Review ArticleMathematical Methods in Biomedical Optics
Macaveiu Gabriela
Academic Center of Optical Engineering and Photonics Polytechnic University of Bucharest 313 Splaiul Independentei060042 Bucharest Romania
Correspondence should be addressed to Macaveiu Gabriela macaveiu gabrielayahoocom
Received 13 August 2013 Accepted 10 September 2013
Academic Editors O Baffa and X Zhang
Copyright copy 2013 Macaveiu Gabriela This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper presents a review of the phenomena regarding light-tissue interactions especially absorption and scattering Themost important mathematical approaches for modeling the light transport in tissues and their domain of application ldquofirst-orderscatteringrdquo ldquoKubelka-Munk theoryrdquo ldquodiffusion approximationrdquo ldquoMonte Carlo simulationrdquo ldquoinverse adding-doublingrdquo and ldquofiniteelement methodrdquo are briefly described
1 Introduction
When tissues are exposed to light reflection refractionabsorption or scattering can occur which lead to energylosses in the incident beam
Refraction is not significant in biomedical applicationsexcept for laser irradiation of transparent media such ascornea tissue in opaque media the most important phenom-ena are scattering and absorption depending on the materialtype of the tissue and the incident wavelength Knowledge ofabsorbing and scattering properties of the tissues is neededfor predicting success of laser surgery treatment
Directmeasurementmethods simply use the Beer attenu-ation law but they need corrections when surface reflectionsoccur due to the mismatched refractive indexes
Indirect techniques use theoretical models for the scat-tering phenomena the indirect noniterative methods needsimple equations to connect optical properties to the mea-sured quantities while the indirect iterative methods candevelop sophisticated models in which the optical propertiesare iterated until the computed reflection and transmissionmatch the measured values
2 Basic Phenomena RegardingLight and Tissues
Reflection means the electromagnetic waves return fromsurfaces upon they are incident generally being boundary
surfaces between twomaterials of different refractive indexessuch as air and tissue The simple law of reflection statesthat the reflection angle equals the incidence angle while thesurface is supposed to be smooth having small irregularitiescompared to the radiation wavelengthThe real tissues do notact like opticalmirrors the roughness of the reflecting surfaceleads to multiple beam reflections (diffuse reflection)
Refractionmeans a displacement of the transmitted beamthrough the surface that separates two media with differentrefractive indexes and it originates from the change of thespeed of light passing through the surface
Refraction usually occurs together with reflection thereflectivity of a surface is ameasure of the amount of reflectedradiation and it is the ratio of the reflected and incidentelectric field amplitudes
The reflectance is the ratio of the reflected and incidentintensities thus it is equal to the square of the reflectivity Atnormal incidence the reflectance of a surface separating airandwater is about 2 [1 2] but in several cases the reflectancecan be important with values that cannot be neglected inthese cases proper eye protectionwhen using lasers is needed
Even one might expect that the intensities of the reflectedand of the refracted beams would be complementary such astheir sum would be equal to the incident intensity which isnot exact the intensity is the ratio between the power and thearea unit while the cross-sections of the incident reflectedand refracted beams do not have the same value except for
2 ISRN Biomedical Engineering
the normal incidence But the total energy in these beams isconserved
Absorption means that the incident beam intensitydecreases when passing through the tissue because the beamenergy is partly converted into heat motion or into vibrationof the absorbent medium molecules
The laws of absorption describe the effect of the tissuethickness (Lambert law)
where 119868(119911) is the beam intensity at the distance 119911 1198680is
the intensity of the incident beam 120572 is the absorptioncoefficient of the medium 119888 is the concentration of theabsorbing medium and 1198961015840 depends on other parameters ofthe substance than concentration The inverse of 120572 119871 =
1120572 is named the length of absorption and it measures thedistance at which the incident intensity drops by 1119890
Absorption in biological tissues is mainly determinedby the water molecules especially in the IR region of thespectrum and by protein and pigments in the UV andvisible range melanin is the basic pigment of the skinand hemoglobin is component of vascularized tissues Mostbiomolecules have complex absorption band structures in the400ndash600 nm range but the spectral range of 600ndash1200 nm(the therapeutic window) is free of absorption phenomenaneither water normacromolecules absorb near IR so that thelight penetrates biological tissues with little loss and enablestreatments of profound structures The skin is the highestabsorber in the visible domain while the cornea is totallytransparent
There is an almost perfect match between the absorptionpeaks of skin melanin and hemoglobin (Figure 1) and thegreen and yellow wavelengths at 531 nm and 568 nm ofkrypton ion lasers meaning these lasers can be used forcoagulating blood vessels Sometimes the original absorptionof the tissue is increased using special dyes and inks beforelaser treatments so that the treatment efficiency would behigher and the neighborhood tissues are less affected
Scattering can be elastic (when the incident photonenergy has the same value as the scattered photon energy)or inelastic (when a fraction of the incident photon energyis converted into forced vibrations of the medium particles)Rayleigh scattering is of elastic type where the scatteringcoefficient decreases with the fourth power of thewavelength
Brillouin scattering is inelastic and is caused by theacoustic waves that induce inhomogeneities of the refractiveindex it can be seen like a kind of optical Doppler Effect asthe scattered photon frequency is shifted up or down whenthe scattering particles move towards or away from the lightsource
The decrease of the intensity during scattering is given bya similar law as absorption
Figure 1 Absorption spectra of melanin in skin and hemoglobin(HbO2) in blood Relative absorption peaks of hemoglobin are at280 nm 420 nm 540 nm and 580 nm Data according to Niemz [1]
Inmost tissues (turbidmedia) both absorption and scatteringoccur simultaneously their entire attenuation coefficient is
120572119905= 120572 + 120572
119878 (4)
21 Unscattered (Coherent) Transmission Considering anunscattered beam with no surface reflections incident ona slab of tissue having the thickness d the transmissionis exponentially attenuated (Beerrsquos law) the unscattered(collimated) transmission 119879
119862is given by
119879119862= 119890minus120572119905119889 (5)
Thus the total attenuation coefficient can be determined
120572119905= minus
1
119889ln119879119862 (6)
In the presence of the mismatched surface reflections cor-rections are required for instance when a tissue sample isplaced between glass slides the collimated beam is reflectedat the air-slide slide-tissue tissue-slide and slide-air surfacesIf the sample is only a few optical depths (8) thick one mustconsider multiple internal reflections Thus a net reflectioncoefficient is given [3] by
and 119903119905are the Fresnel reflections at the air-
glass respectively glass-tissue interfaces then the measuredtransmission 119879
119879 =(1 minus 119903)
2
1 minus 11990321198792
119862
119879119862 (8)
is solved to obtain 119879119862 in order to calculate 120572
119905
ISRN Biomedical Engineering 3
The optical depth is defined by
119889 = int
119904
0
1205721199051198891199041015840 (9)
where 1198891199041015840 is the element of the optical path and 119904 is the totallength of the optical path If the attenuation coefficient 120572
119905is
constant wemeet the case of homogeneous attenuation thus119889 = 120572
119905119904
In the literature [1 3] we also find reduced scattering andattenuation coefficients
1205721015840
119878= 120572119878(1 minus 119892)
1205721015840
119905= 120572 + 120572
1015840
119878
(10)
which include the particular situation when 119892 = 1 (justforward scattering) when the intensity is not attenuated Thelinear transport coefficient 1205721015840
119905describes the inverse of the
mean free path between two interaction events in a strongscattering medium
No theory completely explains why the photons arepreferably scattered in the forward direction nor the wave-length dependencies of the phenomena A more convenientapproach is to work with the photon probability function (orphase function) 119901(120579) to be scattered by an angle 120579 whichcan be fitted to experimental measurements According tothis dependence the scattering can be isotropic which means119901(120579) does not depend on 120579 or else anisotropic One canevaluate the anisotropy of the scattering using the coefficientof anisotropy 119892 defined as the average cosine of the scatteringangle 120579
119892 =
int4120587119901 (120579) cos 120579119889120596int4120587119901 (120579) 119889120596
(11)
where 119889120596 = sin 120579119889120579 119889120593 is the solid angle element Isotropicscattering will use 119892 = 0 while 119892 = 1 means totally forwardscattering and 119892 = minus1means totally backward scattering Formost biological tissues we can assume that 119892 values are inthe range of 07ndash099 [1] because the most frequent scatteringangles range between 8∘ and 45∘
The probability (phase) function is normalized
1
4120587int4120587
119901 (120579) 119889120596 = 1 (12)
The best fit to the experiments (Figure 2) was given by theHenyey-Greenstein phase function
Figure 2Themainly straightforward scattering process is describedby the Henyey-Greenstein phase function for different values of 119892Data according to Niemz [1]
The absorption coefficients are generally obtained sub-tracting transmitted reflected and scattered intensities fromthe incident intensity experimental methods determineeither the total attenuation coefficient or both absorption andscattering coefficients Rotating the detector one can obtainthe angular dependence of the scattered intensity thus havingthe anisotropy coefficient 119892
3 General Physical Model forLight Propagation in Tissue
Mathematical description of absorption and scattering canbe made in two ways The most fundamental approach isthe analytical theory based on Maxwellrsquos equations but theircomplexity and the inhomogeneities of biological tissueslimit the possibility to obtain the exact analytical solutionsthus limiting the applicability of the theory
The second approach is the photon transport theorywhich deals with photon beams passing through absorbingand scattering media without considering Maxwellrsquos equa-tions it was extensively used for laser-tissue interactionswhere its predictions were satisfactory in many cases thoughit is a less strict theory compared to analytical theories
Radiance J is the power flux density in a given direction119904 within the solid angle unit 119889120596 (Wsdotcmminus2sdotsrminus1) it decreasesdue to scattering and absorption but increases by the lightscattered from 119904
where119901(119904 1199041015840) is the phase function of a scattered photon from1199041015840 to 119904 direction 120572
119905= 120572119904+ 120572119886is the attenuation coefficient
which includes scattering and absorption coefficients and119889119904 denotes the infinitesimal path length The differential
4 ISRN Biomedical Engineering
equation (15) for radiance is called the radiative transportequation For symmetric scattering about the optical axis119901(119904 1199041015840) = 119901(120579) where 120579 is the scattering angle
Intensity is the measurable optical property it is obtainedby integrating radiance over the solid angle
119868 (119903) = int4120587
119869 (119903 119904) 119889120596 (16)
Its value can be measured within the experiments so one canexpress radiance by
where 120575(120596 minus 120596119878)means a solid angle delta function oriented
in the 119904 directionWhen laser propagates inside a turbid medium the
radiance can be separated in two terms depicting a coherent(unscattered) component and a diffuse (scattering) compo-nent
119869 = 119869119862+ 119869119889 (18)
The coherent radiance can be calculated from119889119869119862
119889119904= minus120572119905119869119862 (19)
which has the solution
119869119862= 1198680120575 (120596 minus 120596
119878) 119890minus119889 (20)
where 1198680is the incident intensity and the dimensionless
parameter 119889 is given by (5) The unscattered term containsall the light that did not interact with the tissue and ischaracterized by exponential decaying
Evaluation of the second term the diffuse radiance isan important problem as it contains all light that has beenscattered at least once one cannot exactly determine the pathof all the scattered photons so adequate approximations orstatistical approach have to be made depending on whetherabsorption or scattering is dominant Complexity of theapproach is related to its accuracy and to the computing timeit needs
4 Mathematical Methods
Mathematical methods are based upon assumptions regard-ing the incident light sources and boundary conditionsTheyare referred to as ldquofirst-order scatteringrdquo ldquoKubelka-Munktheoryrdquo ldquodiffusion approximationrdquo ldquoMonte Carlo simulationrdquoand ldquoinverse adding-doublingrdquo
41 First-Order Scattering Scatering means that multiplescattering is not considered in some cases the diffuse radi-ance is much smaller than the coherent radiance and one canassume it can be neglected
Although it is a simple solution it is often inapplicable for thetherapeutic window (600ndash1200 nm)
J2
J1
120579
AKM
AKM
dzz
+dJ1
+ dJ2J2
SKMSKM
Figure 3 Geometry in Kubelka-Munk theory [1]
42 Kubelka-MunkTheory Unlike the previous assumptionKubelka andMunk supposed the radiance to be diffuse (119869
119862=
0) and defined two coefficients 119860KM for the absorption and119878KM for scattering of diffuse radiation different from 120572 and120572119904that referred only to coherent radiationConsidering diffuse radiance through a one-dimensional
isotropic slab with no reflection at the boundaries thisapproach is equivalent to the diffusionmodel with two-phasefunctions peaked forward and backward [3]
