# Fast and compact scatterometer for low-scattering biological medium qualification

A Sarrade, J El Allam, V Espinas, L Hoffart, C Picard, T David, G Georges, C Deumié, L Siozade Lamoine, J Fade

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A Sarrade, J El Allam, V Espinas, L Hoffart, C Picard, et al.. Fast and compact scatterometer for low-scattering biological medium qualification. 2026. hal-05490843v2

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# Fast and compact scatterometer for low-scattering biological medium qualification

A. Sarrade,<fnref n="1" />,<fnref n="2" />, a) J. El Allam,<fnref n="1" /> V. Espinas,<fnref n="1" />,<fnref n="3" /> L. Hoffart,<fnref n="2" /> C. Picard,<fnref n="4" />,<fnref n="5" /> T. David,<fnref n="1" /> G. Georges,<fnref n="1" /> C. Deumié,<fnref n="1" />,<fnref n="3" /> L. Siozade Lamoine,<fnref n="1" />,<fnref n="3" /> and J. Fade<fnref n="1" />,<fnref n="3" />

(Dated: 6 February 2026)

We present the design, calibration, qualification, and application of a fast and compact scatterometer designed for the characterization of ex-vivo human corneas. As an alternative to standard qualification processes, and in order to meet the current need for improved reliability of graft qualification, this instrument provides a non-invasive, label-free and contact-free, quantitative mean of characterizing corneal grafts, based on back-scattered light diffusometry. The instrument is shown to be capable of detecting BRDF (bidirectional reflectance distribution function) variations of $1.4 \times 10^{-4} \text{ sr}^{-1}$ with a detectivity threshold of $2.0 \times 10^{-3} \text{ sr}^{-1}$, with an overall dynamics > 27 dB for the Total Integrated Scattering (TIS) parameter used for graft quality assessment. We also demonstrate the feasibility of integrating this new technique into tissue bank qualification processes, through controlled test experiments reported here.

## I. INTRODUCTION

According to current estimates from the World Health Or- ganization (WHO), approximately 45 million people world- wide suffer from blindness<fnref n="1" />. Among them, about 1.8 mil- lion experience visual impairment due to corneal diseases such as keratitis, keratoconus, endothelial dystrophy, corneal trauma, and other pathologies<fnref n="1" />. In 2013, approximately 185,000 corneal transplantations were performed worldwide<fnref n="2" />, with only a limited number of countries, including the United States<fnref n="3" /> and India<fnref n="4" /> considered as self-sufficient in corneal tissues supply. At the same time, about 50% of potential grafts are disqualified due to poor tissue quality<fnref n="5" />. However, it is widely acknowledged that current qualification techniques re- main imperfect⁶,⁷,⁸,⁹. The development of reliable, quantita- tively accurate, and contactless tissue characterization meth- ods therefore represents a major challenge to increase graft availability for patients and improve success rate of corneal transplantation¹⁰,¹¹.

The cornea is the transparent, convex, lens-shaped tissue through which light enters the eye. It is responsible for ap- proximately two-thirds of the eye's refractive power, and its structural and optical integrity is essential for maintaining good visual acuity¹²,¹³. Structurally, the cornea is composed of five layers¹⁴, namely (from outer to inner): the epithe- lium, Bowman's membrane, the stroma, Descemet's mem- brane, and the endothelium. The stroma, which accounts for approximately 90% of the total corneal thickness, is com- posed of layered collagen fibrils and constitutes the primary site of corneal scarring and edema formation. The endothe- lium is a monolayer of cells interfacing the cornea with the

anterior chamber of the eye and play a crucial role in regu- lating corneal hydration¹⁵,¹⁶,¹⁷. Currently, corneal graft qual- ity is assessed using two complementary¹⁸ optical evaluation methods: transparency assessment and endothelial cell den- sity measurement. Transparency is qualitatively evaluated by visual inspection performed by an operator to identify macro- scopic defects or visible scars, while endothelial cell den- sity is estimated through semi-automated microscopic anal- ysis following tissue staining with Trypan blue, a procedure that requires direct graft manipulation. Both techniques are highly dependent on operator expertise, imaging conditions and they do not provide direct, quantitative metrics related to the cornea's optical scattering or microstructural organiza- tion. As a result, these methods exhibit limited reproducibility and sensitivity, particularly for detecting subtle structural or hydration-related alterations of the tissue.

To improve the reliability of corneal graft qualification, sev- eral laboratories have investigated alternative optical meth- ods for assessing endothelial cells and graft transparency. Corneal densitometry provides a quantitative evaluation of graft transparency and stromal edema by analyzing light spec- ularly backscattered within the cornea¹⁹,²⁰,²¹,²²,²³. Optical Coherence Tomography (OCT) is widely used to obtain cross- sectional and topographical information on corneal grafts²⁴, allowing the identification of major stromal scars or lesions²⁵, and enabling reliable measurements of graft thickness²⁶,²⁷,²⁸. More recent studies have also explored the use of OCT- derived metrics to grade graft transparency²⁹,³⁰,³¹. In parallel, significant efforts have been devoted to improving endothe- lial cell density (ECD) assessment, with alternatives to specu- lar microscopy³²,³³, such as Gabor domain optical coherence microscopy and fluorescence-based imaging techniques³⁴,³⁵ to visualize endothelial cells. Other approaches focus on automating ECD counting to reduce operator-dependent variability³⁶,³⁷,³⁸. Additional methods under investigation include transmitted light measurement³⁹,⁴⁰,⁴¹, assessment of

