Report Bachelor Project Physics and Astronomy 15 EC,
conducted between 05-04–2017 and 10-07–2017
The effect of hematocrit on the Near Infrared Spectroscopy
signal at different wavelengths.
A MatLab simulation for measuring hematocrit with Transmission
based Near-Infrared Spectroscopy using the Diffusion Equation.
Rosa Rougoor
Department of Biomedical Engineering and Physics
Supervisors:
Ton van Leeuwen
Abstract
The aim of this research is to find a relation between the hematocrit (Hct) of blood and Near Infrared Spectroscopy (NIRS) signals when performing NIRS on a slab of biological tissue. In order to find a relation, a MatLab simulation of light propagating through biological tissue is made. The used model is based on the Diffusion Approximation. The effect of different variables on this Hct-NIRS signal relation is measured. The different variables are: the thickness of the slab of tissue, the size of the pulse in arterial blood (as a fraction of the total arterial blood volume) and the volume fraction of melanosomes in the skin. This was done for wavelength combinations of 810 and 1310 nm, 940 and 1310 nm and 940 and 1180 nm. The effect of pulse size is negligible for all wavelength combinations. The effect of both slab thickness and melanosome fraction was smallest for the wavelength combination of 940 nm and 1310 nm.
Popular scientific summary in Dutch at 6 VWO level
De hematocriet van je bloed is de volumefractie van je totale bloed die wordt ingenomen door rode bloedcellen. Het is een waarde die veel gebruikt wordt in de medische wereld, onder andere omdat het een indicatie voor uitdroging is. Ben je uitgedroogd, dan zit er minder water in je bloed en neemt het totale volume van het bloed af. Tegelijkertijd blijft het volume rode bloedcellen wel gelijk. De hematocriet neemt daardoor toe. Op dit moment moet om de hematocriet van je bloed te kunnen bepalen een bloedafname genomen worden. In dit onderzoek willen we kijken of dit ook anders kan, namelijk door middel van de analyse van lichtsignalen in het nabij Infrarode lichtspectrum. Licht met een bepaalde golflengte wordt in verschillende mate geabsorbeerd en ver-strooid door verschillende stoffen in je lichaam. Door een analyse van licht met twee verschillende golflengtes dat bijvoorbeeld door een vinger gaat, kan je dus informatie verkrijgen over de samen-stelling van het weefsel en bloed in de vinger. Door middel van een computer simulatie wordt in dit onderzoek onderzocht of door middel van de analyse van de lichtsignalen kunnen bepalen wat de hematocriet waarde van het bloed is.
Contents
Contents iv 1 Introduction 1 2 Theoretical Background 7 2.1 Optical Properties . . . 7 2.1.1 Absorption coefficient . . . 72.1.2 Reduced scattering coefficient . . . 8
2.2 Diffusion Approximation . . . 8
2.2.1 From Radiative Transer Equation to Diffusion Equation . . . 8
3 Method 11 3.1 The diffusive equation model . . . 11
3.2 Volume fractions . . . 12
3.3 Absorption coefficient . . . 13
3.4 Reduced scattering coefficient . . . 13
3.5 The effect of changing volume fractions . . . 15
4 Results 17 4.1 RoR and SpO2 . . . 17
4.2 RoR and Hct . . . 21 4.2.1 SpO2 Value . . . 21 4.2.2 Pulse Size . . . 21 4.2.3 Slab Thickness . . . 21 4.2.4 Melanosome Fraction. . . 22 5 Conclusions 27 Bibliography 29 Appendix 31
Chapter 1
Introduction
The hematocrit (Hct) of blood is the volume percentage of red blood cells in the blood. It is calculated by taking the ratio of the volume of red blood cells over the total blood volume. One of the reasons hematocrit is measured is, among others, that abnormal values are an indication of dehydration or hyper-hydration [9]. When someone is dehydrated, there is less water in the blood and consequently the volume fraction of red blood cells rises compared to the total blood volume. On the other hand, an abnormal high hematocrit value indicates being dehydrated. Both dehydration and hyper-hydration show negative effects for critically ill patients [11]. Therefore, a quick and accurate hydration assessment is of vital importance at for example the intensive care unit. To measure the hematocrit, currently a blood sample is needed. Other current hydration assessment techniques, such as isotope dilution, neutron activation analysis and bio-electrical impedance spectroscopy are expensive, not portable, require a lot of time (2-4 hours) or require technical expertise that is only available in a few laboratories [2]. In this report we investigate the possibilities of a new, fast and non-invasive method to measure Hct using Near Infra-red Spectroscopy (NIRS).
NIR Spectroscopy has already proven to be useful in pulse oximetry, a method for measuring the oxygen saturation of hemoglobins in arterial blood (SpO2). SpO2 is the ratio of oxygen saturated hemoglobins (oxyhemoglobin) over the total hemoglobin (saturated and unsaturated): HbO2/(Hb + HbO2). In order to measure the SpO2, pulse oximetry uses the different optical properties of different tissues and the pulsating character of the arterial blood. Firstly I will give a brief introduction to the important optical properties of biological tissue and then I will discuss how this in combination with the pulsating character of blood can be used to measure the SpO2 value of the blood.
Near-infrared light (light in a range of approximately 650 nm - 2500 nm) propagating through biological tissue is scattered and absorbed by the tissue constituents but can still penetrate up to a few centimetres into the tissue. Absorption characteristics are described by the absorption coefficient, which is described as the power absorbed per unit volume divided by the incident power per unit area. The scattering characteristics are defined by the scattering coefficient and the anisotropy factor g. The scattering coefficient is defined as the ratio between the power scattered in the unit volume and the incident power per unit area and the anisotropy factor is defined as the average cosine of the scattering angle θ: g = < cos θ >. It characterises the direction and therefore the effectiveness of scattering. Combined, µs and g define the reduced scattering
coefficient: µ0s = µs (1 − g) [8]. In figures 1.1 and 1.2 some typical absorption and reduced
scattering spectra are shown.
Although the tissue-light interaction is dominated by scattering, it is the absorption that can provide important physiological information [8]. This is because most of the constituents of biological tissue have a specific absorption spectrum in the near-infrared range of wavelengths, meaning that some wavelengths are more absorbed by a specific constituent than others, as can be seen in figure1.1. The attenuation of light as a result of this absorption is most easily described
CHAPTER 1. INTRODUCTION
Figure 1.1: The absorption spectra of water (H2O, dashed blue line), deoxyhemoglobin (Hb,
blue line) and oxyhemoglobin (HbO2, red line) taken at 100%. Wavelength is given in nm and
absorption coefficients in mm−1. The Y-axis has a logarithmic scale.
Figure 1.2: The reduced scattering coefficient of blood, epidermis, dermis and subcutaneous fat at a 100%. In black the total reduced scattering of a tissue consisting of 8% blood, 38% dermis, 2% epidermis and 52% fat [6] [10] [1].
