rus
Issue #3/2020
M. M. Kugeiko, S. A. Lisenko
Opto-Physical Measurements in Conditions of a Priori Uncertainty: Theoretical Aspects
Opto-Physical Measurements in Conditions of a Priori Uncertainty: Theoretical Aspects
DOI: 10.22184/1993-7296.FRos.2020.14.3.270.280
A new look at the theory of optical and physical measurements in the framework of the concept of “a priori” is presented. Its use for assessing the state of biological tissues and determining the effectiveness of photodynamic therapy is considered. It is shown how to quickly evaluate the parameters of the studied object in the absence of a priori information about the studied object and the impossibility of carrying out calibration procedures. The basis of the solutions is the combination of methods of regression analysis and approximation of the functional relationships of the recorded signals with the optical characteristics of the media.
A new look at the theory of optical and physical measurements in the framework of the concept of “a priori” is presented. Its use for assessing the state of biological tissues and determining the effectiveness of photodynamic therapy is considered. It is shown how to quickly evaluate the parameters of the studied object in the absence of a priori information about the studied object and the impossibility of carrying out calibration procedures. The basis of the solutions is the combination of methods of regression analysis and approximation of the functional relationships of the recorded signals with the optical characteristics of the media.
Теги: approximation relations. biophysical parameters inverse problem optical and physical measurements optical parameters regression relationships scattering media the concept of “a priori” аппроксимационные соотношения биофизические параметры концепция «безаприорности» обратная задача оптико-физические измерения оптические параметры рассеивающие среды регрессионные связи
Opto–Physical Measurements in Conditions of a Priori Uncertainty: Theoretical Aspects
M. M. Kugeiko, S. A. Lisenko
Belorussian State University, Minsk, Belarus
A new look at the theory of optical and physical measurements in the framework of the concept of “a priori” is presented. Its use for assessing the state of biological tissues and determining the effectiveness of photodynamic therapy is considered. It is shown how to quickly evaluate the parameters of the studied object in the absence of a priori information about the studied object and the impossibility of carrying out calibration procedures. The basis of the solutions is the combination of methods of regression analysis and approximation of the functional relationships of the recorded signals with the optical characteristics of the media.
Key words: scattering media, optical and physical measurements, optical parameters, biophysical parameters, inverse problem, the concept of “a priori”, regression relationships, approximation relations.
Received on: 07.02.2020
Accepted on: 15.04.2020
INTRODUCTION
Most optical and physical measurements belong to the indirect class, which consist in determining the desired value of a physical quantity based on the results of direct measurements of other physical quantities that are functionally related to the sought quantity. In this case, the task of processing information in optical and physical measurements implies not only the establishment of functional relationships between the information received and the determined characteristic, but also the interpretation of the obtained indirect information about the object being studied (i. e., solving the inverse problem).
In almost all cases, the task of interpreting the obtained indirect information is a multi-parameter and often incorrect inverse problem. To solve such problems, the use of a priori information about the object of study is required. Usually (especially for location systems) such information is unknown, and obtaining it requires additional measurements. For remote sensing of the atmosphere, non-invasive diagnosis of biological objects, this is difficult to implement. Inaccuracy here is also manifested in the extremely strong dependence of the solution on the measurement error [1–4]. Therefore, the problem of carrying out calibration measurements with high accuracy requires its solution. For many practical cases, such a task is more complicated than creating the measuring system itself.
The amount of required a priori information is significantly reduced when using regularizing algorithms [1, 5]. At the same time, this approach requires a certain selection of regularization parameters. The accuracy of the restoration of the determined parameters depends on them. It should be noted that regularization methods lose their stability with inaccurate selection of regularization parameters and with an increase in the error of the measured values [2, 5].
Examples of modern diagnostic tools are known that do not have metrological support today. The need to use a priori information and assumptions about the object under study did not allow metrologically certifying laser radar systems in the global environmental pollution monitoring networks being created (global, European, CIS, and Belarus). The reason is ignorance of the state of the atmosphere due to its instability. We observe the same picture in the situation with instrumental systems of non-invasive optical diagnostics of biophysical parameters of biological objects.
It was decided to implement the solution of the problem of quantitative assessment of the determined parameters of objects in two stages. At the first stage, the optical parameters of the object are determined from the obtained measurement information. The standard move is a comparison of experimental and calculated data. The calculated data are obtained in the framework of the model of light transfer and interaction [3, 4, 6, 7]. To ensure the required accuracy, we use the methods of radiation transfer theory. Methods use various approximations about the dominant interaction process; it requires large computational costs. Therefore, the use of such methods excludes the possibility of interpreting experimental data in real time. Such a statement relates, for example, to the use of the most accurate Monte Carlo (MC) method [8].
