International Conference on Applied Mathematics and Pharmaceutical Sciences (ICAMPS'2012) Jan. 7-8, 2012 Dubai

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1 NEW MODIFICATION OF LAMBERT-BEER S LAW USING SIMULATION OF LIGHT PROPAGATION IN TISSUE FOR ACCURATE NON-INVASIVE HEMOGLOBIN MEASUREMENTS Ahmad Al Nabulsi,2, Omar Abdallah 2, Lutz Angermann and Armin Bolz 2 Abstract Based on the optical methods the concentrations of the hemoglobin fractions can be calculated in a few seconds. Many theories in near-infrared spectroscopy (NIRS) were treated in this field. A new representation will be exhibited here and verified. This representation governing the light transmittance, the fractional hemoglobin concentration and the change of the blood volume in the diastole and systole of the pulse. This representation has been improved depending mainly on modeling and simulation of the light propagation in tissue and improving Lambert-Beer s law. Diffusion equation has been used to model the light propagation in tissue with different boundary conditions. Finite element code has been developed. The poor modeling of Lambert- Beer s law leads to the idea of improving it by modifying it with other bilinear equation. In our application an earlobe sensor with an array of Light Emitting Diodes LEDs with different wavelengths and a photo detector were used to detect the transmitted light intensities. It was concluded that the new equation gives a good indication in calculating the hemoglobin concentration. After that an inverse problem to a system of linear equation will be ready to solve for the next step. In this system of equation illposed problem will be discussed. Keywords Hemoglobin Concentration, Lambert-Beer s law, light propagation and diffusion equation, finite element method INTRODUCTION TILL now, there is no available device which can measure accurately the glucose concentration, the total hemoglobin (THb) concentration or the fractional oxygen saturation (SaO2) non invasively [2]. Department of Mathematics, TU Clausthal, Germany. 2 Institute of Biomedical Engineering, Karlsruhe Institute of Technology, Karlsruhe, Germany{ahmad.nabulsi, armin.bolz}@kit.edu, abdallah@ibt.uni-karlsruhe.de, angermann@math.tu-clausthal.de The pulse oximeter has proven its effectiveness in clinical practice. An attentive and knowledgeable physician must be aware, however, its proper use and limitations. Only then it can be considered as a useful supplement to the care of patients. Dyshemoglobinemias by methemoglobin, carboxyhemoglobin, and Sulfhmoglobin can cause inaccurate pulse oximeter readings by absorption of one or both wavelengths. The measurement of spectroscopy is based on the interaction between light and tissue. Light propagation in turbid media like human tissue, can be considered as a process of energy diffusion, so that, propagation can be described using transport equation. But due to difficulty to solve it, approximation diffusion equation was derived [5] and [6], where the solution of the diffusion equation is only valid for the approximation µ a µ s, where µ a is the absorbtion coefficient, µ s is the reduced scattering coefficient. Therefore the solution of diffusion equation is expected to break down in high absorbtion turbid media. In the case of hemoglobin fractions, as shown in Fig., the difference between the extenction coefficients of methemoglobin, oxyhebmoglobin, reduced hemoglobin and carboxyhemoglobin allows to calculate its fractions in the blood at different wavelengths in the visible and NIR regions with the modified Lambert-Beer s law or other theories. Fig. 2 shows the principles of the transmission method. The modified Lambert-Beer s law is written in the form: I T = I exp( aµ a d+ G), () where, I is intensity of irradiated light, I T is intensity of transmitted light, µ a is absorption coefficient, d is thickness of the irradiated tissue, a is multiplication factor to consider the optical path length elongation of the photons due to scattering of the medium, G is a factor considering the geometry of the probe and the lost of photons due to scattering. This modification of Lambert-Beer s law introduce a good physical modeling to the light intensity behavior which is used in invasive measurements [], but unfortunately 6