<footnote n="1">1) Aix Marseille Univ, CNRS, Centrale Med, Institut Fresnel, Marseille, France</footnote>

<footnote n="2">2) Ramsay Générale de Santé, Clinique Monticelli-Velodrome, Marseille, France</footnote>

<footnote n="3">3) Centre Européen de Recherche en Imagerie Médicale (CERIMED), Aix-Marseille University, CNRS, Marseille, France</footnote>

<footnote n="4">4) Tissue Bank, French Blood Establishment, Marseille, France</footnote>

<footnote n="5">5) ADES, French Blood Establishment /Aix-Marseille University/CNRS, Marseille, France</footnote>

<footnote>a) Author to whom correspondence should be addressed: adrien.sarrade@fresnel.fr</footnote>
endothelial cell morphology<sup>42</sup>, or evaluation of endothelial cell activity<sup>43</sup>.

Despite these advances, most existing techniques either rely on indirect proxies of optical quality, require complex imaging modalities or tissue manipulation, or remain insufficiently sensitive to subtle, hydration-related microstructural changes that directly affect corneal light scattering. In this paper, we investigate an alternative optical approach based on the quantitative analysis of the light backscattered by the cornea under controlled illumination conditions. Measurements of corneal backscattering provide direct access to optical signatures associated with tissue microstructure and hydration state, and therefore offer a physically grounded means of quantifying corneal transparency in both ex vivo and in vivo contexts<sup>44</sup>. Recent advances in scatterometry have enabled the development of low-flux optical systems that are compatible with fragile biological tissues while remaining compact and cost-effective. Building on these developments, we present the design and qualification of a fast, compact, and precise scatterometer specifically tailored for corneal graft assessment. This scatterometer measures the angularly resolved scattering (ARS) from which the corneal scattering signature is extracted as a dimensionless scalar quantity, namely, the Total Integrated Scattering (TIS), which can be monitored throughout the various stages of the graft preservation or preparation in tissue banks.

This manuscript is structured as follows: Section II introduce the principles of scatterometry measurements principles and describes the design of our scatterometer setup. Section III details the qualification and characterization of the instrument performances. Section IV presents some validation measurements on tissue phantoms followed by representative applications on human corneal grafts that demonstrate the ability of the setup to quantitatively assess corneal transparency. Section V discusses the results and outlines perspectives for future improvements.

Some of the measurements reported in Sections III and IV were performed on human corneal grafts that were disqualified for transplant, that were obtained from donors who did not oppose the scientific use of their tissues. All procedures complied with the principles of the Declaration of Helsinki for biomedical research involving human tissues. Corneal procurement for this study was authorized by the French Biomedicine Agency (PFS 23-003).

## II. DESIGN AND DESCRIPTION OF THE INSTRUMENT

### A. Scatterometry principles

Scatterometry consists in measuring the radiance (in units of $W.m^{-2}.sr^{-1}$) emitted in many directions by a sample under controlled illumination (see Fig.1). The radiance distribution of the scattered light can inform on the scatterers properties, geometry and/or distribution in the sample, as well as on the anisotropy factor and on the optical properties of the medium (scattering and absorption coefficients). In this study, let us consider a steady state monochromatic coherent laser

source illuminating a sample (here a corneal graft) at normal incidence. The instrument designed focuses on the reflected half-space to measure backscattered light, in the perspective of potential in-vivo operation in future developments.

Conventionally in scatterometry, either the detector or the sample is mobile to achieve the highest possible angular resolution along the azimuthal angle $\phi$ and elevation angle $\theta$, albeit at the expense of acquisition speed. However, for the sake of stability, ease of sample manipulation and acquisition speed, both sample and sensors are fixed in our instrument. In the clinical context of this work, we assume a radial symmetry of the sample which allows us to set $\phi = 0$ without loss of generality. The incident plane of the setup is therefore defined with the vertical laser illumination, along with the meridian arc of the hemisphere at $\phi = 0$. A set of $N$ photodiodes, distributed along this arc at a constant radius $R$ from the sample, make it possible to sample the scattered radiance at various values of the elevation angle $\theta$, to retrieve the angularly resolved scattering (ARS). Each photodiode is angularly separated from its neighbours by an angle denoted by $\delta\theta$ (radians).

<figure><img src="image_2.png" /><figcaption>FIG. 1. Sketch of the scatterometry principle.</figcaption></figure>

Once acquired, the ARS is normalized by the incident illumination radiance (through a reference photodiode, see below) to get the bidirectional reflectance distribution function (BRDF) in units of $sr^{-1}$. The BRDF is commonly defined as the function relating incident radiance along a direction $\hat{n}_{in}$ with the scattered radiance along direction $\hat{n}_{out}$. Due to the assumed symmetry properties of the sample, the BRDF will be expressed in this work as a function of the scattering angle $\theta$ only, i.e., $BRDF(\hat{n}_{in}, \hat{n}_{out}) = BRDF(\theta)$.

