Basic 3-dimensional UV Exposure Model

This page is part of the UV Exposure Model (UVEM) documentation.
It contains a documentation about the basic parameters of the exposure model.
For an overview of all UVEM-related pages, see the UV Exposure Model main page.

Seckmeyer et al. (2013) defined the biologically weighted exposure Ex_weighted [J] as the radiant energy received by exposed body surfaces of a human who stands on a horizontal plane.

Ex_weighted can be calculated by integrating the spectral radiance weighted with a biological action spectrum and the geometry of a human over all azimuth and zenith angles of the upper hemisphere and the exposure time. The complete equation of Ex_weighted, considering all dependencies, is complex and the calculation can be quite demanding. Therefore, multiple simplifying assumptions have been made, which are described in detail in Seckmeyer et al. 2013. The simpliffed equation of Ex_weighted, used in this model, is given as:

where L is the radiance, S the biological action spectrum and A_proj the human geometry. The solid angle dΩ represents the different directions in the sky and depends on the azimuth angle and the zenith angle.

Definition of Radiance and Irradiance

The spectral radiance L is defined as:

which depends on the wavelength λ, the time t and the incident angle which is defined as the angle between incident light beam and horizontal plane. In addition, dQ represents the radiant energy, dA the area element and dΩ the solid angle. For a receiver that is not orientated normal to the source, the area element dA must be weighted with the cosine of the angle between the direction of the beam and the normal to the area dA (WMO, 2008; CIE, 2011). The radiance can be a source or a receiver based quantity. However, in this model the radiance is used as a receiver based quantity only. This can be best visualized by a reversed cone with the given solid angle dΩ as its base and the vertex on the area dA. See schematic diagram in Figure 1. In comparison the spectral irradiance E_lambda is defined as the radiant energy dQ arriving per time interval dt, per wavelength λ and per area dA from any origin incident onto a horizontally oriented area element (Seckmeyer et al., 2010) The quantity irradiance is sometimes also referred to as radiative flux.

Figure 1: Schematic diagram of the quantities radiance (a) and (b) and irradiance (c). In (a) the receiving area dA is oriented normal to the source, while in (b) the angle between the normal of the area and the incident beam is 45°. The diagram (c) visualizes the irradiance, where radiation of any origin is received by the area element dA.

Radiation Input for Exposure Model To calculate the biologically-weighted UV exposure the spectral radiance L is weighted with a biological action spectrum. In this model, the action spectra for erythema, defined by the CIE (1998),and the action spectra for the vitamin D3 synthesis is used. In Figure 2, the simulated (diffuse) sky radiance weighted with the vitamin D3 action spectrum is shown for Hannover on 21 March at solar noon. The radiance in Figure 2 is visualized as a polar plot, where the zenith is located in the center and the azimuth angles are marked around the plot. It should be noted, that similar to an astronomical map, the directions of east and west are inverted. Distribution of vitamin D3-weighted sky radiance in Hannover on 21 March at solar noon. The position of the sun is marked with a black asterisk. Please note that the high values around the sun position are caused by the high diffuse (scattered) radiation.

Human Geometry For the calculation of the exposure, the human geometry is taken into account. This is done by using projection areas of all uncovered surface areas of the human. For the calculation of the projection areas a, 3D voxel (volumetric pixel) model, segmented from data of a whole-body computed tomography scan of a patient, is used. The person was 38 years old, 176 cm of height and had a weight of 68.9 kg, thus approximately representing an average male adult (Valentin, 2002). In the Figure 3 (on the right), a two-dimensional projection of the voxel model is shown for three different viewpoints. Additionally, the projection areas of a human with no clothing, summer clothing and winter clothing are visualized as function of the azimuth and zenith angle in form of polar plots. Figure 3: (a) Projection of the 3D voxel model with winter clothing, visualized for incident angles 30°, 60° and 85°, with the front turned by 30° in azimuth direction. Only hands and face are exposed to UV radiation, which is shown in light gray color and clothing shown in dark gray. (b)-(d) projection areas of the 3D voxel model oriented towards 180° (south), as a function of incident and azimuth angles. The minimal projection area in each plot is located in the middle of each picture, representing a view from the zenith at an incident angle of 90°. (b) Projection area of a human with winter clothing. The three asterisks mark the projections shown in (a). (c) Projection area of a human without any clothing, which results in nearly identical projection areas between the front and back of the human. (d) Projection area of a human with summer clothing, where face, hands, neck and arms are exposed.

Calculation of exposure

To calculate the human exposure the weighted radiances must be integrated over all directions of the upper hemisphere and the exposure period t. In order to receive the total human exposure, the biologically-weighted direct normal irradiance multiplied with the projection area of the corresponding sun position is added to the diffuse exposure Ex_weighted. In order to convert the calculated human exposure from Joule into IU, a conversion factor of 70.97 [IU J^-1] (i.e. IU per Joule of vitamin D₃-weighted UV) is used (Seckmeyer et al. 2013)

Exposure Model Assumptions

For the calculation of the biologically-weighted human exposure using the exposure model described in Seckmeyer et al. 2013 and Schrempf et al. 2017a & b, various assumptions were made. In the following some general assumptions, adopted from these publications, are listed.

• The exposure from solar radiation of a human in vertical posture experiencing a UV Index of 10 is equivalent to 1000 IU per minute.
• 1000 IU per day are sufficient for a healthy vitamin D status.
• The conversion factor is applicable for all solar zenith angles and ozone values.
• The conversion factor is derived for a skin type II person.
• Different parts of the human skin have the same spectral transmission and the same sensitivity towards the incoming energy.
• The accumulated vitamin D increases linearly with exposure (a linear dose effect relationship is assumed).
• In the case of a clothed person, the model wears skintight clothes with no transmission of radiation.
• UV radiation reflected from the ground can be neglected due to low albedo in this wavelength region.
• The reflectivity of the surface materials of the detected obstructions is low in the UV wavelength region and can be neglected

References:

• Seckmeyer, G., Schrempf, M.Wieczorek, A., Riechelmann, S., Graw, K., Seckmeyer, S., and Zankl, M. 2013. A Novel Method to Calculate Solar UV Exposure Relevant to Vitamin D Production in Humans, Photochem. Photobiol., 89(4), 974-983, DOI: 10.1111/php.12074.
• Schrempf, M., Thuns, N., Lange, K., and Seckmeyer, G. 2017a. Einuss der Verschattung auf die Vitamin-D-gewichtete UV-Exposition eines Menschen, Aktuelle Derm, DOI: 10.1055/s-0043-105258.
• Schrempf, M., Thuns, N., Lange, K., and Seckmeyer, G. 2017b. Impact of Orientation on the Vitamin D Weighted Exposure of a Human in an Urban Environment, Int. J. Environ. Res. Public Health, 14(8), 920, DOI: 10.3390/ijerph14080920.