2 Figure : Extinction coefficient of the different hemoglobin fractions. this equation suffers in our application from different weaknesses. The nonhomogeneous nature of the earlobe tissue is a big challenge which needs a practical manner to find meaningful approach to solve this problem. The photoplesmography (PPG) signal when it has been detected, has carried the whole important information from the interior tissue, but the main influence to construct it is coming from the diastole and systole of the blood. Therefore due to the difficulty to implement this equation on the blood from the surface of the tissue without removing the influence of the surrounding tissue we need in fact a special treatment of this problem. We have two terms to be computed and to be valid in each used wavelength in the equation of modified Lambert-Beer s law (a, G), so we lake the mathematical modeling of these terms. In the following chapters we will introduce a new methodology to solve these problems and develop a new version of Lambert-Beer s law to compute the hemoglobin concentration accurately. 2 MATHEMATICAL MODEL Our application is relaying on using a light sensor, which is constructed as two parts, the first one carrying different LED s with different wavelengths as light sources. The other side is carrying the detector to detect the transmitted PPG signal. The light spreads out from the light source to distribute in the earlobe tissue with a different behaviors. In the following work we noted that the light propagation in the earlobe tissue in radial distribution acting as Gauss function Fig. 3. In the forward distribution the light decaying exponentially. Tissue is a scattering and absorbing medium, and obviously has a higher refractive index that air, so a big portion of light will be reflected back on the air-tissue interface owing to a Fresnel boundary condition [9]. Havlin, Nossal, Weiss and co-workers examined properties of the surface emission profiles of particles (photons) injected into a turbid medium consisting of two layers [4] and [3]. The two layers differ in the coefficient that appears when internal absorption is modeled in terms of Beer s law. Therefore in the following we will model the earlobe as a different layers, and model the light propagation in the respective layers, and then compute the transmitted light intensity from the blood in the systole and diastole. A. Earlobe model The earlobe tissue will be modeled as described below. The earlobe tissue consists mainly of epidermis, dermis and subcutaneous. In the middle spreads mainly the blood which can be split into arterial and venous blood. We have also the additional arterial blood in the systole pulse, so that our aim is to build a mathematical model describing blood layers influence. To imagine the model see Fig. 2. In Fig. 2 Figure 2: Multi-layers Earlobe in the diastole case (left) and systole case (right). d,d 2,d 3,d B,d B,d 3,d 2,d are the thickness of the epidermis, dermis, subcutaneous, blood layer in the diastole and the systole case, subcutaneous, dermis and epidermis, respectively. B.Source Model u (x,z)= x on z=. (2) The light emitted from the source I = a φ rdr and spreads in some way until reaching the tissue surface with a different values. We modeled its behavior as Gauss function: u (x)=i exp( 2 (x m σ )2 ), (3) in my case m=, σ 2 is the variance. C. Light Modeling The first layer which simulates the epidermis is thin and a big ratio of the light is absorbed and scattered there. So the light propagation is modeled there like 62

3 D i u i + µ ai u i = S i (x,z) in i, u i =( n i n i ) 2 u i on Γ i, + 2b 2i 3µ ti u i (x,z)= on Γ 3i, u i (x,z) z (6) where i=2,3,...,7, Figure 3: Source and earlobe modeling, and light intensities portions the following, find u V such that: D u + µ a u = S (x,z) in, u = on Γ, (4) u (x,z) z + 2b 2 3µ t u (x,z)= on Γ 3, where = [ a,a ] [,z ] and u is the fluence W rate distribution in the first layer per, and the mm source function inside the layer S (x,y)= { u (x)µ s exp( µ t z)when (x,z),, otherwise, where a in mm is big enough to ensure that the light does not spread a way and a in mm is the source radius, D is the diffusion constant in mm, µ a is the absorption coefficient in mm, µ s is the scattering coefficient in mm, µ t is the attenuation coefficient in mm, n is the refractive index. The Fresnel boundary condition is included in b 2 : b 2 = (n2 + n2 2 ) + sin n ( 2 n ) 2n n 2 sin ( n 2 n ). (5) The whole optical properties and the other parameters are also have the same symbols in the respective layers. Figure 4: First layer with boundary conditions and internal sources in 2d, Γ,Γ 3,Γ 4 refers to the layers boundaries. The other six layers have the same model, so we have found u i V such that: S i (x,y)= { u (x)µ si exp( µ t i z i )exp( µ ti z) in i,, otherwise, (7) where i =[ a,a ] [z i,z i+ ], b 2i = (n2 i + n2 i+ ) + sin n ( i+ n i ) 2n i n i+ sin ( n i+ n i ). (8) D. Boundary Condition: Different kinds of boundary conditions in the boundary value problem of interest