FEM computer exercise in Comsol MultiPhysics

Transcription

1 FEM computer exercise in Comsol MultiPhysics 1 Introduction MultiPhysics is computer software that makes it possible to numerically solve partial differential equations. The numerical solution relies on the Finite Element Method (FEM), in which the geometry studied is divided into a finite element mesh. Thus, instead of trying to solve a highly non-linear problem on the entire geometry, an approximate solution is sought in each element. If this element is considerably small the physical problem is assumed to vary linearly. For more information about the Finite Element Method see for example Ottosen & Petersson Introduction to the Finite Element Method, In this computer exercise light propagation in biological media will be treated. Light induced processes, such as fluorescence and heat transfer, will also be investigated. Previously you have worked with the diffusion approximation to the radiative transfer equation. The diffusion equation is valid when the scattering is large compared to the absorption and when studying diffuse light propagation, i.e. at sufficient distances from any light sources. In a previous computer exercise the method ADI was used to solve the diffusion equation. In this computer exercise the approach is basically the same but you are now provided with a graphical user interface where arbitrary geometries can be defined as well as inhomogeneities inside the geometry. Besides allowing for arbitrary geometries, MultiPhysics also provides the multiphysics possibility, i.e. different physical phenomena that depend on each other can be solved iteratively. In tissue optics this can be utilized when heating of tissue is considered since the light itself will act as a heat source when absorbed by the tissue. Fluorescence is another photo-physical phenomenon that will be studied. In the sections that follow, we repeat the diffusion equation and illustrate how it is represented in MultiPhysics. The reader should be familiar with the diffusion approximation for light propagation in biological media from lectures and previous computer exercises. Following this, the theory of fluorescence and heat transfer in biological media is given. Practical guidelines of modeling using MultiPhysics and its graphical user interface are provided as a help for this exercise and for your projects. Finally, the exercises to be conducted during the classroom session are given. To download software used in this exercise, go to the homepage of the Tissue optics course; Download the zip-file MultiPhysicsExercise.zip and unzip it in your computer. 2 Light propagation using the diffusion equation The diffusion equation, as stated in the diffusion computer exercise, is: 1 c ' d dt φ ( r, t) D( r, t) φ( r, t) + µ φ( r, t) = S( r t) (1) a, 1 (24)

2 In MultiPhysics we will use the Helmholtz representation of Equation 1, in steady state. A translation of the diffusion equation can be found in Optical tomtography in medical imaging, Arridge, Inverse Problems, In MultiPhysics, the Helmholtz equation is represented by: ( c u) + au = f (2) Identifying the parameters in Equation (1) with those in (2) yields: u = φ c = D = 3 a = µ a f = S ' ( µ + µ ) a 1 s Hence you solve for the fluence rate represented by the variable u after specifying the diffusion coefficient c, the absorption coefficient a and the source term f. 3 Theory of fluorescence in biological media Fluorescence is a photo-physical process that can occur when light is absorbed by different tissue chromophores. The absorption excites the chromophore, which in turn relaxes to a lower state and from this state emits light when de-excited to the ground state, see Figure 1. Figure 1 Energy levels and relaxation processes involved following the absorption of a photon. Fluorescence occurs in endogenous and exogenous chromophores. The most important endogenous chromophores, naturally present inside biological media, are elastin, collagen, FAD, NADH, keratin and tryptophan. When excited by a short wavelength light source, in the UV- or blue wavelength bands, these give rise to so called autofluorescence, see Figure 2. 2 (24)

3 Figure 2 Autofluorecence induced by 337 nm N2-laser. A hot research field within fluorescence spectroscopy and fluorescence imaging is Molecular Imaging. The aim of the research is to assess concentrations and spatial locations of exogenous fluorescent markers. Another name of these markers is fluorophores. The fluorophores are systematically applied to small animals, such as mice. The fluorophores attach to certain molecules that in the presence of a certain disease are built up at the morbid location. Hence the location of the disease can be determined as well as concentration changes over time. This approach closely resembles PET, MRI or CT in clinical diagnostics. In contrast to current practical methods, which only determine the spatial location of the disease, molecular imaging can also assess absolute concentrations of different fluorophores. There are of course a wide range of fluorophores, each suited for a different application. Here we only present the family of cyanine dyes. The spectra in Figure 3a-b) are taken from the website Amersham Biosciences website. Figure 3 a) Absorption spectra of cyanine-dyes 3 (24)