CHAPTER 1. INTRODUCTION in the classic Lambert-Beer law [7]:
T = I I0
= e−d∗Pi(µa,i∗ci), (1.1)
where
• T = transmittance
• I = light intensity after passing through the tissue • I0 = initial light intensity
• d = path length of the light through the tissue • µa,i = molar absorption coefficient of constituent i
• ci = molar concentration of constituent i
It shows us that the intensity of light decays exponentially with the absorption coefficient and path length. This looks pretty simple, but the problem is that the path length is not simply the thickness of the tissue. Due to scattering not all detected photons travel the same distance. Some get barely scattered and follow almost a direct path from the emitter to detector, while others get highly scattered and travel longer distances and consequently have a higher chance to get absorbed [7]. That is why there should also be accounted for scattering as well. As mentioned before, pulse oximetry also makes use of the pulsating character of the arterial blood. With pulse oximetry the transmittance of light of two different wavelengths is measured, usually through a finger. During a pulse, the light has to travel through more arterial blood compared to right after a pulse. Because of the volume change of the blood during a pulsation, the light is differently absorbed. Consequently the transmittance signal is a pulsating signal. This means the transmittance can be divided into a baseline part and a pulsating part on top of that, as is shown in figure1.3. The bigger the pulse, the bigger the absorption and thus the smaller the total power of the light that is transmitted through the tissue. Because absorption is wavelength dependent, the transmittance signal (both the total transmittance as the pulsating part only) will be different for different wavelengths too.
As said before, one should account for the fact that path lengths travelled by the photons are changed as a result of scattering. In a simple model this is done by using the so-called differential path length factor (DPF), which is different for different wavelengths. The Lambert-Beer equation then looks as follows:
I I0
= e(−d·µa·DP F ), (1.2)
where µa is the total absorption coefficient (Pi(µa,i∗ ci) in equation1.1). The pulsating part of
the intensity can be described using this equation as: ∆I I0 = e−d·(µa+∆µa)·DP F − e−d·µa·DP F = e−d·µa·DP F e−d·∆µa·DP F − e−d·µa·DP F = e−d·µa·DP F × (e−d·∆µa·DP F − 1) = e−d·µa·DP F × (−d · ∆µ a· DP F ) Now we have, I = e−d·µa·DP F I
0 and ∆I = e−d·µa·DP F × (−d · ∆µa· DP F ) I0. One can take the
ratio of the pulsating signal dI over the total baseline intensity I, resulting in: ∆I
I =
e−d·µa·DP F × (−d · ∆µ
a· DP F ) I0
CHAPTER 1. INTRODUCTION
Figure 1.3: The transmittance signal is pulsating as a result of the pulsation of the arterial blood. During a pulse, light is absorbed differently because of the increased amount of arterial blood compared to directly after a pulse.
When this is done for two different wavelengths, one can take the so called ’Ratio of Ratio’s’ (RoR): RoR = ∆I I wl1 ∆I I wl2 = (−d · ∆µa,wl1· DP Fwl1) (−d · ∆µa,wl2· DP Fwl2) =(·∆µa,wl1· DP Fwl1) (·∆µa,wl2· DP Fwl2) (1.4) When one takes the Ratio of Ratio’s, and that is it’s power, the effects of variables such as slab size, pulse size or arterial blood fraction cancel out. The Ratio of Ratio’s is especially useful when one wants to measure the fraction of a constituent in the arterial blood. This is because the arterial blood is the part that causes the pulsating signal used in the RoR. Therefore the difference in absorption as result of this pulse (∆µa) can be expressed in terms of the fractions of
constituents of the arterial blood and their absorption coefficients: ∆µa = [arterial blood fraction] · [pulse size] ∗
X
i
(µa,i∗ fractioni), (1.5)
where pulse size is given as a fraction of the total arterial blood volume and i represents the different constituents. This gives a direct relation between the measured Ratio of Ratio’s and the fraction of the constituents. In figure1.4 is shown how this Ratio of Ratio’s relates to the SpO2 value. This indicates that there is a relation between the hematocrit value and the RoR as well. The aim of this research is to see what this relation is for different wavelength combinations and what the effect is of pulse size, slab thickness and melanosome (light absorbing pigment) fraction on this relation. When a relation is found that does not change with these variables (pulse size, slab thickness and melanosome fraction), NIR spectroscopy is very likely to be a good method for measuring hematocrit. Despite from being a non-invasive method it will be fast method that can be incorporated into a small appliance, as has been proven with pulse oximetry appliances.
The model with DPF factor showed above is really useful for showing the idea of the Ratio of Ratio’s. However, for accurate calculations, a more complex model is needed to account for 4 The effect of hematocrit on the Near Infrared Spectroscopy signal at different wavelengths.
CHAPTER 1. INTRODUCTION
Figure 1.4: The Ratio of Ratio’s (Modulation Ratio) of red (660 nm) and near-infrared light (940 nm) plotted against the oxygen saturation (SpO2) [7].
the effect of scattering more accurately. In this report a solution of the Diffusion Equation as described by Martelli et all [8] will be used. The fact that the model is more complex does not change the power of the Ratio of Ratio’s in combination with NIR Spectroscopy.
As said before, the aim of this research is to find a relation between the Ratio of Ratio’s and the hematocrit of the blood. In order to find a relation, a MatLab simulation of light propagat-ing through biological tissue will be made. In this simulation it is possible to change different parameters, such as the thickness of the slab of tissue and the fractions of different constituents in the tissue. The simulation will be valuated by comparing it to the better known effect of oxy-gen saturation of blood on the Ratio of Ratio’s. Next, the effect of different variables (i.e. slab thickness, change in pulse volume and the melanosome concentration) will be measured with the simulation. Last, the relation between RoR and Hct will be evaluated for different combinations of wavelengths using the same changes in variables as for the RoR-SpO2 measurements.
Chapter 2
Theoretical Background
As said in the introduction, the first step of this research is to make a simulation of light propagat-ing through biological tissue. Therefore, we need a model to describe the propagation of light. Describing the propagation of light through scattering and absorbing tissue is not an easy thing to do. But with a few assumptions it is possible to make models that approximate the beha-viour accurately enough to do calculations with. In the introduction the Lambert-Beer model was already mentioned, but due to the effects of scattering describing the propagation of light through biological tissue requires a more complex model. This chapter will elaborate on the used model and the relevant optical properties of biological tissues. Eventually, this will be linked to calculating the Ratio of Ratio’s.
2.1
Optical Properties
2.1.1
Absorption coefficient
Absorption characteristics of a tissue are expressed in the absorption coefficient. The total ab-sorption coefficient of a tissue is given by the sum of the contributions of all the constituents of the tissue:
µa=
X
i
fv,i µa,i (2.1)
where fv,i (L L−1) is the volume fraction of the chromophore i and µa,i (mm−1) the absorption
coefficient for that pure component.
The absorption coefficient of the pure component in equation 2.1can be calculated using the extinction coefficient i (mm−1 M−1), the mass concentration Cm,i (g L−1) and the molecular
weight MW (g mole−1) of that component [6]: µa,i=
ln(10) i Cm,i
M W (2.2)
Using the units given here, this will result in an absorption coefficient in mm−1. Since the extinction coefficient is dependent on wavelength, the absorption coefficient will be as well.
