A mathematical model for evaluation of power characteristics of the optical link of the fluorimeter is developed. The main objective of fluorimeter is measurement of intensity of fluorescent radiation of human skin in vivo. The model is realized in a packet of computer mathematics Mathcad and consists of the units modeling energetic characteristics of passive optical elements, radiators and photodetectors by analytic functions - laws of photometry. For creation of models elements reference, literary and experimental data on them are used. Basic purpose of model - operational quantitative comparing of constructive solutions for instrument by energetic criterion - photodiode`s output signal. The given obviously mathematical functions provide openness of model and accessibility for modification by the user.
I f (λλ)d AU 420 600 420 300 where If() – a range of intensity of fluorescent radiation of skin in the range of lengths of waves (420 – 600) nanometers; Ibs() – a range of intensity is elastic the radiation of excitation of fluorescence reflected by skin in the range of lengths of waves (300 – 420) nanometers. The experimental ranges of If() and Ibs() register the spectrometer which is an AGE Reader part.
Due to the lack of the AGE Reader equipment available to most medical institutions in Russia similar operations are carried out only at the research level [4-6, 9]. Researches are conducted on original universal spectrofluorometers [10-14] which operation assumes an involvement in it a highly qualified staff, there was no consensus also by diagnostic criterion [9].
In the Samara university with the assistance of authors of the real operation the diagnostic fluorimeter capable to solve the
problem of measurement of AFR caused by AGE and the implementing integral diagnostic criterion (1) is created. From
original, universal, research spectrofluorometers it shall differ in compactness, simplicity of construction and operation at the
expense of optimized under the decision of the task set above by optical, electronic and algorithmic structures, and from
instruments of the AGE Reader family - the budgetary element basis and easy replicability. The on-stage performance group
managed to create two prototypes of the fluorimeter [
As development is in a stage of optimization of designer decisions now, creation of simple mathematical models of the principal structural components of the diagnostic fluorometer is necessary for operational comparing of possible modifications of optical and optical-electronic elements and their relative positioning.
The optical circuit of the diagnostic fluorometer [16] is provided in fig. 1. Excitation of fluorescence of AGE which are
contained in skin – a research object 5, is carried out by the radiation of a short-range ultra-violet or violet LED or the junction
laser 1 which passes through the collimating optics 2 and the clearing light filter 3. The range of radiation of excitation of AFR
of 350 - 415 nanometers is caused by the AGE fluorescent properties [
4 5
Let us demonstrate the structure and operation of the energy characteristic model on the problem of optimizing the b and h design parameters. We propose using a laser module that includes the SLD3233VF semiconductor laser [21] and a built-in adjustable collimator as a radiation emitter in the design under development. Its peak emission wavelength is 405 nm, and the peak emission power in a continuous mode is equal to 65 mW. Due to the adjustable collimator, the emission power density in the examined skin area and the effective characteristic beam size can be varied within an order of magnitude. Therefore, the radiation emitter exciting SAF (skin autofluorescence) in the present paper is modeled by distributing the E (x, y) power densityirradiance-over the object of investigation.
We expect to use as the emission detector the BPW21R silicon photodiode with an integrated light filter that shifts the peak of its spectral sensitivity to a wavelength of 560 nm, which matches better with the AGE (advanced glycation endproducts) fluorescence spectrum. Since the photodiode manufacturers provide its S() sensitivity indicatrix [22], it is logical to consider a model of a point-contact photodetector the photodiode model to analyze the fluorimeter energy. The point-contact photodetector model features a defined direction of the sensitivity indicatrix axis, along which the sensitivity reaches its peak Smax = 1value and the angle, which determines the direction to the radiation emitter, is measured from it as well. The analysis of the angular dependence of S(), given in graphical form in [22], showed that it is not differentiated from cosine: S ( ) cos , (2)
The geometry of an optical system model is shown in Fig. 3. The object of investigation is considered flat and located in the xOy plane. The skin test area has the shape of a square with side. The origin of the reference system coincides with the geometric center of the examined area. The photodiode is located at a P point belonging to the yOz plane and is at a b distance from the z axis that is equal to the length of PH or OB segments and at a h height from the xOy plane equal to the length of PB or HO segments. The axis of the sensitivity indicatrix is directed vertically down along the PB segment. xOy x уОz z Н
О A a/2 С
D b r
Θ a/2 Р
h В y
Fig. 3. Geometry of mathematical model of power characteristics.
The element of the dxdy scattering surface with the center at the C point, the boundary point of the CP segment that connects the surface element and the photodiode serves as an elementary radiation emitter for the photodiode. The A and D points are projections of the C point onto the corresponding axes of reference. The CB segment is a projection of the CP segment onto the xOy plane.