4 Figure 3 b) Fluorescence spectra of cyanine-dyes. To model fluorescence in MultiPhysics, one first solves the diffusion equation (a Helmholtz equation) for the excitation light, which will act as the source term for the fluorescence. The fluorescence light distribution is then modeled by adding another Helmholtz equation and solving for the fluence rate (automatically given a different variable name than the excitation light since two physics modes are present) using another set of optical properties (since these are bound to vary between excitation and fluorescence wavelength) and the absorbed excitation light as source term. The following equations represent this in a formal way: ( D φ ) + µ φ S (3) ex a ex = i i i i ex i ( D φ em ) + µ a φem = µ a Y φex (4) Eq. (3), same as Eq. (2), describes the excitation light propagation while Eq. (4) models the emission light at wavelength i. Looking carefully on the second equation s source term may cause nuisance. µ a ex is the absorption coefficient at the excitation wavelength, ф ex is the excitation fluence rate solved for in the first equation and Y i is the fluorescence yield fraction. To understand the yield fraction a closer look at Figure 3b) might ease. All fluorescence is not emitted equally for all wavelengths. For DsRed the most probable wavelength is 580 nm. When not dealing with quantitatively simulations as here one set the yield parameter, Y i, to unity for the wavelength 580 nm. Further the yield parameter for the wavelength 600 nm, for DsRed, is set to (24)

5 4 Theory of heat transfer in tissue 4.1 Biological basis for heat treatment The effectiveness of various therapeutic methods such as radiotherapy and chemotherapy has been shown to increase by inducing a temperature elevation of a few degrees above body temperature. At even higher temperatures cells are rapidly killed. Heat treatment has also been utilised as a surgical method to coagulate or vaporise tumours in many parts of the body. With the invention of the laser in 1960, a new device to induce heat for medical purposes appeared. Using thin optical fibres the surgeon can in a minimally invasive way access the whole gastrointestinal canal and the urogenital tract. Thermal contraction of the treated area results in sealing of small blood vessels, which arrests haemorrhage (i.e., bleeding). Hyperthermia is defined as temperatures in the interval of about o C. In cancer therapy it is generally recognised that hyperthermia should be combined with another therapeutic method, such as ionising radiation, drugs or photodynamic therapy, in order to fully exploit the benefits of the treatment. At least three distinct effects of hyperthermia contribute to the result, namely the direct cell kill, the indirect cell kill due to the destruction of microvasculature, and the heat sensitisation to another treatment modality. Tumours show an apparent, albeit slight, increase in sensitivity to heat in the range o C as compared with normal tissue. Above 44 o C, the difference in heat sensitivity is lost. Clinical hyperthermia has therefore been applied in the above temperature range for about one hour to try to cause tumour cell death while preserving normal tissue. Thermotherapy employs temperatures, which enable coagulation of the treated tissue. Coagulation means a denaturation of proteins causing cell death. The process results in an increase in the scattering properties of tissue (compare with the boiled egg white). Rapid tissue destruction is accomplished by elevating the tissue temperature above 55 o C. No difference in heat sensitivity exists at this temperature; tumour and normal tissue are destroyed at the same high rate. Therefore, heat must be applied at precisely the right location not to induce gross normal tissue damage. Lasers as well as other light sources are used to produce thermal effects in tissue. For heating and ablation purposes, infrared radiation is used since it is absorbed by water. The infrared region extends roughly from 300 GHz THz, which corresponds to wavelengths between 1 mm and 750 nm. Examples of light sources in clinical use are the CO 2 (10 µm) and the Nd:YAG lasers (1064 nm). Visible light is commonly used in ophthalmology and dermatology, the wavelength of visible light ranging from about 400 nm to 750 nm. Another parameter, which determines the treatment result, is the pulse length. For heating of large volumes, as in hyperthermia and thermo-therapy, continuous wave mode is preferentially employed as thermal dissipation due to conduction is the principal determinant of tissue damage. 5 (24)