In figure1.1the absorption spectra of water, oxyhemoglobin and deoxyhemoglobin are shown. As one can see, the difference between absorption coefficients for oxyhemoglobin and deoxyhemo-globin at a 660 nm wavelength is big, with deoxyhemodeoxyhemo-globin being the biggest absorber. Bearing in mind the logarithmic scale on the y-axis, the absorption due to water is negligible compared to the absorption due to both oxy- and deoxyhemoglobin at those points. At 810 nm, the absorption coefficients of both hemoglobins are equal and at 940 the absorption coefficients the absorption coefficient of oxyhemoglobin is bigger than the one of deoxyhemoglobin (in contrast to 660 nm).
CHAPTER 2. THEORETICAL BACKGROUND
This makes 660, 810 and 940 nm interesting wavelengths when one wants information about the oxygen saturation of the blood. Furthermore, from approximately 1150 nm the absorption coef-ficient of water is bigger than that of oxy- and deoxyhemoglobin, with the biggest peak at 1440 nm. At 1310 nm, the absorption coefficients of Hb and HbO2 are approximately equal again. All these interesting wavelengths lie within the range of 500 nm - 1500 nm.
2.1.2
Reduced scattering coefficient
In an event of scattering the photon is deflected in a different direction of propagation, but the energy and therefore the wavelength and frequency stays unchanged [8]. The reduced scattering spectra look different than the absorption spectra. No characteristic peaks, but rather a smooth decaying graph. Data of reduced scattering coefficients at different wavelengths can be fitted to the following equation:
µ0s= a λ
500 (nm) −b
(2.3) where wavelength λ is normalized by a reference wavelength of 500 nm to get a dimensionless value. That term is raised to a power b, the ’scattering power’. The factor a in front is the value of the reduced scattering coefficients at 500 nm [6].
Alternatively, the data can be fitted to an equation where the wavelength dependency is de-scribed in separate contribution for Rayleigh and Mie scattering. However, for predicting beha-viour of light diffusion within the range of 400 - 1300 nm wavelengths, both equations are equally good [6]. Since this is the most interesting range of wavelengths for this research, the first most simple equation2.3is used. Although the reduced scattering coefficient cannot tell us that much about the physiological properties of the tissue, it is really important and cannot be neglected. As said before, it is the reason that not all the photons travel the same distance and why a more complicated model is needed.
2.2
Diffusion Approximation
As a result of scattering, the final detected photo signal is a mixture of photons having travelled different path lengths [7]. Therefore, this simple Lambert-Beer equation is not sufficient to describe light travelling through highly scattering media as biological tissue.
2.2.1
From Radiative Transer Equation to Diffusion Equation
There are multiple ways to describe light propagating through biological tissue, one of them being the radiative transfer equation (RTE). This equation that describes the physics of light propagation is based on the principle of conservation of energy. I will not discuss the RTE in all completeness, but rather give a general overview of what it comprises. Basically, the RTE is a balance where all mechanisms that make the radiance at a certain wavelength decrease or increase in a volume element dV of the medium are represented. The RTE can be written as [8]:
∂
v∂tI(~r, ˆs, t) + ∇ · [ˆsI(~r, ˆs, t)] + µtI(~r, ˆs, t) = µs Z
4π
p(ˆs, ˆs0)I(~r, ˆs, t)dΩ0+ (~r, ˆs, t), (2.4)
where v is the speed of light in the medium and µt is the sum of the absorption and scattering
coefficient. is the power emitted at time t per unit volume and unit solid angle along ˆs, also called the source term and dΩ0 is the unit solid angle in the direction ˆs0. Considering a volume
element dV at position ~r, the terms in equation2.4represent the following: • ∂
v∂tI(~r, ˆs, t)dV dΩdt is the total temporal change of energy propating in direction ˆs within
dV , dΩ and dt. This term equals zero for steady-state sources.
CHAPTER 2. THEORETICAL BACKGROUND • ∇ · [ˆsI(~r, ˆs, t)]dV dΩdt = ˆs · [∇I(~r, ˆs, t)]dV dΩdt = ∂I(~∂sr,ˆs,t)dV dΩdt represents the spacial change or net flux of the energy of the photons propagating along ˆs within dV , dΩ and dt. • µtI(~r, ˆs, t)dV dΩdt represents the fraction of the energy moving along ˆs within dΩ and dt
that is extracted by scattering and absorption.
Because of the principle of conservation of energy, this energy changes must equal the sum of: • µsR
4π
p(ˆs, ˆs0)I(~r, ˆs, t)dΩ0dV dΩdt, which represent the energy coming from any direction ˆs0,
scattered in direction ˆs.
• (~r, ˆs, t)dV dΩdt representing the energy generated by a source inside dV along ˆs within dΩ and dt.
Getting solutions for the RTE is a highly computational process, usually based on numerical methods. There are no general analytic solutions and that is why usually approximate models are used [8]. The diffusion approximation is the most widely and successfully used approximation to get solutions for both time-dependent as steady-state sources [8].
The diffusion approximation consists of two simplifying assumptions. The first one being that the radiance inside the diffusive medium is almost isotropic. With ’almost isotropic’ is meant that only an isotropic term and a linear anisotropic term of a series expansion of the radiance is taken into account: I(~r, ˆs, t) = 1 4πφ(~r, t) + 3 4π ~ J (~r, t) · ˆs (2.5) where φ(~r, t) is the fluence rate. The fluence rate is defined as the radiance integrated over the whole solid angle: R
4π
I(~r, ˆs, t)dΩ. Therefore φ(~r, t) equals the average radiance when divided by 4π (the whole solid angle) as in2.5. ~J (~r, t) in the second term of equation2.5 is the flux vector. The flux vector is defined as R
4π
I(~r, ˆs, t)ˆs which represents both the amount and direction of the net flux of power.
Secondly, only ’slow time variation’ of the flux of power is assumed. This means that the time variation of the flux vector ~J (~r, t) over a time interval ∆t = 1/(vµ0s) is negligible with respect to the vector itself. In formula this can be written as:
1 vµ0 s ∂ ~J (~r, t) ∂t ~ J (~r, t) (2.6)
These assumptions can only be made when the direction of light propagation is random. In other words, these assumptions can be done when photons have undergone many scattering events, since scattering tends to randomize the direction of light propagation [8]. However, this so-called diffusive regime can be obstructed when there is too much absorption. Since photons travelling a longer path length have a bigger chance to get absorbed, only photons that have undergone a few scattering events will not be absorbed. In conclusion, the diffusion approximation is valid when scattering events are predominant on absorption [8].