are used. Homogenous Dirichlet BC on Γ 2 Robin BC on Γ 3 Homogenous (Natural) Neumann BC on Γ 4 3 FINITE ELEMENT PROGRAMM Our FE programm is built practically depending on the open source (deal.ii), theoretically consists of the following parts [7], [8]: 3. Preprocessor Every finite element computation starts with partitioning the domain into a finite element mesh, which is called meshing. In this step the nodes and its coordinates, as well as the element and the boundary edges in 2d for example can be produced. Mesh generation is a research field on its own, and a lot of software has been generated which makes mesh generation an (almost completely) automated procedure. However in this context, we must point out to the regularity of the shape of the elements. In the case of triangles the ratio of the diameter of the element to the radius of the largest circle which fit in the triangle should be less than some constant. In the case that the generated mesh satisfies the ratio of the maximum diameter of the elements to the radius of the inner circle of an element must be less than some constant, then the mesh is quasi uniform. 63

4 A. Conforming Finite Element Method Let R 2 be decomposed into a set of elements T h so that = T T h T. (9) This set of elements are assumed to be closed, T T 2 = / for all T,T 2 T h, T T 2. The elements can be of various types:. Interior elements lying completely in the interior of the domain. Regarding to the element computation, we get the contribution of each element in the global matrix and the right hand side vector, element by element. 2. Boundary elements that share at least a side of the boundary of the domain. This type of elements has a great importance when we want to apply the boundary conditions, where we apply the boundary condition along the whole corresponding edge, not along the nodes as in d problems. The end points of each element are called vertices (or nodes), each vertex of T T h is also a vertex of any neighbor of T. T h is then called a triangulation. The space of piecewise polynomials of degree p N is P p (T h ) := { v : R : v T P p for all T T h }. The space of conforming finite elements of degree p is S p h :=P p(t h ) C () H (). () B. Order of Approximation Regarding to this concept the question arises what is the criterion to choose the order of approximation? The two standards which we depend on are the accuracy and the economy, so that we have to change the order of approximation with respect to them. C. Linear Approximation In this case p = and our piecewise linear finite element space consists of linear basis functions of the form φ I (x,y) = a + a x+a 2 y and has the following properties: Local support: For each node I the respective shape function is φ I (x,y) has a value at I, and zero otherwise. The graph of φ I (x,y) is connected to the other five neighboring nodes around I by linear lines. Linear independence: N i= α i φ i (x,y)= α i =. () Completeness: The linear combination of φ i should be able to exactly represent any linear polynomial in the domain. D. Quadratic Approximation In this case p = 2, the shape functions are quadratic polynomials and they should be linearly independent, complete and have a local support property that is only defined in one or two elements and the rest of elements has a zero value. E. Finite Element Solution Representation The approximated solution will be represented as follows: u f e = N i= u i φ i (x,y), (2) where N is the number of degrees of freedom. 3.2 Processor u i =( n i n i ) 2 u i on Γ i, (3) In this part of the code, element calculations are the heart of it, which include assembling element by element the matrix and right hand side, then the contribution of all elements build the structure of the global matrix and right hand side vector, after that we have to impose the boundary conditions. At the end the modified global matrix and right hand side vector will be created. A system of linear equation has to be solved, so that we have to find a suitable sparse matrix solver like CG iteration, and to deal with ill-posed problems and to find some optimization methods like Schur complement method. A. Element Calculation Using Galarkin method we can find the following weak formulation form D u h v h da+µ a u h v h da= S v h da+ u h Γ nvds+ u h Γ 3 nvds, (4) for all v h V h. To use finite element tools, the integral on in the continuous representation is summed up from the first element in to the last one in the triangulation. After that we do the integration for each element and sum up all these integrations, which is called looping on the whole element in programming language. For element T T, the corresponding matrix and right hand side are, for i, j=,2,3 M T [i, j]=d k Ni T N T j da+µ ak Ni T N T j da T - F Γ 3 Ni T N T j ds, (5) where F = 2b 2 3µ t, k=,...,7, R T [i]= S Ni T da+ u (x)ni T ds. (6) Γ T 64