We propose using the concept of “apriority-free” in solving problems of remote sensing using methods and systems based on optical-physical measurements. The concept of “a priori” is based on several principles:
the maximum possible exclusion of a priori information or assumptions about the studied object;
the maximum exclusion of the influence of hardware parameters, environmental parameters on the measurement results when solving the problem of calibration and metrological support;
receiving information in real time.
Let us turn to the substantiation of the theoretical aspects of the concept of “a priori” for the use of optical-physical measurements1 under conditions of uncertainty.
THEORETICAL ASPECTS OF OPTICAL AND PHYSICAL MEASUREMENTS IN CONDITIONS OF INFORMATION UNCERTAINTY
Earlier, we developed a regression approach to solving inverse problems of optical sensing of biological media. It consists in the fact that linearly independent components are extracted from the recorded optical signals. The procedure is based on extracting the projections of the signals onto the space from the eigenvectors of their covariance matrix. Further, the desired medium parameters are found by establishing their stable regression relationships with linearly independent signal components.
The use of linearly independent values in regressions corresponds to the extraction of the “useful signal” from the initial (inverted) data and the rejection of “noise”. Such a procedure allows us to construct solutions to inverse problems that are resistant to random “disturbances” (fluctuations) of these data. To search for linearly independent quantities, we construct the averaged dependences by statistical modeling of optical signals with maximum variability of the medium parameters. We accept only those parameters of the medium that affect the process of radiation transfer in it. Next, we construct eigenvectors and obtain regression solutions of inverse problems. Subsequently, this allows one to perform operational processing of the measured optical signals without solving the radiation transfer equation in the medium under study, and to determine a priori information as in regularization methods for the inverse problem [6, 7].
To decompose the inverted data into linearly independent components, the basis of the covariance matrix is used. The optimal dimension of the basis is determined based on closed numerical experiments to restore the parameters of the medium from the characteristics of its light scattering. To do this, initially on the basis of a simulated ensemble of realizations of the measurement vector , a “test” ensemble is formed. In it, each realization is obtained by superimposing on the components of the initial vector random deviations within the measurement error δ. Next, all implementations of the “test” ensemble are selected. For each of them, a solution to the inverse problem is found by the regression formulas between the model parameters and linearly independent quantities composed of the components of .
As a result, the obtained values of the model parameters * are compared with their actual values and the errors of their recovery are calculated. In accordance with a given measurement error, we select the number of linearly independent components of the reversed data and the light scattering characteristics of the medium. We stop the selection on those characteristics by which it is possible to determine specific environmental parameters under the conditions of a priori uncertainty of all others most accurately. Such a solution also makes it possible to evaluate the information content of the measured data, get an idea of the theoretically achievable accuracy of restoring the environmental parameters from them, and investigate the influence of the quantity and accuracy of optical measurements on the accuracy of solving the inverse problem.
In particular, in [6, 7, 9, 10], solutions to the inverse problems of spectroscopy of biological tissues were presented. The goal was to determine the optical and structural-morphological parameters of tissues. The optical parameters included the absorption coefficient, transport scattering coefficient, and the anisotropy factor of the scattering indicatrix. The structural and morphological parameters are the concentrations of melanin, total hemoglobin and bilirubin in the tissue, the degree of blood oxygenation, the average diameter of blood vessels, the concentration and size of effective scatterers. The solutions were based on the use of optical models of the skin and mucous membranes of a person, as well as the Monte Carlo (MC) method, and on this basis, obtaining stable regression solutions to inverse problems of spectroscopy of biological tissues with spatial resolution.
The technical implementation is proposed and the efficiency of determining the optical and structural-morphological parameters of tissues from the measured signals is evaluated. The backscattering signals of tissues were recorded using fiber-optic technology with spatial diversity of the channels of sending and receiving radiation. The spectral region of strong light absorption by chromophores of biological tissue is in the range of λ < 600 nm. The functional diagram of the measurements of backscattering signals of the mucous membranes is shown in Fig. 1. Radiation is injected into the medium through a fiber with a diameter of 0.2 mm, and the backscattered radiation is collected by receiving fibers located at distances L = 0.23; 0.46; 0.69; 0.92; 1.15 mm from the center of the light-transmitting fiber (diameter of the receiving fibers 0.2 mm).