Since the surface of an examined object – skin - can be assumed to be Lambertian [23] in the first approximation with the p reflectance at the angles of radiation incidence and radiation scattering up to 70o, its L brightness will not depend on the scattering angle, and a brightness value of the surface element will be determined by its irradiance [24]: L(x, y) E(x, y) , (3)
I 0 (x, y) L(x, y)dxdy ( )E(x, y)dxdy , and the radiation intensity itself will vary according to the cosine law: I ( x, y, ) I 0 ( x, y) cos ( )E( x, y) cos dxdy , where is the angle between the CP segment and the normal to the xOy plane. Denoting the length of the CP segment by r, we obtain the irradiance created by the dxdy element at the location of the P photodiode: E p (x, y, ) I (x, y, ) /[r(x, y)]2 ,
Denoting the irradiance transfer ratio into an electrical signal at the k output of the photodiode, and taking into consideration its S() sensitivity indicatrix, we will have the following dependence of the U output signal on the design parameters: a / 2 a / 2 U (k ) {E( x, y) /[r( x, y)]2}cos cos dxdy , (7) a / 2 a / 2 Since the axis of the sensitivity indicatrix is perpendicular to the xOy plane, then we have = and from СРВ: cos РВ РС h / r .
r(x, y)
CB2 PB2
DB2 CD2 PB2
x2 (b y)2 h2 , a / 2 a / 2 that is convenient for calculations using the Mathcad software package. which substituted in (8) and (7), results in the expression for the U output signal of the photodiode: a / 2 a / 2 U (k ) {E( x, y) h 2 /[ x 2 (b y) 2 h 2 ]2 }dxdy
Let us demonstrate the model optimization potential by solving the following problems: 1) using a wide or narrow beam to
excite SAF, i.e. to select the window a size; 2) the influence of the form of a spatial distribution of the E (x, y) power density
over an object per an output signal magnitude; 3) how sharply the output signal varies depending on a distance change between
the 2b photodiodes and 4) the h height of their location above an investigated object. Since the optimization refers to a particular
design shown in Fig. 2 with the selected type of BPW21R photodiode whose case diameter [22] is 9 mm, we have b 4 mm. It
is not feasible to reduce the h height to values less than 10 mm without complicating the optical system by using beam splitters,
mirrors, etc. as well, i.e. we have h 10 mm. Clinically tested devices [
Mathematical Modeling / V.N. Grishanov, V.S. Kulikov, K.V. Cherepanov
The optimization goal is to obtain a set of a, b and h geometric parameters with the design constraints discussed above that do not significantly reduce the U output signal, all other things being equal. Then the value of the constant factor before the (k/) integral (10) is taken equal to 1000 in order to obtain single-valued integers along the ordinate axis, the value of the Р и emission power of the SAF exciting source is assumed equal to 1, the U output signal is measured in nominal units and the normalization condition is used: a / 2 a / 2
E ( x, y) dxdy 1. (11) a / 2 a / 2
The uniform distribution of power density is the simplest one with simulation results easily verified physically: E(x, y) Е0 Ри / а2 Const (12) Fig. 4. Results of modeling (parameter in a frame has identical value for all curves of the schedule): a) - influence of the cross size of a bunch on the size of an output signal; b) – dependence of an output signal from photo diode arrangement height over a research object; c) – influence of distance of the photo diode from an optical axis of the probing bunch; d) – influence of the law of distribution of density of power of radiation of excitement of AFK on an object:: uniform distribution; - Gaussian distribution.
The simulation results are shown in Fig. 4. It follows from Fig. 4a that, in terms of the device energy, the formation of a small aperture beam does not provide any tangible advantage. The dependence on the height of the photodiode above the object of investigation appears to be more significant (Fig. 4b). It can be seen that this distance should be minimized; on the other hand, small ~ 1 mm height variations caused, for example, by the need to replace a light filter with a light filter of a different thickness or errors in manufacturing optical element holders should not significantly affect the magnitude of an output signal. The last remark is related to the variation of the b design parameter (Fig. 4c).
Fig. 4d illustrates the effect of the law of the Е(х, у) power density distribution over the object. In addition to the uniform distribution, a Gaussian distribution is introduced into the model, as it is characteristic of laser radiation emitters: E(x, y) М 0 exp[2 (x2 y 2 ) w2 ] , (13) where М0 is the power density on the Gaussian beam axis ; w is the radius (the distribution parameter) of the beam. The normalization to the full power of the SAF exciting source is carried out with the help of the expression: М 0 2Ри / w2 . (14)
The comparison in Fig. 4d was carried out for the following values of the parameters: a = 5 mm and w = 1.25 mm. The choice of the value of a Gaussian beam radius was determined by the condition that the total radiation power of the emitter should almost completely fall on the object without masking it with an output window with a characteristic size of 5 mm. The condition should be taken into consideration since it is known [25] that only 86.5% of the total power passes through the cross section of a Gaussian beam of 2 w diameter and to increase the total power to 99.99%, the 4 w cross section diameter is required. As could be expected from physical considerations, according to the results of running the model with a uniform distribution of the power density of different cross sections (Figure 4a) the output signal is insensitive to the law of power density distribution, at least in the category of radially symmetric distributions. The (Ugaussian – Uuniform)/Ugaussian relative difference of values does not exceed 2.5%, which confirms the correctness of the mathematical apparatus used.
A mathematical model has been developed that enables to predict the energy characteristics of a device by automating a calculating component of designing according to the manufacturer's specifications, literature or experimental data on the parameters of optoelectronic system components of the designed diagnostic fluorimeter and the optical properties of a diagnosed object.
Simulation showed that the most significant contribution to the device energy is made by the distance between the surface of a diagnosed object and the photosensitive pad of the photodetector. To obtain a maximum output signal of the photodetector, it is required to minimize the distance within the range of permissible design constraints. A small effect of the size of a diagnosed area on the output signal provides an additional degree of freedom for medical applications.
The model is implemented in the Mathcad software package to which mathematical models of the optical system components described above, the number of components and their parameters and expressions that the spectrum transfer over propagating radiation through an optical component obeys are introduced. The explicitly defined mathematical functions ensure the openness of the model and the feasibility of its modifying by a user.