A flux 1198691in the incident direction and a backscattered
flux 1198692are shown in Figure 3 two differential equations can
be written each states that in both directions the radianceencounters two losses (scattering and absorption) and a gainfrom opposite direction scattered photons
1198891198691
119889119911= minus119878KM1198691 minus 119860KM1198691 + 119878KM1198692
1198891198692
119889119911= minus119878KM1198692 minus 119860KM1198692 + 119878KM1198691
(23)
where 119911 is the mean direction of the incident radiationTheirgeneral solutions are
1198691 (119911) = 11986211119890
minus120574119911+ 11986212119890+120574119911
1198692(119911) = 119862
21119890minus120574119911
+ 11986222119890+120574119911
(24)
with
120574 = radic1198602
KM + 2119860KM119878KM (25)
Considering average values of scattered and coherent pathlengths [1] and since diffuse scattering implies that 119869 does notdepend on the scattering angle one can find the connectionbetween the absorption and scattering coefficients
119860KM = 2120572
119878KM = 120572119878(26)
The Kubelka-Munk expressions for reflection and trans-mission of diffuse radiance on a slab of thickness 119889 are [3]
119877 =sinh (119878KM119910119889)
119909 cosh (119878KM119910119889) + 119910 sinh (119878KM119910119889)
119879 =119910
119909 cosh (119878KM119910119889) + 119910 sinh (119878KM119910119889)
(27)
ISRN Biomedical Engineering 5
so that the scattering and absorption coefficients can beexpressed in terms of the measured values of 119877 and 119879
119878KM =1
119910119889ln[
1 minus 119877 (119909 minus 119910)
119879]
119860KM = (119909 minus 1) 119878KM
(28)
where the parameters 119909 and 119910 can be found using
119909 =1 + 1198772minus 1198792
2119877
119910 = radic1199092 minus 1
(29)
This theory is a simple model to measure the optical proper-ties of the tissues but it is restricted by the assumption thatthe incident light is already diffuse the isotropic scatteringand matched boundary refractive indexes which are atypicalfor laser-tissue interaction It can be extended consideringmore fluxes but an important disadvantage is the extendedcomputing time
43 Diffusion Approximation When the scattering phenom-ena dominate absorption the diffuse term in (18) can beexpanded in a series
with 119860 + 119861 = 1198680 Different sets of values for 120572
119905 120572119904
and 119892 provide similar radiances in diffusion approximationcalculus
0
500
1000
1500
2000
minus1500 minus1000 minus500 0 500 1000 1500
Position x (120583m)
Dep
thz
(120583m
)
lowast
Figure 4 Monte Carlo simulated movement of a photon through ahomogeneous medium Data according to Wang and Jacques [4]
44 Monte Carlo Simulations The Monte Carlo methodessentially runs a computer simulation based upon a numer-ical approach to the transport equation (15) The statisticalapproach implies the simulation of a number of 119873 photonsrandom walk the statistical accuracy of the results is propor-tional to radic119873 so that a valuable approximation has to takeinto account a large number of photons This method hasbecome a powerful tool for many disciplines and it requireslarge computers or networks
The main idea of Monte Carlo method for the absorp-tion and scattering phenomena is to follow the opticalpath of a photon through a turbid medium The dis-tance between two collisions is chosen from a logarith-mic distribution using a random computer generated num-ber
Absorption is depicted as a decrease of a weight attributedto the photon during propagation scattering is providedby choosing a new direction of propagation according toa given phase function and another random number Thewhole procedure continues until the photon weight reachesa minimum cut-off value or the photon escapes from theconsidered region
Monte Carlo simulation (Figure 4) needs five steps [1]
(1) Photons are generated at a surface of the consideredregion so that their distribution can be fitted to agiven light source (ie Gaussian beam)
(2) Pathway generation the distance to the first collisionis computed supposing that absorbing and scatteringparticles are randomly distributed a random number0 lt 1205851lt 1 is generated so the distance
119871 (1205851) = minus
ln 1205851
120588120590119878
(36)
where 120588 is the particle density and 120590119878is their scatter-
ing cross-section [1] thus 1120588120590119878represents the mean
6 ISRN Biomedical Engineering
free path and a scattering point is obtained Then asecond random number 120585
2is generated to determine
the scattering angle according to the phase functionthen the third random number is generated to get theazimuth angle
120593 = 21205871205853 (37)
(3) Absorption makes the photon weight decrease by119890minus120572119871(120585
1) 120572 being the absorption coefficient
(4) Elimination of the photon when its attributed weightreaches a certain value then a new photon is launchedand proceeds with step 1
(5) Detection after repeating steps 1ndash4 for a sufficientnumber of photons the computer has stored a mapof pathways so one can make statistical predictionsupon the fraction of the incident photons beingabsorbed by the medium and the spatial and angulardistribution of the transmitted photons
As the simulation accuracy increases with the largernumbers of photons the necessity of extending calculationsmakes them time consuming If earlier results are stored inthe computermemory andused again if neededwith the samephase function computing time will be saved
45 Inverse Adding-Doubling Method The name of themethod comes from reversing the usual process when calcu-lating reflectance and transmittance from the optical prop-erties this method assumes that reflection and transmissionof incident light at a certain angle are known If we need thesame properties for a layer twice as thick we divide it intotwo equal slabs and then add the reflection and transmissioncontributions of either slabThese properties can be obtainedfor an arbitrary slab of tissue starting with a thin known slaband doubling it until the necessary thickness is achieved
The adding method can be extended to simulate layeredtissues with different optical properties
46 The Finite Element Method Boundary and Source Con-ditions The model to calculate light transport in stronglyscattering materials was needed as a component in imagereconstruction schemes that localize the optical propertiesof the tissues from boundary measurements by solving theinverse problem Diffusion approximation to the radiativetransfer equation (15) is an adequate model for scatter-dominant materials for it assumes that scattering dominatesabsorption (generally true for biological tissues) and theanisotropic propagation is weak that is not the case nearsources and boundaries
When the only sources of light aremonochromatic lasersa frequency-independent model of light transport is suitablemulti-wavelength systems will be applied sequentially so thatthe change of frequency is expressed by the change in theoptical properties
The finite element method for the propagation of lightin scattering media aims [5 6] to find the photon density Φand the radiance in a strong scattering domain Ω together
with the outward current (existence) Γ through the boundary120597Ω using the diffusion approximation model to the radiativetransport equation (15) The existence Γ through boundary120597Ω at the point 120585 isin 120597Ω is defined as
where Φ(119903) is the photon density at 119903 isin Ω and 119896(119903) denotesthe diffusion coefficient (34) The solution we seek is acontinuous linear approximation Φℎ of Φ The considereddomainΩ is partitioned into119863nonoverlapping elements 120591 ineach element Φℎ is assumed linear Nodes 119873
119895(119895 = 1 119901)
are attached to the element vertices and Φℎ at each point119903 within the element 120591
119894is the linear interpolation of nodal
values Φ119895
Φℎ(119903 119905) = sum
119895119873119895isin120591119894
Φ119895 (119905) 120595119895 (119903) (40)
where 120595119895are linear nodal shape functions [5] with support
over all elements which have the node 119873119895in the position 119903
119895
as a vertex and 120595119894(119903119894) = 120575119894119895 Equation (39) becomes
The solution of (32) requires appropriate boundary condi-tions The Dirichlet condition (DBC)
Φ (120585) = 0 forall120585 isin 120597Ω (42)
means that the medium around Ω is a perfect absorberphotons are absorbed when crossing 120597Ω so that outside thedomain the photon density equals zero A more realisticboundary condition would be
which is named [5] a Robin boundary condition (RBC) thatconstrains a linear combination of the photon densities andthe current at 120597Ω It can be modified to incorporate amismatch of the refractive indexes 119899 within Ω and 1198991015840 in thesurrounding medium we assume 1198991015840 = 1 so that (41) becomes
where 119877 is the parameter governing internal reflection at theboundary 120597Ω that can be fitted from experimental curves Amodified Robin condition [5] can be
with the critical angle 120579119862= arcsin (1119899) and 119877
0= (119899 minus 1)
2(119899 + 1)
2 (ie for 119899 = 14 we get 119860 = 325)While RBC is a more accurate physical situation than the
DBC the mathematical approach is simple with DBC To geta compromise we can introduce an extrapolated boundaryat a certain distance 119889ext from the physical boundary atwhich DBC apply For an arbitrary two-dimensional domainΩ the extrapolated boundary condition is obtained simplyby adding a border of thickness 119889ext around Ω Differentexpressions [5] of extrapolating 119889ext can be applied to arbi-trary geometries if the radius of the boundary curvature issmall compared to the mean free path both for matched andmismatched refractive index
There are two possibilities to model the light sourcesincident at a point on the boundary collimated or diffusesources
(1) The diffusion equation cannot describe correctly colli-mated sources so we can represent a collimated pencilbeam by an isotropic source at the 11205721015840
119878depth that is
accurate at distances larger than the mean free pathfrom the source but breaks down close to the sourceThe implementation of the collimated source needsa delta-shaped term 119902
0in (39) or other models have
been using the analogy to nuclear engineering withcylinder sources and exponentially decaying
(2) Diffuse sources on the surface can be regarded as aninward directed diffuse photon current distributedover the illuminated boundary segment 120597Ω
2sub 120597Ω
The Dirichlet condition (40) is written for diffusesources
Experimental data [5 6] lead to the conclusion that the Robinmodel and the extrapolated boundary models are equivalentbut the Dirichlet model shows significant lack of accuracythat should prevent its use in most applications
Figure 5 shows intensity distributions calculated [1] witheither method inside a turbid medium assuming isotropicscattering 119886 denotes the ratio of absorption and scatteringcoefficients (optical albedo)
119886 =120572119878
120572119905
=120572119878
120572 + 120572119878
(49)
1
01
0010 2 4 6 8 10
Relat
ive d
iffus
e int
ensit
y
Optical depth
Kubelka-MunkTransport theory
Monto CarloDiffusion oppraximation
a = 09
a = 099
Figure 5 Intensity distributions inside turbidmedium comparisonData according to Niemz [1]
The values 119886 = 09 and 119886 = 099 mean that scattering ispredominantThe ordinate shows diffuse intensity in units ofincident intensity
5 Conclusions
Different methods for solving the transport equation werediscussed the most important are the Kubelka-Munk theorythe diffusion approximation and Monte Carlo simulations
The Kubelka-Munk theory deals only with diffuse radi-ation and it is limited to the cases where scattering dom-inates absorption an important disadvantage is the one-dimensional geometry
The diffusion approximation is not restricted to diffuseradiation but it also applies to predominant scattering phe-nomena which is a powerful tool
Monte Carlo simulations provide most accurate solu-tions sincemany input parametersmay be taken into accountin specially developed computer programs the methodallows two-dimensional and three-dimensional evaluationsalthough they require a long computing time
The finite element method was compared [5] to MonteCarlo corresponding methods and it was found to be in goodagreement with Robin boundary conditions the diffusionapproximation although not strictly valid near boundariesis still able to produce correct data when using appropriateboundary conditions whichmatches best a given experimen-tal situation It can be used when theMonte Carlo computingspeed is significantly reduced
8 ISRN Biomedical Engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] M H Niemz Laser-Tissue Interactions Springer Berlin Ger-many 2007
[2] P E Sterian Photonics vol 1 Printech Publishing HouseBucharest Romania 2000 (Romanian)
[3] W F Cheong S A Prahl and A J Welch ldquoA review ofthe optical properties of biological tissuesrdquo IEEE Journal ofQuantum Electronics vol 26 no 12 pp 2166ndash2185 1990
[4] L Wang and S L Jacques Monte Carlo Modeling of LightTransport in Multi-Layered Tissues in Standard C University ofTexas MD Anderson Cancer Center 1992
[5] M Schweiger S R Arridge M Hiraoka and D T Delpy ldquoThefinite element method for the propagation of light in scatteringmedia boundary and source conditionsrdquo Medical Physics vol22 no 11 part 1 pp 1779ndash1792 1995
[6] S Arridge and M Schweiger ldquoPhoton measurement densityfunctions Part 2 finite-element method calculationsrdquo AppliedOptics vol 34 no 34 pp 8026ndash8037 1995
the normal incidence But the total energy in these beams isconserved
Absorption means that the incident beam intensitydecreases when passing through the tissue because the beamenergy is partly converted into heat motion or into vibrationof the absorbent medium molecules
The laws of absorption describe the effect of the tissuethickness (Lambert law)
where 119868(119911) is the beam intensity at the distance 119911 1198680is
the intensity of the incident beam 120572 is the absorptioncoefficient of the medium 119888 is the concentration of theabsorbing medium and 1198961015840 depends on other parameters ofthe substance than concentration The inverse of 120572 119871 =
1120572 is named the length of absorption and it measures thedistance at which the incident intensity drops by 1119890