Such a geometry of measurements provided an optical signal value acceptable for practice (not less than 10–4 of the power of the probing light beam) with a maximum distance between the illuminating and receiving fibers not exceeding the diameter of the instrument channel of the endoscope.
The errors in the restoration of tissue parameters caused by their statistical scatter and optical measurement errors are estimated. Combined processing of the spectral and spatial characteristics of backscattering tissue using the obtained regressions allows one to reduce the error in the restoration of optical parameters of tissue to ~3%, while their recovery from spectral parameters is possible only with an error of ~9%, and from backscattering spatial characteristics ~ 7% [7].
A solution to the inverse problem of reconstructing two-dimensional distributions of structural-morphological parameters (SMP) of human tissues from their multispectral images has been proposed [7]. To eliminate the influence of uneven illumination and the geometry of the tissue survey on the results of the restoration of its parametric maps, tissue images normalized to one of its spectral layers are used. The inverse problem is to restore the tissue SMP from the spectral values of each pixel of the normalized image. To solve it, regression relations are used, previously obtained on the basis of a representative sample of the diffuse reflection spectrum of the medium simulating the tissue under study. The stability of the regression operators for converting tissue images into parametric maps to measurement errors is ensured by using linearly independent image components for solving the inverse problem, obtained by projecting the spectral values of each pixel onto the eigenvectors of the covariance matrix of the diffuse reflection tissue spectrum.
On this basis, “calibration-free” measuring systems have been created that are resistant to changes in the hardware constants of receiving-emitting and recording units, environmental parameters, and optical element contamination. Using regression relationships, methods for the operative reconstruction of two-dimensional distributions of SMP of the skin, mucous membranes, and human fundus have been developed and patented (Patents BY10918 C1, 2008; BY19144 C1, 2015; RU2510506 C, 2014; RU2506567 C1, 2014; RU2511747 C2, 2014; BY18652 C1, 2014; RU2501522 C2, 2013; BY18653 C1, 2014; RU2517155 C1, 2014; RU2536217 C1, 2014; RU2539367 C1, 2014). The working wavelengths of the methods were selected that provide the greatest accuracy and stability for solving inverse problems under conditions of general variability of all tissue parameters that affect their diffuse reflection spectra.
At the same time, the regression method does not allow a subtle analysis of the light scattering characteristics of the tissue, for example, to reveal in them the features caused by small variations in the component composition of the blood. In particular, this refers to the content of dyshemoglobins in the blood (carboxy-, met- and sulfhemoglobin), the absorption spectra of which substantially overlap with similar spectra of the main forms of hemoglobin (oxy- and deoxyhemoglobin). Small variations in the spectrum of tissue backscattering associated with variations in hemoglobin composition are largely averaged over the statistical material used to obtain the regressions. This reduces the sensitivity of the method to the concentration of each specific form of hemoglobin.
Under these conditions, a significantly higher accuracy in estimating the desired parameters can be achieved by measuring the backscattering spectrum of the tissue with high spectral resolution and then solving the inverse problem. The essence of which is to model the theoretical spectrum with respect to the experimental one by selecting model parameters. High spectral resolution of the initial data ensures the stability of the inverse problem to optical measurement errors and eliminates the ambiguity of its solution within the framework of the used biological tissue model. The difficulty here lies in an adequate theoretical calculation of the characteristics of tissue backscattering.
In our previous works, expressions were obtained for solving inverse problems in the optics of skin biotissues, mucous membranes, bulbar conjunctiva, and the human fundus [7, 11–14]. We especially note the developed approximation analogues of the MC method for fast calculations of the characteristics of radiation transfer in a homogeneous and two-layer multi-scattering media.
For example, the calculation of the diffuse reflection coefficient of a homogeneous, semi-infinite medium with an absorption coefficient k = 0.004–7.0 mm–1, a transport scattering coefficient β′ = 0,3–5,1 mm–1, and an average cosine of the scattering indicatrix g = 0,5–0,97 and the refractive index η = 1.35–1.45 based on the following approximation formula [7.11]:
where is the diffusion depth of light penetration into the medium, A, am, bm, cm and dm are numerical coefficients calculated in advance by numerically simulating the radiation transfer into environment. The maximum error in calculating R according to this formula for the indicated values of the optical parameters of the medium is 4%, and the average error is 0.9%. The analytical approximation allows us to simulate the diffuse reflection coefficient of most biological tissues and many media of non-biological origin in the spectral regions of strong and weak light absorption in them.