6 4.2 Heat transfer in tissue In the context of tissue heat treatment the four principal modes of heat transfer are conduction, convection, radiation and evaporation. This section briefly describes the different modes. Conduction is an intrinsic property of the medium, whereas evaporation acts as a boundary condition at tissue-air boundaries. Radiation and convection are present to varying degrees both inside the bulk tissue and at surfaces. Conduction Thermal conduction is transfer of energy from the more energetic particles in a substrate to the adjacent less energetic ones as a result of interactions between the particles. This mode of tissue heat transfer is described by Fourier s law, which states that the heat flow at a given point in the tissue is proportional to the gradient of the temperature. An energy balance for a volume element gives the following relation for transient conduction T ρ c = ( λ T ) (5) t where ρ (kg m -3 ) is the density, c (J kg -1 K -1 ) is the specific heat capacity, λ (W m -1 K -1 ) is the thermal conductivity, and T (K) is the tissue temperature. The ratio λ/(ρc) is called the thermal diffusivity, α (m 2 s -1 ), which describes the dynamic behaviour of the thermal process. The thermal properties depend on the type of tissue and temperature. Since most tissues can be considered as being composed of a combination of water, proteins and fat, the magnitude of the conductivity can be estimated with knowledge of the proportions of these components. Table 1 shows selected data on the thermo-physical properties of tissue measured at room or body temperature. It can be observed that of the different components constituting tissue, water has the highest values of the thermal conductivity and diffusivity. Therefore, tissues with high water content are the best thermal conductors and respond the fastest to a thermal disturbance. Material Conductivity (W m -1 K -1 ) Density (kg m -3 ) 10-3 Specific heat (kj kg -1 K -1 ) Diffusivity (m 2 s -1 ) 10 7 Muscle Fat Kidney Heart Liver Brain o C Table 1. Thermophysical properties of human tissue and water Convection Convection is the term applied to heat transfer between a surface and a fluid, e.g., air or blood, moving over the surface. Convection problems involve heat transfer between the surface of the body and the surrounding air and also between blood and the vessel wall. The heat transfer between a conducting solid, which in this context is tissue, and a 6 (24)

7 convecting fluid is usually described using Newton s law of cooling which states that the heat flow q (W m -2 ) normal to the surface is proportional to the temperature difference between the surface T (K) of the tissue and the bulk temperature of the fluid T : q = h( T T ) (6) where the proportionality constant, h (W m -2 K -1 ), is called the local heat convection coefficient. As shown in Figure 4, there exists a thermal boundary layer if the fluid free stream temperature and the temperature of the tissue differ. Figure 4 Schematic picture of the thermal boundary layer. At the tissue surface there is no fluid motion and energy transfer occurs only by conduction. The magnitude of the heat convection coefficient therefore depends on the temperature gradient at the surface but also on the conditions in the boundary layer. Equations can be set up to account for the physical processes in the boundary layer but the solutions readily become very complex. Here it suffices just to give examples of the convection coefficient in a few cases; Mode h (W/m 2 /K) Free convection 5-25 (tissue air) Forced convection (tissue air, i.e. a windy day) convection with phase change (boiling) Table 2. Values of the convection coefficient for a few cases Radiation Every body emits electromagnetic radiation proportional to the fourth power of the absolute temperature. The maximum heat flow, q (W m -2 ) emitted from a black body is expressed by: ( 4 4 q = T T ) εσ (7) s where σ (W m -2 K -4 ) is the Stefan-Boltzmann constant, T s and T (K) are the tissue surface and environmental temperature, respectively and ε is the emissivity. Since tissue is not a perfect black body the emissivity is less than one. For human skin, the value of the emissivity is in the range When modelling internal biological tissue, intrinsic radiative heat transfer processes constitute a negligible contribution and the process of radiation is often considered only a boundary effect. 7 (24)

8 Evaporation The heat loss from moist tissue in contact with air will be dominated by heat loss associated with evaporation of water. The driving force for the loss of tissue moisture by diffusion is the difference in water concentration between the tissue and the surrounding air. For ideal gas mixtures, such as air and water vapour, the concentration of a component is proportional to the partial pressure. The water at the tissue surface is in thermodynamic equilibrium with the water vapour in air, i.e., at the surface the pressure of the water vapour is equal to the saturated water vapour pressure at the surface temperature. Therefore, water will migrate from the tissue surface to the air if the saturated vapour pressure at the surface is greater than the partial pressure of vapour in the surrounding air. Evaporation acts as a loss term in the bio-heat equation at tissue surfaces. Bio-heat equation Inside tissue, the heat transfer is described by the bio-heat equation taking into account conduction, convection by blood and possible heat sources (Q): T ρ c = ( λ T ) + Q + h( T T ) (8) t blood Convection between tissue surface and surrounding air, radiation and evaporation are processes that have to be modelled as appropriate boundary conditions. In MultiPhysics, the bio-heat equation is modelled by a physics mode called Heat Transfer by Conduction. The time dependent mode should be utilised. Why? In MultiPhysics, this physics mode is represented by: T 4 4 δ ρc λ T = Q + h T T + C T T (9) ts t ( ) ( ) ( ) blood trans ambient After setting δ ts identical to 1 and ignoring radiation within the sample, Equations (8) and (9) are easily identified as equal. 8 (24)