With the diffusion approximation, the diffusion equation (DE) can be derived from the RTE. The diffusion equation is a simpler equation, with several analytical solutions available. This deriv-ation is described by Martelli, Del Bianco, Ismaelli and Zaccanti in their book ’Light Propagderiv-ation through Biological Tissue’ [8]. The derivation is also shown in Appendix A. We end up with a solution for the Diffusion equation in the form of the time-resolved Transmittance for a slab geometry: T (t) = +∞ Z 0 T (ρ, t)2πρdρ = exp(−µavt) 2(4Dvt)1/2t3/2 × m=+∞ X m=−∞ z1mexp − z 2 1m 4Dvt − z2mexp − z 2 2m 4Dvt , (2.7) where:
CHAPTER 2. THEORETICAL BACKGROUND
Figure 2.1: Time-dependent total transmittance (dashed line) from a slab with s = 40mm, mua =
0, mu0s= 1mm−1, ni = 1.4 and nr= 1.4 [8]
• v = speed of light in tissue • D = 1/(3µ0s),
• z1m= (1 − 2m)s − 4mze− zs,
• z2m= (1 − 2m)s − (4m − 2)ze+ zs and
• zs= 1/µ0s.
The z1m and z2m terms are a result of describing the right boundary conditions for the slab
geometry using the ”method of images”. In addition to the real source of light, an infinite number (hence the sum over m from −∞ to ∞) of positive and negative sources in an infinite medium with the same optical properties as the slab are used. In appendix A this is shown graphically as well. A plot of this the transmittance T(t) from a slab of thickness s = 40mm, mua = 0,
mu0s= 1mm−1 and nr= 1.4 is shown in figure2.1.
Lastly, the sum in equation 2.7 over m is an infinite sum. For a practical application, for example a simulation, this sum should be truncated [8]. The total transmittance can be calculated by integrating equation2.7over the time, or over distance when the time is converted to distance firstly. This is the same as taking the area under the graph of T(t) vs. time, a specific sample of which is shown in figure2.1. As mentioned in the introduction, the transmittance is changing as result of the pulsation of blood. One can calculate the total transmittance for the minimum of the pulse and the maximum of the pulse to calculate dI and I (the pulsating intensity signal and baseline intensity signal). Although equation2.7is way more complex than the Lambert-Beer equation with DPF factor which was shown in the introduction, we can still take the Ratio of Ratio’s when we calculate dI and I for two different wavelengths.
Chapter 3
Method
The aim of this research is to make a simulation where the volume fractions of different components can be changed in order to see what the effect of these changes is on the Ratio of Ratio’s. In this chapter is described how this is done.
3.1
The diffusive equation model
The function used in the simulation to model the transmittance is the time-resolved solution for transmittance of the Diffusive Equation 2.7. For the simulation, the sum over m is truncated at m = -100 at the lower limit and m = 100 at the upper limit. The transmittance is written in this equation as a function of absorption coefficient µa, reduced scattering coefficient µ0s (hidden in
the Diffusion Coefficient) and time t. The values or definitions of the other variables in equation
2.7 are shown in table 3.1. The time is converted to path length by multiplication with the speed of light within the tissue v, while the time-dependent transmittance is converted into a path length dependent transmittance bij dividing by v. Next, the total transmittance of all photons together can be calculated by integrating over all path lengths. Because absorption and reduced scattering coefficients are wavelength dependent, the transmittance is as well. Therefore, the total transmittance can be calculated for every wavelength. In this simulation, wavelengths ranging from 500 to 1750 nm with steps of 5 nm are used.
Variable Quantity Units
Speed of light in vacuum (c) 2.99792458e+8 mm/ms Thickness of the slab (s) 10 mm Refractive index diffusive medium (ni) 1.4
-External refractive index (ne) 1
-Speed of light in medium (v) c/ni mm/ms
Diffusion Coefficient (D) 1/(3 · µ0s) mm Coefficient for calculating extrapolation distance (A) 3.0 -Extrapolation Distance (ze) 2 · A · D mm;
Source Distance (zs) 1/µ0s mm;
Table 3.1: Variables and their values or definitions and units as used in the simulation, unless stated otherwise.
CHAPTER 3. METHOD
3.2
Volume fractions
To use the diffusion equation for slab geometry, the slab of tissue must be treated as a homogen-eous slab. As absorption and reduced scattering coefficients for the slab we use the sum of the absorption and reduced scattering coefficients of the different components. These are calculated by multiplying the coefficients of pure components with the associated fraction volume. The used volume fractions for calculating the absorption coefficient, unless stated differently, are shown in table 3.2. The ones for calculating the reduced scattering coefficients are shown in table3.3. In this research we distinguish arterial and venous blood and treat the rest of the tissue as skin. These three distinguishable tissues consist of different tissues themselves. If a volume fraction consist of multiple sub-volumes itself, these sub-volumes are given as volume fraction of the former volume fraction in the table. So to get the volume fraction of the total tissue, the successive relative volume fractions in the table need to be multiplied with each other.
Comp. Vol. frac. Comp. Vol. frac. Comp. Vol. frac. Comp. Vol. frac. Comp. Vol. frac.
Arterial blood 0.02 + 0.02 * pulse RBC 0.45 Dry RBC 0.8 HbX 0.95 HbO2 SpO2 Hb (1 - SpO2) Lipids 1-HbX Water 1-Dry RBC WBC 0.01 Plasma 1-RBC-WBC Water 0.95 Albumin 1-PlasmaWater Venous blood 0.06 RBC 0.45 Dry RBC 0.8 HbX 0.95 HbO2 SvO2 Hb (1 - SvO2) Lipids 1-HbX Water 1-Dry RBC WBC 0.01 Plasma 1-RBC-WBC Water 0.95 Albumin 1-PlasmaWater Skin 0.92 Melanosome 0.001 Fat 0.30 Water 0.70
Table 3.2: Volume fractions (Vol. frac.) of every component (Comp.) used as basis for the calculation of total absorption coefficient. These values can be adjusted in the simulation, by which the absorption coefficients get adjusted. pulse = fraction of total arterial blood volume that is added during pulsation, RBC = red blood cells, WBC = white blood cells, Dry RBC = dry constituents in red blood cells (assumed to be hemoglobins and lipids), PlasmaWater = water of plasma, HbX = all hemoglobins.
Comp. Vol. frac Comp. Vol. frac. Blood 0.06 + 0.02 + pulse * 0.02
Skin 0.92
Dermis 0.41 Epidermis 0.02 Subcutaneous Fat 0.57
Table 3.3: Volume fractions used as basis for the calculation of total reduced scattering coefficient. These values can be adjusted in the simulation, by which the reduced scattering coefficients get adjusted.
Furthermore, some components are present in multiple sub-volumes. Water for example, is present in red blood cells, in blood plasma of both arterial and venous blood and in skin. The volume fraction of the total volume can be calculated by adding volume fractions of different 12 The effect of hematocrit on the Near Infrared Spectroscopy signal at different wavelengths.
