p pen dix a - center for biomedical imaginged as l r t n x l l m lm y a an d s r t n x l l m q lm y...

46

Upload: others

Post on 13-Oct-2020

0 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix A

PN Approximation Method

I follow the PN approximation to reduce the general transport equation to a di�usion

equation� The method is simply to expand all angular dependent quantities in a

spherical harmonic series and truncate the series at the N th moment ���� ��� ���� I

rst review the method for the photon transport equation and then apply it to the

correlation transport equation�

A�� Photon Transport Equation

The linear transport equation for photons propagating in media that scatter and absorb

photons is ���� ��� ���

v

�L�r� � � t�

�t�r�L�r� � � t�� � �tL�r� � � t� � �s

ZL�r� � �� t�f�� � � ��d� ��S�r� � � t� �

�A��

This equation is described in detail in section ��� L�r� � � t� is the radiance at position

r� traveling in direction � � at time t� with units of Wm�� sr�� �sr�steradian�unit solid

angle�� The normalized phase function is f�� � � �� which represents the probability of

scattering into an angle � � from angle � � v is the speed of light in the medium and

�t � �s � �a is the transport coe�cient where �s is the scattering coe�cient and �a

is the absorption coe�cient� S�r� � � t� is the spatial and angular distribution of the

��

Page 2: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

source with units of W m�� sr��� The photon �uence is given by

��r� t� �Zd� L�r� � � t� � �A���

while the photon �ux� or current density� is given by

J�r� t� �Zd� L�r� � � t�� � �A���

Both the �uence and the �ux have units of W m���

Within the PN approximation the radiance and source distribution are expanded

as

L�r� � � t� �NXl��

lXm��l

�l�m�r� t�Yl�m�� � � �A���

and

S�r� � � t� �NXl��

lXm��l

ql�m�r� t�Yl�m�� � � �A���

The photon �uence ��r� t� is given by ���� �see eq� �A���� and the components of the

�ux J�r� t� are given by ���m �see eq� �A������ The ql�m�r� t� are the amplitudes of the

di�erent angular moments of the source at position r and time t�

For the phase function� we make the reasonable assumption that the probability

amplitude is only dependent on the change in direction of the photon and thus

f�� � � �� ��Xl��

�l �

��glPl�� � � ��

��Xl��

lXm��l

glY�

l�m�� ��Yl�m�� � � �A���

where Pl�x� is a Legendre Polynomial and the second line is obtained using the angular

addition rule ����� The phase function is normalized and therefore g� � �

Substituting these expansions into the photon transport equation� eq� �A��� we

obtain

NXl��

lXm��l

��

v

�t� � � r� �t

��l�m�r� t�Yl�m�� �� ql�mYl�m�� �

��sZd� ��l�m�r� t�Yl�m��

���Xl���

lXm���l

gl�Y�

l��m��� ��Yl��m��� �

�� � � �A���

Page 3: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix A� PN Approximation Method ��

Here �t � �s��a is the transport coe�cient� The integral over � � is calculated using

the orthogonality relation for the spherical harmonics� i�e�

Zd� Yl�m�� �Y

l��m��� � � �l�l��m�m� � �A���

The transport equation then becomes

NXl��

lXm��l

��

v

�t� � � r� �

�l�t

��l�m�r� t�� ql�m

�Yl�m�� � � � � �A���

where ��l�t � �s�� gl� � �a is the reduced transport coe�cient�

Next� we multiply eq� �A��� by Y �

����� � and integrate over � � We can use the

orthogonality relation for spherical harmonics �eq� �A���� on all the terms except the

term with � � r�l�m�r� t�� The result is

v

�t���� � �

���t ���� �

NXl��

lXm��l

Zd� � � r�l�mYl�m�� �Y �

����� � � q��� � �A���

The r and t dependence of �l�m� and ql�m is assumed�

We now focus on the remaining integral� First take the dot product between �

and the gradient operator so that we have the components � x��x� � y

��y� and � z

��z� The

components of � Yl�m�� � can be written in terms of spherical harmonics� specically