The model for a two-layer medium is based on the representation of diffuse reflection coefficient in the form of the product of effective light transmission by layers, taking into account its multiple scattering and re-reflection between them. The dependences of the effective transmission of each layer on the optical parameters of the medium for the cases of its collimated and diffuse illumination are described by polynomial functions with given numerical coefficients [13]. The proposed formulas approximate with high accuracy the results of numerical calculations of diffuse reflection coefficient (Fig. 2). The standard error of the calculation of the diffuse reflection coefficient of the medium in the range R = 0.01–0.60 is 0.65%.
For the correct solution of the problems of laser radiation dosimetry and a comprehensive analysis of the effectiveness of photodynamic therapy, knowledge of a whole complex of parameters is necessary. These include absolute concentrations of FS and molecular oxygen in the tissue; volume concentration of capillaries and their average diameter; the ratio of the concentrations of various forms of hemoglobin; the depth of penetration of light into the fabric; distribution of radiation density in tissue and light power absorbed by endogenous (natural) and exogenous (specially introduced) tissue chromophores [15]. Moreover, to select the optimal spectral and energy characteristics of laser radiation, all parameters characterizing the processes of radiation transfer in the medium and its interaction with the substance of the medium should be presented as a functional dependence on the wavelength of the probe radiation.
A quantitative measure of the effectiveness of photophysical and photochemical processes in an irradiated tissue associated with the absorption of light by its chromophore X can be the spectrum of the action of light on a given chromophore [7, 16]:
,
where CX and kX are the concentration and absorption coefficient of the chromophore, Φ (z, λ) is the radiation density spectrum at the tissue depth z (spatial illumination), W / m2.
The idea put by the authors as a basis for the noninvasive definition of the functions KX (z, λ) consists in the following. It is known that the light field inside the biological tissue and the radiation field scattered by it in the opposite direction are determined by the same tissue parameters. The characteristics of the backscattered radiation field are measured directly during the experiment. We establish optically significant tissue parameters by solving the corresponding inverse problem. Further, using the theory of radiation transfer, we calculate all the necessary spectral and spatial characteristics of the light field inside the tissue, as well as the deep distributions of the light action spectra on its chromophores. Thus, the parameters of photo exposure, providing the optimal therapeutic effect, can be selected according to the characteristics of radiation backscattered by the tissue. However, this requires fast and reliable methods for calculating the characteristics of light fields inside and outside the irradiated tissue [7].
The radiation field in the medium is described by the superposition of the incident collimated, incident diffuse and reflected diffuse flows. Luminous fluxes in neighboring elementary layers of the medium and on its surface are connected by simple matrix operators. We obtain them by combining the asymptotic formulas of the theory of radiation transfer with the single-scattering approximation. A comparison of the results of calculating light fluxes in a five-layer medium (modeling human skin) by the engineering method and the MC method shows that they are comparable in accuracy with a significant difference in computational costs [7].
Examples of determining the parameters of photo-exposure were demonstrated in [18, 19] when studying the spectra of the action of light on a photosensitizer and oxyhemoglobin during photodynamic therapy, as well as the rate of photoisomerization of bilirubin during phototherapy of jaundice in newborns.
Patents have been obtained for methods that allow photodynamic therapy of human oncological diseases (Patents RU2521838 C1, 2013; BY19557 C1, 2014; Eurasian Patent 031413 B1, 2018).
Developed and patented: a device for the diagnosis of malignant tumors in the human mucosa by determining the concentration of hemoglobin and the degree of blood oxygenation in it (Patent BY19558 C1, 2015); a device for determining the concentration of hemoglobin and the degree of blood oxygenation in the mucous membranes (Patent RU2528087 C1, 2014). This device can be successfully used during endoscopic examinations of the mucous membranes of the oral cavity, esophagus, organs of the gastrointestinal tract and lungs.
CONCLUSION
Visual inspection (examination) of human organs and tissues is the first-line method for diagnosing a number of diseases and is widely used in oncology, surgery, ophthalmology, and in a number of other areas of medicine. In modern medicine, such an examination is performed using specialized systems: dermatoscopes, endoscopes, fundus cameras, etc. However, a visual examination of the patient’s tissues is entirely based on the experience, qualifications and color perception of the doctor and does not provide an objective diagnosis of the disease.