CHAPTER 3. METHOD Component Volume fraction of total tissue volume
HbO2 Arterial Blood * RBC * Dry RBC * HbX * (SpO2) + Venous Blood * RBC * Dry RBC * HbX * (SvO2) Hb Arterial Blood * RBC * Dry RBC * HbX * (1- SpO2)
+ Venous Blood * RBC * Dry RBC * HbX * (1- SvO2) Lipids Arterial Blood * RBC * Dry RBC * Lipids
+ Venous Blood * RBC * Dry RBC * Lipids WBC Arterial Blood * WBC + Venous Blood * WBC
Albumin Arterial Blood * Plasma * Albumin + Venous Blood * Plasma * Albumin Melanosome Skin * Melanosome
Fat Skin * Fat Water
Arterial Blood * RBC * Water (RBC) + Arterial Blood * Plasma * Water (Plasma) + Venous Blood * RBC * Water (RBC) + Venous Blood * Plasma * Water (Plasma) + Skin * Water (Skin)
Table 3.4: how to calculate the volume fraction of the total volume of the tissue for every com-ponent of the tissue. The names in this table represent the volume fractions as indicated in figure
3.2when not indicated otherwise.
sub-volumes. This is done for all components in table3.4. The same can be done for the volume fractions for calculating the reduced scattering coefficient.
3.3
Absorption coefficient
In order to get the total absorption coefficient of the slab, one needs to multiply the volume fractions with the absorption coefficients of the pure components and add all those together. The data for absorption coefficients was not always provided in the right units. So firstly, everything had to be converted to absorption coefficients in mm−1 using equation 2.2. The conversion is
shown in the attached data files, including the used values for molar weight and concentration. For eumelanin and pheomelanin no complete data set was found for the desired wavelengths and although the concentration of melanins is relatively low, their absorption is high enough to make a fair contribution to the total absorption. Therefore we cannot neglect them. But as Steven L. Jacques states on his website [5], the concentration of melanins in melanosomes and consequently the total absorption and scattering properties can vary ten-fold. So using separate absorption coefficients for different types of melanins and multiplying those with their concentrations does not necessarily give accurate values for the absorption coefficients of all melanins together. Jacques suggests to use a formula that approximate the general shape of the melanosome absorption spectrum in skin: µa,melanosome= 1.70e11 ∗ λ3.48, where λ is the wavelength in nm [5]. The total
absorption coefficients are calculated using the total volume fractions given in table3.4. The total absorption spectra of arterial blood, venous blood and skin are shown in figure3.1. Figure3.2is a more zoomed-in picture of figure3.1.
3.4
Reduced scattering coefficient
For the wavelength dependent reduced scattering coefficients equation2.3was used. The wavelength, normalised by reference wavelength 500 nm is raised with the scattering power b. Next, the equa-tion was scaled by substituting the value of the reduced scattering coefficient at the reference wavelength into a: µ0
s(λ = 500nm). The scaling factor and scattering power for dermis, epidermis
and subcutaneous fat were given by experiments of Salomatima et al. (2006) [10]. For blood the values were from Alexandrakis et al. (2005) [1]. The spectra are shown in figure1.2.
CHAPTER 3. METHOD
Figure 3.1: The absorption coefficient of arterial blood, venous blood, and skin taken at a 100%. In cyan the total absorption coefficient is shown, where every component is multiplied by the associated volume fraction.
Figure 3.2: A zoom of figure3.1. The absorption coefficient of arterial blood, venous blood, and skin taken at a 100%. In cyan the total absorption coefficient is shown, where every component is multiplied by the associated volume fraction.
CHAPTER 3. METHOD
3.5
The effect of changing volume fractions
When the total absorption and reduced scattering , one can pick the wavelengths to calculate the Ratio of Ratio’s with. First, as a validation of the simulation, the RoR-SpO2 relation will be examined at the usual wavelengths in pulse oximetry 660 nm and 940 nm for SpO2 values ranging from 0.8 to 1. The relation will be examined at different slab thicknesses, pulse sizes and melanosome concentrations. The ranges of change of the variables are given in table3.5. Secondly,
Quantity/Component Range of change Slab thickness s 8 - 16 mm [2 mm steps]
Pulse 0.02 - 0.10 [fraction of arterial blood volume, steps of 0.02] Melanosome fraction 0.0002 - 0.001 [0.0002 steps]
Table 3.5: The ranges of change in slab thickness, pulse and melanosome concentration for ex-amining the RoR-SpO2 and RoR-Hct relation.
the RoR-Hct relation will be examined. This will be done by plotting the RoR for a Hct ranging from 0.3 to 0.6, the range that is expected to be found in human blood [9]. When changing the Hct values over this range, we assume that this does not affect the ratio of components inside the red blood cells. The effect of pulse size, slab thickness and melanosome concentration is examined by applying the same ranges of change as in table3.5. Next to this changes, the RoR-Hct relation will also be evaluated at different SpO2 values and different wavelengts combinations. The used SpO2 values and wavelengths combinations are shown in table3.6.
Quantity Used values
SpO2 0.80, 0.85, 0.90, 0.95, 0.98 Wavelength combination
810 nm & 1310 nm 940 nm & 1310 nm 940 nm & 1180 nm
Table 3.6: The used values for SpO2 and the used wavelength combinations for examining the relation between RoR and Hct.
The wavelength combination of 810 & 1310 nm is chosen because here the effect of SpO2 on the RoR-Hct relation is expected to be negligible. This is expected because the absorption coefficients of oxyhemoglobin and deoxyhemeglobin are the same at these two wavelengths, so changing the ratio between these two should not effect the total absorption. For practical reasons the other two wavelength combinations contain the 940 nm wavelength. This is because for other wavelength combinations than the 810 & 1310 nm combination, the SpO2 value will most probably have an effect. By including one of the wavelengths that is already used for SpO2 measurements, only one extra wavelength should be added to measure SpO2 at the same time and to be able to correct for the effect of SpO2. The 1310 nm wavelength is then chosen to minimize the SpO2 effect, since oxyhemoglobin and deoxyhemoglobin have the same absorption coefficient at this wavelength. The 1180 nm is chosen because it is closer to the 940 nm wavelengths, so the difference in scattering is smaller and at 1180 water has a higher absorption coefficient than both oxyhemoglobin and deoxyhemoglobin.
Chapter 4
Results
4.1
RoR and SpO2
The Ratio of Ratio’s was plotted for a SpO2 range of 0.8 to 1.0. The RoR was calculated using the commonly used wavelengths for SpO2 assessments: 660 nm and 940 nm. This was done for the different pulse sizes, slab thicknesses and melanosome volume fractions from table 3.5. The graphs are respectively shown in 4.1, 4.2 and 4.3. In Appendix B a zoom-in of these graphs is depicted. The RoR-SpO2 graphs for different pulse sizes and different slab thicknesses overlap
Figure 4.1: RoR (660 & 940 nm) vs. SpO2 graph for a pulse of 0.02, 0.04, 0.06, 0.08, 0.10. almost entirely. The effect of the melanosome fraction in skin is far more noticeable. To get a better view of the differences, graphs of RoR vs. pulse size, slab thickness and melanosome fraction are plotted for different SpO2 values. These are shown in respectively figure4.4,4.5and4.6. The mean values and the standard deviations of the RoR for the different SpO2 values in figure 4.4) are given in table4.1. The same is done for figure4.5in table 4.2and for figure4.6in table4.3. The deviation of the mean as result of different pulse sizes gets bigger as the SpO2 value gets higher. Nevertheless, the biggest deviation (at SpO2 = 0.98) is still less than 0.2 percent of the mean. The effect of the different slab sizes on the RoR-SpO2 relation is bigger than the effect of pulse. The biggest deviation here (again at the highsest SpO2 value) is 0.51 percent. Clearly, the effect of the melanosome fraction is the biggest. In contrast to the effect of pulse size and slab thickness, the effect of different melanosome fractions gets smaller with higher SpO2 values. For a common healthy SpO2 value of 0.98, the deviation of the mean RoR is 2.8 percent. Furthermore,
CHAPTER 4. RESULTS
Figure 4.2: RoR (660 & 940 nm) vs. SpO2 graph for a slab thickness of 8, 10, 12, 14 and 16 mm. SpO2: 0.8 0.85 0.90 0.95 0.98
Mean RoR 0.98318 0.85016 0.72020 0.59313 0.51821 Stand. Dev. 3.3E-05 0.00037 0.00065 0.00082 0.00087
Table 4.1: Mean values of RoR (660 & 940 nm) over different pulse sizes for different SpO2 values. the higher the melanosome fractions get, the smaller their relative effect on the Ratio of Ratio’s.