� xY�

����� � � sin � cos�Y �

����� �

� ��

�� � � �� � � ��

�� � ��� � ��

����Y �

��������� �

�� � ��� � ��� � ��� � �

����Y �

��������� �

�� � � �� � � ��

�� � ��� � ��

����Y �

��������� �

��

�� � ��� � ��� � ��� � �

����Y �

��������� � � �A��

� yY�

����� � � sin � sin�Y �

����� �

�i

��� � �� � � ��

�� � ��� � ��

����Y �

��������� �

Page 4: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

� �i

��� ��� � ��� � ��� � �

����Y �

��������� �

�i

��� � ��� � ��

�� � ��� � ��

����Y �

��������� �

� �i

��� ��� � ���� ��� � �

����Y �

��������� � � �A���

� zY�

����� � � cos �Y �

����� �

� �

��� � ��� � �

�� � ��� � ��

����Y �

������� �

��� ��� �

�� � ��� � �

����Y �

������� � � �A���

� is the direction of the photon where � and � are respectively the polar and azimuthal

angles of the photon direction� These equations are derived with the help of the

recurrence relations for associated Legendre polynomials using the Condon�Shortley

phase convention� Refer to section ��� of Arfken ���� for more details� The book by

Rose entitled Elementary Theory on Angular Momentum is also helpful �����

Upon substituting eqs� �A����A��� into eq� �A��� the integral of three spherical

harmonics becomes an integral of two spherical harmonics which is easily calculated

using the orthogonality relation eq� �A���� After straight�forward algebra the transport

equation nally becomes

v

�t���� � �

���t ���� �

vuut�� � ���� � �

�� � ��� � ��

��

�x� i

�y

���������

vuut�� � �� � �

�� � ��� � �

��

�x� i

�y

���������

vuut�� � �� � � ��

��� ��� � ��

��

�x� i

�y

���������

vuut�� ��� � ���� ��� � �

��

�x� i

�y

���������

vuut�� � �� � � �

�� � ��� � ��

�z������

Page 5: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix A� PN Approximation Method ���

vuut �� ��� �

��� ��� � ��

�z������ � q��� � �A���

This is a coupled set of linear di�erential equations for ����� The PN approximation

is obtained by neglecting all ���� and g� for � N �

A���� P� and Di�usion Approximation

Within the P� approximation� the � � equation gives

v

�t������a�����

s�

��

�x� i

�y

�������

s�

��

�x� i

�y

������

s

�z���� � q��� �

�A���

and the � equations give

v

�t���� � �

���t ���� �

s

�z���� � q��� � �A���

v

�t���� � �

���t ���� �

s�

��

�x� i

�y

����� � q��� � �A���

v

�t����� � �

���t ����� �

s�

��

�x� i

�y

����� � q���� � �A���

We now transform ���� and the ���m�s in eqs� �A�����A��� into the �uence ��r� t�

and net �ux J�r� t�� The �uence is

��r� t� �Zd� L�r� � � t� �

�Xl��

lXm��l

�l�m

Zd� Yl�m�� � �

p�� ���� � �A���

The ���m�s can be gathered in appropriate linear combinations to give the components

of the net �ux J� The appropriate linear combinations are obtained from the denition

of the net �ux �recall eq� �A����� They are

J�r� t� �Zd� L�r� � � t��

��Xl��

lXm��l

�l�m

Zd� �sin � cos��x� sin � sin��y � cos ��z�Yl�m�� �

Page 6: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

s��

�Xl��

lXm��l

�l�m

Zd�

�s

��Y �

����� � � Y �

������ ���x

�is

��Y �

����� � � Y �

������ ���y

�Y �

����� ��z

�Yl�m�� �

s��

�s

������� � ������ �x� i

s

������ � ������ �y � �����z

�� �A����

Eqs� �A�����A��� are thus equivalent to

v

�t��r� t� � �a��r� t� �r � J�r� t� � S��r� t� � �A���

v

�tJ�r� t� � �

���t J�r� t� �

�r��r� t� � S��r� t� � �A����

Here S��r� t� and S��r� t� are respectively the monopole and dipole moments of the

source at position r and time t as dened in eq� ������

Eq� �A��� and eq� �A���� constitute the P� approximation to the photon transport

equation� The derivation of the di�usion equation from these equations is discussed

in section �� starting with eq� ����� and eq� ������ which correspond to eq� �A���

and eq� �A���� respectively�

A���� P� Approximation

With the P� approximation we had four coupled di�erential equations to decouple�

The P� approximation will give us � coupled di�erential equations� In general the