Absorption in biological tissues is mainly determinedby the water molecules especially in the IR region of thespectrum and by protein and pigments in the UV andvisible range melanin is the basic pigment of the skinand hemoglobin is component of vascularized tissues Mostbiomolecules have complex absorption band structures in the400ndash600 nm range but the spectral range of 600ndash1200 nm(the therapeutic window) is free of absorption phenomenaneither water normacromolecules absorb near IR so that thelight penetrates biological tissues with little loss and enablestreatments of profound structures The skin is the highestabsorber in the visible domain while the cornea is totallytransparent
There is an almost perfect match between the absorptionpeaks of skin melanin and hemoglobin (Figure 1) and thegreen and yellow wavelengths at 531 nm and 568 nm ofkrypton ion lasers meaning these lasers can be used forcoagulating blood vessels Sometimes the original absorptionof the tissue is increased using special dyes and inks beforelaser treatments so that the treatment efficiency would behigher and the neighborhood tissues are less affected
Scattering can be elastic (when the incident photonenergy has the same value as the scattered photon energy)or inelastic (when a fraction of the incident photon energyis converted into forced vibrations of the medium particles)Rayleigh scattering is of elastic type where the scatteringcoefficient decreases with the fourth power of thewavelength
Brillouin scattering is inelastic and is caused by theacoustic waves that induce inhomogeneities of the refractiveindex it can be seen like a kind of optical Doppler Effect asthe scattered photon frequency is shifted up or down whenthe scattering particles move towards or away from the lightsource
The decrease of the intensity during scattering is given bya similar law as absorption
Figure 1 Absorption spectra of melanin in skin and hemoglobin(HbO2) in blood Relative absorption peaks of hemoglobin are at280 nm 420 nm 540 nm and 580 nm Data according to Niemz [1]
Inmost tissues (turbidmedia) both absorption and scatteringoccur simultaneously their entire attenuation coefficient is
120572119905= 120572 + 120572
119878 (4)
21 Unscattered (Coherent) Transmission Considering anunscattered beam with no surface reflections incident ona slab of tissue having the thickness d the transmissionis exponentially attenuated (Beerrsquos law) the unscattered(collimated) transmission 119879
119862is given by
119879119862= 119890minus120572119905119889 (5)
Thus the total attenuation coefficient can be determined
120572119905= minus
1
119889ln119879119862 (6)
In the presence of the mismatched surface reflections cor-rections are required for instance when a tissue sample isplaced between glass slides the collimated beam is reflectedat the air-slide slide-tissue tissue-slide and slide-air surfacesIf the sample is only a few optical depths (8) thick one mustconsider multiple internal reflections Thus a net reflectioncoefficient is given [3] by
and 119903119905are the Fresnel reflections at the air-
glass respectively glass-tissue interfaces then the measuredtransmission 119879
119879 =(1 minus 119903)
2
1 minus 11990321198792
119862
119879119862 (8)
is solved to obtain 119879119862 in order to calculate 120572
119905
ISRN Biomedical Engineering 3
The optical depth is defined by
119889 = int
119904
0
1205721199051198891199041015840 (9)
where 1198891199041015840 is the element of the optical path and 119904 is the totallength of the optical path If the attenuation coefficient 120572
119905is
constant wemeet the case of homogeneous attenuation thus119889 = 120572
119905119904
In the literature [1 3] we also find reduced scattering andattenuation coefficients
1205721015840
119878= 120572119878(1 minus 119892)
1205721015840
119905= 120572 + 120572
1015840
119878
(10)
which include the particular situation when 119892 = 1 (justforward scattering) when the intensity is not attenuated Thelinear transport coefficient 1205721015840
119905describes the inverse of the
mean free path between two interaction events in a strongscattering medium
No theory completely explains why the photons arepreferably scattered in the forward direction nor the wave-length dependencies of the phenomena A more convenientapproach is to work with the photon probability function (orphase function) 119901(120579) to be scattered by an angle 120579 whichcan be fitted to experimental measurements According tothis dependence the scattering can be isotropic which means119901(120579) does not depend on 120579 or else anisotropic One canevaluate the anisotropy of the scattering using the coefficientof anisotropy 119892 defined as the average cosine of the scatteringangle 120579
119892 =
int4120587119901 (120579) cos 120579119889120596int4120587119901 (120579) 119889120596
(11)
where 119889120596 = sin 120579119889120579 119889120593 is the solid angle element Isotropicscattering will use 119892 = 0 while 119892 = 1 means totally forwardscattering and 119892 = minus1means totally backward scattering Formost biological tissues we can assume that 119892 values are inthe range of 07ndash099 [1] because the most frequent scatteringangles range between 8∘ and 45∘
The probability (phase) function is normalized
1
4120587int4120587
119901 (120579) 119889120596 = 1 (12)
The best fit to the experiments (Figure 2) was given by theHenyey-Greenstein phase function
Figure 2Themainly straightforward scattering process is describedby the Henyey-Greenstein phase function for different values of 119892Data according to Niemz [1]
The absorption coefficients are generally obtained sub-tracting transmitted reflected and scattered intensities fromthe incident intensity experimental methods determineeither the total attenuation coefficient or both absorption andscattering coefficients Rotating the detector one can obtainthe angular dependence of the scattered intensity thus havingthe anisotropy coefficient 119892
3 General Physical Model forLight Propagation in Tissue
Mathematical description of absorption and scattering canbe made in two ways The most fundamental approach isthe analytical theory based on Maxwellrsquos equations but theircomplexity and the inhomogeneities of biological tissueslimit the possibility to obtain the exact analytical solutionsthus limiting the applicability of the theory
The second approach is the photon transport theorywhich deals with photon beams passing through absorbingand scattering media without considering Maxwellrsquos equa-tions it was extensively used for laser-tissue interactionswhere its predictions were satisfactory in many cases thoughit is a less strict theory compared to analytical theories
Radiance J is the power flux density in a given direction119904 within the solid angle unit 119889120596 (Wsdotcmminus2sdotsrminus1) it decreasesdue to scattering and absorption but increases by the lightscattered from 119904
where119901(119904 1199041015840) is the phase function of a scattered photon from1199041015840 to 119904 direction 120572
119905= 120572119904+ 120572119886is the attenuation coefficient
which includes scattering and absorption coefficients and119889119904 denotes the infinitesimal path length The differential
4 ISRN Biomedical Engineering
equation (15) for radiance is called the radiative transportequation For symmetric scattering about the optical axis119901(119904 1199041015840) = 119901(120579) where 120579 is the scattering angle
Intensity is the measurable optical property it is obtainedby integrating radiance over the solid angle
119868 (119903) = int4120587
119869 (119903 119904) 119889120596 (16)
Its value can be measured within the experiments so one canexpress radiance by
where 120575(120596 minus 120596119878)means a solid angle delta function oriented
in the 119904 directionWhen laser propagates inside a turbid medium the
radiance can be separated in two terms depicting a coherent(unscattered) component and a diffuse (scattering) compo-nent
119869 = 119869119862+ 119869119889 (18)
The coherent radiance can be calculated from119889119869119862
119889119904= minus120572119905119869119862 (19)
which has the solution
119869119862= 1198680120575 (120596 minus 120596
119878) 119890minus119889 (20)
where 1198680is the incident intensity and the dimensionless
parameter 119889 is given by (5) The unscattered term containsall the light that did not interact with the tissue and ischaracterized by exponential decaying
Evaluation of the second term the diffuse radiance isan important problem as it contains all light that has beenscattered at least once one cannot exactly determine the pathof all the scattered photons so adequate approximations orstatistical approach have to be made depending on whetherabsorption or scattering is dominant Complexity of theapproach is related to its accuracy and to the computing timeit needs
4 Mathematical Methods
Mathematical methods are based upon assumptions regard-ing the incident light sources and boundary conditionsTheyare referred to as ldquofirst-order scatteringrdquo ldquoKubelka-Munktheoryrdquo ldquodiffusion approximationrdquo ldquoMonte Carlo simulationrdquoand ldquoinverse adding-doublingrdquo
41 First-Order Scattering Scatering means that multiplescattering is not considered in some cases the diffuse radi-ance is much smaller than the coherent radiance and one canassume it can be neglected
Although it is a simple solution it is often inapplicable for thetherapeutic window (600ndash1200 nm)
J2
J1
120579
AKM
AKM
dzz
+dJ1
+ dJ2J2
SKMSKM
Figure 3 Geometry in Kubelka-Munk theory [1]
42 Kubelka-MunkTheory Unlike the previous assumptionKubelka andMunk supposed the radiance to be diffuse (119869
119862=
0) and defined two coefficients 119860KM for the absorption and119878KM for scattering of diffuse radiation different from 120572 and120572119904that referred only to coherent radiationConsidering diffuse radiance through a one-dimensional
isotropic slab with no reflection at the boundaries thisapproach is equivalent to the diffusionmodel with two-phasefunctions peaked forward and backward [3]
A flux 1198691in the incident direction and a backscattered
flux 1198692are shown in Figure 3 two differential equations can
be written each states that in both directions the radianceencounters two losses (scattering and absorption) and a gainfrom opposite direction scattered photons
1198891198691
119889119911= minus119878KM1198691 minus 119860KM1198691 + 119878KM1198692
1198891198692
119889119911= minus119878KM1198692 minus 119860KM1198692 + 119878KM1198691
(23)
where 119911 is the mean direction of the incident radiationTheirgeneral solutions are
1198691 (119911) = 11986211119890
minus120574119911+ 11986212119890+120574119911
1198692(119911) = 119862
21119890minus120574119911
+ 11986222119890+120574119911
(24)
with
120574 = radic1198602
KM + 2119860KM119878KM (25)
Considering average values of scattered and coherent pathlengths [1] and since diffuse scattering implies that 119869 does notdepend on the scattering angle one can find the connectionbetween the absorption and scattering coefficients
119860KM = 2120572
119878KM = 120572119878(26)
The Kubelka-Munk expressions for reflection and trans-mission of diffuse radiance on a slab of thickness 119889 are [3]
119877 =sinh (119878KM119910119889)
119909 cosh (119878KM119910119889) + 119910 sinh (119878KM119910119889)
119879 =119910
119909 cosh (119878KM119910119889) + 119910 sinh (119878KM119910119889)
(27)
ISRN Biomedical Engineering 5
so that the scattering and absorption coefficients can beexpressed in terms of the measured values of 119877 and 119879
119878KM =1
119910119889ln[
1 minus 119877 (119909 minus 119910)
119879]
119860KM = (119909 minus 1) 119878KM
(28)
where the parameters 119909 and 119910 can be found using
119909 =1 + 1198772minus 1198792
2119877
119910 = radic1199092 minus 1
(29)
This theory is a simple model to measure the optical proper-ties of the tissues but it is restricted by the assumption thatthe incident light is already diffuse the isotropic scatteringand matched boundary refractive indexes which are atypicalfor laser-tissue interaction It can be extended consideringmore fluxes but an important disadvantage is the extendedcomputing time
43 Diffusion Approximation When the scattering phenom-ena dominate absorption the diffuse term in (18) can beexpanded in a series
with 119860 + 119861 = 1198680 Different sets of values for 120572
119905 120572119904
and 119892 provide similar radiances in diffusion approximationcalculus
0
500
1000
1500
2000
minus1500 minus1000 minus500 0 500 1000 1500
Position x (120583m)
Dep
thz
(120583m
)
lowast
Figure 4 Monte Carlo simulated movement of a photon through ahomogeneous medium Data according to Wang and Jacques [4]
44 Monte Carlo Simulations The Monte Carlo methodessentially runs a computer simulation based upon a numer-ical approach to the transport equation (15) The statisticalapproach implies the simulation of a number of 119873 photonsrandom walk the statistical accuracy of the results is propor-tional to radic119873 so that a valuable approximation has to takeinto account a large number of photons This method hasbecome a powerful tool for many disciplines and it requireslarge computers or networks
The main idea of Monte Carlo method for the absorp-tion and scattering phenomena is to follow the opticalpath of a photon through a turbid medium The dis-tance between two collisions is chosen from a logarith-mic distribution using a random computer generated num-ber
Absorption is depicted as a decrease of a weight attributedto the photon during propagation scattering is providedby choosing a new direction of propagation according toa given phase function and another random number Thewhole procedure continues until the photon weight reachesa minimum cut-off value or the photon escapes from theconsidered region
Monte Carlo simulation (Figure 4) needs five steps [1]
(1) Photons are generated at a surface of the consideredregion so that their distribution can be fitted to agiven light source (ie Gaussian beam)
(2) Pathway generation the distance to the first collisionis computed supposing that absorbing and scatteringparticles are randomly distributed a random number0 lt 1205851lt 1 is generated so the distance
119871 (1205851) = minus
ln 1205851
120588120590119878
(36)
where 120588 is the particle density and 120590119878is their scatter-
ing cross-section [1] thus 1120588120590119878represents the mean
6 ISRN Biomedical Engineering
free path and a scattering point is obtained Then asecond random number 120585
2is generated to determine
the scattering angle according to the phase functionthen the third random number is generated to get theazimuth angle