Previous works<sup>45,46</sup> highlighted that reflected photons from the epithelial surface and multi-scattered photons from the stromal volume of the cornea can be discriminated by considering different angular sections of the BRDF. Typically, surfacic interaction mainly contributes to the BRDF at low $\theta$ values (< 30°) while volumic scattering contributes at higher $\theta$ values (>30°). In order to quantify the transparency of the graft we need to consider the photons backscattered by the stroma<sup>47,48</sup> (i.e., $\theta > 30°$). Moreover, due to our setup footprint, we can only measure BRDF for angles $\theta$ below 75°.

From the voltage $V(\theta)$ measured at an angle $\theta$, we can
compute the “BRDFcos”, denoted in the following BRDFc and defined as

$$BRDF_c(\theta) = BRDF(\theta) \times \cos(\theta) = C_e \frac{V(\theta)}{V_0}, \quad (1)$$

where $V_0$ is the voltage measured on the reference photodiode (used to ensure normalization by the laser optical power in case of laser fluctuations) and where $C_e$ is a calibration constant, whose computation will be detailed in the next subsection.

In this study, the approximate integral of the BRDFc between elevation angles $\theta_{min} = \theta_{imin} = 35^\circ$ (photodiode index $i_{min} = 7$) and $\theta_{max} = \theta_{imax} = 70^\circ$ (photodiode index $i_{max} = 14$) is referred to as the total integrated scattering (TIS[$\theta_{min}:\theta_{max}$], which will be abbreviated as TIS in the remainder of the manuscript for the sake of conciseness). Since our measurements are performed at a fixed value of $\phi = 0$ across a single meridian arc, the extrapolation of the integration over the solid angle defined between $\theta_{min}$ and $\theta_{max}$ yields the following expression for the TIS:

$$TIS = 2\pi \delta\theta \sum_{j=i_{min}}^{i_{max}} BRDF_c(\theta_j) \sin(\theta_j). \quad (2)$$

This dimensionless quantity will be used in our studies to quantify the overall scattering properties of the graft through a single scalar parameter for some clinical applications (see Section IV). The TIS can be used to detect abnormal scattering signals due to scars or lesions in the stroma$^{49}$. It can also be used to monitor the evolution of graft transparency during its life-cycle from harvest to transplant$^{50}$.

## B. Radiometric calibration

The calibration constant $C_e$ can be obtained from a calibrated sample with known albedo, following Lambert's scattering law. We use a Spectralon calibration target (Spectralon SRS-99-010, Labsphere, USA) with calibrated albedo $\rho = 0.99$ and with orthotropic (Lambertian) scattering profile, i.e., with a BRDFc of the form

$$BRDF_c^L(\theta) = \frac{\rho}{\pi} \cos(\theta), \quad (3)$$

where the superscript L denotes measurements on the Lambertian target.

Denoting by $\underline{Y} = [\frac{V^L(\theta_1)}{V_0}, \dots, \frac{V^L(\theta_N)}{V_0}]$ the vector of the normalized voltage measurements with the calibration target, and by $\underline{X} = \frac{\rho}{\pi}[\cos(\theta_1), \dots, \cos(\theta_N)]$, Eq. (1) applied at each measurement angle can be rewritten in a vector form as $\underline{Y} = \kappa\underline{X} = \frac{1}{C_e}\underline{X}$, and therefore the calibration constant can be inferred by a least-square minimization approach $\hat{\kappa} = \underset{\kappa}{\operatorname{argmin}} \|\underline{Y} - \kappa\underline{X}\|^2 = (X^T X)^{-1} X^T Y$, which leads to the following closed

form expression for $\hat{C}_e$

$$\hat{C}_e = \frac{1}{\hat{\kappa}} = \frac{\rho}{\pi} \frac{\sum_{k=1}^{N} \cos^2(\theta_k)}{\sum_{k=1}^{N} \frac{V^L(\theta_k)}{V_0} \cos(\theta_k)}. \quad (4)$$

Due to its fast and easy implementation, the calibration is repeated every day on the instrument, to ensure comparable measurements across a long-term experimental campaign.

## C. Instrument design and description

The scatterometer is a rather compact system (97.5 × 68 × 34.5 cm) suitable to be used in a tissue bank, such as the French Blood Establishment (EFS PACA-Corse, Marseille, France, see Fig.2), were our setup is currently used for experimental ex-vivo corneal graft qualification$^{51}$. The instrument is controlled by a laptop computer via a Python software routine.

<figure><img src="image_3.png" /><figcaption>FIG. 2. Photograph of the experimental scatterometer setup (a), and diagram of the optical setup (b).</figcaption></figure>

The optical path diagram is presented in Fig.2 and follows the prescription of the scatterometry principles detailed in the previous section. A 640 nm laser with an output power of 40 mW (BioRay, Coherent, USA) is directed into the instrument with a pair of mirrors, and is attenuated using an optical neutral density filter (OD 1.0, Thorlabs, USA). It then passes through a beam-splitter (BSN10, Thorlabs, USA), which directs 10% of the laser intensity towards a reference photodiode (SM1PD1A, Thorlabs, USA). The transmitted light is made to illuminate vertically at normal incidence a sample (calibration lambertian diffuser or cornea) that can be placed in an open Petri glass dish above a beam dump. The backscattered light is collected by 15 photodiodes (SM1PD1A, Thor-
labs, US) evenly distributed along a meridian arc, with the first photodiode positioned to 5° (photodiode index i = 1) and the last to 75° (photodiode index i = N = 15). The mechanical arch designed (which can be seen in Fig. 2, (a)) maintains a constant radius of R=30 cm, and each photodiode is fixed along it and separated from the next one with an angle of δθ = 5° = 5π/180 rad. The photodiodes voltages are then processed by an acquisition chain detailed below.