PN approximation will have �N � �� coupled di�erential equations� To simplify

the derivation� we can approximate the one�dimensional transport equation� obtain

a di�erential equation for the �uence� and then extend the one�dimensional result to

three dimensions� In one�dimension we need only concern ourselves with the m � �

equations and therefore the PN approximation will have N � coupled di�erential

equations� This simplication works because the photon transport equation does

not have a prefered direction� and thus all di�erential operators that appear in the PN

Page 7: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix A� PN Approximation Method ���

equations will be symmetric under rotations �e�g� r� and r�� We can thus transform

the one�dimensional equation to a higher dimension equation by replacing ��n

�z�nto r�n

where n is an integer and r is the di�erential operator for the desired dimensional

space� In three dimensions r � ��x�x� �

�y�y � �

�z�z� This simplication does not work

if the optical properties are spatially varying� In this case� spatial derivatives of the

optical properties appear that are not spherically symmetric�

In one dimension the � � equation is

v

�t���� � �a���� �

s

�z���� � q��� � �A����

The � equation is

v

�t���� � �

���t ���� �

s

�z���� �

s�

�z���� � q��� � �A����

The � � equation is

v

�t���� � �

���t ���� �

s�

�z���� �

s�

��

�z���� � q��� � �A����

The � � equation is

v

�t���� � �

���t ���� �

s�

��

�z���� � q��� � �A����

We can decouple these equations to nd an equation for ���� by back�substitution�

First� re�write the equations as

A���� � B���� � q���

C���� � D���� � E���� � q���

F���� � G���� � H���� � q���

I���� � J���� � q��� �

�A����

The coe�cientsA through J are given by eq� �A���� through eq� �A����� First multiply

the third line by J � switch the order of J and H� and substitute for J���� from line

Page 8: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

�� Transposing J and H is okay since we are assuming the optical properties are

spatially uniform� The third line of eq� �A���� becomes

JF����� JG���� �HJ���� � Jq���

JF����� JG���� �Hq��� �HI���� � Jq���

JF����� �JG�HI����� � Jq��� �Hq���

�A����

Next� multiply the second line of eq� �A���� by �JG�HI�� transpose �JG�HI� and

E� and substitute for �JG�HI����� from the third line of eq� �A����� The second line

of eq� �A���� becomes

�JG�HI�C����� �JG�HI�D���� � E�JG �HI����� �

�JG�HI�q���

�JG�HI�C����� �JG�HI�D���� � EJq���� EHq��� � EJF���� �

�JG�HI�q���

�JGC �HIC������ �JGD �HID � EJF ����� �

�JG�HI�q��� �EJq��� � EHq���

�A����

Finally� multiply the rst line of eq� �A���� by L � �JGD�HID�EJF �� transpose

L and B� and substitute for L���� from the third line of eq� �A����� The rst line of

Page 9: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix A� PN Approximation Method ���

eq� �A���� becomes

LA���� �BL���� �

Lq���

LA���� � �BJG�BHI�q����BEJq����BEHq���� �BJGC �BHIC����� �

Lq���

�JGDA �HIDA� EJFA�BJGC �BHIC����� �

�JGD �HID � EJF �q���� �BHI �BJG�q����BEJq����BEHq���

�A����

Finally� replace the coe�cients with their denitions from eq� �A����� change ��

�z�

to r� and ��

�z�to r� and we obtain�

�r � r�� �������r� � � Wq����r� � �Xq����r� � � Y q����r� � � Zq����r� � �

�A���

where

� �� �

v�� i

v

���a����

���t �������t �������t

����a�

���t ����a�

���t �������t �

���t

�A����

� � ����i v� �a���i

v� �

���t ���i

v� �

���t ���i

v� �

���t � � �A����

The right�hand�side of eq� �A��� contains the moments of the source distribution

where

W � ���i

v� �

���t

�i

v� �

���t

�i

v� �

���t

� ��

�i

v� �

���t

r�

����i

v� �

���t

r� �A����

X � ��

s

��

�z�� ��

s

�i

v� �

���t

�i

v� �

���t

�z�A����

Y � ��

s�

��

�i

v� �

���t

r� �A����

Z � ��

s��

� � � � ����

�z��A����

Page 10: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

Solutions to the P� equation and comparisons with the di�usion equation are made

in section ����

A�� Correlation Transport Equation

The PN approximation as described for the photon transport equation can be applied

to the correlation transport equation with only a few modications� The correlation

transport equation is

r�GT� �r� � � � �� � �tG

T� �r� � � � � � �s

ZGT� �r� �

�� � �gs��� � � �� � �f�� � � ��d� �� S�r� � � �

�A����

Here� GT� �r� � � � � is the unnormalized temporal eld correlation function which is

a function of position r� direction � � and correlation time � � The scattering and

absorption coe�cients are respectively �s and �a� and �t � �s � �a is the trans�

port coe�cient� Furthermore� gs��� � � �� � � is the normalized temporal eld correlation

function for single scattering� f�� � � �� is the normalized di�erential cross�section� and