120593 = 21205871205853 (37)
(3) Absorption makes the photon weight decrease by119890minus120572119871(120585
1) 120572 being the absorption coefficient
(4) Elimination of the photon when its attributed weightreaches a certain value then a new photon is launchedand proceeds with step 1
(5) Detection after repeating steps 1ndash4 for a sufficientnumber of photons the computer has stored a mapof pathways so one can make statistical predictionsupon the fraction of the incident photons beingabsorbed by the medium and the spatial and angulardistribution of the transmitted photons
As the simulation accuracy increases with the largernumbers of photons the necessity of extending calculationsmakes them time consuming If earlier results are stored inthe computermemory andused again if neededwith the samephase function computing time will be saved
45 Inverse Adding-Doubling Method The name of themethod comes from reversing the usual process when calcu-lating reflectance and transmittance from the optical prop-erties this method assumes that reflection and transmissionof incident light at a certain angle are known If we need thesame properties for a layer twice as thick we divide it intotwo equal slabs and then add the reflection and transmissioncontributions of either slabThese properties can be obtainedfor an arbitrary slab of tissue starting with a thin known slaband doubling it until the necessary thickness is achieved
The adding method can be extended to simulate layeredtissues with different optical properties
46 The Finite Element Method Boundary and Source Con-ditions The model to calculate light transport in stronglyscattering materials was needed as a component in imagereconstruction schemes that localize the optical propertiesof the tissues from boundary measurements by solving theinverse problem Diffusion approximation to the radiativetransfer equation (15) is an adequate model for scatter-dominant materials for it assumes that scattering dominatesabsorption (generally true for biological tissues) and theanisotropic propagation is weak that is not the case nearsources and boundaries
When the only sources of light aremonochromatic lasersa frequency-independent model of light transport is suitablemulti-wavelength systems will be applied sequentially so thatthe change of frequency is expressed by the change in theoptical properties
The finite element method for the propagation of lightin scattering media aims [5 6] to find the photon density Φand the radiance in a strong scattering domain Ω together
with the outward current (existence) Γ through the boundary120597Ω using the diffusion approximation model to the radiativetransport equation (15) The existence Γ through boundary120597Ω at the point 120585 isin 120597Ω is defined as
where Φ(119903) is the photon density at 119903 isin Ω and 119896(119903) denotesthe diffusion coefficient (34) The solution we seek is acontinuous linear approximation Φℎ of Φ The considereddomainΩ is partitioned into119863nonoverlapping elements 120591 ineach element Φℎ is assumed linear Nodes 119873
119895(119895 = 1 119901)
are attached to the element vertices and Φℎ at each point119903 within the element 120591
119894is the linear interpolation of nodal
values Φ119895
Φℎ(119903 119905) = sum
119895119873119895isin120591119894
Φ119895 (119905) 120595119895 (119903) (40)
where 120595119895are linear nodal shape functions [5] with support
over all elements which have the node 119873119895in the position 119903
119895
as a vertex and 120595119894(119903119894) = 120575119894119895 Equation (39) becomes
The solution of (32) requires appropriate boundary condi-tions The Dirichlet condition (DBC)
Φ (120585) = 0 forall120585 isin 120597Ω (42)
means that the medium around Ω is a perfect absorberphotons are absorbed when crossing 120597Ω so that outside thedomain the photon density equals zero A more realisticboundary condition would be
which is named [5] a Robin boundary condition (RBC) thatconstrains a linear combination of the photon densities andthe current at 120597Ω It can be modified to incorporate amismatch of the refractive indexes 119899 within Ω and 1198991015840 in thesurrounding medium we assume 1198991015840 = 1 so that (41) becomes
where 119877 is the parameter governing internal reflection at theboundary 120597Ω that can be fitted from experimental curves Amodified Robin condition [5] can be
with the critical angle 120579119862= arcsin (1119899) and 119877
0= (119899 minus 1)
2(119899 + 1)
2 (ie for 119899 = 14 we get 119860 = 325)While RBC is a more accurate physical situation than the
DBC the mathematical approach is simple with DBC To geta compromise we can introduce an extrapolated boundaryat a certain distance 119889ext from the physical boundary atwhich DBC apply For an arbitrary two-dimensional domainΩ the extrapolated boundary condition is obtained simplyby adding a border of thickness 119889ext around Ω Differentexpressions [5] of extrapolating 119889ext can be applied to arbi-trary geometries if the radius of the boundary curvature issmall compared to the mean free path both for matched andmismatched refractive index
There are two possibilities to model the light sourcesincident at a point on the boundary collimated or diffusesources
(1) The diffusion equation cannot describe correctly colli-mated sources so we can represent a collimated pencilbeam by an isotropic source at the 11205721015840
119878depth that is
accurate at distances larger than the mean free pathfrom the source but breaks down close to the sourceThe implementation of the collimated source needsa delta-shaped term 119902
0in (39) or other models have
been using the analogy to nuclear engineering withcylinder sources and exponentially decaying
(2) Diffuse sources on the surface can be regarded as aninward directed diffuse photon current distributedover the illuminated boundary segment 120597Ω
2sub 120597Ω
The Dirichlet condition (40) is written for diffusesources
Experimental data [5 6] lead to the conclusion that the Robinmodel and the extrapolated boundary models are equivalentbut the Dirichlet model shows significant lack of accuracythat should prevent its use in most applications
Figure 5 shows intensity distributions calculated [1] witheither method inside a turbid medium assuming isotropicscattering 119886 denotes the ratio of absorption and scatteringcoefficients (optical albedo)
119886 =120572119878
120572119905
=120572119878
120572 + 120572119878
(49)
1
01
0010 2 4 6 8 10
Relat
ive d
iffus
e int
ensit
y
Optical depth
Kubelka-MunkTransport theory
Monto CarloDiffusion oppraximation
a = 09
a = 099
Figure 5 Intensity distributions inside turbidmedium comparisonData according to Niemz [1]
The values 119886 = 09 and 119886 = 099 mean that scattering ispredominantThe ordinate shows diffuse intensity in units ofincident intensity
5 Conclusions
Different methods for solving the transport equation werediscussed the most important are the Kubelka-Munk theorythe diffusion approximation and Monte Carlo simulations
The Kubelka-Munk theory deals only with diffuse radi-ation and it is limited to the cases where scattering dom-inates absorption an important disadvantage is the one-dimensional geometry
The diffusion approximation is not restricted to diffuseradiation but it also applies to predominant scattering phe-nomena which is a powerful tool
Monte Carlo simulations provide most accurate solu-tions sincemany input parametersmay be taken into accountin specially developed computer programs the methodallows two-dimensional and three-dimensional evaluationsalthough they require a long computing time
The finite element method was compared [5] to MonteCarlo corresponding methods and it was found to be in goodagreement with Robin boundary conditions the diffusionapproximation although not strictly valid near boundariesis still able to produce correct data when using appropriateboundary conditions whichmatches best a given experimen-tal situation It can be used when theMonte Carlo computingspeed is significantly reduced
8 ISRN Biomedical Engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] M H Niemz Laser-Tissue Interactions Springer Berlin Ger-many 2007
[2] P E Sterian Photonics vol 1 Printech Publishing HouseBucharest Romania 2000 (Romanian)
[3] W F Cheong S A Prahl and A J Welch ldquoA review ofthe optical properties of biological tissuesrdquo IEEE Journal ofQuantum Electronics vol 26 no 12 pp 2166ndash2185 1990
[4] L Wang and S L Jacques Monte Carlo Modeling of LightTransport in Multi-Layered Tissues in Standard C University ofTexas MD Anderson Cancer Center 1992
[5] M Schweiger S R Arridge M Hiraoka and D T Delpy ldquoThefinite element method for the propagation of light in scatteringmedia boundary and source conditionsrdquo Medical Physics vol22 no 11 part 1 pp 1779ndash1792 1995
[6] S Arridge and M Schweiger ldquoPhoton measurement densityfunctions Part 2 finite-element method calculationsrdquo AppliedOptics vol 34 no 34 pp 8026ndash8037 1995
where 1198891199041015840 is the element of the optical path and 119904 is the totallength of the optical path If the attenuation coefficient 120572
119905is
constant wemeet the case of homogeneous attenuation thus119889 = 120572
119905119904
In the literature [1 3] we also find reduced scattering andattenuation coefficients
1205721015840
119878= 120572119878(1 minus 119892)
1205721015840
119905= 120572 + 120572
1015840
119878
(10)
which include the particular situation when 119892 = 1 (justforward scattering) when the intensity is not attenuated Thelinear transport coefficient 1205721015840
119905describes the inverse of the
mean free path between two interaction events in a strongscattering medium
No theory completely explains why the photons arepreferably scattered in the forward direction nor the wave-length dependencies of the phenomena A more convenientapproach is to work with the photon probability function (orphase function) 119901(120579) to be scattered by an angle 120579 whichcan be fitted to experimental measurements According tothis dependence the scattering can be isotropic which means119901(120579) does not depend on 120579 or else anisotropic One canevaluate the anisotropy of the scattering using the coefficientof anisotropy 119892 defined as the average cosine of the scatteringangle 120579
119892 =
int4120587119901 (120579) cos 120579119889120596int4120587119901 (120579) 119889120596
(11)
where 119889120596 = sin 120579119889120579 119889120593 is the solid angle element Isotropicscattering will use 119892 = 0 while 119892 = 1 means totally forwardscattering and 119892 = minus1means totally backward scattering Formost biological tissues we can assume that 119892 values are inthe range of 07ndash099 [1] because the most frequent scatteringangles range between 8∘ and 45∘
The probability (phase) function is normalized
1
4120587int4120587
119901 (120579) 119889120596 = 1 (12)
The best fit to the experiments (Figure 2) was given by theHenyey-Greenstein phase function
Figure 2Themainly straightforward scattering process is describedby the Henyey-Greenstein phase function for different values of 119892Data according to Niemz [1]
The absorption coefficients are generally obtained sub-tracting transmitted reflected and scattered intensities fromthe incident intensity experimental methods determineeither the total attenuation coefficient or both absorption andscattering coefficients Rotating the detector one can obtainthe angular dependence of the scattered intensity thus havingthe anisotropy coefficient 119892
3 General Physical Model forLight Propagation in Tissue
Mathematical description of absorption and scattering canbe made in two ways The most fundamental approach isthe analytical theory based on Maxwellrsquos equations but theircomplexity and the inhomogeneities of biological tissueslimit the possibility to obtain the exact analytical solutionsthus limiting the applicability of the theory
The second approach is the photon transport theorywhich deals with photon beams passing through absorbingand scattering media without considering Maxwellrsquos equa-tions it was extensively used for laser-tissue interactionswhere its predictions were satisfactory in many cases thoughit is a less strict theory compared to analytical theories
Radiance J is the power flux density in a given direction119904 within the solid angle unit 119889120596 (Wsdotcmminus2sdotsrminus1) it decreasesdue to scattering and absorption but increases by the lightscattered from 119904
where119901(119904 1199041015840) is the phase function of a scattered photon from1199041015840 to 119904 direction 120572
119905= 120572119904+ 120572119886is the attenuation coefficient
which includes scattering and absorption coefficients and119889119904 denotes the infinitesimal path length The differential
4 ISRN Biomedical Engineering
equation (15) for radiance is called the radiative transportequation For symmetric scattering about the optical axis119901(119904 1199041015840) = 119901(120579) where 120579 is the scattering angle
Intensity is the measurable optical property it is obtainedby integrating radiance over the solid angle
119868 (119903) = int4120587
119869 (119903 119904) 119889120596 (16)
Its value can be measured within the experiments so one canexpress radiance by
where 120575(120596 minus 120596119878)means a solid angle delta function oriented
in the 119904 directionWhen laser propagates inside a turbid medium the
radiance can be separated in two terms depicting a coherent(unscattered) component and a diffuse (scattering) compo-nent
119869 = 119869119862+ 119869119889 (18)
The coherent radiance can be calculated from119889119869119862
119889119904= minus120572119905119869119862 (19)
which has the solution
119869119862= 1198680120575 (120596 minus 120596
119878) 119890minus119889 (20)
where 1198680is the incident intensity and the dimensionless
parameter 119889 is given by (5) The unscattered term containsall the light that did not interact with the tissue and ischaracterized by exponential decaying
Evaluation of the second term the diffuse radiance isan important problem as it contains all light that has beenscattered at least once one cannot exactly determine the pathof all the scattered photons so adequate approximations orstatistical approach have to be made depending on whetherabsorption or scattering is dominant Complexity of theapproach is related to its accuracy and to the computing timeit needs
4 Mathematical Methods
