Refractive Index Formula of Blood as a Function of Temperature and Concentration

Abstract

The analysis of blood parameters is main procedure for defining the patient’s condition. The refractive index of blood was calculated using the experimental data for a medical application and diagnostic purposes. There is small change in the refractive indices with changing with the personal conditions, illness, parasitization, temperature and others. Theoretical simulations of the refractive index of blood are difficult, and it is unpredictable in different conditions. We proposed a new formula for the refractive index of blood as a function of wavelength, concentration, and temperature by using the genetic programming method. Input parameters were temperature (°C), concentration (g/L), and wavelength (nm). The refractive index values of blood were the output parameters. A total of 492 training and testing sets were selected in the spectral range of 436 to 1550 nm, in the temperature range of 20-45 °C and the concentration of 0-200 g/L HbC. The model proposes the refractive index formula blood for all the input parameters given in the range without need of extra parameters. The results are good agreement with experimental measurements in the literature and compared to Sellmeier equation.

Key words
Artificial intelligence; refractive index; human blood; genetic programming

Introduction

The optical properties of blood provide important information for diagnostic applications. The health care professionals have looked for medical conditions and signs of disease in the blood parameters. This connective liquid within the body are used to determine the condition with non-invasively. The cells are suspended in plasma of heterogeneous medium of blood. Blood plasma contains water and various water-soluble proteins and low molecular compounds (Tuchin 2015). The blood cells consist of erythrocytes, leukocytes, and platelets with different ratios and sizes depending on demographic properties and diet (Friebel et al. 2006). The determination of refractive index of blood is attractive and have important applications involve sensing of blood groups (Rakhshani 2020, Zhernovaya et al. 2011), oxygenated and deoxygenated hemoglobin samples (Alade et al. 2018), optical diagnosis and laser treatments (Jin et al. 2006) and other applications. Researchers have proposed different methods for deriving the real and complex part of the refractive index (Oduncuoğlu 2020, Tuchin 2015, 2016, Zhernovaya et al. 2011). The refractive index is a defined as $n=n_{\mathrm{Real}}+n_{\mathrm{Imaginary}}$ . The $n=n_{\mathrm{rea}}l$ is the real part and $n=n_{\mathrm{Imaginary}}$ is the imaginary part of complex number, respectively. The real part represents the change in the speed of propagated light through a medium, compared to the speed of light in a vacuum and an imaginary part is the related to absorption parameter of material. The structure of blood is complex, and this attractive research subject is mostly studied to develop a mathematical model. The optical parameters of blood depend on the optical properties of its components. Blood carries oxygen and nutrients to the trillions of cells is a very complex medium, which makes the theoretical simulation of its properties considerably difficult. In a simplified model, the blood can be presented as (Heller 1965, Tuchin 2016)

$$n=\sum _{i=1}^N{n}_i{f}_i$$(1)

where $n_i$ and $f_i$ are the refractive indices and the volume fractions of N individual components of cells suspended in plasma, respectively. In this study, we mainly focus on real part of refractive index values of blood. The optical properties of blood were different from the other human tissues (Lazareva & Tuchin 2018). The optical properties of cells and blood are determined both experimentally and theoretically to interpret their interaction with light. Artificial intelligence modeling is used to determine parameters and solve the unknown mechanism of the systems. AI aims to generate results with high accuracy to simulate the data. The main role of this program is the use of information (Kurt & Oduncuoğlu 2015a, b, Kurt et al. 2015, Verdegay et al. 2008, Yilmaz et al. 2018). An important trend for the future of health technology will be the increasing use of artificial intelligence for medical applications with increased efficiency and effectiveness. The fact that the intelligence of healthcare in the future must come both from artificial and conventional forms (Mitchell & Haroun 2016). The new medical technologies are developed by using this infrastructure. The artificial intelligence in healthcare has the potential to save costs, broadening accessibility, and can be an asset for patients and providers. We proposed a formula to determine the refractive index of blood by using genetic programming. Genetic programing uses the given experimental data to find desired parameter of given problems (Yilmaz et al. 2018). The genetic programming as a powerful modeling technique related to the statistical approach seems very logical method for predicting the parameter (Koza 1992).