5 B. Linear Mapping It is computationally easier to work on the unit element rather than to work on the physical one. If we have a generic triangle T T, then the linear mapping looks like in Fig. 5. Because of the linear map- Figure 5: Linear mapping ping, the piecewise linear polynomials N T i (x,y) which are defined on the generic physical element T, will be transformed to a piecewise linear polynomials on the unit element ˆN i (ε,η), where i=,2,3. Nˆ (ε,η) = ε η, (7) Nˆ 2 (ε,η) = ε, (8) Nˆ 3 (ε,η) = η. (9) Returning to (5), the following two parts NT i N T i are needed: N T i N T i = ˆN i = ˆN i Here the jacobian arises: J = ( and + ˆN i, (2) + ˆN i. (2) ) (22) and the determinant of the jacobian which will be a constant in the linear approximation, but no longer constant in higher order approximation. J = (23) x and y are functions of ε and η, thus ( ) ( ) dx dε = J dy dη (24) In order to compute dε,dη, we need to perform the inverse transformation ( ) ( ) dε = J dη dx, (25) dy where ( J = J ) (26) Therefore, = J, (27) = J, (28) = J, (29) = J. (3) (3) Substituting all above into in (2), the computation of this part of the matrix is done. M T [i, j]=d k ˆN i Nˆ j J da+µ ak ˆN i Nˆ j J da T T 2b 2 - ˆN i J ds, (32) 3µ t Γ }{{ 3 } to be computed only if Γ 3 T R T [i]= S ˆN i J da+ u (x) ˆN i J ds. Γ C. Assembling. Initialize M, R, the matrix and the right-hand side vector. 2. Loop over the elements. 3. Loop over the local degrees of freedom in each element. 4. Local to global enumeration. 5. Implementation of the Boundary Conditions: The job here is to identify the edges. After generating the mesh, we loop over the whole element edges, and by imposing some conditions related to the domain coordinate where a specific type of boundary condition is applied. After gathering these information, which element edges are laying on e.g. homogenous Dirichlet, non-homogenous Dirichlet and homogenous Robin boundary conditions can be decided, and from there we can find out which degrees of freedom of a domain has a signed values. We used standard deviation to observe the simulation performance. The default performance is mean square error, and this error is the residual in our boundary value problems. Our approximated error is as follows: Err= N N err 2, (34) 65

6 where err= D i u i + µ ai u i S i (x,z) L 2 (35), and u i is the corresponding approximate solution Layer.5 x Layer Layer 2 6 x Layer Figure 6: Convergence history layers, 2, 3, 4, linear approximation q, quadratic approximation q2, λ=66 nm Layer 3 x Layer Layer 2 2 x Layer Figure 7: Convergence history layers, 2, 3, 4, linear approximation q, quadratic approximation, λ=76 nm We note that the error of the solution is decreased whenever we do more refinements, and the quadratic approximation solution (q2) is better than the linear approximation solution (q), therefore the quadratic approximation solution will be considered in our future work. 3.3 Post-Processor After the numerical solution has been obtained, a new and a very important step of the work has to start to compute the transmitted intensity in each layer. Relaying on the resulted light intensities the transmittance term will be defined and used in the term of modifying Lambert-Beer s law tansmitted intensity Algorithm will compute the transmitted light intensity in each layer see the transmitted light intensities from the blood layer in Fig. 8. Algorithm. Find the transmitted light intensity in each layer,. Compute u i =( n i n i ) 2 u i on Γ i, (36) Where i=2,3,...,7. 2. Compute u z i. 3. The transmitted light intensity from each layer would be computed through the following relation: I T Ri = D i 2π a a u i z i dx. (37) 4. In the same way the transmitted light intensity would be computed on the tissue surface through the following relation: I T R = D 2π a MODIFICATION OF LAMBERT-BEER S LAW u dx. (38) a z As a good start we will implement Lambert-Beer s law on the blood layer in the systole and diastole cases as follows: I 4 = I 3 exp( µ a d), (39) I 4 = I 3 exp( µ a d ). (4) Where I 4,I 4 are the transmitted light intensity from the blood layer in the diastole and systole cases respectively, d,d = d + δd are the thickness of the blood layers in the diastole and systole cases respectively as well Fig. 8. The transmittance is the ratio of Figure 8: Blood layer in the diastole case (left) and systole case (right). the transmitted light intensity in the diastole case to the transmitted light intensity of the systole case, as seen in Fig. 8. T = I 4 I 4, (4) applying that to (39), we will get the following: T = exp( µ ad) exp( µ a d ), (42) 66