### D. Acquisition chain

The acquisition chain, which is sketched in Fig.3, comprises a microcontroller, a timer, photodiodes, a multiplexer, a linearization stage, and an amplification/inversion stage. The system is controlled by a microcontroller (PIC18F4553, Microchip, USA), which controls the other components and interfaces with the laptop. The microcontroller uses a timer (LTC6993CDCB, Analog Devices, USA) set at 240 Hz to modulate the laser beam and trigger the analog-to-digital conversion (ADC).

<figure><img src="image_4.png" /><figcaption>FIG. 3. Sketch of the acquisition chain.</figcaption></figure>

According to the datasheet, the 16 photodiodes used have a responsivity of 0.34 A/W, a dark current of 600 nA, a noise equivalent power (NEP) of $2 \times 10^{-13}$ W/Hz, with an active area of 63.6 mm², corresponding to $5.6 \times 10^{-5}$ sr of solid angle in this configuration. Each photodiode is connected via coaxial cables to a custom-made electronic board for signal processing and acquisition.

The acquisition of the 16 photocurrents from the photodiodes is multiplexed in time through a multiplexer (MUX36S16, Texas Instruments, USA) that sequentially selects each channel. Following the multiplexer, the signal passes through a current-voltage converter stage and an inverter stage, where it is linearized and amplified. The amplified signal is then digitized using an analog-to-digital conversion stage, integrated in the micro-controller chip, which digitizes the input range 0–5 V with a 12-bit resolution (4096 values), corresponding to a voltage discretization step of 1.22 mV per bit. From the resulting digital square-wave signal (due to the 240 Hz modulation, see Fig. 4 (a)), the voltage value $V(\theta)$ retrieved at the photodiode at angle $\theta$ is obtained by averaging 168 voltage measurements during the onset time of the laser, over 8 consecutive periods (see Fig.4).

Overall, this configuration results in a total acquisition time of approximately 25 seconds for all 16 channels. Finally for each observation angle, the mean voltage value $V(\theta)$ is converted to a $BRDF_c$ value (see Fig. 5) using the reference voltage $V_0$ and the calibration constant $C_e$ according to Eq. (1).

We evaluate the measurement uncertainty in voltage by

$$ \Delta V = 1.97 \frac{\sigma}{\sqrt{n}}, \qquad (5) $$

with $\sigma$ the standard deviation of the signal over n measures (using Student's t-quantile for a 95% confidence interval). It has been checked that $\Delta V$ was not dependent on the signal level on the photodiodes.

<figure><img src="image_5.png" /><figcaption>FIG. 4. Two periods of the raw voltage signal collected from each diode (a). Concatenating the voltage measurements during the onset time of the laser over seven periods provides a 168-sample voltage signal at the photodiode at angle $\theta$, the average of which yields the final measurement $V(\theta)$ (b).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 5. Photograph of a corneal graft (a) and corresponding $BRDF_c$ (b). $BRDF_c$ of a cornea (crosses symbols), $BRDF_c$ of the lambertian calibration target (dots symbols) and detectivity threshold of the instrument (triangles).</figcaption></figure>

### III. QUALIFICATION OF THE INSTRUMENT

This section highlights and quantifies the limits and performances of the instrument. First, we assess the detection sensitivity threshold. Then, we provide an analysis of the measurement uncertainty and reproducibility errors to determine overall precision and robustness of the instrument.
## A. Sensitivity and detection threshold

At a wavelength of 640 nm, the SM1PD1A photodiodes saturate when illuminated by an optical power of 1.4 µW. Due to the very low reflectivity of the corneal samples to be characterized with this instrument, the strongest level of optical signal is obtained with the Lambertian calibration target. Therefore, the illumination power has been adjusted to avoid saturation on the photodiodes using an appropriate neutral density filter under this calibration configuration. The values of $BRDF_c$ of the Lambertian target range from $0.35 \text{ sr}^{-1}$ near normal incidence, to $0.07 \text{ sr}^{-1}$ on the photodiode at 75°. The TIS of the Lambertian target is 0.584.

To determine the detectivity threshold of the system, we measured the optical power vs voltage characteristic curve of the system, which is displayed in Fig.6 (a). In this plot, the triangles correspond to the measured data, while the line is a linear interpolation of the data points. We observed that the linear relationship cannot be achieved below approximately 15 mV, thereby fixing the detectivity limit at approximately 7 nW in terms of optical power impinging on a photodiode. This loss of linearity at low light level can be explained by examining the example raw voltage data displayed in Fig.4: below an optical power of 8 nW, the fluctuations of the voltage signal would, after digitization of the signal, lead to a significant number of samples at a voltage value of 0 V. Those zero-valued samples tend to strongly bias the mean voltage estimation, and hence explain the deviation from the linear behaviour of the acquisition chain.