SpO2: 0.8 0.85 0.90 0.95 0.98 Mean RoR 0. 97456 0.84985 0.72054 0.59378 0.51888 Stand. Dev. 0.0019 0.00215 0.00231 0.00251 0.00266
Table 4.2: Mean values of RoR (660 & 940 nm) over different slab thicknesses for different SpO2 values.
CHAPTER 4. RESULTS
Figure 4.3: RoR (660 & 940 nm) vs. SpO2 graph for a melanosome fraction of skin of 0.0005, 0.0010, 0.0015, 0.0020, 0.0025, 0.0030
SpO2: 0.8 0.85 0.90 0.95 0.98 Mean RoR 0.94319 0.82124 0.70210 0.58564 0.5169 Stand. Dev. 0.06948 0.05376 0.03819 0.02298 0.01441
Table 4.3: Mean values of RoR (660 & 940 nm) over different melanosome fractions for different SpO2 values.
CHAPTER 4. RESULTS
Figure 4.5: RoR (660 & 940 nm) vs. Slab thickness for different SpO2 values.
Figure 4.6: RoR (660 & 940 nm) vs. Melansome fraction for different SpO2 values.
CHAPTER 4. RESULTS
4.2
RoR and Hct
Now we have seen what the effect of different pulse sizes, slab thicknesses and melanosome fractions is on the RoR-SpO2 relation, we will do the same for the RoR-Hct relation. The effect of different SpO2 values will also be assessed. This will be done at three different wavelength combinations for which there is a linear relation between the RoR and Hct value: 810 and 1310 nm, 940 and 1310 nm and 940 and 1180 nm.
The effect of SpO2 values, pulse sizes, slab thicknesses and melanosome fractions on the RoR-Hct for wavelengths 810 nm and 1310 nm are shown in figures4.7, 4.10. 4.13and4.16relatively. In figures4.8, 4.11,4.14and 4.17these effects are shown for the wavelength combination of 940 nm and 1310 nm. Lastly, figures 4.9, 4.12, 4.15 and 4.18 show the effects with a wavelength combination of 940 nm and 1180 nm.
4.2.1
SpO2 Value
The effect of different SpO2 values is negligible for the wavelength combination of 810 nm and 1310 nm. This makes sense, since the absorption of deoxyhemoglobin and oxyhemoglobin at these wavelengths are equal. Therefore, a change in ratio between these two hemoglobins does not effect the absorption and consequently the Ratio of Ratio’s. At the other two combinations, the effect of the SpO2 value was bigger. On the positive side though, these combinations both include a wavelength of 940 nm. This means only adding the 660 nm wavelength to the combination gives the opportunity to measure the SpO2 value at the same time and to get rid of this uncertainty.
Figure 4.7: RoR (810 and 1310 nm) vs. Hct for different SpO2 values.
4.2.2
Pulse Size
For all wavelength combinations the pulse size does not have an effect on the relation of RoR-Spo2, as can be concluded from the graphs for different pulse sizes overlapping each other. A small deviation can be seen for the higher RoR values, but this deviation is negligible compared to the mean Hct values.
4.2.3
Slab Thickness
The effect of slab thickness is can be seen in the graphs of all three wavelength combinations. In figure4.13, 4.14and4.15 a line is drawn at a RoR value of 2.0. The difference in Hct value for
CHAPTER 4. RESULTS
Figure 4.8: RoR (940 and 1310 nm) vs. Hct for different SpO2 values.
Figure 4.9: RoR (940 and 1180 nm) vs. Hct for different SpO2 values.
the outermost slab thicknesses (8 mm and 16 mm) can be read off those graphs. The smaller the difference, the better. Furthermore, a bigger slope of the graph also gives a better result, since a small change in Hct then gives a bigger change in RoR. The difference in outermost Hct values at a RoR of 2.0 and the mean slope are given in table4.4.
4.2.4
Melanosome Fraction
Also for melanosome fraction the difference in outermost Hct values (at melanosome fractions of 0.0005 and 0.003 of the skin) at a RoR value of 2.0 and the mean slope of the RoR-Hct graphs are calculated. The results are shown in table4.5.
Figure 4.10: RoR (810 and 1310 nm) vs. Hct for different pulse sizes.
Figure 4.13: RoR (810 and 1310 nm) vs. Hct for different slab thicknesses.
Figure 4.14: RoR (940 and 1310 nm) vs. Hct for different slab thicknesses.
Wavelength combination Difference in Hct at RoR of 2.0 Mean slope 810 nm & 1310 nm 0.030 4.23
940 nm & 1310 nm 0.020 4.47
940 & 1180 nm 0.026 3.57
Table 4.4: The difference in outermost Hct values (at slab thickness of 8 mm and 16 mm) at a RoR value of 2.0 and the mean slope of the RoR-Hct graph of the different slab thicknesses.
Figure 4.15: RoR (940 and 1180 nm) vs. Hct for different slab thicknesses.
Figure 4.18: RoR (940 and 1180 nm) vs. Hct for different melanosome fractions.
Wavelength combination Difference in Hct at RoR of 2.0 Mean slope 810 nm & 1310 nm 0.132 3.967
940 nm & 1310 nm 0.061 4.300
940 & 1180 nm 0.072 3.467
Table 4.5: The difference in outermost Hct values (at melanosome fractions of 0.0005 and 0.003 of the skin) at a RoR value of 2.0 and the mean slope of the RoR-Hct graph of the different melanosome fractions.