Thus, the development of operational high-precision methods for the quantitative diagnosis of biological objects according to the spectral and spatial characteristics of their multiple scattering and their multispectral images is extremely relevant; methods of non-invasive control of light-induced processes in biological tissues during phototherapy. The basis for creating a new generation of optoelectronic systems using the minimum amount of a priori information, additional measurements, calibration procedures is the development of methods that meet the conditions of the concept of “a priori”.
REFERENCES
M. M. Kugeiko, S. A. Lisenko
Belorussian State University, Minsk, Belarus
A new look at the theory of optical and physical measurements in the framework of the concept of “a priori” is presented. Its use for assessing the state of biological tissues and determining the effectiveness of photodynamic therapy is considered. It is shown how to quickly evaluate the parameters of the studied object in the absence of a priori information about the studied object and the impossibility of carrying out calibration procedures. The basis of the solutions is the combination of methods of regression analysis and approximation of the functional relationships of the recorded signals with the optical characteristics of the media.
Key words: scattering media, optical and physical measurements, optical parameters, biophysical parameters, inverse problem, the concept of “a priori”, regression relationships, approximation relations.
Received on: 07.02.2020
Accepted on: 15.04.2020
INTRODUCTION
Most optical and physical measurements belong to the indirect class, which consist in determining the desired value of a physical quantity based on the results of direct measurements of other physical quantities that are functionally related to the sought quantity. In this case, the task of processing information in optical and physical measurements implies not only the establishment of functional relationships between the information received and the determined characteristic, but also the interpretation of the obtained indirect information about the object being studied (i. e., solving the inverse problem).
In almost all cases, the task of interpreting the obtained indirect information is a multi-parameter and often incorrect inverse problem. To solve such problems, the use of a priori information about the object of study is required. Usually (especially for location systems) such information is unknown, and obtaining it requires additional measurements. For remote sensing of the atmosphere, non-invasive diagnosis of biological objects, this is difficult to implement. Inaccuracy here is also manifested in the extremely strong dependence of the solution on the measurement error [1–4]. Therefore, the problem of carrying out calibration measurements with high accuracy requires its solution. For many practical cases, such a task is more complicated than creating the measuring system itself.
The amount of required a priori information is significantly reduced when using regularizing algorithms [1, 5]. At the same time, this approach requires a certain selection of regularization parameters. The accuracy of the restoration of the determined parameters depends on them. It should be noted that regularization methods lose their stability with inaccurate selection of regularization parameters and with an increase in the error of the measured values [2, 5].
Examples of modern diagnostic tools are known that do not have metrological support today. The need to use a priori information and assumptions about the object under study did not allow metrologically certifying laser radar systems in the global environmental pollution monitoring networks being created (global, European, CIS, and Belarus). The reason is ignorance of the state of the atmosphere due to its instability. We observe the same picture in the situation with instrumental systems of non-invasive optical diagnostics of biophysical parameters of biological objects.
It was decided to implement the solution of the problem of quantitative assessment of the determined parameters of objects in two stages. At the first stage, the optical parameters of the object are determined from the obtained measurement information. The standard move is a comparison of experimental and calculated data. The calculated data are obtained in the framework of the model of light transfer and interaction [3, 4, 6, 7]. To ensure the required accuracy, we use the methods of radiation transfer theory. Methods use various approximations about the dominant interaction process; it requires large computational costs. Therefore, the use of such methods excludes the possibility of interpreting experimental data in real time. Such a statement relates, for example, to the use of the most accurate Monte Carlo (MC) method [8].
We propose using the concept of “apriority-free” in solving problems of remote sensing using methods and systems based on optical-physical measurements. The concept of “a priori” is based on several principles:
the maximum possible exclusion of a priori information or assumptions about the studied object;
the maximum exclusion of the influence of hardware parameters, environmental parameters on the measurement results when solving the problem of calibration and metrological support;
receiving information in real time.
Let us turn to the substantiation of the theoretical aspects of the concept of “a priori” for the use of optical-physical measurements1 under conditions of uncertainty.
THEORETICAL ASPECTS OF OPTICAL AND PHYSICAL MEASUREMENTS IN CONDITIONS OF INFORMATION UNCERTAINTY
Earlier, we developed a regression approach to solving inverse problems of optical sensing of biological media. It consists in the fact that linearly independent components are extracted from the recorded optical signals. The procedure is based on extracting the projections of the signals onto the space from the eigenvectors of their covariance matrix. Further, the desired medium parameters are found by establishing their stable regression relationships with linearly independent signal components.