9 5 Guidelines for MultiPhysics modeling This section can be used as a guideline when solving the exercises during the classroom session and also for your future projects. 5.1 Model navigator The model navigator is where you define the physical phenomena of your problem. It is also here one selects if the geometry is two- or three-dimensional. If the problem is a combination of coupled differential equations, i.e. heat transfer due to incident light or fluorescence, one selects Multiphysics. Here each physical phenomenon is defined by first marking the equation in the left window and then pressing Add. 5.2 Main window After selection of the physical equation(s) the user is redirected to the main window. Here the geometry, static constants, physical constants, boundary conditions, mesh resolution and solver parameters are defined. Defining the geometry The geometry is essentially like a drawing-program. The geometry is built up by rectangles and circles in 2D and blocks, spheres, ellipsoids and cylinders in 3D. Note that MultiPhysics uses SI-units and that the ruler is in meters. When creating several geometric objects (e.g. a circle inside a rectangle) the geometric objects are assigned a subdomain number (in this particular case the rectangle would constitute subdomain 1 and the circle subdomain 2). Static constants When modeling, it is always time-saving to define all constants (such as the absorption and scattering coefficients) with a variable name. This can be done via the menus Options > Expressions > Subdomain Expressions, see Figure 5. Figure 5 Defining static constants for subdomain 1. In Figure 5, the optical properties are defined for subdomain 1. Note that the subdomains can have different optical properties. 9 (24)

10 Physical settings To define the physical problem the constants has to be put in to context. This is done in Physics > Subdomain Settings. For each multiphysics mode the parameters are specified. For instance in the case of light propagation with µ a = 0.5 cm -1 and µ s =9 cm -1 (as specified in Subdomain Expressions) the equation parameters are defined as shown in Figure 6. Figure 6 Defining the physical problem in Physics > Subdomain Settings. The case shown represents modelling of the diffusion equation where the optical properties have already been defined via Options > Expressions > Subdomain Expressions. The source term is set to zero since the point source is defined via the menu Physics > Point Settings. Point settings & point sources Point sources are often used when modeling light propagation in infinite homogeneous media. To create a point, simply click the Point icon. The position of the source is set via dialog box or by clicking inside the geometry depending on the number of dimensions you work with. The same holds for squares, circles etc. A point source is defined through the menu Physics > Point settings. Select the point number that represents the source, the point is highlighted in red in the main window if active. In the weak box input P 0 *u_test, where P 0 is the laser beam power in Watts. u_test is the MultiPhysics representation of the dirac-function. Boundary conditions settings The boundary conditions are specified in Physics > Boundary Settings. Each boundary can have a different boundary condition of Dirichlet- or Neumann-type. Default is to set the fluence rate to zero at the boundary which is feasible if the geometry is sufficiently large. Research in the field of mathematical modeling of light propagation has come up with more accurate boundary conditions for solving the diffusion equation using FEM, that also simulate Fresnel reflection on a boundary including refractive index mismatch. One commonly used boundary condition that accounts for the refractive index mismatch between tissue (n~1.4) and air (n~1) is the Robin-type boundary condition. To achieve 10 (24)

11 this in MultiPhysics, choose the Neumann-type boundary condition with q=1/(2*2.74) and g=0. Mesh Before attempting to solve the problem a mesh has to be initiated. This can be done by pressing the Initialize Mesh button located in the tool bar, see Figure 7. A standard computer should manage up to elements. Figure 7 Initialize mesh. In the case of a point source, one might want to make the mesh more detailed closer to the source. This can be achieved by limiting the maximum element size close to the point corresponding to the point source, see Figure 8. Figure 8 Refining the mesh close to point source (point 5) under Mesh > Free Mesh Parameters. Solver The solver is activated by pressing the Solve button in the tool bar, see Figure 9. Solution times vary depending on the problem ( minutes) and the solver usually comes to a temporary pause at 89%. Figure 9 Solver activation. Post-processing To facilitate plotting useful results in MultiPhysics, explore Cross-section Plot Parameters or Domain Plot Parameters in the menu Postprocessing. These menus 11 (24)

12 offer relatively many options for cross-section plots, slices etc in beautiful colors. In addition to the variable(s) solved for, one can also plot different mathematical expressions containing parameters solved for, see Figure 10. If you prefer creating your own Matlab plots, it is also possible to export data as text files via File > Export > Postprocessing data. Figure 10 Plotting the absorbed fluence rate (instead of fluence rate (u)) on a line cross-section. Coupled physics modes When modeling fluorescence and heat induced by light it is necessary to perform two sets of simulations. The first step is to model the light distribution by solving the diffusion equation (implemented by the Helmholtz equation) on the appropriate geometry for a set of optical properties valid at the excitation wavelength. The result from this simulation constitutes the source term when modeling fluorescence or heat propagation following absorption of light. The next step is to add the second mode (diffusion or bio-heat equation) via the menu Multiphysics > Model Navigator. Appropriate constants (such as absorption and scattering coefficients at the fluorescence wavelength or thermal conductivity, heat capacity and density) need to be defined in Subdomain Expressions. The physical problem, i.e. the diffusion or bio-heat equation, need to be specified in Subdomain Settings. The source term should equal the fluence rate already solved for times the absorption coefficient. Before solving for either fluorescence or heat, we need to specify that the previous solution (i.e. the excitation light) should be used as initial value and that we intend to solve for the newly introduced variable. This is achieved by opening the Solver Manager 12 (24)