Mathematical methods are based upon assumptions regard-ing the incident light sources and boundary conditionsTheyare referred to as ldquofirst-order scatteringrdquo ldquoKubelka-Munktheoryrdquo ldquodiffusion approximationrdquo ldquoMonte Carlo simulationrdquoand ldquoinverse adding-doublingrdquo
41 First-Order Scattering Scatering means that multiplescattering is not considered in some cases the diffuse radi-ance is much smaller than the coherent radiance and one canassume it can be neglected
Although it is a simple solution it is often inapplicable for thetherapeutic window (600ndash1200 nm)
J2
J1
120579
AKM
AKM
dzz
+dJ1
+ dJ2J2
SKMSKM
Figure 3 Geometry in Kubelka-Munk theory [1]
42 Kubelka-MunkTheory Unlike the previous assumptionKubelka andMunk supposed the radiance to be diffuse (119869
119862=
0) and defined two coefficients 119860KM for the absorption and119878KM for scattering of diffuse radiation different from 120572 and120572119904that referred only to coherent radiationConsidering diffuse radiance through a one-dimensional
isotropic slab with no reflection at the boundaries thisapproach is equivalent to the diffusionmodel with two-phasefunctions peaked forward and backward [3]
A flux 1198691in the incident direction and a backscattered
flux 1198692are shown in Figure 3 two differential equations can
be written each states that in both directions the radianceencounters two losses (scattering and absorption) and a gainfrom opposite direction scattered photons
1198891198691
119889119911= minus119878KM1198691 minus 119860KM1198691 + 119878KM1198692
1198891198692
119889119911= minus119878KM1198692 minus 119860KM1198692 + 119878KM1198691
(23)
where 119911 is the mean direction of the incident radiationTheirgeneral solutions are
1198691 (119911) = 11986211119890
minus120574119911+ 11986212119890+120574119911
1198692(119911) = 119862
21119890minus120574119911
+ 11986222119890+120574119911
(24)
with
120574 = radic1198602
KM + 2119860KM119878KM (25)
Considering average values of scattered and coherent pathlengths [1] and since diffuse scattering implies that 119869 does notdepend on the scattering angle one can find the connectionbetween the absorption and scattering coefficients
119860KM = 2120572
119878KM = 120572119878(26)
The Kubelka-Munk expressions for reflection and trans-mission of diffuse radiance on a slab of thickness 119889 are [3]
119877 =sinh (119878KM119910119889)
119909 cosh (119878KM119910119889) + 119910 sinh (119878KM119910119889)
119879 =119910
119909 cosh (119878KM119910119889) + 119910 sinh (119878KM119910119889)
(27)
ISRN Biomedical Engineering 5
so that the scattering and absorption coefficients can beexpressed in terms of the measured values of 119877 and 119879
119878KM =1
119910119889ln[
1 minus 119877 (119909 minus 119910)
119879]
119860KM = (119909 minus 1) 119878KM
(28)
where the parameters 119909 and 119910 can be found using
119909 =1 + 1198772minus 1198792
2119877
119910 = radic1199092 minus 1
(29)
This theory is a simple model to measure the optical proper-ties of the tissues but it is restricted by the assumption thatthe incident light is already diffuse the isotropic scatteringand matched boundary refractive indexes which are atypicalfor laser-tissue interaction It can be extended consideringmore fluxes but an important disadvantage is the extendedcomputing time
43 Diffusion Approximation When the scattering phenom-ena dominate absorption the diffuse term in (18) can beexpanded in a series
with 119860 + 119861 = 1198680 Different sets of values for 120572
119905 120572119904
and 119892 provide similar radiances in diffusion approximationcalculus
0
500
1000
1500
2000
minus1500 minus1000 minus500 0 500 1000 1500
Position x (120583m)
Dep
thz
(120583m
)
lowast
Figure 4 Monte Carlo simulated movement of a photon through ahomogeneous medium Data according to Wang and Jacques [4]
44 Monte Carlo Simulations The Monte Carlo methodessentially runs a computer simulation based upon a numer-ical approach to the transport equation (15) The statisticalapproach implies the simulation of a number of 119873 photonsrandom walk the statistical accuracy of the results is propor-tional to radic119873 so that a valuable approximation has to takeinto account a large number of photons This method hasbecome a powerful tool for many disciplines and it requireslarge computers or networks
The main idea of Monte Carlo method for the absorp-tion and scattering phenomena is to follow the opticalpath of a photon through a turbid medium The dis-tance between two collisions is chosen from a logarith-mic distribution using a random computer generated num-ber
Absorption is depicted as a decrease of a weight attributedto the photon during propagation scattering is providedby choosing a new direction of propagation according toa given phase function and another random number Thewhole procedure continues until the photon weight reachesa minimum cut-off value or the photon escapes from theconsidered region
Monte Carlo simulation (Figure 4) needs five steps [1]
(1) Photons are generated at a surface of the consideredregion so that their distribution can be fitted to agiven light source (ie Gaussian beam)
(2) Pathway generation the distance to the first collisionis computed supposing that absorbing and scatteringparticles are randomly distributed a random number0 lt 1205851lt 1 is generated so the distance
119871 (1205851) = minus
ln 1205851
120588120590119878
(36)
where 120588 is the particle density and 120590119878is their scatter-
ing cross-section [1] thus 1120588120590119878represents the mean
6 ISRN Biomedical Engineering
free path and a scattering point is obtained Then asecond random number 120585
2is generated to determine
the scattering angle according to the phase functionthen the third random number is generated to get theazimuth angle
120593 = 21205871205853 (37)
(3) Absorption makes the photon weight decrease by119890minus120572119871(120585
1) 120572 being the absorption coefficient
(4) Elimination of the photon when its attributed weightreaches a certain value then a new photon is launchedand proceeds with step 1
(5) Detection after repeating steps 1ndash4 for a sufficientnumber of photons the computer has stored a mapof pathways so one can make statistical predictionsupon the fraction of the incident photons beingabsorbed by the medium and the spatial and angulardistribution of the transmitted photons
As the simulation accuracy increases with the largernumbers of photons the necessity of extending calculationsmakes them time consuming If earlier results are stored inthe computermemory andused again if neededwith the samephase function computing time will be saved
45 Inverse Adding-Doubling Method The name of themethod comes from reversing the usual process when calcu-lating reflectance and transmittance from the optical prop-erties this method assumes that reflection and transmissionof incident light at a certain angle are known If we need thesame properties for a layer twice as thick we divide it intotwo equal slabs and then add the reflection and transmissioncontributions of either slabThese properties can be obtainedfor an arbitrary slab of tissue starting with a thin known slaband doubling it until the necessary thickness is achieved
The adding method can be extended to simulate layeredtissues with different optical properties
46 The Finite Element Method Boundary and Source Con-ditions The model to calculate light transport in stronglyscattering materials was needed as a component in imagereconstruction schemes that localize the optical propertiesof the tissues from boundary measurements by solving theinverse problem Diffusion approximation to the radiativetransfer equation (15) is an adequate model for scatter-dominant materials for it assumes that scattering dominatesabsorption (generally true for biological tissues) and theanisotropic propagation is weak that is not the case nearsources and boundaries
When the only sources of light aremonochromatic lasersa frequency-independent model of light transport is suitablemulti-wavelength systems will be applied sequentially so thatthe change of frequency is expressed by the change in theoptical properties
The finite element method for the propagation of lightin scattering media aims [5 6] to find the photon density Φand the radiance in a strong scattering domain Ω together
with the outward current (existence) Γ through the boundary120597Ω using the diffusion approximation model to the radiativetransport equation (15) The existence Γ through boundary120597Ω at the point 120585 isin 120597Ω is defined as
where Φ(119903) is the photon density at 119903 isin Ω and 119896(119903) denotesthe diffusion coefficient (34) The solution we seek is acontinuous linear approximation Φℎ of Φ The considereddomainΩ is partitioned into119863nonoverlapping elements 120591 ineach element Φℎ is assumed linear Nodes 119873
119895(119895 = 1 119901)
are attached to the element vertices and Φℎ at each point119903 within the element 120591
119894is the linear interpolation of nodal
values Φ119895
Φℎ(119903 119905) = sum
119895119873119895isin120591119894
Φ119895 (119905) 120595119895 (119903) (40)
where 120595119895are linear nodal shape functions [5] with support
over all elements which have the node 119873119895in the position 119903
119895
as a vertex and 120595119894(119903119894) = 120575119894119895 Equation (39) becomes
The solution of (32) requires appropriate boundary condi-tions The Dirichlet condition (DBC)
Φ (120585) = 0 forall120585 isin 120597Ω (42)
means that the medium around Ω is a perfect absorberphotons are absorbed when crossing 120597Ω so that outside thedomain the photon density equals zero A more realisticboundary condition would be
which is named [5] a Robin boundary condition (RBC) thatconstrains a linear combination of the photon densities andthe current at 120597Ω It can be modified to incorporate amismatch of the refractive indexes 119899 within Ω and 1198991015840 in thesurrounding medium we assume 1198991015840 = 1 so that (41) becomes
where 119877 is the parameter governing internal reflection at theboundary 120597Ω that can be fitted from experimental curves Amodified Robin condition [5] can be
with the critical angle 120579119862= arcsin (1119899) and 119877
0= (119899 minus 1)
2(119899 + 1)
2 (ie for 119899 = 14 we get 119860 = 325)While RBC is a more accurate physical situation than the
DBC the mathematical approach is simple with DBC To geta compromise we can introduce an extrapolated boundaryat a certain distance 119889ext from the physical boundary atwhich DBC apply For an arbitrary two-dimensional domainΩ the extrapolated boundary condition is obtained simplyby adding a border of thickness 119889ext around Ω Differentexpressions [5] of extrapolating 119889ext can be applied to arbi-trary geometries if the radius of the boundary curvature issmall compared to the mean free path both for matched andmismatched refractive index
There are two possibilities to model the light sourcesincident at a point on the boundary collimated or diffusesources
(1) The diffusion equation cannot describe correctly colli-mated sources so we can represent a collimated pencilbeam by an isotropic source at the 11205721015840
119878depth that is
accurate at distances larger than the mean free pathfrom the source but breaks down close to the sourceThe implementation of the collimated source needsa delta-shaped term 119902
0in (39) or other models have
been using the analogy to nuclear engineering withcylinder sources and exponentially decaying
(2) Diffuse sources on the surface can be regarded as aninward directed diffuse photon current distributedover the illuminated boundary segment 120597Ω
2sub 120597Ω
The Dirichlet condition (40) is written for diffusesources
Experimental data [5 6] lead to the conclusion that the Robinmodel and the extrapolated boundary models are equivalentbut the Dirichlet model shows significant lack of accuracythat should prevent its use in most applications
Figure 5 shows intensity distributions calculated [1] witheither method inside a turbid medium assuming isotropicscattering 119886 denotes the ratio of absorption and scatteringcoefficients (optical albedo)
119886 =120572119878
120572119905
=120572119878
120572 + 120572119878
(49)
1
01
0010 2 4 6 8 10
Relat
ive d
iffus
e int
ensit
y
Optical depth
Kubelka-MunkTransport theory
Monto CarloDiffusion oppraximation
a = 09
a = 099
Figure 5 Intensity distributions inside turbidmedium comparisonData according to Niemz [1]
The values 119886 = 09 and 119886 = 099 mean that scattering ispredominantThe ordinate shows diffuse intensity in units ofincident intensity
5 Conclusions
Different methods for solving the transport equation werediscussed the most important are the Kubelka-Munk theorythe diffusion approximation and Monte Carlo simulations
The Kubelka-Munk theory deals only with diffuse radi-ation and it is limited to the cases where scattering dom-inates absorption an important disadvantage is the one-dimensional geometry
The diffusion approximation is not restricted to diffuseradiation but it also applies to predominant scattering phe-nomena which is a powerful tool
Monte Carlo simulations provide most accurate solu-tions sincemany input parametersmay be taken into accountin specially developed computer programs the methodallows two-dimensional and three-dimensional evaluationsalthough they require a long computing time
The finite element method was compared [5] to MonteCarlo corresponding methods and it was found to be in goodagreement with Robin boundary conditions the diffusionapproximation although not strictly valid near boundariesis still able to produce correct data when using appropriateboundary conditions whichmatches best a given experimen-tal situation It can be used when theMonte Carlo computingspeed is significantly reduced
8 ISRN Biomedical Engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] M H Niemz Laser-Tissue Interactions Springer Berlin Ger-many 2007
[2] P E Sterian Photonics vol 1 Printech Publishing HouseBucharest Romania 2000 (Romanian)
[3] W F Cheong S A Prahl and A J Welch ldquoA review ofthe optical properties of biological tissuesrdquo IEEE Journal ofQuantum Electronics vol 26 no 12 pp 2166ndash2185 1990
[4] L Wang and S L Jacques Monte Carlo Modeling of LightTransport in Multi-Layered Tissues in Standard C University ofTexas MD Anderson Cancer Center 1992
[5] M Schweiger S R Arridge M Hiraoka and D T Delpy ldquoThefinite element method for the propagation of light in scatteringmedia boundary and source conditionsrdquo Medical Physics vol22 no 11 part 1 pp 1779ndash1792 1995