Experimental Setup

The optical properties of blood depend on physiological and biochemical parameters are determined. The experimental data for the refractive index of blood are measured using the Abbemat Refractometer (Friebel et al. 2006, Jacques 1965, Lazareva & Tuchin 2018, Liu et al. 2019, Meinke et al. 2006, Nahmad-Rohen et al. 2015, Yahya & Saghir 2016). The collected experimental data in papers were conducted by using dry human hemoglobin Lyophilized powder dissolved in isotonic phosphate-buffered saline to maintain the PH at 7.4 (Sydoruk et al. 2012, Yahya & Saghir 2016, Zhernovaya et al. 2011). The real part of the refractive index of blood were modelled and estimated based on the concentration, temperature, and wavelength. The proposed model was developed using genetic programming. The proposed model is able to predict the refractive index of blood with an acceptable level of accuracy.

The Model

Recently, it is common to use computers and computational methods to solve problems. In this study, the genetic programming for the formulation of the refractive index is used. The input parameters are given in Table I.

Table I
The variables used in model.

Input Variable	Range	Output Variable	Range
Wavelength	430-1550 nm
Temperature	20-45 C	Refractive Index	1.321-1.395
Concentration	0-200 g/L

The thermal and concentration dependence of the refractive index varies with wavelength. There are different formulations for determining the refractive index of blood, but no direct explicit formulations related to temperature, concentration, and wavelength. These ranges show the potential for clinical applications. The formulations were proposed by genetic programming to automatically extract relationships in a system for predicting the refractive index. The genetic programming main goal is to solve problems by searching all possible solutions. A genetic programming is adapted from the Darwinian evolution theory of natural selection based on survival of the fittest resulting from variability crossing over and mutation phenomena. The parameters in this system given in Table II are the chromosomes and expression trees. The simple rules determine the spatial organization of the functions and terminals in the expression trees and the type of interaction between sub-expression trees (Ferreira 2006). The iterative computational process is used to find successive generations for fitness until the fittest solutions are obtained.

Table II
Genetic Programming parameters used for proposed models.

Name	Data
Number of Generations	251511
Function Set	$+’,-,*,/,\sqrt{x},\exp \left(x\right),x^2,x^3,x^{1/3}$
Chromosomes	30
Head Size	7
Number of Genes	3
Training Samples	445
Testing Samples	47
Linking Function	Addition
Training $R^2$	0.9499
Testing $R^2$	0.9434
Fitness Function	RRSE
Mean Squared Error (MSE)	$3.38{10}^{-6}$
Mean Absolute Error (MAE)	$1.43{10}^{-3}$
Relative Absolute Error (RAE)	$2.08{10}^{-1}$
Root Mean Squared Error (RMSE)	$1.84{10}^{-3}$
Correlation Coefficient	0.974648

GP is used for predicting parameter tool in medicine, social sciences, and other disciplines. The genetic programming are explained in books and papers (Ferreira 2006, Goldberg & Holland 1998, Koza 1992, 1994, Koza & Koza 1992). It was aimed to be in the best fitness zone, to minimize errors, to simplify, and to find the best predicted result (Pouraliakbar et al. 2016). A combination of these were investigated the increase performance of the algorithm in predicting output parameter. The optimal simplified equation for refractive index equation is given below:

$$n_{\mathrm{real}}={[1.557037+\frac{T}{\lambda (5.013366+C))}]}^{\frac{2}{3}}+\frac{-29.07134(C+9.98566)}{{\lambda }^2}+\frac{6.34024-T}{{6.902436}^2(C+\lambda )}$$(2)

where C is the concentration (g/L), T is the temperature (C), and $\lambda$ is the wavelength (nm). The refractive index of blood with different parameters $n_{\mathrm{Real}}(T,C,\lambda )$ has been calculated by using this equation. The accuracy of the model was evaluated using statistical parameters. The coefficient of correlation, the root mean square error (RMSE) and the mean absolute error (MAE) and explanations of these parameters have been provided in detail References (Kurt & Oduncuoğlu 2015b, Kurt et al. 2016). The correlation coefficient (R) was used to calculate the model performance. The calculated results were in good agreement with the experimental results.