To define the detectivity threshold on a more quantitative basis, we set the detectivity limit as the voltage under which the noise cannot be considered as gaussian anymore, relying on a Shapiro-Wilk test⁵². The normality of the residual noise of the measures presented in Fig.4 (a) has been tested and the p-value of each test is represented in Fig.4 (b). Below a voltage of 15 mV, the p-value strongly decrease indicating a significant rejection of the null hypothesis of normality of the noise distribution, thereby confirming the 15 mV detectivity voltage limit of our instrument. This detectivity limit corresponds to a lowest $BRDF_c$ of $2 \times 10^{-3} \text{ sr}^{-1}$ and a TIS of $6.8 \times 10^{-3}$.

## B. Measurement uncertainty

The precision of the instrument can be assessed by analyzing the voltage fluctuations in the measured signal (see Fig.4). For a given angle $\theta$, the value $V(\theta)$ is estimated from the average of $n = 168$ temporal samples of the digitized voltage signal delivered by the corresponding photodiode, assumed to remain constant over the acquisition time of the $n$ samples. From the measured standard deviation $\sigma = 5.5$ mV of this temporal voltage signal, we thus extrapolate with Eq. (5) the uncertainty $\Delta V$ of the averaged voltage measurement $V(\theta)$ as $\Delta V = 0.84$ mV. This uncertainty will be considered as constant as a function of the signal level (above detectivity limit), and the same for all photodiodes (including the reference photodiode).

<figure><img src="image_7.png" /><figcaption>FIG. 6. (a): Optical power vs voltage characteristic of the acquisition chain: measured data (triangle) and linear fit (line). Small data points represent the proportion of zero-valued samples detected in the digitized voltage signal. (b): P-value given by the normality Shapiro-Wilk test assessed on the residual noise of each measured data.</figcaption></figure>

From $\Delta V$, it is now quite straightforward to extrapolate the uncertainty in the estimation of the calibration constant, on the values of $BRDF_c$ and on the TIS. The detail of the error propagation calculus is provided in Appendix 1. Concerning the $BRDF_c$, the uncertainty level depends on the amplitude of the signal. It remains however quite constant, ranging from $1.2 \times 10^{-4} \text{ sr}^{-1}$ for the lowest optical power (7 nW), and reaching a maximum of $1.4 \times 10^{-4} \text{ sr}^{-1}$ for the strongest signal measured near normal incidence on the lambertian target. As for the TIS, we propose to upper-bound the estimated uncertainty by considering the case of the strongest optical signal obtained on the lambertian calibration sample, yielding $\Delta TIS \le 1.6 \times 10^{-4}$, from the expression of Eq. (A4) in Appendix 1.

These estimations of the uncertainty on the measured scattering parameters $BRDF_c$ and TIS have been checked experimentally through repeated measurements. For this purpose, we used two humans corneal grafts and varied their thickness 60 times each using swelling and de-swelling liquid media in which the sample was immersed. These experiments provided us with 120 distinct experimental configurations covering the entire range of possible thicknesses and TIS values reachable by a cornea. For each of these configurations, we measured the TIS four consecutive times without altering the setup nor the sample, in order to assess the repeatability of the instrument. The average TIS min-max difference over 4 measurements was $7.3 \times 10^{-4}$, while the average standard deviation of the TIS value over 4 measurements was $3.6 \times 10^{-4}$, in quite
fair agreement with the theoretical derivation discussed above.

These uncertainty evaluations finally lead to a global dynamic range of the measured TIS values above 30 dB (1:1600).

### C. Reproducibility measures on corneal samples

In this section we now quantify how the misalignment of a corneal graft in the instrument can impact the measured TIS value. Defining the X and Y axes on the horizontal plane of the sample (with X the normal to the incidence plane), and defining Z as the vertical direction of the laser illumination beam, we shall refer to X/Y displacements as *in-plane* displacements, and to displacement along the Z axis as *out-of-plane* displacement. In-plane alignment of the sample has a precision of 250 µm along X and Y using micrometric stages, whereas out of plane alignment has a precision estimated to 5 mm. This quite low precision along the Z axis accounts for the strong variability of the cornea grafts geometries, due to high variability of the dissection technique by the surgeon during tissue procuration. During routine use of the instrument, the alignment is performed by placing the cornea in a Petri dish in the instrument, leading to a vertical location of the cornea apex within a 5 mm range from the reference horizontal plane (defined with the Spectralon calibration target). The in-plane centering is then performed by pointing the laser illumination spot on the cornea apex, using the specular reflection in the instrument housing to ensure best vertical alignment.

For each misalignment test presented below, it has been checked that the grafts used did not undergo any significant variation of thickness during a series of measurements. Each of the displacements that were manually performed correspond to realistic potential misalignments of the graft during manipulation of the scatterometer instrument by an operator.

The dependency of in-plane alignment of the graft on the TIS value has been assessed by measuring the TIS after several displacements of 50 µm of the sample over a total range of ±150 µm along the X and Y axes, centered on the normal optimal alignment of the graft. Results are presented in Fig.7. For each axis the experiment is repeated three times, and the average min-max variation in the TIS observed is $7.6 \times 10^{-4}$ for the X axis and $5.7 \times 10^{-4}$ for the Y axis.

As for the Z axis, the influence of the vertical alignment of the graft on the TIS has been assessed by measuring the TIS after several vertical displacements of 2 mm of the graft, over a total range of ±5 mm. The experiment has been repeated 4 times, for various experimental conditions: several orientations of the graft (epithelium facing up or down); and presence or not of a surrounding medium (air or deswelling medium). Results are presented in Fig.7, showing that the average error observed in this case on the TIS was $7.2 \times 10^{-4}$.