The use of linearly independent values in regressions corresponds to the extraction of the “useful signal” from the initial (inverted) data and the rejection of “noise”. Such a procedure allows us to construct solutions to inverse problems that are resistant to random “disturbances” (fluctuations) of these data. To search for linearly independent quantities, we construct the averaged dependences by statistical modeling of optical signals with maximum variability of the medium parameters. We accept only those parameters of the medium that affect the process of radiation transfer in it. Next, we construct eigenvectors and obtain regression solutions of inverse problems. Subsequently, this allows one to perform operational processing of the measured optical signals without solving the radiation transfer equation in the medium under study, and to determine a priori information as in regularization methods for the inverse problem [6, 7].
To decompose the inverted data into linearly independent components, the basis of the covariance matrix is used. The optimal dimension of the basis is determined based on closed numerical experiments to restore the parameters of the medium from the characteristics of its light scattering. To do this, initially on the basis of a simulated ensemble of realizations of the measurement vector , a “test” ensemble is formed. In it, each realization is obtained by superimposing on the components of the initial vector random deviations within the measurement error δ. Next, all implementations of the “test” ensemble are selected. For each of them, a solution to the inverse problem is found by the regression formulas between the model parameters and linearly independent quantities composed of the components of .
As a result, the obtained values of the model parameters * are compared with their actual values and the errors of their recovery are calculated. In accordance with a given measurement error, we select the number of linearly independent components of the reversed data and the light scattering characteristics of the medium. We stop the selection on those characteristics by which it is possible to determine specific environmental parameters under the conditions of a priori uncertainty of all others most accurately. Such a solution also makes it possible to evaluate the information content of the measured data, get an idea of the theoretically achievable accuracy of restoring the environmental parameters from them, and investigate the influence of the quantity and accuracy of optical measurements on the accuracy of solving the inverse problem.
In particular, in [6, 7, 9, 10], solutions to the inverse problems of spectroscopy of biological tissues were presented. The goal was to determine the optical and structural-morphological parameters of tissues. The optical parameters included the absorption coefficient, transport scattering coefficient, and the anisotropy factor of the scattering indicatrix. The structural and morphological parameters are the concentrations of melanin, total hemoglobin and bilirubin in the tissue, the degree of blood oxygenation, the average diameter of blood vessels, the concentration and size of effective scatterers. The solutions were based on the use of optical models of the skin and mucous membranes of a person, as well as the Monte Carlo (MC) method, and on this basis, obtaining stable regression solutions to inverse problems of spectroscopy of biological tissues with spatial resolution.
The technical implementation is proposed and the efficiency of determining the optical and structural-morphological parameters of tissues from the measured signals is evaluated. The backscattering signals of tissues were recorded using fiber-optic technology with spatial diversity of the channels of sending and receiving radiation. The spectral region of strong light absorption by chromophores of biological tissue is in the range of λ < 600 nm. The functional diagram of the measurements of backscattering signals of the mucous membranes is shown in Fig. 1. Radiation is injected into the medium through a fiber with a diameter of 0.2 mm, and the backscattered radiation is collected by receiving fibers located at distances L = 0.23; 0.46; 0.69; 0.92; 1.15 mm from the center of the light-transmitting fiber (diameter of the receiving fibers 0.2 mm).
Such a geometry of measurements provided an optical signal value acceptable for practice (not less than 10–4 of the power of the probing light beam) with a maximum distance between the illuminating and receiving fibers not exceeding the diameter of the instrument channel of the endoscope.
The errors in the restoration of tissue parameters caused by their statistical scatter and optical measurement errors are estimated. Combined processing of the spectral and spatial characteristics of backscattering tissue using the obtained regressions allows one to reduce the error in the restoration of optical parameters of tissue to ~3%, while their recovery from spectral parameters is possible only with an error of ~9%, and from backscattering spatial characteristics ~ 7% [7].
A solution to the inverse problem of reconstructing two-dimensional distributions of structural-morphological parameters (SMP) of human tissues from their multispectral images has been proposed [7]. To eliminate the influence of uneven illumination and the geometry of the tissue survey on the results of the restoration of its parametric maps, tissue images normalized to one of its spectral layers are used. The inverse problem is to restore the tissue SMP from the spectral values of each pixel of the normalized image. To solve it, regression relations are used, previously obtained on the basis of a representative sample of the diffuse reflection spectrum of the medium simulating the tissue under study. The stability of the regression operators for converting tissue images into parametric maps to measurement errors is ensured by using linearly independent image components for solving the inverse problem, obtained by projecting the spectral values of each pixel onto the eigenvectors of the covariance matrix of the diffuse reflection tissue spectrum.
