Optik - International Journal for Light and Electron Optics 250 (2022) 168269 A 0 ( O G J D a b A K G U O R 1 t c c e s m b i s t o t d h R Contents lists available at ScienceDirect Optik - International Journal for Light and Electron Optics journal homepage: www.elsevier.com/locate/ijleo riginal Research Article eometrical limits for UV-C inactivation of pathogens aime Quintana a, Antonio A. Fernández-Balbuena a, Juan Carlos Martínez-Antón a, aniel Vázquez a, Luis Prada b, Luis Estrada b, Javier Alda a,∗ Applied Optics Complutense Group, University Complutense of Madrid, Av. Arcos de Jalón, 118, 28037 Madrid, Spain Luminalia Ingeniería y Fabricación, P.I. de Granda, Siero, 33199 Asturias, Spain R T I C L E I N F O eywords: eometrical optics ltraviolet light ptical disinfection adiometry A B S T R A C T The inactivation of pathogens through the irradiation of ultraviolet light depends on how light propagates within the medium where the microorganism is immersed. A simple geometrical optics analysis, and a fluence evaluation reveal some reservoirs where the pathogen may hide and be weakly exposed to the incoming radiation. This geometrical hide-outs also generate a tail in the plot of the total inactivation plot vs. the incoming fluence. We have analyzed these facts using geometrical optics principles and illumination engineering computational packages. The results obtained from previous biomedical measurements involving SARS-CoV-2 have been used to evaluate the inactivation degree for an spherical geometry applicable to airborne pathogens, and for an spherical cap geometry similar to that used in biomedical experiments. The case presented here can be seen as the worst-case scenario applicable under collimated illumination. . Introduction Triggered by the onset of the Covid19 pandemic, several strategies to disinfect pathogens have been revisited and applied o inactivate the SARS-CoV-2 [1–6]. Among them, the known capability of germicidal UV-C sources [7] has been applied to the ase of SARS-CoV-2 [8,9]. The detailed analysis of the culture medium used for the essays has revealed the importance of optical oncepts under UV irradiation. Optical absorption, parameterized through the absorption coefficient, is one of the main factors when valuating the intrinsic, characteristic fluence for inactivation. Geometry is the other one. Actually, geometry has been analyzed in ome other previous papers where the appearance of menisci at the outer free surface of the culture medium cast shadows on the edium itself [4,7,10–13]. If active homogenization is not possible during UV irradiation, the presence of unexposed volumes should e considered when analyzing the results of the UV inactivation strategies. In practice, some of the UV-inactivation essays require the rradiation of virus-containing droplets deposited on a flat substrate and illuminated with an almost collimated light source. Besides, ome culture media used in these analysis (e.g., Dubelcco’s Modified Eagle Medium) have a non-negligible absorption coefficient hat diminished the available UV-C radiation for inactivation. When the virus is considered air-borne, the medium takes the shape f a sphere, and before desiccation the medium is aqueous. In any case, the propagation of light within the medium is relevant o account for the disinfection level. To properly address these issues, geometrical optics, ray tracing and illumination engineering esign packages may help to understand better how light moves through the sample [9,14]. It is known that the inactivation of RNA virus follows an exponential law [7,15,16]: 𝜂 = exp ( − 𝐹 𝐹𝑖 ) , (1) ∗ Corresponding author. E-mail address: javier.alda@ucm.es (J. Alda). vailable online 11 November 2021 030-4026/© 2021 The Authors. Published by Elsevier GmbH. This is an open access article under the CC BY license http://creativecommons.org/licenses/by/4.0/). ttps://doi.org/10.1016/j.ijleo.2021.168269 eceived 12 October 2021; Accepted 30 October 2021 http://www.elsevier.com/locate/ijleo http://www.elsevier.com/locate/ijleo mailto:javier.alda@ucm.es https://doi.org/10.1016/j.ijleo.2021.168269 http://crossmark.crossref.org/dialog/?doi=10.1016/j.ijleo.2021.168269&domain=pdf https://doi.org/10.1016/j.ijleo.2021.168269 http://creativecommons.org/licenses/by/4.0/ Optik 250 (2022) 168269J. Quintana et al. g f b p b o p g T 2 I l d w o r d t T s t t a m m r 2 𝑟 𝑛 f where 𝜂 is the fraction of surviving pathogens observed for a fluence level, 𝐹 , being 𝐹𝑖 the characteristic fluence of the virus at the iven UV-C wavelength, 𝜆. Some authors refer 𝐹𝑖 as 𝐹D63, meaning that the surviving ratio has dropped till 1∕𝑒 = 0.37 when the luence is 𝐹𝑖. This intrinsic parameter should be independent of the virus environment (geometry, culture medium, etc.) and could e used to evaluate the surviving rate