13 in the menu Solve and using the settings shown in Figure 11. After completing these settings, the solver can be activated. Post-processing can be used for any of the two variables solved for. Figure 11 Setting up a solver that uses the results from a previous simulation (e.g. the fluence rate distribution) as initial value and solves for the second physics mode (e.g. the temperature). 13 (24)

14 6 Classroom exercises There are three classroom exercises. The first governs the diffusion equation and light propagation from a point source. The second is about fluorescence from a deeply lying inclusion and the third is about light induced heat transfer for port-wine stain removal. We suggest that you complete the first exercise and then choose between the fluorescence- or heat-exercise. If you want to do both it is OK but you might be short of time. It is suggested that you choose the exercise according to your project, but this is not required. Start COMSOL MultiPhysics Modeling light propagation from a point source In order to model the light distribution according to the diffusion approximation, choose Helmholtz equation located under MULTIPHYSICS/PDE modes/classical PDEs in the Model Navigator. We choose to work in three dimensions for demonstration purposes. Create your own geometry consisting of a cube, cone, cylinder or ellipsoid and a point located somewhere within the bulk medium. In order for boundary effects not to affect the result, create a bulk medium with minimum volume of 65 cm 3 and place a point source well within this volume. Define the optical properties as static constants in Options > Expressions > Subdomain Expressions. Utilize optical properties relevant for human muscle. Define the physical problem, i.e. the diffusion equation, in Physics > Subdomain Settings. Diffusion and absorption coefficients are most easily set by using static constants as defined above. Set the source term, f, to zero. The initial value for the fluence rate should be set equal to zero. Introduce the point source in Physics > Point Settings. For point source settings, see MultiPhysics Guidelines on the previous pages. Introduce a mesh. The mesh should be made more detailed closer to the light source. This can be achieved via Mesh > Free Mesh Parameters and setting the Maximum Element Size to 1 mm for the point corresponding to the light source. Start the solver (typical solution time 1-2 min). Does your solution agree with the analytical solution in an infinite homogeneous medium? Hint: In Postprocessing > Cross Section Plot Parameters you can create a plot of the fluence rate distribution along a line cross-section through the point source. Once MultiPhysics plots u in a figure you can export this cross-section data to a text file by. Start Matlab and type [data,r] = PlotLineCrossSection(filename.txt); in the Matlab command window in order to read data into Matlab (see also help PlotLineCrossSection, NOTE: make sure the Matlab working directory is /MultiPhysicsExercise/, i.e. where you unzipped the zip-file). The fluence rate and distance vectors become available as data and r. In a new m-file, create a variable describing the fluence rate from a point source according to the analytical solution and compare the two results in a plot. Do your different solutions agree? 14 (24)

15 For those interested in (interstitial) PDT for their project: How would blood surrounding the point source influence the fluence rate distribution and the magnitude of the fluence rate distribution. For optical properties of blood, see Figure. 3D requires a lot more computer capacity and when possible you should take advantage of any symmetry properties of your geometry. How many dimensions were really needed for the exercise you just completed? 6.2 Modeling fluorescence from an inclusion In Molecular imaging one purpose is to estimate the position of a deeply lying fluorophore inside a small animal. We will in this exercise investigate what affects the fluorescence emitted from a fluorophore placed at different depths. To evaluate this task a mouse-model in 3D should be built up preferably with interior organs specified. This is of course time-consuming and not the subject of this exercise, so we simplify. Here the mouse is approximated with an ellipsoid in 3D. A commonly used method in small animal imaging is that the mouse is slightly compressed using two glass plates. This provides the possibilities to images planar surfaces. The inclusion is one or several spheres positioned inside the mouse model. In Figure 12 the modeling geometry is seen with the inclusion inserted. In this case we use homogeneous media. This is definitely not a requirement when modeling using the finite element method. For your project you can insert absorbing or scattering heterogeneities if you want as well as arbitrary geometries. Figure 12 Modeling geometry for fluorescence. 15 (24)