[6] S Arridge and M Schweiger ldquoPhoton measurement densityfunctions Part 2 finite-element method calculationsrdquo AppliedOptics vol 34 no 34 pp 8026ndash8037 1995
equation (15) for radiance is called the radiative transportequation For symmetric scattering about the optical axis119901(119904 1199041015840) = 119901(120579) where 120579 is the scattering angle
Intensity is the measurable optical property it is obtainedby integrating radiance over the solid angle
119868 (119903) = int4120587
119869 (119903 119904) 119889120596 (16)
Its value can be measured within the experiments so one canexpress radiance by
where 120575(120596 minus 120596119878)means a solid angle delta function oriented
in the 119904 directionWhen laser propagates inside a turbid medium the
radiance can be separated in two terms depicting a coherent(unscattered) component and a diffuse (scattering) compo-nent
119869 = 119869119862+ 119869119889 (18)
The coherent radiance can be calculated from119889119869119862
119889119904= minus120572119905119869119862 (19)
which has the solution
119869119862= 1198680120575 (120596 minus 120596
119878) 119890minus119889 (20)
where 1198680is the incident intensity and the dimensionless
parameter 119889 is given by (5) The unscattered term containsall the light that did not interact with the tissue and ischaracterized by exponential decaying
Evaluation of the second term the diffuse radiance isan important problem as it contains all light that has beenscattered at least once one cannot exactly determine the pathof all the scattered photons so adequate approximations orstatistical approach have to be made depending on whetherabsorption or scattering is dominant Complexity of theapproach is related to its accuracy and to the computing timeit needs
4 Mathematical Methods
Mathematical methods are based upon assumptions regard-ing the incident light sources and boundary conditionsTheyare referred to as ldquofirst-order scatteringrdquo ldquoKubelka-Munktheoryrdquo ldquodiffusion approximationrdquo ldquoMonte Carlo simulationrdquoand ldquoinverse adding-doublingrdquo
41 First-Order Scattering Scatering means that multiplescattering is not considered in some cases the diffuse radi-ance is much smaller than the coherent radiance and one canassume it can be neglected
Although it is a simple solution it is often inapplicable for thetherapeutic window (600ndash1200 nm)
J2
J1
120579
AKM
AKM
dzz
+dJ1
+ dJ2J2
SKMSKM
Figure 3 Geometry in Kubelka-Munk theory [1]
42 Kubelka-MunkTheory Unlike the previous assumptionKubelka andMunk supposed the radiance to be diffuse (119869
119862=
0) and defined two coefficients 119860KM for the absorption and119878KM for scattering of diffuse radiation different from 120572 and120572119904that referred only to coherent radiationConsidering diffuse radiance through a one-dimensional
isotropic slab with no reflection at the boundaries thisapproach is equivalent to the diffusionmodel with two-phasefunctions peaked forward and backward [3]
A flux 1198691in the incident direction and a backscattered
flux 1198692are shown in Figure 3 two differential equations can
be written each states that in both directions the radianceencounters two losses (scattering and absorption) and a gainfrom opposite direction scattered photons
1198891198691
119889119911= minus119878KM1198691 minus 119860KM1198691 + 119878KM1198692
1198891198692
119889119911= minus119878KM1198692 minus 119860KM1198692 + 119878KM1198691
(23)
where 119911 is the mean direction of the incident radiationTheirgeneral solutions are
1198691 (119911) = 11986211119890
minus120574119911+ 11986212119890+120574119911
1198692(119911) = 119862
21119890minus120574119911
+ 11986222119890+120574119911
(24)
with
120574 = radic1198602
KM + 2119860KM119878KM (25)
Considering average values of scattered and coherent pathlengths [1] and since diffuse scattering implies that 119869 does notdepend on the scattering angle one can find the connectionbetween the absorption and scattering coefficients
119860KM = 2120572
119878KM = 120572119878(26)
The Kubelka-Munk expressions for reflection and trans-mission of diffuse radiance on a slab of thickness 119889 are [3]
119877 =sinh (119878KM119910119889)
119909 cosh (119878KM119910119889) + 119910 sinh (119878KM119910119889)
119879 =119910
119909 cosh (119878KM119910119889) + 119910 sinh (119878KM119910119889)
(27)
ISRN Biomedical Engineering 5
so that the scattering and absorption coefficients can beexpressed in terms of the measured values of 119877 and 119879
119878KM =1
119910119889ln[
1 minus 119877 (119909 minus 119910)
119879]
119860KM = (119909 minus 1) 119878KM
(28)
where the parameters 119909 and 119910 can be found using
119909 =1 + 1198772minus 1198792
2119877
119910 = radic1199092 minus 1
(29)
This theory is a simple model to measure the optical proper-ties of the tissues but it is restricted by the assumption thatthe incident light is already diffuse the isotropic scatteringand matched boundary refractive indexes which are atypicalfor laser-tissue interaction It can be extended consideringmore fluxes but an important disadvantage is the extendedcomputing time
43 Diffusion Approximation When the scattering phenom-ena dominate absorption the diffuse term in (18) can beexpanded in a series
with 119860 + 119861 = 1198680 Different sets of values for 120572
119905 120572119904
and 119892 provide similar radiances in diffusion approximationcalculus
0
500
1000
1500
2000
minus1500 minus1000 minus500 0 500 1000 1500
Position x (120583m)
Dep
thz
(120583m
)
lowast
Figure 4 Monte Carlo simulated movement of a photon through ahomogeneous medium Data according to Wang and Jacques [4]
44 Monte Carlo Simulations The Monte Carlo methodessentially runs a computer simulation based upon a numer-ical approach to the transport equation (15) The statisticalapproach implies the simulation of a number of 119873 photonsrandom walk the statistical accuracy of the results is propor-tional to radic119873 so that a valuable approximation has to takeinto account a large number of photons This method hasbecome a powerful tool for many disciplines and it requireslarge computers or networks
The main idea of Monte Carlo method for the absorp-tion and scattering phenomena is to follow the opticalpath of a photon through a turbid medium The dis-tance between two collisions is chosen from a logarith-mic distribution using a random computer generated num-ber
Absorption is depicted as a decrease of a weight attributedto the photon during propagation scattering is providedby choosing a new direction of propagation according toa given phase function and another random number Thewhole procedure continues until the photon weight reachesa minimum cut-off value or the photon escapes from theconsidered region
Monte Carlo simulation (Figure 4) needs five steps [1]
(1) Photons are generated at a surface of the consideredregion so that their distribution can be fitted to agiven light source (ie Gaussian beam)
(2) Pathway generation the distance to the first collisionis computed supposing that absorbing and scatteringparticles are randomly distributed a random number0 lt 1205851lt 1 is generated so the distance
119871 (1205851) = minus
ln 1205851
120588120590119878
(36)
where 120588 is the particle density and 120590119878is their scatter-
ing cross-section [1] thus 1120588120590119878represents the mean
6 ISRN Biomedical Engineering
free path and a scattering point is obtained Then asecond random number 120585
2is generated to determine
the scattering angle according to the phase functionthen the third random number is generated to get theazimuth angle
120593 = 21205871205853 (37)
(3) Absorption makes the photon weight decrease by119890minus120572119871(120585
1) 120572 being the absorption coefficient
(4) Elimination of the photon when its attributed weightreaches a certain value then a new photon is launchedand proceeds with step 1
(5) Detection after repeating steps 1ndash4 for a sufficientnumber of photons the computer has stored a mapof pathways so one can make statistical predictionsupon the fraction of the incident photons beingabsorbed by the medium and the spatial and angulardistribution of the transmitted photons
As the simulation accuracy increases with the largernumbers of photons the necessity of extending calculationsmakes them time consuming If earlier results are stored inthe computermemory andused again if neededwith the samephase function computing time will be saved
45 Inverse Adding-Doubling Method The name of themethod comes from reversing the usual process when calcu-lating reflectance and transmittance from the optical prop-erties this method assumes that reflection and transmissionof incident light at a certain angle are known If we need thesame properties for a layer twice as thick we divide it intotwo equal slabs and then add the reflection and transmissioncontributions of either slabThese properties can be obtainedfor an arbitrary slab of tissue starting with a thin known slaband doubling it until the necessary thickness is achieved
The adding method can be extended to simulate layeredtissues with different optical properties
46 The Finite Element Method Boundary and Source Con-ditions The model to calculate light transport in stronglyscattering materials was needed as a component in imagereconstruction schemes that localize the optical propertiesof the tissues from boundary measurements by solving theinverse problem Diffusion approximation to the radiativetransfer equation (15) is an adequate model for scatter-dominant materials for it assumes that scattering dominatesabsorption (generally true for biological tissues) and theanisotropic propagation is weak that is not the case nearsources and boundaries
When the only sources of light aremonochromatic lasersa frequency-independent model of light transport is suitablemulti-wavelength systems will be applied sequentially so thatthe change of frequency is expressed by the change in theoptical properties
The finite element method for the propagation of lightin scattering media aims [5 6] to find the photon density Φand the radiance in a strong scattering domain Ω together
with the outward current (existence) Γ through the boundary120597Ω using the diffusion approximation model to the radiativetransport equation (15) The existence Γ through boundary120597Ω at the point 120585 isin 120597Ω is defined as
where Φ(119903) is the photon density at 119903 isin Ω and 119896(119903) denotesthe diffusion coefficient (34) The solution we seek is acontinuous linear approximation Φℎ of Φ The considereddomainΩ is partitioned into119863nonoverlapping elements 120591 ineach element Φℎ is assumed linear Nodes 119873
119895(119895 = 1 119901)
are attached to the element vertices and Φℎ at each point119903 within the element 120591
119894is the linear interpolation of nodal
values Φ119895
Φℎ(119903 119905) = sum
119895119873119895isin120591119894
Φ119895 (119905) 120595119895 (119903) (40)
where 120595119895are linear nodal shape functions [5] with support
over all elements which have the node 119873119895in the position 119903
119895
as a vertex and 120595119894(119903119894) = 120575119894119895 Equation (39) becomes
The solution of (32) requires appropriate boundary condi-tions The Dirichlet condition (DBC)
Φ (120585) = 0 forall120585 isin 120597Ω (42)
means that the medium around Ω is a perfect absorberphotons are absorbed when crossing 120597Ω so that outside thedomain the photon density equals zero A more realisticboundary condition would be
which is named [5] a Robin boundary condition (RBC) thatconstrains a linear combination of the photon densities andthe current at 120597Ω It can be modified to incorporate amismatch of the refractive indexes 119899 within Ω and 1198991015840 in thesurrounding medium we assume 1198991015840 = 1 so that (41) becomes
where 119877 is the parameter governing internal reflection at theboundary 120597Ω that can be fitted from experimental curves Amodified Robin condition [5] can be
with the critical angle 120579119862= arcsin (1119899) and 119877
0= (119899 minus 1)
2(119899 + 1)
2 (ie for 119899 = 14 we get 119860 = 325)While RBC is a more accurate physical situation than the
DBC the mathematical approach is simple with DBC To geta compromise we can introduce an extrapolated boundaryat a certain distance 119889ext from the physical boundary atwhich DBC apply For an arbitrary two-dimensional domainΩ the extrapolated boundary condition is obtained simplyby adding a border of thickness 119889ext around Ω Differentexpressions [5] of extrapolating 119889ext can be applied to arbi-trary geometries if the radius of the boundary curvature issmall compared to the mean free path both for matched andmismatched refractive index
There are two possibilities to model the light sourcesincident at a point on the boundary collimated or diffusesources
(1) The diffusion equation cannot describe correctly colli-mated sources so we can represent a collimated pencilbeam by an isotropic source at the 11205721015840
119878depth that is
accurate at distances larger than the mean free pathfrom the source but breaks down close to the sourceThe implementation of the collimated source needsa delta-shaped term 119902
0in (39) or other models have
been using the analogy to nuclear engineering withcylinder sources and exponentially decaying
(2) Diffuse sources on the surface can be regarded as aninward directed diffuse photon current distributedover the illuminated boundary segment 120597Ω
2sub 120597Ω
The Dirichlet condition (40) is written for diffusesources
Experimental data [5 6] lead to the conclusion that the Robinmodel and the extrapolated boundary models are equivalentbut the Dirichlet model shows significant lack of accuracythat should prevent its use in most applications
Figure 5 shows intensity distributions calculated [1] witheither method inside a turbid medium assuming isotropicscattering 119886 denotes the ratio of absorption and scatteringcoefficients (optical albedo)
119886 =120572119878
120572119905
=120572119878
120572 + 120572119878
(49)
1
01
0010 2 4 6 8 10
Relat
ive d
iffus
e int
ensit
y
Optical depth
Kubelka-MunkTransport theory
Monto CarloDiffusion oppraximation
a = 09
a = 099
Figure 5 Intensity distributions inside turbidmedium comparisonData according to Niemz [1]
The values 119886 = 09 and 119886 = 099 mean that scattering ispredominantThe ordinate shows diffuse intensity in units ofincident intensity
5 Conclusions