On this basis, “calibration-free” measuring systems have been created that are resistant to changes in the hardware constants of receiving-emitting and recording units, environmental parameters, and optical element contamination. Using regression relationships, methods for the operative reconstruction of two-dimensional distributions of SMP of the skin, mucous membranes, and human fundus have been developed and patented (Patents BY10918 C1, 2008; BY19144 C1, 2015; RU2510506 C, 2014; RU2506567 C1, 2014; RU2511747 C2, 2014; BY18652 C1, 2014; RU2501522 C2, 2013; BY18653 C1, 2014; RU2517155 C1, 2014; RU2536217 C1, 2014; RU2539367 C1, 2014). The working wavelengths of the methods were selected that provide the greatest accuracy and stability for solving inverse problems under conditions of general variability of all tissue parameters that affect their diffuse reflection spectra.
At the same time, the regression method does not allow a subtle analysis of the light scattering characteristics of the tissue, for example, to reveal in them the features caused by small variations in the component composition of the blood. In particular, this refers to the content of dyshemoglobins in the blood (carboxy-, met- and sulfhemoglobin), the absorption spectra of which substantially overlap with similar spectra of the main forms of hemoglobin (oxy- and deoxyhemoglobin). Small variations in the spectrum of tissue backscattering associated with variations in hemoglobin composition are largely averaged over the statistical material used to obtain the regressions. This reduces the sensitivity of the method to the concentration of each specific form of hemoglobin.
Under these conditions, a significantly higher accuracy in estimating the desired parameters can be achieved by measuring the backscattering spectrum of the tissue with high spectral resolution and then solving the inverse problem. The essence of which is to model the theoretical spectrum with respect to the experimental one by selecting model parameters. High spectral resolution of the initial data ensures the stability of the inverse problem to optical measurement errors and eliminates the ambiguity of its solution within the framework of the used biological tissue model. The difficulty here lies in an adequate theoretical calculation of the characteristics of tissue backscattering.
In our previous works, expressions were obtained for solving inverse problems in the optics of skin biotissues, mucous membranes, bulbar conjunctiva, and the human fundus [7, 11–14]. We especially note the developed approximation analogues of the MC method for fast calculations of the characteristics of radiation transfer in a homogeneous and two-layer multi-scattering media.
For example, the calculation of the diffuse reflection coefficient of a homogeneous, semi-infinite medium with an absorption coefficient k = 0.004–7.0 mm–1, a transport scattering coefficient β′ = 0,3–5,1 mm–1, and an average cosine of the scattering indicatrix g = 0,5–0,97 and the refractive index η = 1.35–1.45 based on the following approximation formula [7.11]:
where is the diffusion depth of light penetration into the medium, A, am, bm, cm and dm are numerical coefficients calculated in advance by numerically simulating the radiation transfer into environment. The maximum error in calculating R according to this formula for the indicated values of the optical parameters of the medium is 4%, and the average error is 0.9%. The analytical approximation allows us to simulate the diffuse reflection coefficient of most biological tissues and many media of non-biological origin in the spectral regions of strong and weak light absorption in them.
The model for a two-layer medium is based on the representation of diffuse reflection coefficient in the form of the product of effective light transmission by layers, taking into account its multiple scattering and re-reflection between them. The dependences of the effective transmission of each layer on the optical parameters of the medium for the cases of its collimated and diffuse illumination are described by polynomial functions with given numerical coefficients [13]. The proposed formulas approximate with high accuracy the results of numerical calculations of diffuse reflection coefficient (Fig. 2). The standard error of the calculation of the diffuse reflection coefficient of the medium in the range R = 0.01–0.60 is 0.65%.
For the correct solution of the problems of laser radiation dosimetry and a comprehensive analysis of the effectiveness of photodynamic therapy, knowledge of a whole complex of parameters is necessary. These include absolute concentrations of FS and molecular oxygen in the tissue; volume concentration of capillaries and their average diameter; the ratio of the concentrations of various forms of hemoglobin; the depth of penetration of light into the fabric; distribution of radiation density in tissue and light power absorbed by endogenous (natural) and exogenous (specially introduced) tissue chromophores [15]. Moreover, to select the optimal spectral and energy characteristics of laser radiation, all parameters characterizing the processes of radiation transfer in the medium and its interaction with the substance of the medium should be presented as a functional dependence on the wavelength of the probe radiation.