in other situations. The results obtained from the geometry and computational evaluation rovide a local value of the fluence within the volume of interest. By doing so, 𝐹 = 𝐹 (𝑥, 𝑦, 𝑧) and the surviving ratio 𝜂 = 𝜂(𝑥, 𝑦, 𝑧) ecomes local [17]. This local inactivation is used to analyze the exposure level of each region of the volume of interest. In this contribution, we have considered a simple geometrical optics model to understand better the behavior of light for two f the geometries involving curved surfaces (Section 2). In this section, the results from an illumination engineering computational ackage are used to determine the local fluence. In Section 3, we calculate the global surviving ratio and we provide a simple eometrical explanation for the presence of significant tails in the inactivation curves, revealing physical hide-outs for the pathogen. he main conclusions of this manuscript are given in Section 4. . Geometrical model Geometrical optics is likely one of the most powerful tools when sketching how light propagates and behaves in optical systems. ts simplicity makes possible to perform back-of-the-envelope calculations and drawings that show the first approach to find the ocation and size of objects and images transformed by diopters, mirrors and lenses. Geometrical optics is also the backbone of optical esign packages widely used by optical engineers on a daily basis. In particular, illumination engineering merges geometrical optics ith optical energy evaluations to understand how radiant flux moves through materials. This is why we have chosen geometrical ptics as the first step in our model. Even more, the paraxial approach of geometrical optics makes possible to locate points of elevance in the propagation of the energy from the source, as focal points [18–20]. In this analysis we consider two basic geometries: a sphere filled with water, and a droplet filled with culture medium and eposited on a flat substrate. The sphere is a first approximation to an aqueous airborne pathogen’s container. Although the size of he sphere changes depending on environmental variables, in this analysis we assume that the spherical shape remains invariant. he droplet geometry (modeled as an spherical cap) appears in biological essays where the culture medium is deposited on a flat ubstrate. Some other geometries include cylindrical vessels and Petri dishes [10]. A optical artifact involved in cylindrical vessel is he formation of menisci at the air/liquid interface. Their importance at the outer region of the free surface of the liquid depends on he ratio between the menisci’s radius and the diameter of the cylindrical vessel. Some interesting calculations have been made to nalyze the effect of menisci formation in biological experiments [10]. Our calculation does not include this effect because we are ore interested in the validation of computational tools for the analysis of the UV-C interaction with pathogens immersed in optical edia [13]. In this section, once the geometry is fixed, we will use an illumination engineering package, TracePro, to evaluate the elative local fluence that will be used to model the global surviving ratio of the sample. .1. The sphere From a geometrical optics point of view (see Fig. 1a), we can see the sphere as the composition of two diopters having a radius 1 = −𝑟2 = 𝑟, that interface between the surrounding medium 𝑛1 = 𝑛′2 = 𝑛𝑎, that is assumed to be air (𝑛𝑎 = 1), and the inside medium, ′ 1 = 𝑛2 = 𝑛𝑤, that we consider as water (𝑛𝑤 = 1.35 at 𝜆 = 254 nm). After some algebra using the paraxial approach, we obtain the ocal length of the water sphere as 𝑓 ′ sphere = 𝑟 𝑛𝑤 2 ( 𝑛𝑤 − 1 ) , (2) meaning that light will focus behind the sphere at a distance 𝑠′2,sphere = 𝑓 ′ sphere − 𝑟 from the back vertex of the sphere. The large increase in flux associated with the focal point happens outside the sphere and has no influence on the inactivation process. On the other hand, part of the light entering the sphere is reflected from the back surface of the sphere, that acts as a mirror. Now, the calculation involves a refraction to enter the sphere plus a reflection. Therefore, an inner focal point appears at 𝑠′2,sphere,mirror = −𝑟 2 − 𝑛𝑤 3 − 𝑛𝑤 . (3) This effect generates a focusing point within the sphere that could be of help with considering the local inactivation factor. Actually, the reflected light propagates towards the input surface #1, and generate an additional focal point in reflection that occurs outside the sphere. This paraxial calculation already shows large inhomogeneities of the irradiance within the drops. Successive reflections can be considered that contribute less and less to the local irradiance within the sphere. In this contribution, we have evaluated these reflections using TracePro (TracePro is a product of Lambda