteric of the exponential phase of microalgae. The much slower increase in OD682 and decrease in OD220 after the 7th day of the reinoculation indicated the stationary phase had been reached. S. abundans cells were subjected to dielectrophoresis at the second, fourth and eighth days after the reinoculation.

The DEP velocities of microalgae shown in Figure 1c indicated that DEP velocity of microalgae in the stationary phase is slower than the one of microalgae in the exponential phase. Moreover, the movement of microalgae in the exponential phase changed from negative DEP velocity to positive DEP velocity (i. e. the transition frequency) at much lower frequency (253 kHz) compared with microalgae in the stationary phase (436 kHz). The differences in the above dielectric behavior were the results of changing cellular compositions during growth and can be applied to separate cells with different physiological status.

CONCLUSION

This study demonstrated a straightforward method for discriminating microalgal cells in different growth phases using dielectrophoresis. When DEP force is balanced by the drag force, the variation in dielectric properties of microalgal cells at different physiological states lead to different DEP velocities. The DEP velocity of microalgal cells in the stationary phase are slower than those in the exponential phase, making DEP a straightforward method for separating microalgal cells in different growth phases.

Figure 1 :

 1 (a) Single microalgal cell trapped and moved to the center of the electrode set by positive DEP force. (b) Growth curve of S. abundans (OD682) and the nitrate concentration in BG-11 medium (OD220). (c) DEP velocity of S. abundans at different days after reinoculation.

ACKNOWLEDGEMENTS

This study is supported by the Ministry of Science and Technology of Taiwan under grant number MOST 108-2221-E-007-071, and LABEX Lasips (ANR-10-LABX-0040-LaSIPS) managed by the French National Research Agency under the "Investissements d'avenir" program (n°ANR-11-IDEX-0003-02).