16 The inclusion consists of the fluorophore DsRed, see Figure 3a-b) for absorption and emission properties. Further we use a collimated laser beam to excite the fluorophore, modeled by a point source placed at a distance 1/µ s from the bottom of the model, [x y z]=[0 0 1/µ s ]. The induced fluorescence is detected on the upper side of the model. This is done by extracting a line plot, as before, using Postprocessing > Cross Section Plot Parameters. Choose the coordinates [x 0 y 0 z 0 ] = [ ] and [x 1 y 1 z 1 ] = [ ]. Each emission wavelength requires an extra coupled diffusion equation. Hence it is not feasible to model a full fluorescence spectrum within this computer exercise. For the first exercise we will model the excitation light at 550 nm and fluorescent emission light at 580 and 600 nm Single inclusion in 3D mouse model To model the fluorescent emission from one inclusion, do as follows: Load the model file named 3Dmousemodel.mph. This loads the mouse model seen in Figure 12 with one application mode, i.e. Helmholtz equation. This means that only one equation can be solved for. We need a set of coupled diffusion equations since we are going to model fluorescence. Press the Multiphysics menu and click Model navigator. Here you see one Helmholtz equation (hzeq) which the one modeling the excitation light. Mark helmholtz equation in the menu and click Add two times. This adds two Helmholtz equations (hzeq2 and hzeq3) to the model that will model the two emission wavelengths. Click OK. Before we start defining the parameters we should select which application mode that should be active. Do this by selecting Multiphysics > Helmholtz equation (hzeq), i.e. the excitation light mode. Define the optical properties as static constants in Options > Expressions > Subdomain Expressions. Utilize optical properties for DsRed stated below for excitation (550 nm), emission 1 (580 nm) and emission 2 (600 nm). Excitation (λ = 550nm) Emission (λ = 580nm) Emission (λ = 600nm) Emission (λ = 620nm) Emission (λ = 650nm) µ a (m -1 ) (Tissue) µ s (m -1 ) (Tissue) µ a (m -1 ) (DsRed) µ s (m -1 ) (DsRed) Yield (DsRed) Table 3. Optical properties for DsRed. 16 (24)

17 It is noted that the same optical properties holds for the tissue as for the fluorophore except for the excitation absorption coefficient. The reason is that the fluorophore absorption coefficient is directly related to the concentration (c), through the relation µ a =c. ε. ε is the extinction coefficient of the fluorophore. Hence a larger concentration yields a larger absorption coefficient. The fluorophore is here modeled as an additional absorption coefficient to the excitation wavelength. Figure 13 An example for the tissue subdomain (SD1). Figure 14 Corresponding subdomain settings for the fluorophore (SD2) As can be seen the only thing that differs is the excitation absorption coefficient and the yield. The absorption coefficient is higher for the inclusion since we have an additional fluorophore in this subgeometry. Define the physical problem, i.e. the coupled diffusion equations. Since we are dealing with an equation system, as opposed to the previous problem, the physical constants for the emission wavelength, in addition to the excitation wavelength, should be defined. This is easiest done using Physics > Equation Systems > 17 (24)

18 Subdomain Settings. This is exactly the same as for one Helmholtz equation but all coupled equations are shown at the same time. We are using three wavelengths with the variables u (excitation), u2 (emission 580 nm) and u3 (emission 600 nm). Define the constants according to the example below, (use your own constants if you have chosen to use other names and remember to do it for all subdomains). Figure 15 The diffusion coefficient Figure 16 The absorption coefficient 18 (24)

19 Figure 17 The source term for the inclusion geometry Above you see that we have inserted the yield for the two emission wavelengths which basically represents how much fluorescence that will be generated in each spectral band. Further the constant 7 is the absorption coefficient for the fluorophore. Remember that the source term for the tissue geometry (SD1) should all be set to zero since we are using a point source. Introduce the point source in Physics > Point Settings. For point source settings, see MultiPhysics Guidelines on the previous pages. Since there are several equations and only the excitation diffusion equation should be affected by the point source click Multiphysics > Helmholtz Equation (hzeq) (if not already done). This makes the excitation light diffusion equation active. Introduce the point source as before using P o =0.1 W. Select boundary conditions. Make hzeq active (if not already done), select Physics > Boundary conditions and set all boundaries to Neumann with g=0 and q=1/(2*2.74). Do this for hzeq2 and hzeq3 as well. Introduce a mesh. The mesh should be made more detailed closer to the light source. This can be achieved via Mesh > Mesh Parameters and setting the Maximum Element Size to 0.5 mm for the point corresponding to the light source. Additionally we want the detector boundary to be more detailed set the maximum element size to 2 mm for the upper planar boundary. Start the Solver Manager, Solve for excitation light and Store solution. Then Solve for the emission light using the Current solution as the initial value. The surface plot is shown, but extract the detection side line plot, i.e. the values of the fluence rate for -0.03<x<0.03, y=0 and z=0.02. Plot and export both emission wavelengths as a txt-file (be sure to incorporate the depth and emission wavelength in the file name). 19 (24)