Chapter 5
Conclusions
When looking at the results for the RoR-SpO2 relation (for validating the simulation) we see that the effects of pulse size and slab thickness on the RoR-SpO2 relation at wavelengths 660 nm and 940 nm are negligible. This is in line with known results of pulse oximetry. It is the charm of the Ratio of Ratio’s: the effect of a lot of variables cancel out. However, we do see a clear effect of the melanosome fraction. With a high concentration of melanosomes, one measures a relatively low SpO2 value. This effect gets bigger for lower SpO2 values. The melanosome fraction here is a volume fraction of the total skin, but melanosomes are only found in the epidermis. The epidermis takes only about 2 percent of the total tissue volume [3]. Therefore, the changes in fraction made here are relatively big: The melanosome fraction ranges from 0.05 to 0.30 percent of the total skin. That means a multiplication of 6 times. When this is all concentrated in 2 percent of the skin, it makes a big difference in concentration. For example, 0.05% of the total skin correspond to 0.05/2.0 ∗ 100 = 2.5% of the epidermis and 0.30% to 0.30/2.0 ∗ 100% = 15% of the epidermis. That is a difference of 12.5% of the volume of the epidermis. This effect however might be smaller in real life. Since the melanosomes are only present in the epidermis, the path length will be changed only on a small part of the total path length as a result of a change in melanosome fraction. For the biggest part of the tissue (everything except for the epidermis) the path length will stay the same. To get a better understanding of how big the effect is in real tissue, a better estimation of melanosome concentration is needed. When at the RoR-Hct relation, we see that the melanosome fraction is of importance here as well. Again, this big effect might be due to a disproportional increase in melanosome fraction compared to real differences in different skin types. However, we can still compare the three wavelength combination to see which one is best. A RoR-Hct relation is best when the slope of the graph is the biggest (so only a small difference in Hct gives a big difference in RoR value) and the difference in Hct value for a certain RoR value as a result of different SpO2 values, pulse sizes, slab thicknesses or melanosome concentration is the smallest. For all three wavelength combinations, the effect of the pulse size was negligible. The effect of SpO2 values was only negligible for the wavelength combination of 810 nm and 1310 nm as a consequence of deoxyhemoglobin and oxyhemoglobin having the same absorption coefficiens at these wavelengths. The effect was bigger for the other two wavelength combinations, but these combinations both contained the 940 nm wavelength. This wavelength is already used to measure SpO2 value (together with 660 nm), so one could add a 660 nm wavelength photo emitter to measure the SpO2 value at the same time. For both slab thickness and melanosome concentration, the best used wavelength was not directly clear. Therefore, in table 4.4and 4.5 the mean slope and the difference in Hct value at an RoR value of 2.0 as a result of change in respectively slab thickness and melanosome concentration are shown. For both slab thickness and melanosome fraction change, the wavelength combination of 940 and 1310 gives the best result: biggest slope and smallest difference in Hct value for a fixed RoR value. So the wavelength combination of 940 and 1310 nm gives the smallest effect of slab thickness and melanosome fraction. The effect of melanosome fraction can be, as already discussed, unproportional to real differences in skin types. For the slab thickness, an example of the effect can be read from table 4.4. When measuring a
CHAPTER 5. CONCLUSIONS
RoR value of 2.0, doubling the slab thickness from 8 mm to 16 mm gives a difference in Hct value of 0.020. This is a relative difference of 0.020/0.362 ∗ 100 = 5.5% of the lowest boundary Hct value. Usually, the effect of most variables are cancelled out when taking the Ratio of Ratio’s. Therefore, one can wonder whether the effect of slab size is an artefact of the simulation or something that is happening physiological as well. The biggest limitation of the simulation is that it is treating the slab of tissue as a homogeneous slab. Therefore, a change in a specific part of the slab is physiologically in real life only changing the absorption and scattering of some photons (the ones travelling through that specific part), while in the simulation it is effecting all photons. But as the slab thickness is something that is effecting all photons, both in real life as in the simulation, chances are small that this effect is because of this limitation of the simulation. Since a thicker slab means a longer path length, meaning more scattering and consequently a bigger chance of absorption, it is plausible that the effect of slab thickness is something that is actually measurable in real life as well. The possibilities for an integrated slab thickness measurement in a future appliance is something that could be explored in following research. Another important topic that I would suggest for future research is the effect of different scattering regimes. The further apart the used wavelengths, the more different the scattering behaviour of the light. So using two wavelengths that are far apart from each other does not give the right results when using the same scattering coefficients for the whole range of wavelengths.
In conclusion, for the chosen scattering coefficients in equation2.3, the wavelength combination of 940 and 1310 nm gave the best results for measuring Hct. However, in future research the effect of different scattering regimes at these and possibly other wavelength combination should be examined. Of course, although the results shown here look promising, also comparison studies with NIR spectroscopy with real tissue should be performed. Lastly, more research has to be done in order to get a better understanding of how Hct is related to the hydration status of a patient.
Bibliography
[1] G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou. Tomographic bioluminescence imaging by use of an optical-pet (opet) system, a computer simulation feasibility study. Physics in Medicine & Biology, 50(17):4225–4241, 2005. 2,13
[2] L.E. Armstrong. Hydration Assessment Techniques. Nutrition Reviews, 36(6):S40 – S54, 2005. 1
[3] A N Bashkatov, E A Genina, V I Kochubey, and V V Tuchin. Optical properties of human skin, subcutaneous and mucous tissues in the wavelength range from 400 to 2000 nm. Journal of Physics D: Applied Physics, 38(15):2543, 2005. 27
[4] T.J. Farrel, M.S. Patterson, and B. Wilson. A diffusion theory model for spatially resolved, steady state diffusive reflectance for the noninvasive determination of tissue optical properties in vivo. Med.Phys, 19:879–888, 1992. 34
[5] S. L. Jacques. Melanosome absorption coefficient, July 1998. 13
[6] S. L. Jacques. Optical properties of biological tissues: a review. Physics in Medicine and Biology, 58:R37 – R61, 2013. 2,7,8
[7] P. D. Mannheimer. The Light–Tissue Interaction of Pulse Oximetry. Anesthesia & Analgesia, 105(6):S10–S17, 2007. 3,5,8
[8] F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti. Light Propagation through Biological Tissue and Other Diffusive Media. Bellingham, Washington USA: Spie Press, 2010. 1, 5, 8,
9,10,32,33
[9] L. J. Martin. Hematocrit, March 2016. 1,15
[10] E. Salomatina, B. Jiang, J. Novak, and N. Yaroslavsky, A. Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range. Journal of Biomedical Optics, 11(6):064026–1 – 064026–9, 2006. 2,13
[11] S. Samoni, V. Vigo, L. I. B. Res´endiz, G. Villa, S. De Rosa, F. Nalesso, ..., and F. Forfori. Impact of hyperhydration on the mortality risk in critically ill patients admitted in intensive care units: comparison between bioelectrical impedance vector analysis and cumulative fluid balance recording. Critical Care, 20(1):95, 2016. 1
Appendix A: From RTE to a
matlab implementation of
Diffusion Equation
After intergrating both sides of the RTE over all directions and rewriting it by changing order of derivatives and integrals a new version of the RTE is obtained. When this is combined with the assumptions of the diffusion approximation, one obtains Fick’s law:
~ J ( ~r, t) = −D ∇φ(~r, t) − 3 Z 4π (~r, ˆs, t)ˆsdΩ , (1)
with the physical meaning that photons tend to migrate towards those regions in the medium where the photon density is smaller. D in this equation is the diffucient coefficient and is defined as:
D = 1 3(µa+ µ0s)
. (2)
However, since the DE is only applicable in the diffusive regime where µa µ0s, the diffusive
coefficient used here will be defined as:
D = 1 3µ0
s
. (3)
All the terms discussed above are defined with reference to the more general case of a time-dependent source. In many application however, the medium is illuminated by Continuous Wave (CW) sources, which emit constant power (a steady-state source). In that case the radiance I, the fluence φ and the flux ~J are not time dependent and can be written as I(~r, ˆs), φ(~r) and ~J (~r). So, after a few more steps, the DE for a homogeneous medium with isotropic source can be written as: −D∇2+ µ a φ(~r) = q0(~r), (4) with q0(~r) = Z 4π (~r, ˆs)dΩ = 4π(~r). (5)
Boundary Conditions
The physics of the problem investigated is what determines the boundary conditions of this partial differential equation. For this research we are looking at a slap of biological tissue with finite boundaries. Therefore we will be looking for the boundary conditions for the DE at the interface between diffusive media (inside the tissue) and non-scattering media (outside the tissue slab). Important to note here is that we do not look at different layers of diffusive media. The slab of biological tissue is treated as being homogeneous.