The impact of out-of-plane misalignment on TIS uncertainty has been also theoretically determined in Appendix 1. For a typical corneal graft, evaluation of Eq. (A5) yields a $\Delta TIS = 1.4 \times 10^{-4}$, wheras for the lambertian target one obtains $\Delta TIS^L = 2.5 \times 10^{-3}$. This evaluation is quite similar in order of magnitude to the experimental evaluation reported

<figure><img src="image_8.png" /><figcaption>FIG. 7. Evolution of TIS with controlled misalignment of a corneal graft. (a) provides the influence on the TIS of in-plane misalignment by axis. (b): same for vertical misalignment for different configurations of the graft : Up for epithelium facing upward, Down for epithelium facing downward, Air for open air graft, Liquid for immersed graft. Circles stand for the mean of each serie.</figcaption></figure>

above. In order to retain a safe estimation of the uncertainty on cornea sample measurements when misalignment errors are accounted for, we shall retain in the following the value $\Delta TIS \le 10^{-3}$. This uncertainty value yields an overall measurement dynamics of more than 27 dB (1:580).

## IV. APPLICATION TO QUALITY ASSESSMENT OF CORNEAL GRAFTS

In this section, we provide some examples of use of this instrument for the quality assessment of corneal grafts. These examples, along with the instrument calibration and qualification processes provided in previous sections, allows us to ground our experimental results$^{49,50}$ and potential clinical conclusions$^{51}$ on a solid metrologic basis. We first consider below a validation experiment on controlled phantom of scattering media, before we provide some experimental results on human corneas.

### A. Validation tests on tissue phantoms

To highlight the potential range of use of our instrument, especially on low turbidity samples such as edema-free, physiological corneas, artificial tissue phantoms with very scattering levels have been designed and tested in the instrument. These tissue phantoms have been designed in our laboratory, based on various dilutions ( 0.3%, 1% and 2%) of an agar so-
lution (Agar Agar, Jeulin, FRANCE) in water, which were let to gelify in a petri dish (Falcon petri dish, 5 cm diameter), providing us with 10 mm thickness test disk that have been used as test samples. Photographs of the three disks on a printed text are provided in Fig. 8.a, showing the low level of scattering reached with these samples, but highlighting the increase of scattering with agar concentration.

Fig.8.b presents the scatterometry results obtained on the three disks. The *BRDF*<sub>c</sub> of each sample has been plotted (colour circle symbols), as well as the detectivity threshold (triangle symbols), and the *BRDF*<sub>c</sub> of the lambertian orthotropic scatterer used for calibration (black circles). In agreement with the photographs showing a slight increase of the turbidity with agar concentration, the same evolution of transparency is observed in the *BRDF*<sub>c</sub> graph for angles above 35°, which corresponds to the conditions of use of our instrument for TIS evaluation. The levels of *BRDF*<sub>c</sub> measured here correspond to typical diffusion levels observed in human cornea grafts. This controlled experiment illustrated the ability of the instrument to quantify low levels of scattered light, the analysis of which will be used below to qualify corneal grafts.

<figure><img src="image_9.png" /><figcaption>FIG. 8. Photographs of the agar scattering tissue phantoms laid on a text (a). The figures indicate the agar concentration used in the phantom preparation. (b) *BRDF*<sub>c</sub> of scattering tissue phantoms with varying agar concentration.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIG. 9. Evolution of TIS over time for 3 corneal grafts during deswelling phase (a). Evolution of TIS versus thickness during graft deswelling (b).</figcaption></figure>

## B. Application to real corneal grafts

In France, corneal grafts are stored in tissue banks before qualification and before delivery in an organoculture medium, at a temperature of 31°C during several days. In this medium the graft swells up to typically twice its physiological thickness due to water uptake by the tissue. After graft qualification in Eye Bank, and prior to their delivery to the surgeon, grafts are removed from the organoculture medium and placed in a deswelling medium in order to recover their physiological thickness before surgery. The deswelling phase may last between 12 hours to 96 hours. At regular intervals during this deswelling stage, we measured the TIS of the selected grafts with our scatterometer instrument, along with their thickness with a commercial OCT instrument (Oqlabscope 2.0, Lumedica, USA). In addition, photographs of the grafts were acquired to provide a qualitative assessment of their transparency. Studies of the swelling and deswelling behavior of corneal grafts, as well as of their scattering properties during these phases, are currently under investigation<sup>51</sup>, as they may provide tissue banks with a complementary, contact-free optical diagnostic tool for assessing tissue physiological quality.

In the following, we present representative examples of the use of the instrument in the context of cornea graft characterization, and we illustrate typical corneal graft behaviors in this setup. Fig.9 depicts the behavior of 3 corneal grafts placed in Tissue-C deswelling medium (Tissue-C, Alchimia, ITALY). Fig.9. (a) represents the variation of the TIS over time during deswelling of the graft. The typical levels of TIS measured on cornea grafts lie above the detectivity threshold of our instrument, and the low measurement uncertainty allows reliable comparison between different samples and deswelling kinetics. In Fig.9. (a), the decrease in TIS can be accurately approximated by an exponential decay, from which a characteristic decay time τ (min) is extracted as a quantitative descrip-
tor of the deswelling kinetics. The uncertainty associated with the estimation of the value τ is evaluated in Appendix B, as a function of the TIS measurement precision estimated above. This analysis shows that in the typical situations illustrated in Fig. 9. (a), the precision on the estimation of τ is sufficient to clearly discriminate between different characteristic decay times. Previous studies⁵⁰ have addressed the possibility to use τ as a qualification parameter for corneal graft, an hypothesis currently investigated on a wider cohort of samples⁵¹.