20 Assignment fluo1 Solve and plot the excitation light and both emission wavelengths when the inclusion is positioned in the center of the mouse. Look at the intensity distribution at the boundary of the two emission wavelengths. Explain the difference! Assignment fluo2 Move the inclusion to different depths between 7 mm and 15 mm. Solve and plot the intensity distributions (along the line) for the emission wavelengths. What happens when the inclusion is positioned deeper? Compare both of the emission wavelengths. Can you say anything about the relative change of the emission wavelengths relative the depth of the inclusion? Here you can make a short Matlab-script that loads all the txt-files for the different inclusion depths and plots the ratio (580 nm/600 nm) as a function of the inclusion depth Multiple inclusions in 3D mouse model Now we want to study two inclusions inside the mouse model. We want to study what happens when the fluorophores are not the same, in this case we will model fluorophores of typd DsRed and HcRed, see Figure 3a-b). Add a sphere-shaped, or another inclusion of your choise, to the geometry. This inclusion will be assigned Subdomain 3. What is the difference between HcRed and DsRed? Optional: We are only modeling two emission wavelengths which is sufficient for this exercise but if you want you can add application modes (i.e. emission wavelengths) in the Model Navigator, in the Multiphysics menu. Here just hzeq4 and hzeq5 respectively. Note that you have to insert all optical parameters as before for the two new application modes. Excitation (λ = 550nm) Emission (λ = 580nm) Emission (λ = 600nm) Emission (λ = 620nm) Emission (λ = 650nm) µ a (m -1 ) (Tissue) µ s (m -1 ) (Tissue) µ a (m -1 ) (HcRed) µ s (m -1 ) (HcRed) Yield (HcRed) Table 4. Optical properties for HcRed. Make a mesh and solve first for the excitation and then for the emission wavelengths, as above. Study how the surface plot, at the detector boundary, changes when plotting the different wavelengths. Use the Slice-plot in Postprocessing mode. Can you determine which of the fluorophores that is HcRed and DsRed respectively? 20 (24)

21 Suppose that you use a filtered CCD-camera to detect the emission at boundary. If the fluorophores are positioned far from each other it is evident that we can detect two different intensity distributions from the inclusions. But what happens with the intensity distribution when the fluorophores are positioned close to each other? If we only want to detect one of the fluorophores how should we proceed in an experiment using the CCD-camera? The intensity distribution gives us some information about the x- and y- coordinates about the fluorophore. But what about the depth? Could the depth be estimated using multispectral data? Discuss with your partner how this should be done also incorporate the instructor in the discussions! Hint: think of what happened when you moved a single inclusion in previous exercise. Optional: Also try to move the point source along the boundary (keep the z- coordinate fixed and change x- and y-coordinates of the point source). Does this make it easier to discriminate the inclusions? Optional: What if you change the concentration of the fluorophore. How do you do this? What is the effect as compared to the movement of the source? Optional: In a real experiment we also have autofluorescence inside the mouse. How can we model this? Discuss with your partner, and if you will try to extend your model to also incorporate autofluorescence. Hint: autofluorescence can be treated as a fluorophore with high yield for short wavelengths of low yield for longer wavelengths, as is the case for DsRed. Further the concentration is much smaller that in the fluorophore, typically a contrast 1:100 can be used. The autofluorescence fluorophore is positioned in the whole geometry and not only in a small inclusion. The fluorescence exercise has formed an introduction to what you can do when modeling fluorescence using the diffusion equation. Based on the knowledge you have acquired here you can model fluorescence in a specific model for your particular project. When doing that start to think of what the geometry should look like, how many application modes (spectral bands) you should use, how to model the excitation light source. Within the field of Molecular Imaging there are of course many optical techniques where Fluorescence Molecular Tomography is one important modality. The modeling performed above should introduce you to the problems arising here. Problems that are of particular interest are; How can we get rid of all autofluorescence in a measurement and only detect fluorescence from the inclusion? What happens when the mouse tissue is heterogeneous? Where should we place the excitation source and where should we detect the emitted light? Should the excitation source position be fixed or should we move it over the boundary? Is the fluorophore size large and close to the boundary or is it small and far from the boundary? All of these problems make the reconstruction of the fluorescent inclusion a highly ill-posed mathematical problem but even so based on the diffusion equation you can reconstruct both fluorophore position and estimate the concentration of the fluorophore. This is out-of-scope for this exercise. 21 (24)