Different methods for solving the transport equation werediscussed the most important are the Kubelka-Munk theorythe diffusion approximation and Monte Carlo simulations
The Kubelka-Munk theory deals only with diffuse radi-ation and it is limited to the cases where scattering dom-inates absorption an important disadvantage is the one-dimensional geometry
The diffusion approximation is not restricted to diffuseradiation but it also applies to predominant scattering phe-nomena which is a powerful tool
Monte Carlo simulations provide most accurate solu-tions sincemany input parametersmay be taken into accountin specially developed computer programs the methodallows two-dimensional and three-dimensional evaluationsalthough they require a long computing time
The finite element method was compared [5] to MonteCarlo corresponding methods and it was found to be in goodagreement with Robin boundary conditions the diffusionapproximation although not strictly valid near boundariesis still able to produce correct data when using appropriateboundary conditions whichmatches best a given experimen-tal situation It can be used when theMonte Carlo computingspeed is significantly reduced
8 ISRN Biomedical Engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] M H Niemz Laser-Tissue Interactions Springer Berlin Ger-many 2007
[2] P E Sterian Photonics vol 1 Printech Publishing HouseBucharest Romania 2000 (Romanian)
[3] W F Cheong S A Prahl and A J Welch ldquoA review ofthe optical properties of biological tissuesrdquo IEEE Journal ofQuantum Electronics vol 26 no 12 pp 2166ndash2185 1990
[4] L Wang and S L Jacques Monte Carlo Modeling of LightTransport in Multi-Layered Tissues in Standard C University ofTexas MD Anderson Cancer Center 1992
[5] M Schweiger S R Arridge M Hiraoka and D T Delpy ldquoThefinite element method for the propagation of light in scatteringmedia boundary and source conditionsrdquo Medical Physics vol22 no 11 part 1 pp 1779ndash1792 1995
[6] S Arridge and M Schweiger ldquoPhoton measurement densityfunctions Part 2 finite-element method calculationsrdquo AppliedOptics vol 34 no 34 pp 8026ndash8037 1995
so that the scattering and absorption coefficients can beexpressed in terms of the measured values of 119877 and 119879
119878KM =1
119910119889ln[
1 minus 119877 (119909 minus 119910)
119879]
119860KM = (119909 minus 1) 119878KM
(28)
where the parameters 119909 and 119910 can be found using
119909 =1 + 1198772minus 1198792
2119877
119910 = radic1199092 minus 1
(29)
This theory is a simple model to measure the optical proper-ties of the tissues but it is restricted by the assumption thatthe incident light is already diffuse the isotropic scatteringand matched boundary refractive indexes which are atypicalfor laser-tissue interaction It can be extended consideringmore fluxes but an important disadvantage is the extendedcomputing time
43 Diffusion Approximation When the scattering phenom-ena dominate absorption the diffuse term in (18) can beexpanded in a series
with 119860 + 119861 = 1198680 Different sets of values for 120572
119905 120572119904
and 119892 provide similar radiances in diffusion approximationcalculus
0
500
1000
1500
2000
minus1500 minus1000 minus500 0 500 1000 1500
Position x (120583m)
Dep
thz
(120583m
)
lowast
Figure 4 Monte Carlo simulated movement of a photon through ahomogeneous medium Data according to Wang and Jacques [4]
44 Monte Carlo Simulations The Monte Carlo methodessentially runs a computer simulation based upon a numer-ical approach to the transport equation (15) The statisticalapproach implies the simulation of a number of 119873 photonsrandom walk the statistical accuracy of the results is propor-tional to radic119873 so that a valuable approximation has to takeinto account a large number of photons This method hasbecome a powerful tool for many disciplines and it requireslarge computers or networks
The main idea of Monte Carlo method for the absorp-tion and scattering phenomena is to follow the opticalpath of a photon through a turbid medium The dis-tance between two collisions is chosen from a logarith-mic distribution using a random computer generated num-ber
Absorption is depicted as a decrease of a weight attributedto the photon during propagation scattering is providedby choosing a new direction of propagation according toa given phase function and another random number Thewhole procedure continues until the photon weight reachesa minimum cut-off value or the photon escapes from theconsidered region
Monte Carlo simulation (Figure 4) needs five steps [1]
(1) Photons are generated at a surface of the consideredregion so that their distribution can be fitted to agiven light source (ie Gaussian beam)
(2) Pathway generation the distance to the first collisionis computed supposing that absorbing and scatteringparticles are randomly distributed a random number0 lt 1205851lt 1 is generated so the distance
119871 (1205851) = minus
ln 1205851
120588120590119878
(36)
where 120588 is the particle density and 120590119878is their scatter-
ing cross-section [1] thus 1120588120590119878represents the mean
6 ISRN Biomedical Engineering
free path and a scattering point is obtained Then asecond random number 120585
2is generated to determine
the scattering angle according to the phase functionthen the third random number is generated to get theazimuth angle
120593 = 21205871205853 (37)
(3) Absorption makes the photon weight decrease by119890minus120572119871(120585
1) 120572 being the absorption coefficient
(4) Elimination of the photon when its attributed weightreaches a certain value then a new photon is launchedand proceeds with step 1
(5) Detection after repeating steps 1ndash4 for a sufficientnumber of photons the computer has stored a mapof pathways so one can make statistical predictionsupon the fraction of the incident photons beingabsorbed by the medium and the spatial and angulardistribution of the transmitted photons
As the simulation accuracy increases with the largernumbers of photons the necessity of extending calculationsmakes them time consuming If earlier results are stored inthe computermemory andused again if neededwith the samephase function computing time will be saved
45 Inverse Adding-Doubling Method The name of themethod comes from reversing the usual process when calcu-lating reflectance and transmittance from the optical prop-erties this method assumes that reflection and transmissionof incident light at a certain angle are known If we need thesame properties for a layer twice as thick we divide it intotwo equal slabs and then add the reflection and transmissioncontributions of either slabThese properties can be obtainedfor an arbitrary slab of tissue starting with a thin known slaband doubling it until the necessary thickness is achieved
The adding method can be extended to simulate layeredtissues with different optical properties
46 The Finite Element Method Boundary and Source Con-ditions The model to calculate light transport in stronglyscattering materials was needed as a component in imagereconstruction schemes that localize the optical propertiesof the tissues from boundary measurements by solving theinverse problem Diffusion approximation to the radiativetransfer equation (15) is an adequate model for scatter-dominant materials for it assumes that scattering dominatesabsorption (generally true for biological tissues) and theanisotropic propagation is weak that is not the case nearsources and boundaries
When the only sources of light aremonochromatic lasersa frequency-independent model of light transport is suitablemulti-wavelength systems will be applied sequentially so thatthe change of frequency is expressed by the change in theoptical properties
The finite element method for the propagation of lightin scattering media aims [5 6] to find the photon density Φand the radiance in a strong scattering domain Ω together
with the outward current (existence) Γ through the boundary120597Ω using the diffusion approximation model to the radiativetransport equation (15) The existence Γ through boundary120597Ω at the point 120585 isin 120597Ω is defined as
where Φ(119903) is the photon density at 119903 isin Ω and 119896(119903) denotesthe diffusion coefficient (34) The solution we seek is acontinuous linear approximation Φℎ of Φ The considereddomainΩ is partitioned into119863nonoverlapping elements 120591 ineach element Φℎ is assumed linear Nodes 119873
119895(119895 = 1 119901)
are attached to the element vertices and Φℎ at each point119903 within the element 120591
119894is the linear interpolation of nodal
values Φ119895
Φℎ(119903 119905) = sum
119895119873119895isin120591119894
Φ119895 (119905) 120595119895 (119903) (40)
where 120595119895are linear nodal shape functions [5] with support
over all elements which have the node 119873119895in the position 119903
119895
as a vertex and 120595119894(119903119894) = 120575119894119895 Equation (39) becomes
The solution of (32) requires appropriate boundary condi-tions The Dirichlet condition (DBC)
Φ (120585) = 0 forall120585 isin 120597Ω (42)
means that the medium around Ω is a perfect absorberphotons are absorbed when crossing 120597Ω so that outside thedomain the photon density equals zero A more realisticboundary condition would be
which is named [5] a Robin boundary condition (RBC) thatconstrains a linear combination of the photon densities andthe current at 120597Ω It can be modified to incorporate amismatch of the refractive indexes 119899 within Ω and 1198991015840 in thesurrounding medium we assume 1198991015840 = 1 so that (41) becomes
where 119877 is the parameter governing internal reflection at theboundary 120597Ω that can be fitted from experimental curves Amodified Robin condition [5] can be
with the critical angle 120579119862= arcsin (1119899) and 119877
0= (119899 minus 1)
2(119899 + 1)
2 (ie for 119899 = 14 we get 119860 = 325)While RBC is a more accurate physical situation than the
DBC the mathematical approach is simple with DBC To geta compromise we can introduce an extrapolated boundaryat a certain distance 119889ext from the physical boundary atwhich DBC apply For an arbitrary two-dimensional domainΩ the extrapolated boundary condition is obtained simplyby adding a border of thickness 119889ext around Ω Differentexpressions [5] of extrapolating 119889ext can be applied to arbi-trary geometries if the radius of the boundary curvature issmall compared to the mean free path both for matched andmismatched refractive index
There are two possibilities to model the light sourcesincident at a point on the boundary collimated or diffusesources
(1) The diffusion equation cannot describe correctly colli-mated sources so we can represent a collimated pencilbeam by an isotropic source at the 11205721015840
119878depth that is
accurate at distances larger than the mean free pathfrom the source but breaks down close to the sourceThe implementation of the collimated source needsa delta-shaped term 119902
0in (39) or other models have
been using the analogy to nuclear engineering withcylinder sources and exponentially decaying
(2) Diffuse sources on the surface can be regarded as aninward directed diffuse photon current distributedover the illuminated boundary segment 120597Ω
2sub 120597Ω
The Dirichlet condition (40) is written for diffusesources
Experimental data [5 6] lead to the conclusion that the Robinmodel and the extrapolated boundary models are equivalentbut the Dirichlet model shows significant lack of accuracythat should prevent its use in most applications
Figure 5 shows intensity distributions calculated [1] witheither method inside a turbid medium assuming isotropicscattering 119886 denotes the ratio of absorption and scatteringcoefficients (optical albedo)
119886 =120572119878
120572119905
=120572119878
120572 + 120572119878
(49)
1
01
0010 2 4 6 8 10
Relat
ive d
iffus
e int
ensit
y
Optical depth
Kubelka-MunkTransport theory
Monto CarloDiffusion oppraximation
a = 09
a = 099
Figure 5 Intensity distributions inside turbidmedium comparisonData according to Niemz [1]
The values 119886 = 09 and 119886 = 099 mean that scattering ispredominantThe ordinate shows diffuse intensity in units ofincident intensity
5 Conclusions
Different methods for solving the transport equation werediscussed the most important are the Kubelka-Munk theorythe diffusion approximation and Monte Carlo simulations
The Kubelka-Munk theory deals only with diffuse radi-ation and it is limited to the cases where scattering dom-inates absorption an important disadvantage is the one-dimensional geometry
The diffusion approximation is not restricted to diffuseradiation but it also applies to predominant scattering phe-nomena which is a powerful tool
Monte Carlo simulations provide most accurate solu-tions sincemany input parametersmay be taken into accountin specially developed computer programs the methodallows two-dimensional and three-dimensional evaluationsalthough they require a long computing time
The finite element method was compared [5] to MonteCarlo corresponding methods and it was found to be in goodagreement with Robin boundary conditions the diffusionapproximation although not strictly valid near boundariesis still able to produce correct data when using appropriateboundary conditions whichmatches best a given experimen-tal situation It can be used when theMonte Carlo computingspeed is significantly reduced
8 ISRN Biomedical Engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] M H Niemz Laser-Tissue Interactions Springer Berlin Ger-many 2007
[2] P E Sterian Photonics vol 1 Printech Publishing HouseBucharest Romania 2000 (Romanian)
[3] W F Cheong S A Prahl and A J Welch ldquoA review ofthe optical properties of biological tissuesrdquo IEEE Journal ofQuantum Electronics vol 26 no 12 pp 2166ndash2185 1990
[4] L Wang and S L Jacques Monte Carlo Modeling of LightTransport in Multi-Layered Tissues in Standard C University ofTexas MD Anderson Cancer Center 1992
[5] M Schweiger S R Arridge M Hiraoka and D T Delpy ldquoThefinite element method for the propagation of light in scatteringmedia boundary and source conditionsrdquo Medical Physics vol22 no 11 part 1 pp 1779ndash1792 1995
[6] S Arridge and M Schweiger ldquoPhoton measurement densityfunctions Part 2 finite-element method calculationsrdquo AppliedOptics vol 34 no 34 pp 8026ndash8037 1995
free path and a scattering point is obtained Then asecond random number 120585
2is generated to determine
the scattering angle according to the phase functionthen the third random number is generated to get theazimuth angle
120593 = 21205871205853 (37)
(3) Absorption makes the photon weight decrease by119890minus120572119871(120585