Research Corporation, MA, USA). Fig. 2 shows the irradiance maps obtained from TracePro when using a collimated source. These maps already reveal how a non- negligible portion of the sphere only received a faint contribution from the successive and weaker reflections within the sphere. For the spheric geometry, the caustic defines a region that is not illuminated by the incident light refracting at the input surface of the sphere. Analytically, this caustic can be obtained by evaluating the height of the ray at the output surface, ℎ′, as a function of the height at the input surface, ℎ, that is: ℎ′ = 𝑟 sin [ 2 sin−1 ( ℎ ) − sin−1 (ℎ) ] . (4) 2 𝑛𝑤𝑟 𝑟 Optik 250 (2022) 168269J. Quintana et al. Fig. 1. Geometrical arrangement of the incidence on the sphere (left) and the droplet (right). Fig. 2. Relative irradiance maps at a meridional plane for the collimated incidence on the sphere (a), and the droplets on stainless steel (b), and BK7 glass (c). The maximum relative irradiance reaches large values in the spherical case because its focusing properties. For the case treated here, ℎ′ reaches a maximum of ℎ′ = 034𝑟 when ℎ = 0.85𝑟. The region defined by the caustic defines a geometrical hide-out for the pathogen that helps to explain the tail of the inactivation curve (see Section 3). When including the geometry of the sphere in a computational ray-tracing package (TracePro), we can obtain the distribution of the optical flux within the sphere for specific illumination conditions. In our case, we have chosen a collimated source generating parallel rays impinging on the spherical sample. The results are shown as a dimensionless relative irradiance distribution, 𝑒(𝑥, 𝑦, 𝑧) in Fig. 2a. 2.2. The spherical cap The geometrical analysis of the droplet having a spherical cap shape is quite simple (see Fig. 1b). Now, the medium is the culture solution where the pathogen is living. Although important from the radiometric point of view, the presence of absorption does not affect the location of the focal points of the system. From a geometrical optics point of view, this case corresponds with a plane convex lens made of culture media. As it happens with the sphere, the flat interface between culture medium and substrate reflects light towards the droplet, increasing the total flux that propagates through the sample. Moreover, the reflectivity of the substrate may be relevant when calculating the local fluence. The optical parameters of the sample can be calculated through the paraxial approach to obtain a focal distance 𝑓 ′ droplet = 𝑟 ⋅ 𝑛𝑤 𝑛𝑤 − 1 , (5) that is located at a distance 𝑠′2,droplet = 𝑓 ′ droplet −ℎ from the flat surface of the droplet. To complete the paraxial analysis of the sample, we have calculated the location of the focal point obtained after light reflects at the flat surface: 𝑓 ′ droplet,mirror = −𝑠′2,droplet . (6) Again, the algebraic analysis of successive reflections does not contribute significantly to the explanation of the inactivation process. Moreover, these effects are included in the computational evaluation of the relative irradiance, 𝑒(𝑥, 𝑦, 𝑧) (see Fig. 2b and c). 3 Optik 250 (2022) 168269J. Quintana et al. Fig. 3. (a) Plot of log10 𝜂 vs. fluence for the three cases considered in this paper. (b) Map of the local inactivation level (log scale) across a meridional plane of the sphere for an input fluence of 𝐹D99 = 21.6 J/m2. 3. UV inactivation tails and limits In Section 1 we described the inactivation process as an exponential decay of 𝜂 in terms of the fluence arriving to the sample (see Eq. (1). In the previous section we have plotted the relative distribution of irradiance within the sample. This dimensionless distribution can be transformed into irradiance just by fixing the input irradiance, 𝐹0. Then, after multiplying it by the exposure time, we obtain the fluence value. Therefore, the maps in Fig. 2 can be also interpreted as a relative fluence distributions. The calculation of the total inactivation requires the average of the local inactivation map within the sample 𝜂 = 1 𝑉 ∫𝑉 𝜂(𝑥, 𝑦, 𝑧)𝑑𝑉 , (7) where 𝑉 is volume of the sample, and 𝜂(𝑥, 𝑦, 𝑧) = exp(−𝐹 (𝑥, 𝑦, 𝑧)∕𝐹𝑖), where the local fluence is defined as 𝐹 (𝑥, 𝑦, 𝑧) = 𝐹0𝑒(𝑥, 𝑦𝑧) = 𝐸0𝑡exp𝑒(𝑥, 𝑦, 𝑧), (8) where 𝐸0 is the irradiance reaching the sample, and 𝑒(𝑥, 𝑦, 𝑧) is the dimensionless relative irradiance distribution. The value of 𝐹0 depends on the irradiance arriving to the sample, 𝐸0, and the time exposure, 𝑡exp, as 𝐹0 = 𝐸0𝑡exp In Fig. 3a we have plotted the inactivation (in log10 scale) of the pathogen using a value for the characteristic fluence 𝐹𝑖 = 4.7 J/m2 that is valid for the SARS-CoV-2 at 𝜆 = 254 nm. This value has been obtained after exposing the virus immersed in an absorbing culture medium (DMED) to UV-C irradiation produced by low pressure Hg lamps [13]. The case of