BIBLIOGRAPHY
Figure 1: Interface of diffusive medium bounded by a non-scattering (external) medium, with the used notation [8]
Looking at a surface Σ between a diffusive medium bounded by a non-scattering medium as shown in figure 1 Martelli et al. set the following boundary condition for the radiance I [8]: There should be no diffusive light entering the diffusive medium from the external non-scattering medium (since the light cannot be scattered in the external medium). This means that the diffusive radiance I(~r, ˆs, t)(~r ∈ Σ) can only originate from reflections at the boundary. With reference to figure 1, I(~r, ˆs0, t) is the radiance along ˆs0 and I(~r, ˆs, t) is the radiance along the mirror image
direction along ˆs. The given boundary condition result in the following equation:
I(~r, ˆs, t) = RF( ˆs0· ˆq)I(~r, ˆs0, t), (6)
where RF is the Fresnel reflection coefficient that is dependent on nr= ni/ne, the relative
refract-ive index. It is assumed that equation 6 is true on average for all directions ˆs inwardly directed [8]. Now the boundary condition for the total radiance coming from Σ van be obtained:
− Z ˆ s·ˆq<0 I(~r, ˆs, t)(ˆs · ˆq)dΩ = Z ˆ s·ˆq>0 RF(ˆs · ˆq)I(~r, ˆs, t)(ˆs · ˆq)dΩ (7)
Now we can substitute equation2.5for I(~r, ˆs, t) into7 and solve the integrals. After making use of Fick’s law1 we get a boundary condition in terms of the fluence rate φ(~r, t) =R
4π I(~r, ˆs, t)dΩ: φ(~r, t) − 2AD ∂ ∂qφ(~r, t) ~r∈Σ= 0, (8) where coefficient A is dependent of RF and therefore dependent on nr. The dependency of A is
shown in figure2. The equation shows us that the derivative of φ(~r, t) along the direction ˆq normal to the surface is proportional to φ(~r, t) itself. By assuming that ∂q∂ φ(~r, t) remains constant in the non-scattering medium to the value on the boundary, we can extrapolate φ(~r, t). The extrapolated distance Ze is defined as the distance from the boundary where φ(~r, t) is extrapolated to zero.
This definition gives us [8]:
Ze= 2AD. (9)
BIBLIOGRAPHY
Figure 2: The dependency of coefficient A on nr, the relative refractive index. A = 1 if nr = 1
and A > 1 if nr6= 1. Where nr= 1.4, A equals 3 as indicated with the red lines.
Solutions For Diffusion Equation for The slab Geometry
To get the final solutions of the DE, we make use of the ’method of images’. By the method of images, the correct boundary conditions are obtained by using, in addition to the real source an infinite number of positive and negative sources in an infinite diffusive medium that has the same optical properties as the slab. This image sources are shown in figure3. In this way, the solution for the slab geometry can be written as a superstition of solutions of a infinite diffusive medium (solution of infinite medium given by Martelli et al [8]). The solution obtained are for a spatial and temporal Dirac delta source: q(~r, t) = ∂3(~r − ~r
s)∂(t) at ~rs= (0, 0, zs) with 0 < Zs< s. The
sources are placed along the z-axis at the distances listed below: • z+
m= 2m(s + 2ze) + zs
• z−
m= 2m(s + 2ze) − 2ze− zs
• m = 0, ±1, ±2, ..., ±∞
Adding all contributions of the source terms, the Green’s function for the fluence rate at point ~ r = (x, y, z) results in: φ(~r, t) = v (4πDvt)3/2exp − ρ 2 4Dvt − µavt × m=+∞ X m=−∞ exp −(z − z + m)2 4Dvt − exp −(z − z − m)2 4Dvt (10) Referring to the used terms of figure3, the time-resolved transmittance T (ρ, t) is the power crossing the surface at z = s, per unit area, at distance ρ =px2+ y2 from the z-axis with any exit angle.
This can, using1and2.5, being written as [8]:
T (ρ, t) = ~J (ρ, z = s, t) · ˆq. (11) Using Fick’s law1again, this can easily be rewritten in terms of the fluence rate solution10:
T (ρ, t) = −D ∂
BIBLIOGRAPHY
Figure 3: The method of image for the slab geometry. s is the thickness of the slab. z0+ is the real (spatial and temporal Dirac delta) source, the rest are all image sources of a infinite diffusive medium.
Substituting the Green’s function for fluence rate of equation10 and integrating over the entire exit surface gives the final equation for the total time-resolved Transmittance:
T (t) = +∞ Z 0 T (ρ, t)2πρdρ = exp(−µavt) 2(4Dvt)1/2t3/2 × m=+∞ X m=−∞ z1mexp − z 2 1m 4Dvt − z2mexp − z 2 2m 4Dvt , (13) where: • z1m= (1 − 2m)s − 4mze− zsand • z2m= (1 − 2m)s − (4m − 2)ze+ zs.
A plot of this the transmittance T(t) from a slab of thickness s = 40mm, mua= 0, mu0s= 1mm−1
and nr= 1.4 is shown in figure2.1.
The last task before implementing this final equation into a matlab script, is defining the distance zs, the coordinate of the source. Usually this is done by [4]:
zs= 1/(µa+ µ0s) (14)
As said before, from a practical point of view, the DE is only applicable with good accuracy in a diffusive regime. This means that µa µ0s, so the distance zsin equation 14can be written as:
zs= 1/µ0s (15)
[h]
Figure 4: Zoom-in of figure4.1. RoR vs. SpO2 graph for a pulse of 0.02, 0.04, 0.06, 0.08, 0.10. SpO2 values in the range 0.95-0.98.
Figure 5: Zoom-in of figure4.2. RoR vs. SpO2 graph for a slab thickness of 8, 10, 12, 14 and 16 mm. SpO2 values in the range 0.95-0.98.