Results and Discussion

The generalization performance of genetic programming by comparing the experimental and the predicted values are given in Figures 1-2. It is clearly observed that the formulation proposed by the genetic programming formula for the refractive index is very successful. The Sellmeier equation is used to compare the results of our predicted data and is given in Equation (3) as

$$n^2\left(\lambda \right)=1+\frac{A_1{\lambda }^2}{{\lambda }^2-B_1}+\frac{A_2{\lambda }^2}{{\lambda }^2-B_2}$$(3)

The empirically determined coefficients of the Sellmeier equation for whole blood are listed in Table III.

Table III
The coefficients of Sellmeier equations of blood (Liu et al. 2019).

$A_1$	$A_2$	$B_1$	$B_1$
0.7960	5.1819	1.0772 ${10}^4$	-7.8301 ${10}^5$

The wavelength dependence of the refractive index of water is determined by the expression (Kohl et al. 1994).

$$n_{\mathrm{water}}\left(\lambda \right)=1.31848+\frac{6.662}{\lambda -129.2}$$ (4)

Figure 1
The experimental and calculated refractive index of blood ${𝐧}_{\mathrm{𝐫𝐞𝐟𝐫𝐚𝐜𝐭𝐢𝐯𝐞}}(𝐓,𝐂,𝛌)$ with different temperatures (T) and concentration (C).

The wavelength dependences of the refractive index were calculated by using the proposed formula and compared with refractive index values of blood calculated using Sellmeier equations and water obtained from Equation (4). These equations just depend on wavelength. The blood plasma contains up to 90 % water.

Figure 2
The calculated refractive index of blood as a function of temperature and concentration for wavelengths of 500 and 1000 nm. The former is in the visible and the latter is in the near-infrared spectral range of the electromagnetic spectrum. The new model validated within these wavelength ranges and the trends are shown in the graph.

The proposed equation is in good agreement with experimental and calculated results. The refractive index is increasing with increasing concentration (the blood contains low water) and decreasing with increasing temperature accordingly. The average values of temperature rate is

$$\frac{d{n}_{\mathrm{real}}}{dT}=-1.05{10}^{-4}(1/°C)$$(5)

and the average value of concentration is

$$\frac{d{n}_{\mathrm{real}}}{dC}=1.54{10}^{-4}\left(g/L\right)$$(6)

The proposed formula is more sensitive in trend but at low wavelength range, there is a small difference between experimental to predicted data. The linear model is obtained using the following equations

$$n_{\mathrm{refractive}}=n_{\mathrm{real}}(\lambda )+\frac{d{n}_{\mathrm{real}}}{\mathrm{dT}}T(°C)+\frac{d{n}_{\mathrm{real}}}{\mathrm{dC}}C(g/L)$$(7)

The non-linear behavior variations in refractive indices concerning wavelength were observed. The refractive index increases in a non-linear fashion as the wavelength decreases. This is due to electromagnetic interactions between the particles of light and particles within the blood.

Conclusion

In this study, we proposed a new formulation for predicting the refractive index of blood as a function of wavelength, low concentration of protein, and temperature using genetic programming. The genetic programming works well and provides optimized parameters of real part of refractive index for blood. The formula transforms the experimental results easily to the computational media, making them useful for computational simulation of refractive index of blood used in the medical sciences. The refractive index formula is across a wide range of hemoglobin concentrations, temperatures, and wavelengths. It gives us an opportunity to compare all the parameters with each other. The predictive capacity of the program is 98.73% of the training and 98.37% of the testing results. The proposed equations are user-friendly. The statistical parameters ($R^2$, MAPE, and RMS) of the proposed formulation show high potential for predicting the refractive index and use the results for diagnostic purposes. In this study, a simple linear refractive index formula with respect to parameters are also developed and analyzed. The proposed model used key parameters, such as the concentrations, temperature, and wavelength to accurately estimate the refractive index from experimental data.

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Publication Dates

• Publication in this collection
  10 Dec 2021
• Date of issue
  2021

History

• Received
  15 Oct 2020
• Accepted
  25 Sept 2021