7 and this leads to T = exp( µ a d+ µ a (d+ δd)), (43) and this gives the following formula: lnt = µ a δd. (44) 4 RESULTS AND CONCLUSION 4. Modeling Lambert-Beer law is incorrect when applying to our application for finding the hemoglobin concentration non-invasively, so an expected error has to be modeled, Err=ln(T) µ a δd. (45) Our point here is to model this error as a function of µ a,δd in each wavelength. Depending on our Algorithm to describe the light propagation in tissue using the developed finite element code, we computed in the beginning the hemoglobin concentration of different ten persons invasively. So running the code by taking λ=66 and δd =.5,.,.5,...,.4 we found that in each time for a fixed µ a and the different values of δd a linear behavior. By changing the values of µ a we found the same behavior with a small shift, so each line will represent the error function with fixing the absorbtion coefficient and varying δd, in other way we can model the error as a function of δd and a fixed µ a, Err(.,δd) like in Fig. 9. Therefore we can model each line as Figure : The slop as a function of the absorption coefficient, λ=66. Figure : H as a function of δd, λ=66. From Fig. we also noted that with a good approximation by taking the mean value the following: H = m H δd+ h H (48) Substituting (47) and (48) in (54) yields a bilinear equation: Err(µ a,δd)=n µ a δd+ n 2 δd+ n 3. (49) Where n,n 2,n 3 R. Substituting (49) in (45) and dividing with δd > we get the next formula: Figure 9: function versus δd, λ=66. linear function like the following: Err(.,δd)=Mδd+ H. (46) In order to model the whole lines in Fig. 9, we model M and H as the following: From Fig. we noted that M = m M µ a + h M (47) (n + )µ a +(n 3 lnt)d= n 2, (5) where D = δd. We can write (5) as the representation: N (ε C + ε 2 C 2 + ε 3 C 3 + ε 4 C 4 )+(n 3 lnt)d= n 2, (5) where N = n +, and C,C 2,C 3,C 4 are the hemoglobin concentration fractions (methemoglobin, oxyhemoglobin, reducedhemoglobin, carboxyhemoglobin), and ε,ε 2,ε 3,ε 4 are the respective extinction coefficients Fig.. As seen in Table for a given µ a,δd and inserting them in the simulation as an input data for the blood layer, as a result from the simulation we get the term lnt, then the error (Err app ) in (45) is computed. Substituting n,n 2,n 3 in (49) and (5) for a given δd the approximated absorbtion coefficient (µ app ) has been obtained. To see the effectiveness of the error modeling and the modified equation 67

8 represented in the terms in (52). Err µa = µ a µ app, (52) Err Err = err Err app. (53) Table : Example µ a δd err ln T n n n µ aapp Err app.... Err µa Err Err We repeat the same process as we did in the case of the wavelength λ = 66 depending on the same code but with other wavelengths like λ = 68 we found the following results: Figure 2: function versus δd, λ=68. Err(.,δd)=Mδd+ H. (54) Figure 4: function versus δd, λ=88. REFERENCES [] A. Abdallah and A. Bolz. Concentrations of Hemoglobin Fractions Calculation Using Modified Lambert-Beer Law and Solving of an Ill-Posed System of Equations. SPIE Photonics Europe, Biophotonics; Photonic Solution for better Health Care, Bruessel, Belgien, 2. [2] A. A. Abdallah and A. Bolz. Towards Noninvasive Monitoring of Total Hemoglobin Concentration and Fractional Oxygen Saturation Based on Earlobe Pulse, volume [3] S. H. Bonner, R. Nossal and G. Weiss. Model for photon migration in turbid biological media. Journal of the Optical Society of America A, 4: , 987. [4] S. H. Dayan and G. H. Weiss. Photon migration in a two-layer turbid medium a diffusion analysis. MOD- ERN OPTICS,, 39(7), 992. [5] J. J. Duderstadt and L. J. Hamilton. Nuclear reactor analysis. Ph.D. dissertation, 976. [6] A. Ishimaru. Diffusion approximation wave propagation and scattering in random. Academic,, pages 75 9, 978. [7] P. Knabner and L. Angermann. Numerical methods for elliptic and parabolic partial differential equations, volume 44 of Texts in Applied Mathematics. Springer, New York, 22. [8] S. Larsson and V. Thomée. Partial differential equations with numerical methods, volume 45 of Texts in Applied Mathematics. Springer, Berlin, 23. [9] D. Volz. Modeling of light propagation in skin, and an application to noninvasive diagnostics. Ph.D. dissertation, 22. Figure 3: function versus δd, λ=76. Using different wavelengths as in Fig. 2, Fig. 3 and Fig. 4 we will get a system of linear equation. This system of equation suffers from an illposed problem, and this will be the topic of the next work. 68