Similarly, Fig.9.b shows the relationship between the TIS value and the graft thickness during the same deswelling stage, showing a non trivial link between transparency and thickness, illustrating the relevance of studying scattering properties of cornea grafts for tissue quality qualification⁴⁹⁻⁵¹.

## V. CONCLUSION

In this manuscript, we presented a fast and compact scatterometer instrument dedicated to the characterization of low-scattering level biological tissues, namely ex-vivo human cornea grafts. We reported the detailed design of the instrument, of the acquisition chain, as well as its detectivity/saturation limits and its calibration procedure. We demonstrated through analytical error propagation calculus that the main source of error lies in the exact positioning of the graft in the instrument which need to be finely controlled. This theoretical derivation has been confirmed through reproductibility tests allowing us to assess the overall measurement dynamics of the instrument above 23 dB. We also highlighted that the orientation of the graft as well as its surrounding medium has a moderate impact on the amplitude of the signal, but with negligible influence on the repeatability error. Finally, we presented some results obtained on test scattering tissue phantoms and on real corneas, illustrating the potential use of this instrument for clinical purposes in tissue banks.

Future developments of this instrument will consist in the adaptation of the instrument to the sanitary constraints of Tissue banks, as well as improving its ergonomics and level of automation in order to enable routine use during corneal graft qualification. In particular, automated of graft positioning is also needed to reduce the measurement uncertainty and extend the dynamic range of the instrument. Further developments could also include the use of appropriately selected multiple wavelengths and/or polarization-sensitive measurements in order to gain further insight in the sample microstructure from those scatterometry measurements. Finally, ongoing collaborations with ophthalmologists addressing clinical aspects of corneal graft preservation and selection are also considered and under progress in order to further confirm the relevance of such instrument for secured corneal graft qualification.

## ACKNOWLEDGMENTS

The authors would like to express their gratitude to the Tissue Bank of the Établissement Français du Sang-PACA-

Corse (French Blood Institute) for providing the corneal grafts used in this study as well as the CERIMED for technical support. We also express our thanks to the Clinique Monticelli-Vélodrome and Centrale Méditerranée for funding A. Sarrade's work on this study, and to the Institut Imaging (AMIDEX) for funding J. El Allam's internship. Special thanks are due to Emmanuel Bergeret for his expertise and work in assembling the instrument.

## AUTHOR DECLARATION

### Conflict of interest

The authors have no conflicts of interest to disclose.

### Authors contribution

**A. Sarrade:** Conceptualization (supporting); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (equal); Validation (lead); Visualization (lead); Writing - original draft (lead); **J. El Allam:** Conceptualization (supporting); Data curation (supporting); Investigation (supporting); Validation (supporting); Writing - review & editing (supporting); **V. Espinas:** Resources (equal); Software (equal); Validation (supporting); **L. Hoffart:** Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing - review & editing (supporting); **C. Picard:** Resources (equal); Supervision (equal); Writing - review & editing (supporting); **T. David:** Funding acquisition (equal); Resources (equal); Supervision (equal); Writing - review & editing (supporting); **G. Georges:** Conceptualization (supporting); Resources (equal); Software (equal); **C. Deumié:** Conceptualization (equal); Funding acquisition (equal); **L. Siozade Lamoine:** Conceptualization (equal); Formal analysis (supporting); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Validation (supporting); Writing - review & editing (supporting); **J. Fade:** Conceptualization (equal); Formal analysis (supporting); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Validation (supporting); Writing - original draft (supporting); Writing - review & editing (lead);

## DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Appendix A: Evaluation of the theoretical uncertainty on TIS measurement

Assuming gaussian, independent fluctuations in unknown parameters, we apply below an uncertainty propagation cal-
culation in order to evaluate the theoretical uncertainty on TIS measurements with the setup described in this paper.

## 1. Uncertainty of the calibration constant

From the expression of the BRDFc in equation (1), we see that the uncertainty of BRDFc measurements depends on the uncertainty of the evaluation of the measured voltage on the photodiodes, and on the potential uncertainty in the determination of the calibration constant. Let us first determine the uncertainty error on the calibration constant $C_e$. For that purpose, let us rewrite equation (4) as a ratio

$$C_e = \frac{\frac{\rho}{\pi} \sum_k \cos^2(\theta_k)}{\sum_k \frac{V^L(\theta_k)}{V_0^L} \cos(\theta_k)} = \frac{A}{Q}, \quad (A1)$$

where, unless otherwise stated, the summation index $\sum_k$ will denote summation over the whole set of N photodiodes. We assume that $A = \frac{\rho}{\pi} \sum_k \cos^2(\theta_k)$, which does not depend on the voltage measurements, can be determined without uncertainty. Indeed, the positioning of the sample Lambertian target being very precise, no variability of the angles $\theta_k$ from one calibration to another is expected.