A quantitative measure of the effectiveness of photophysical and photochemical processes in an irradiated tissue associated with the absorption of light by its chromophore X can be the spectrum of the action of light on a given chromophore [7, 16]:
,
where CX and kX are the concentration and absorption coefficient of the chromophore, Φ (z, λ) is the radiation density spectrum at the tissue depth z (spatial illumination), W / m2.
The idea put by the authors as a basis for the noninvasive definition of the functions KX (z, λ) consists in the following. It is known that the light field inside the biological tissue and the radiation field scattered by it in the opposite direction are determined by the same tissue parameters. The characteristics of the backscattered radiation field are measured directly during the experiment. We establish optically significant tissue parameters by solving the corresponding inverse problem. Further, using the theory of radiation transfer, we calculate all the necessary spectral and spatial characteristics of the light field inside the tissue, as well as the deep distributions of the light action spectra on its chromophores. Thus, the parameters of photo exposure, providing the optimal therapeutic effect, can be selected according to the characteristics of radiation backscattered by the tissue. However, this requires fast and reliable methods for calculating the characteristics of light fields inside and outside the irradiated tissue [7].
The radiation field in the medium is described by the superposition of the incident collimated, incident diffuse and reflected diffuse flows. Luminous fluxes in neighboring elementary layers of the medium and on its surface are connected by simple matrix operators. We obtain them by combining the asymptotic formulas of the theory of radiation transfer with the single-scattering approximation. A comparison of the results of calculating light fluxes in a five-layer medium (modeling human skin) by the engineering method and the MC method shows that they are comparable in accuracy with a significant difference in computational costs [7].
Examples of determining the parameters of photo-exposure were demonstrated in [18, 19] when studying the spectra of the action of light on a photosensitizer and oxyhemoglobin during photodynamic therapy, as well as the rate of photoisomerization of bilirubin during phototherapy of jaundice in newborns.
Patents have been obtained for methods that allow photodynamic therapy of human oncological diseases (Patents RU2521838 C1, 2013; BY19557 C1, 2014; Eurasian Patent 031413 B1, 2018).
Developed and patented: a device for the diagnosis of malignant tumors in the human mucosa by determining the concentration of hemoglobin and the degree of blood oxygenation in it (Patent BY19558 C1, 2015); a device for determining the concentration of hemoglobin and the degree of blood oxygenation in the mucous membranes (Patent RU2528087 C1, 2014). This device can be successfully used during endoscopic examinations of the mucous membranes of the oral cavity, esophagus, organs of the gastrointestinal tract and lungs.
CONCLUSION
Visual inspection (examination) of human organs and tissues is the first-line method for diagnosing a number of diseases and is widely used in oncology, surgery, ophthalmology, and in a number of other areas of medicine. In modern medicine, such an examination is performed using specialized systems: dermatoscopes, endoscopes, fundus cameras, etc. However, a visual examination of the patient’s tissues is entirely based on the experience, qualifications and color perception of the doctor and does not provide an objective diagnosis of the disease.
Thus, the development of operational high-precision methods for the quantitative diagnosis of biological objects according to the spectral and spatial characteristics of their multiple scattering and their multispectral images is extremely relevant; methods of non-invasive control of light-induced processes in biological tissues during phototherapy. The basis for creating a new generation of optoelectronic systems using the minimum amount of a priori information, additional measurements, calibration procedures is the development of methods that meet the conditions of the concept of “a priori”.
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ABOUT AUTHORS
Kugeiko Mikhail Mikhailovich, e-mail: kugeiko@bsu.by, Dr.of Scien. (Phys.-Math), Professor, Department of Quantum Radiophysics and Optoelectronics, Faculty of Radiophysics and Computer Technology, Belarusian State University, Minsk, Republic Belarus.
ORCID: ORCID:0000-0002-9462-9533
Lisenko Sergey Aleksandrovich, e-mail: lisenko@bsu.by, Dr.of Scien. (Phys.-Math), Professor, Department of Informatics and Computer Systems, Faculty of Radiophysics and Computer Technology, Belarusian State University, Minsk, Republic of Belarus.
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