22 6.3 Modeling light induced heat transfer for port-wine stain removal Port-wine stains arise as a result of a thin blood layer positioned some distance below the epidermis and mostly constitute a cosmetic handicap. Permanent removal of port-wine stains can be achieved by coagulating the blood vessels, where coagulation is induced by heat formed from light absorbed by the blood layer. By carefully choosing the wavelength of the incoming light and the pulse duration, it is possible to coagulate only blood layer and leave surrounding healthy tissue minimally damaged. For this exercise we have prepared a txt-file containing the light distribution as calculated by a Monte Carlo-simulation on the following geometry. Optical constants have been chosen to match those of muscle and blood around 580 nm, representing realistic output wavelengths of for example dye lasers. This wavelength has been chosen to match the absorption of oxy-hemoglobin in order to achieve differential light absorption and thereby heat induction between the blood layer and surrounding tissue. Using Matlab, execute the m-file FluenceRateFromMC.m in order to illustrate the fluence rate distribution resulting from a Monte Carlo simulation (already executed) on the geometry shown in Figure. The file portwine2006.mci was used as input for this simulation. Do you anticipate the treatment to be successful? Would you be able to better differentiate between blood layer and normal tissue if observing the absorbed light dose? The light distribution resulting from irradiation with the 2 cm diameter beam constitutes the source term in the bio-heat equation. Now you are about to model the light-induced temperature distribution. To model the bio-heat equation in two dimensions, choose MultiPhysics/Heat transfer/conduction/transient analysis in the Model Navigator. Create a geometry corresponding to Figure 18. Observe that MultiPhysics utilizes SI-units and the orientation of the coordinate systems in Multiphysics and Figure 18! x- and y-scales and grid can be altered in Options/Axes,Grid Settings. 22 (24)

23 Figure 18 Geometry and various constants for port-wine stain geometry and Monte Carlo simulation. Constants useful for temperature simulation are also indicated. Why don t we use MultiPhysics and the diffusion approximation to simulate the light distribution in this thin and superficially located blood layer? The light distribution, which will act as the source term, can be loaded into MultiPhysics via the menu Options > Functions, choosing New, Interpolation and Browse for the file MCCollimatedBeam.txt. The function name could be set to u, to match previous exercises, use linear interpolation method and constant extrapolation method. Define the physical problem in Physics > Subdomain Settings. Here we need to specify appropriate physical constants for each of the subdomains constituting or geometry, see Figure 18. The heat source should equal the imported variable, u(x,y) (write exactly like this!), multiplied by the absorption coefficient and the laser power (100 W not unrealistic in a clinical setting!). A realistic heat convection coefficient assuming convection by normal blood flow is 5 W/m 3 K. Disregard radiation and set the initial temperature to match the body temperature. Set appropriate boundary conditions via Physics > Boundary Settings. Here we need to differentiate between tissue air (assume free convection with h = 20 W/m 2 K and ignore radiation and inward heat flux) and tissue tissue (assume a constant temperature equal to the body temperature) boundaries. Introduce a mesh Observe that the physics mode chosen is a time dependent problem (why?) and that we need to specify how long the laser pulses are. This is done via Solve > Solver Parameters and Times, which should be defined according to Matlab syntax. Try first 0:0.01:0.1 s. Start the solver. 23 (24)

24 By looking at a vertical line cross-section (utilize Postprocessing > Crosssection Plot Parameters and Line/Extrusion Plot), can you determine whether the blood vessels are coagulated? The requirements are to coagulate the blood layer while sparing surrounding tissue. Repeat the same simulation with a 1 second pulse. Can you say what is to prefer from a therapeutic point of view; prolonging the laser pulse or increasing the pulse power? What can be done in order to save the skin surface? In a clinical setting the doctor cools the skin before treating. Perform the same simulation assuming a constant temperature of 10 o C at the skin surface. How does this influence the treatment volume? What happens to the temperature distribution during the 5 seconds following a laser pulse? For this simulation, revert to free convection at the tissue surface and use the shorter pulse duration. To study the temperature distribution after the laser pulse you need to perform the simulation in two steps; first, simulate the time interval during irradiation as done previously. The next step is to simulate only the time period following the laser pulse. That means that you need to remove the heat source (be careful to type a zero in the box, only deleting the previous source term has no effect) previously defined in Physics > Subdomain Settings, specify the simulation time interval to 0:0.5:5 seconds and tell MultiPhysics to utilize the current solution as the initial value via Solve > Solver Manager. What happens to the heat distribution following a laser pulse? 24 (24)