On the other hand, $Q = \sum_k \frac{V^L(\theta_k)}{V_0^L} \cos(\theta_k)$ does depend on the voltage measurement uncertainty since it corresponds to the normalized voltage measurement on the Lambertian target. In this appendix, we shall assume that the laser has a stable power, ensuring constant $V_0 = V_0^L$ throughout all the measurements and still assuming $\Delta V = \Delta V_0$ (reference photodiode is identical to the others). The squared uncertainty (variance) on $C_e$ can be obtained by $(\Delta C_e)^2 = (A^2/Q^4)(\Delta Q)^2$, with $\Delta Q$ verifying: $(\Delta Q)^2 = (\Delta V/V_0)^2 \sum_k \beta_k$, with

$$\beta_k = \cos^2(\theta_k) \left( 1 + \frac{(V^L(\theta_k))^2}{V_0^2} \right) \quad (A2)$$

yielding finally $(\Delta C_e)^2 = \frac{C_e^2 (\Delta V)^2}{Q^2 V_0^2} \sum_k \beta_k$.

## 2. Uncertainty of $BDRF_c$ measurements

We can now determine $\Delta BRDF_c(\theta)$ from its definition in Eq.(1): $(\Delta BRDF_c(\theta))^2 = \frac{V(\theta)^2}{V_0^2} (\Delta C_e)^2 + \frac{C_e^2}{V_0^2} (\Delta V)^2 + \frac{V^2(\theta)}{V_0^4} C_e^2 (\Delta V_0)^2$. One obtains $(\Delta BRDF_c(\theta))^2 = C_e^2 \frac{(\Delta V)^2}{V_0^2} \epsilon_\theta$, where we set

$$\epsilon_\theta = \left[ 1 + \frac{V(\theta)^2}{V_0^2} + \frac{V(\theta)^2}{Q^2 V_0^2} \sum_k \beta_k \right] \quad (A3)$$

## 3. Uncertainty on TIS measurement

Finally, from the definition of the TIS given in equation (2), the uncertainty on the TIS measurement due to voltage

fluctuations and calibration uncertainty can be extrapolated to:

$$(\Delta TIS)^2 = 4\pi^2(\delta\theta)^2 C_e^2 \frac{(\Delta V)^2}{V_0^2} \sum_{j=i_{min}}^{i_{max}} \epsilon_{\theta_j} \sin^2 \theta_j \quad (A4)$$

## 4. TIS uncertainty due to graft positioning

Let assume now that we do not precisely know the altitude of the sample. This unknown altitude induces an error $\Delta\theta$ on the angles $\theta_i$ that enter the TIS calculation. The contribution of such error to the TIS uncertainty can be expressed as:

$$(\Delta TIS)^2 = 4\pi^2(\delta\theta)^2 \sum_{j=i_{min}}^{i_{max}} \epsilon_{\theta_j} \cos^2 \theta_j [BRDF_c(\theta_j)]^2 (\Delta\theta)^2, \quad (A5)$$

where $\delta\theta$ still denotes the angular sampling along the meridian arc where the photodiodes are located.

## 5. Total TIS uncertainty

Combining the two uncertainty contributions finally yields:

$$(\Delta TIS)_{total}^2 = 4\pi^2(\delta\theta)^2 \frac{C_e^2}{V_0^2} \left\{ (\Delta V)^2 \sum_{j=i_{min}}^{i_{max}} \left[ \epsilon_{\theta_j} \sin^2 \theta_j \right] + (\Delta\theta)^2 \sum_{j=i_{min}}^{i_{max}} \left[ \left[V(\theta_j)\right]^2 \cos^2 \theta_j \right] \right\}. \quad (A6)$$

## Appendix B: Uncertainty on the determination of the decay time $\tau$ of a cornea deswelling kinetics

During a deswelling measurement series, we evaluate the TIS of a corneal graft at L distinct times $t_i$, $i \in [1; L]$. We assume that we can fairly approximate the temporal evolution of the TIS with an exponential model $f(t; A, \tau, B) = Ae^{-t/\tau} + B$, where $\tau$ is the characteristic decay time. Assuming that the TIS data is affected by a centered gaussian noise of with variance $(\Delta TIS)^2$, optimal estimation of the model parameters is obtained by standard least-square minimization. We can evaluate the precision of the estimated parameters $(\hat{A}, \hat{\tau}, \hat{B})$ by computing the covariance matrix $\Sigma_{\hat{A},\hat{\tau},\hat{B}}$ of the estimators

$$\Sigma_{\hat{A}, \hat{\tau}, \hat{B}} = (\Delta TIS)^2 (J^T J)^{-1}, \quad (B1)$$

with the matrix $J \in \mathbb{R}^{L \times 3}$ being the following Jacobian matrix

$$J = \begin{pmatrix} \frac{\partial f}{\partial A}(t_1) & \frac{\partial f}{\partial \tau}(t_1) & \frac{\partial f}{\partial B}(t_1) \\ \vdots & \vdots & \vdots \\ \frac{\partial f}{\partial A}(t_L) & \frac{\partial f}{\partial \tau}(t_L) & \frac{\partial f}{\partial B}(t_L) \end{pmatrix}. \quad (B2)$$

Finally, the estimated variance of $\hat{\tau}$ can be extracted from the second diagonal element of the matrix $\Sigma$, i.e., $\text{var}(\hat{\tau}) = (\Sigma)_{22}$, from which the uncertainty values on the estimated $\tau$ values in Fig. 9.(a) have been obtained.
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