the sphere shows a quite significant feature where the inactivation rate drops very sharply for low fluences and generates a quite high inactivation tail above -2, meaning that for collimated illumination conditions the inactivation level of 99% would require a large amount of fluence (large irradiance or large exposure time). In a practical case, the collimated illuminated conditions are not fulfilled and the UV-C sources are arranged as extended sources. Moreover, some solutions used in UV-C disinfection strategies include reflecting surfaces. Therefore, the collimated source should be taken as the worst case scenario. For the droplet geometry, in Fig. 3 we can distinguish the inactivation evolution for the dielectric and metallic substrates. It is clear that the metallic substrate is able to redirect the energy again towards the sample and generates a larger inactivation than the dielectric case. The behavior of the spherical geometry is illustrated in Fig. 3b where we have plotted the local inactivation for a value of fluence of 𝐹𝐷99 = 21.6 J/cm2 that would correspond to an inactivation level of 99% (−2 in Log scale) if the naked SARS-CoV-2 were directly irradiated by UV-C at 𝜆 = 254 nm. This fluence level is also represented in Fig. 3a as a vertical line We can see that there is a region around the inner focal point where the local inactivation is larger than 99.9% (−3 in Log scale). However, there is a quite significant volume where the inactivation is lower than 50%. After some calculations, for the case presented in Fig. 3b, this volume portion with an irradiance lower than 𝐹𝐷50 is around 7.13%. This portion can be seen as a geometrical reservoir for the pathogen [13]. 4. Conclusions The capabilities of the UV-C irradiation techniques has been positively proved for the inactivation of pathogens in a wide variety of circumstances. Previous studies have determined the characteristic fluence for inactivation of the SARS-CoV-2 at 𝜆 = 254 nm. The use of geometrical optics principles in the paraxial approach helps to understand some basic mechanisms driving the propagation of light through materials and geometries that appear in the airborne propagation of pathogens, and in the biochemical techniques that measure the capabilities of UV-C disinfection strategies. The geometrical model is also present in the computational evaluation of the propagation of radiant energy. This modeling defines a local fluence distribution that helps to determine the inactivation curve as a measurement of the surviving ratio vs. the energy impinging on the medium. 4 Optik 250 (2022) 168269J. Quintana et al. n c e s t l t t p m h a m A g R We have shown some geometric reservoirs of pathogen where the irradiation does not reach a significant level, meaning that a on-negligible fraction of microorganisms remain under-exposed. This fact explain the tail of the inactivation curve. This situation an be considered as the worst-case-scenario that can be improved by an appropriate use of non-collimated light coming from xtended sources, or by the use of reflective surfaces that modifies the direction of incidence of the UV-C radiation. This is specially ignificant for spherical geometries. Also, the behavior of droplets deposited on a plane substrate depends on the characteristics of he substrate. We have found that a metallic surface inactivates more pathogens than a dielectric substrate. This is caused by the arger reflection coefficient of the metallic material in comparison with the dielectric one. As a summary, we have shown how geometrical optics and illumination engineering packages can be of use when determining he overall inactivation of pathogens. We have revealed the presence of geometrical reservoirs that could jeopardize the use of this echnique for collimated sources. The existence of tails in the inactivation curve is merely explained by the geometry of the light ropagating through the media. Moreover, we have evaluated the effect of the optical properties of the substrate, and the culture edium, that is used in biomedical experiments involving droplets deposited on plane substrates. Therefore, this contribution may elp researchers to understand the limits of the UV-C pathogen’s inactivation methods, and propose a way to surpass those limits nd include the propagation of light within the analysis of the inactivation data. Although applied to the case of SARS-CoV-2, this ethodology can be extended for any other case of pathogen, geometry or culture medium. cknowledgments This work has been partially supported by Project COV20-01244-CM funded by the Comunidad de Madrid, and by a research rant (#171-2020) between Luminalia Ingeniería 𝑦 Fabricación and the University Complutense of Madrid. eferences [1] M. Weiss, M.C. 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