1) 120572 being the absorption coefficient
(4) Elimination of the photon when its attributed weightreaches a certain value then a new photon is launchedand proceeds with step 1
(5) Detection after repeating steps 1ndash4 for a sufficientnumber of photons the computer has stored a mapof pathways so one can make statistical predictionsupon the fraction of the incident photons beingabsorbed by the medium and the spatial and angulardistribution of the transmitted photons
As the simulation accuracy increases with the largernumbers of photons the necessity of extending calculationsmakes them time consuming If earlier results are stored inthe computermemory andused again if neededwith the samephase function computing time will be saved
45 Inverse Adding-Doubling Method The name of themethod comes from reversing the usual process when calcu-lating reflectance and transmittance from the optical prop-erties this method assumes that reflection and transmissionof incident light at a certain angle are known If we need thesame properties for a layer twice as thick we divide it intotwo equal slabs and then add the reflection and transmissioncontributions of either slabThese properties can be obtainedfor an arbitrary slab of tissue starting with a thin known slaband doubling it until the necessary thickness is achieved
The adding method can be extended to simulate layeredtissues with different optical properties
46 The Finite Element Method Boundary and Source Con-ditions The model to calculate light transport in stronglyscattering materials was needed as a component in imagereconstruction schemes that localize the optical propertiesof the tissues from boundary measurements by solving theinverse problem Diffusion approximation to the radiativetransfer equation (15) is an adequate model for scatter-dominant materials for it assumes that scattering dominatesabsorption (generally true for biological tissues) and theanisotropic propagation is weak that is not the case nearsources and boundaries
When the only sources of light aremonochromatic lasersa frequency-independent model of light transport is suitablemulti-wavelength systems will be applied sequentially so thatthe change of frequency is expressed by the change in theoptical properties
The finite element method for the propagation of lightin scattering media aims [5 6] to find the photon density Φand the radiance in a strong scattering domain Ω together
with the outward current (existence) Γ through the boundary120597Ω using the diffusion approximation model to the radiativetransport equation (15) The existence Γ through boundary120597Ω at the point 120585 isin 120597Ω is defined as
where Φ(119903) is the photon density at 119903 isin Ω and 119896(119903) denotesthe diffusion coefficient (34) The solution we seek is acontinuous linear approximation Φℎ of Φ The considereddomainΩ is partitioned into119863nonoverlapping elements 120591 ineach element Φℎ is assumed linear Nodes 119873
119895(119895 = 1 119901)
are attached to the element vertices and Φℎ at each point119903 within the element 120591
119894is the linear interpolation of nodal
values Φ119895
Φℎ(119903 119905) = sum
119895119873119895isin120591119894
Φ119895 (119905) 120595119895 (119903) (40)
where 120595119895are linear nodal shape functions [5] with support
over all elements which have the node 119873119895in the position 119903
119895
as a vertex and 120595119894(119903119894) = 120575119894119895 Equation (39) becomes
The solution of (32) requires appropriate boundary condi-tions The Dirichlet condition (DBC)
Φ (120585) = 0 forall120585 isin 120597Ω (42)
means that the medium around Ω is a perfect absorberphotons are absorbed when crossing 120597Ω so that outside thedomain the photon density equals zero A more realisticboundary condition would be
which is named [5] a Robin boundary condition (RBC) thatconstrains a linear combination of the photon densities andthe current at 120597Ω It can be modified to incorporate amismatch of the refractive indexes 119899 within Ω and 1198991015840 in thesurrounding medium we assume 1198991015840 = 1 so that (41) becomes
where 119877 is the parameter governing internal reflection at theboundary 120597Ω that can be fitted from experimental curves Amodified Robin condition [5] can be
with the critical angle 120579119862= arcsin (1119899) and 119877
0= (119899 minus 1)
2(119899 + 1)
2 (ie for 119899 = 14 we get 119860 = 325)While RBC is a more accurate physical situation than the
DBC the mathematical approach is simple with DBC To geta compromise we can introduce an extrapolated boundaryat a certain distance 119889ext from the physical boundary atwhich DBC apply For an arbitrary two-dimensional domainΩ the extrapolated boundary condition is obtained simplyby adding a border of thickness 119889ext around Ω Differentexpressions [5] of extrapolating 119889ext can be applied to arbi-trary geometries if the radius of the boundary curvature issmall compared to the mean free path both for matched andmismatched refractive index
There are two possibilities to model the light sourcesincident at a point on the boundary collimated or diffusesources
(1) The diffusion equation cannot describe correctly colli-mated sources so we can represent a collimated pencilbeam by an isotropic source at the 11205721015840
119878depth that is
accurate at distances larger than the mean free pathfrom the source but breaks down close to the sourceThe implementation of the collimated source needsa delta-shaped term 119902
0in (39) or other models have
been using the analogy to nuclear engineering withcylinder sources and exponentially decaying
(2) Diffuse sources on the surface can be regarded as aninward directed diffuse photon current distributedover the illuminated boundary segment 120597Ω
2sub 120597Ω
The Dirichlet condition (40) is written for diffusesources
Experimental data [5 6] lead to the conclusion that the Robinmodel and the extrapolated boundary models are equivalentbut the Dirichlet model shows significant lack of accuracythat should prevent its use in most applications
Figure 5 shows intensity distributions calculated [1] witheither method inside a turbid medium assuming isotropicscattering 119886 denotes the ratio of absorption and scatteringcoefficients (optical albedo)
119886 =120572119878
120572119905
=120572119878
120572 + 120572119878
(49)
1
01
0010 2 4 6 8 10
Relat
ive d
iffus
e int
ensit
y
Optical depth
Kubelka-MunkTransport theory
Monto CarloDiffusion oppraximation
a = 09
a = 099
Figure 5 Intensity distributions inside turbidmedium comparisonData according to Niemz [1]
The values 119886 = 09 and 119886 = 099 mean that scattering ispredominantThe ordinate shows diffuse intensity in units ofincident intensity
5 Conclusions
Different methods for solving the transport equation werediscussed the most important are the Kubelka-Munk theorythe diffusion approximation and Monte Carlo simulations
The Kubelka-Munk theory deals only with diffuse radi-ation and it is limited to the cases where scattering dom-inates absorption an important disadvantage is the one-dimensional geometry
The diffusion approximation is not restricted to diffuseradiation but it also applies to predominant scattering phe-nomena which is a powerful tool
Monte Carlo simulations provide most accurate solu-tions sincemany input parametersmay be taken into accountin specially developed computer programs the methodallows two-dimensional and three-dimensional evaluationsalthough they require a long computing time
The finite element method was compared [5] to MonteCarlo corresponding methods and it was found to be in goodagreement with Robin boundary conditions the diffusionapproximation although not strictly valid near boundariesis still able to produce correct data when using appropriateboundary conditions whichmatches best a given experimen-tal situation It can be used when theMonte Carlo computingspeed is significantly reduced
8 ISRN Biomedical Engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] M H Niemz Laser-Tissue Interactions Springer Berlin Ger-many 2007
[2] P E Sterian Photonics vol 1 Printech Publishing HouseBucharest Romania 2000 (Romanian)
[3] W F Cheong S A Prahl and A J Welch ldquoA review ofthe optical properties of biological tissuesrdquo IEEE Journal ofQuantum Electronics vol 26 no 12 pp 2166ndash2185 1990
[4] L Wang and S L Jacques Monte Carlo Modeling of LightTransport in Multi-Layered Tissues in Standard C University ofTexas MD Anderson Cancer Center 1992
[5] M Schweiger S R Arridge M Hiraoka and D T Delpy ldquoThefinite element method for the propagation of light in scatteringmedia boundary and source conditionsrdquo Medical Physics vol22 no 11 part 1 pp 1779ndash1792 1995
[6] S Arridge and M Schweiger ldquoPhoton measurement densityfunctions Part 2 finite-element method calculationsrdquo AppliedOptics vol 34 no 34 pp 8026ndash8037 1995
with the critical angle 120579119862= arcsin (1119899) and 119877
0= (119899 minus 1)
2(119899 + 1)
2 (ie for 119899 = 14 we get 119860 = 325)While RBC is a more accurate physical situation than the
DBC the mathematical approach is simple with DBC To geta compromise we can introduce an extrapolated boundaryat a certain distance 119889ext from the physical boundary atwhich DBC apply For an arbitrary two-dimensional domainΩ the extrapolated boundary condition is obtained simplyby adding a border of thickness 119889ext around Ω Differentexpressions [5] of extrapolating 119889ext can be applied to arbi-trary geometries if the radius of the boundary curvature issmall compared to the mean free path both for matched andmismatched refractive index
There are two possibilities to model the light sourcesincident at a point on the boundary collimated or diffusesources
(1) The diffusion equation cannot describe correctly colli-mated sources so we can represent a collimated pencilbeam by an isotropic source at the 11205721015840
119878depth that is
accurate at distances larger than the mean free pathfrom the source but breaks down close to the sourceThe implementation of the collimated source needsa delta-shaped term 119902
0in (39) or other models have
been using the analogy to nuclear engineering withcylinder sources and exponentially decaying
(2) Diffuse sources on the surface can be regarded as aninward directed diffuse photon current distributedover the illuminated boundary segment 120597Ω
2sub 120597Ω
The Dirichlet condition (40) is written for diffusesources
Experimental data [5 6] lead to the conclusion that the Robinmodel and the extrapolated boundary models are equivalentbut the Dirichlet model shows significant lack of accuracythat should prevent its use in most applications
Figure 5 shows intensity distributions calculated [1] witheither method inside a turbid medium assuming isotropicscattering 119886 denotes the ratio of absorption and scatteringcoefficients (optical albedo)
119886 =120572119878
120572119905
=120572119878
120572 + 120572119878
(49)
1
01
0010 2 4 6 8 10
Relat
ive d
iffus
e int
ensit
y
Optical depth
Kubelka-MunkTransport theory
Monto CarloDiffusion oppraximation
a = 09
a = 099
Figure 5 Intensity distributions inside turbidmedium comparisonData according to Niemz [1]
The values 119886 = 09 and 119886 = 099 mean that scattering ispredominantThe ordinate shows diffuse intensity in units ofincident intensity
5 Conclusions
Different methods for solving the transport equation werediscussed the most important are the Kubelka-Munk theorythe diffusion approximation and Monte Carlo simulations
The Kubelka-Munk theory deals only with diffuse radi-ation and it is limited to the cases where scattering dom-inates absorption an important disadvantage is the one-dimensional geometry
The diffusion approximation is not restricted to diffuseradiation but it also applies to predominant scattering phe-nomena which is a powerful tool
Monte Carlo simulations provide most accurate solu-tions sincemany input parametersmay be taken into accountin specially developed computer programs the methodallows two-dimensional and three-dimensional evaluationsalthough they require a long computing time
The finite element method was compared [5] to MonteCarlo corresponding methods and it was found to be in goodagreement with Robin boundary conditions the diffusionapproximation although not strictly valid near boundariesis still able to produce correct data when using appropriateboundary conditions whichmatches best a given experimen-tal situation It can be used when theMonte Carlo computingspeed is significantly reduced
8 ISRN Biomedical Engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] M H Niemz Laser-Tissue Interactions Springer Berlin Ger-many 2007
[2] P E Sterian Photonics vol 1 Printech Publishing HouseBucharest Romania 2000 (Romanian)
[3] W F Cheong S A Prahl and A J Welch ldquoA review ofthe optical properties of biological tissuesrdquo IEEE Journal ofQuantum Electronics vol 26 no 12 pp 2166ndash2185 1990
[4] L Wang and S L Jacques Monte Carlo Modeling of LightTransport in Multi-Layered Tissues in Standard C University ofTexas MD Anderson Cancer Center 1992
[5] M Schweiger S R Arridge M Hiraoka and D T Delpy ldquoThefinite element method for the propagation of light in scatteringmedia boundary and source conditionsrdquo Medical Physics vol22 no 11 part 1 pp 1779ndash1792 1995
[6] S Arridge and M Schweiger ldquoPhoton measurement densityfunctions Part 2 finite-element method calculationsrdquo AppliedOptics vol 34 no 34 pp 8026ndash8037 1995
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] M H Niemz Laser-Tissue Interactions Springer Berlin Ger-many 2007
[2] P E Sterian Photonics vol 1 Printech Publishing HouseBucharest Romania 2000 (Romanian)
[3] W F Cheong S A Prahl and A J Welch ldquoA review ofthe optical properties of biological tissuesrdquo IEEE Journal ofQuantum Electronics vol 26 no 12 pp 2166ndash2185 1990
[4] L Wang and S L Jacques Monte Carlo Modeling of LightTransport in Multi-Layered Tissues in Standard C University ofTexas MD Anderson Cancer Center 1992
[5] M Schweiger S R Arridge M Hiraoka and D T Delpy ldquoThefinite element method for the propagation of light in scatteringmedia boundary and source conditionsrdquo Medical Physics vol22 no 11 part 1 pp 1779ndash1792 1995
[6] S Arridge and M Schweiger ldquoPhoton measurement densityfunctions Part 2 finite-element method calculationsrdquo AppliedOptics vol 34 no 34 pp 8026ndash8037 1995
![Aerodynamics of Airships [MAX M. MUNK]](https://static.documents.pub/doc/80x56/543c4389afaf9fe1338b4711/aerodynamics-of-airships-max-m-munk.jpg)