S�r� � � is the source distribution� The scattering coe�cient is the reciprocal of the

scattering length� �s � �l� and the absorption coe�cient is the reciprocal of the ab�

sorption length� �a � �la� The time dependence �not to be confused with correlation

time� has been left out of the equation since I am only considering measurements with

CW sources� The time dependence can be included by adding a time�derivative of

GT� �r� � � � � �i�e� v

�� ��tGT� �r� � � � �� to the left�hand side of eq� �A�����

In analogy to photon transport� the correlation �uence is

G��r� t� �Zd� GT

� �r� � � � � � �A����

while the correlation �ux is given by

Jg�r� t� �Zd� GT

� �r� � � � �� � �A����

The main di�erence between the correlation transport equation and the photon

transport equation is the appearance of gs��� � � �� � � in the integral� as discussed in

Page 11: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix A� PN Approximation Method ���

section ������ The appearance of this angular dependent quantity will result in integ�

rals of three spherical harmonics which are handled in a fashion similar to the handling

of � � r�l�m�r� t� in the photon transport equation �see section A���Within the PN approximationGT

� �r� � � � � and the source distribution are expanded

as

GT� �r� � � � � �

NXl��

lXm��l

�l�m�r� � �Yl�m�� � � �A���

and

S�r� � � �NXl��

lXm��l

ql�m�r�Yl�m�� � � �A����

For the phase function� we make the reasonable assumption that the amplitude is

only dependent on the change in direction of the photon and thus

f�� � � �� ��Xl��

�l �

��glPl�� � � ��

��Xl��

lXm��l

glY�

l�m�� ��Yl�m�� � � �A����

where Pl is a Legendre Polynomial and the second line is obtained using the angular

addition rule ����� The phase function is normalized and therefore g� � �

The single scattering temporal eld correlation function is

gs��� � � �� � � � exp

��q�D�r��� �

E� exp

��k�oD�r��� �

E � � � � � �

�� �A����

where h�r��� �i is the mean square displacement of the scattering particles and q �

�ko sin����� is the momentum transfer for the scattered photon where � is the angle

between � and � �� For particles undergoing Brownian motion� h�r��� �i � �DB�

whereDB is the Brownian di�usion coe�cient� When � � ��DBk�o���� then eq� �A����

is Taylor expanded to

gs��� � � �� � � � � �DBk

�o� � �DBk

�o��� � � �

� � �DBk�o� � �DBk

�o���

�Xm���

Y �

��m�� ��Y��m�� � � �A����

Page 12: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

�� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

The second line is obtained using the angular addition rule ����� Note that ��DBk�o���

� ��� s when DB � x�� cm� s�� and the wavelength of light is �� nm�

Substituting these expansions into the correlation transport equation� eq� �A�����

we obtain

NXl��

lXm��l

�h� � r� �t

i�l�m�r� � �Yl�m�� �� ql�mYl�m�� �

��sZd� ��l�m�r� � �Yl�m��

��NXl���

lXm���l

gl�Y�

l��m��� ��Yl��m��� �

� � �DBk

�o� � �DBk

�o���

�Xm�����

Y �

��m���� ��Y��m���� �

���� � ��A����

The integral over � � can be calculated for the rst two terms between the ������s on the

third line using the spherical harmonic orthogonality relation �eq� �A����� Integrating

gives

NXl��

lXm��l

h� � r� �

�l�t � glkc

i�l�m�r� � �Yl�m�� �� ql�mYl�m�� �

�kcZd� ��l�m�r� � �Yl�m��

��NXl���

lXm���l

gl�Y�

l��m��� ��Yl��m��� �

� ���

�Xm�����

Y �

��m���� ��Y��m���� �

�� � � � �A����

New notation is introduced to simplify eq� �A����� ��l�t � �s��gl���a is the reduced

transport coe�cient �note ����t � �a� and kc � ��sDBk�o� is a dynamic absorption

coe�cient�

The Y �

��m���� ��Y �

l��m��� �� and Y��m���� �Yl��m��� � can be rewritten in terms of single

spherical harmonics� given �see Arfken section ��� �����

Yl�m�� �Y������ � �

s�

��Bm��l�� Yl���m���� ��

s�

��B�ml Yl���m���� � � �A����

Yl�m�� �Y����� � �

s�

��Aml��Yl���m�� � �

s�

��Aml Yl���m�� � � �A����

Yl�m�� �Y����� � �

s�

��B�m��l�� Yl���m���� ��

s�

��Bml Yl���m���� � � �A����

Page 13: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix A� PN Approximation Method �

The coe�cients Aml and B

ml are given by

Aml �

��l�m��l �m�

��l � ���l � �

����

�A���

Bml �

��l�m��l�m� ���l � ���l � �

����

�A����

The product of spherical harmonics in the integral of eq� �A���� can be rewritten

as

��

Xl�m�

gl�Ym��

l� �� ��Y m�

l� �� ��X

m�����

Y m���

� �� ��Y m��

� �� �

���

Xl�m�

gl�

��

��

�Bm�

��l��� Y m�

���l��� �� ���B�m�

l� Y m����

l��� �� ��

��Bm�

��l��� Y m�

��l��� �� � �B�m�

l� Y m���

l��� �� �

��

��

�Am�

l���Ym��

l����� �� �Am�

l� Ym��

l��� �� ��

��Am�

l���Ym�

l����� � �Am�

l� Ym�

l����� �

� ���

�B�m�

��l��� Y m����

l��� �� ���Bm�

l� Ym����l��� �� ��

��B�m�

��l��� Y m���

l��� �� ��Bm�

l� Ym���l��� �

� �

��

�A����

Doing the nal integral over � � we obtain

NXl��

lXm��l

h� � r� �

�l�t � glkc

i�l�m�r� � �Yl�m�� �� ql�mYl�m�� �

�kc�l�m ���

��

��gl��B

ml

hBml Y

ml �� ��B�m��

l�� Y ml���� �

i

� �

��gl��B

�m��l��

hBml��Y

ml���� ��B�m��

l�� Y ml �� �

i

��

��gl��A

ml

hAml Y

ml �� � �Am

l��Yml���� �

i

��

��gl��A

ml��

hAml��Y

ml���� � �Am

l��Yml �� �

i

� �

��gl��B

�ml

h�B�m

l Y ml �� � �Bm��

l�� Y ml���� �

i

��

��gl��B

m��l��

h�B�m

l��Yml���� � �Bm��

l�� Y ml �� �

i �� � ��A����

Page 14: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

�� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

Next� we multiply eq� �A���� by Y �

����� � and integrate over � � Using the ortho�

gonality relations for the spherical harmonics� eq� �A���� we arrive at

����t ���� � g�kc����

�kc��

hg���B

��B

������ � g���B

����B

������� ������

i

hg���B

������� B�

������� � g���B������� B����

��� ����i

�hg���A

��A

������ � g���A

����A

����������

i�

hg���A

����A

�������� � g���A

����A

��������

i�

h�g���B��

� B��� ���� � g���B

�����B

������������

i

h�g���B���

���B��� ������ � g���B

������B

����������

i �

�B������

��

�x� i

�y

��������� �

�B���

��

�x� i

�y

���������

�B�������

��

�x� i

�y

��������� �

�B��

��

�x� i

�y

���������

� A����

�z������ �A�

�z������ � q��� � �A����

This is a coupled set of linear di�erential equations for �����

To obtain the correlation di�usion equation� we neglect all ���� and g� for � �

The � � equation is

�a���� � kc� � g������ �

s�

��

�x� i

�y

������

��

s�

��

�x� i

�y

����� �

s

�z���� � q��� � �A����

and the � equations are

����t ����� �

g� �

kc����� �

s�

��

�x� i

�y

����� � q���� � �A����

����t ���� �

g� �

kc���� �

s

�z���� � q��� � �A����

Page 15: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix A� PN Approximation Method ��

����t ���� �

g� �

kc���� �

s�

��

�x� i

�y

����� � q��� � �A����

Using the denition for the correlation �uence G��r� � � �eq� �A����� and the cor�

relation �ux Jg�r� � � �eq� �A������ the � � and � equations can be rewritten

as

�aG��r� � � �

���sk

�o

D�r��� �

EG��r� � � �r � Jg�r� � � � S��r� � �A����

����t Jg�r� � � �

��s

g� �

k�oD�r��� �

EJg�r� � � �

�rG��r� � � � S��r� � �A���

Recall that we have assumed that the correlationGT� �r� � � � � is nearly isotropic and that

the scattering particles have moved a distance that is much smaller than a wavelength

of light� The rst assumption is satised when �a � �s and the scattering is not

too anisotropic� The second assumption is satised when k�o h�r��� �i � � To keep

our equations consistent with these approximations it is necessary to drop terms from

eq� �A���� in particular

��sJg�r� � � �

�rG��r� � � � S��r� � �A����

Decoupling eq� �A���� and eq� �A���� for G��r� � �� we arrive at the correlation di�usion

equation

��r �

vD�r

� �a �

���sk

�o

D�r��� �

E�G��r� � � � S��r��r �

vD�S��r�

�A����

where D� � v�����s� is the photon di�usion coe�cient�

Page 16: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

�� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

Page 17: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix B

Henyey�Greenstein Phase Function

The normalized Henyey�Greenstein phase function is ��� ��� ���

f�� � � �� � f�cos �� � � g�

� � � g� � �g cos ������ �B��

where � is the angle between the input direction � � and the output direction � � and g

is the scattering anisotropy� The angle � is in the interval ��� ��� This phase function

is unique because the average cosine of the scattering angle is g�

hcos �i �Z �

�f�cos �� cos � sin �d� � g � �B���

and higher moments of the scattering angle are gl�

Z �

�f�cos ��Pl�cos �� sin �d� � gl � �B���

Pl�cos �� is a Legendre polynomial of order l�

��

Page 18: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

�� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

Page 19: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix C

Monte Carlo Code

Following is the listing of the Monte Carlo code that I used to simulate the propagation

of di�use photon density waves and temporal correlation through turbid media� The

code is written to handle innite and semi�innite media which is either homogeneous

or contains a spherical inhomogeneity with di�erent optical and dynamical properties�

When running simulations with a spherical object the optical properties of the back�

ground and object can be anything� The background is assumed static and the object

is dynamic�

This code is compiled using

gcc �o monte monte�c ran�c ran��c ran��c �lm

or an equivalent� Parameters for the simulation are entered on the command line using

the following format�

unix prompt� monte nphotons musp mua musp� mua� radius sz dz DeltaYmax

DeltaYstep output file

nphotons is the number of photons to run in the simulation� musp and mua are

the reduced scattering and absorption coe�cent of the background medium� musp��

mua�� and radius are the reduced scattering and absorption coe�cent and radius

of the spherical object �if it exists�� sz and dz are the z coordinates of the source

and detector� DeltaYmax and DeltaYstep determine the range over which the total

momentum transfer from moving particles is histogrammed� output file is the le

name for the data generated by the simulation�

��

Page 20: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

�� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

The code can be compiled in many di�erent modes� each of which runs a sim�

ulation for a di�erent geometry� Which mode the code runs in is determined by

the dene statements for HOMOGENEOUS� INFINITE� COLLIMATED� SPHER�

ICAL DETECTOR� and PARTIAL DYNAMIC in the beginning of the program� If

HOMOGENEOUS is dened then the medium is homogeneous� otherwise a spher�

ical object is centered at the origin� If INFINITE is dened then the medium is

innite otherwise it is semi�innite� The source is collimated in the �z direction

if COLLIMATED is dened� otherwise the source is isotropic� If INFINITE and

HOMOGENEOUS are dened and COLLIMATED is not dened then the spherical

symmetry of the geometry can be exploited by using spherical detectors� This is

done by dening SPHERICAL DETECTOR� otherwise ring detectors which lie in

a plane are used� For ring detectors� the detection plane is in the xy plane at the

z�coordinate dz and the source is at x�y�� and z�sz� For spherical detectors the

source is at x�y�z��� Finally� the medium can be homogeneous with a uniformly

distributed static and dynamic component� This is a model of capillary blood �ow�

This geometry is set by dening PARTIAL DYNAMIC where VOL FRAC is dened

as the volume fraction �actually Pblood� see section ������ of the dynamic component�

If PARTIAL DYNAMIC is not dened then the system is ��! dynamic�

Two di�erent sets of output are generated� The rst set contains the time domain

data� A le is generated for each detector position� The le names are

output file ������tdd where the ����� indicates the detector position and �tdd

indicates time domain data� The second set contains the total momentum transfer

data� Once again a le is generated for each detector position� The le names are

output file ������DelY where the ����� indicates the detector position and �DelY

indicates momentum transfer data� The formats are self explanatory�

C�� monte�c

�include�stdio�h��include�stdlib�h�

�include�math�h�

Page 21: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix C� Monte Carlo Code ��

�define HOMOGENEOUS �� DEFINE IF THE MEDIUM IS HOMOGENEOUSOTHERWISE THERE IS A SPHERE AT THE ORIGIN ��

�define INFINITE �� DEFINE IF MEDIUM IS INFINITE OTHERWISE SEMI�INFINITE�����define COLLIMATED �� DEFINE IF SOURCE IS COLLIMATED OTHERWISE ISOTROPIC ��

�ifdef INFINITE�ifdef HOMOGENEOUS�ifndef COLLIMATED�define SPHERICAL�DETECTOR �� DEFINE IF THE DETECTORS ARE SPHERICAL

OTHERWISE RING DETECTORS ARE USED ���endif�endif�endif

�ifdef HOMOGENEOUS���define PARTIAL�DYNAMIC �� DEFINE IF SYSTEM IS UNIFORM BUT PART STATIC

AND PART DYNAMIC� ONLY WORKS IF DEF HOMOGENEOUS���define VOL�FRAC � �� VOLUME FRACTION OD DYNAMIC COMPONENT

USED ONLY IF PARTIAL�DYNAMIC IS DEFINED ���endif

�define pi ��� ���� ��define TRUE �define FALSE

�define MAX�TBIN � �� MAXIMUM NUMBER OF TEMPORAL BINS ���define MAX�DeltaYBIN �� �� MAXIMUM NUMBER OF MOMENTUM TRANSFER BINS ���define MAX�RHOBIN � �� MAXIMUM NUMBER OF RADIAL�RHO BINS ��

main� argc� argv �int argc�char �argv����

int i�j�q�int idum� �� FOR RANDOM NUMBER GENERATOR ��int ncall� �� NUMBER OF CALLS TO RANDOM NUMBER GENERATOR � USEDBY THE GENERATOR ��int ABSORBED� �� FLAG IF PHOTON HAS BEEN ABSORBED ��int b� �� TOTAL NUMBER OF PHOTONS TO RUN ��int p� �� NUMBER OF PHOTONS RUN SO FAR ��int OUTSIDE� �� FLAG IF OUTSIDE SPHERE ��int rhoindex�tindex�DeltaYindex� �� INDICES FOR RADIAL�RHO� TIME�

AND MOMENTUM TRANSFER SCORING ��int nt�nDeltaY� �� NUMBER OF TIME AND MOMENTUM BINS NEEDED ��float rho� �� rho�sqrt�x�x�y�y� POSITION OF PHOTON ��float rhostep� �� WIDTH OF DETECTOR ��float t�tgate�tbin� �� CURRENT TIME� GATE TIME� WIDTH OF TEMPORAL BINS ��float DeltaY� �� CURRENT MOMENTUM TRANSFER ��float DeltaYmax� DeltaYstep�

�� MAXIMUM MOMENTUM TRANSFER� WIDTH OF MOMENTUM BINS ��float velocity� �� PHOTON VELOCITY IN THE MEDIUM ��float g��� �� ANISOTROPY FACTOR ��float foo� �� TEMPORARY VARIABLE ��float mua�musp�mus� �� OPTICAL PROPERTIES OF THE BACKGROUND ��float muainv�musinv�float mua��musp��mus�� �� OPTICAL PROPERTIES OF THE SPHERE ��float muainv��musinv��float radi��radisq� �� RADIUS OF THE SPHERE� RADIUS SQUARE ��float phi�theta�sphi�cphi�stheta�ctheta�

�� SCATTERING ANGLES ��float x�y�z� �� CURRENT PHOTON POSITION ��float xold�yold�zold��� LAST POSITION OF PHOTON ��float xm�ym�zm�lm� �� x�y� and z OF PHOTON INTERSECTION WITH A PLANEDETECTOR� lm IS THE LENGTH FROM LAST SCATTERINGEVENT TO DETECTOR INTERSECTION ��float r�rold� �� CURRENT AND LAST RADIAL POSITION OF PHOTON ��float c�c��c�� �� DIRECTION COSINES ��float co� c�o� c�o� �� OLD DIRECTION COSINES ��float la� �� ABSORPTION DISTANCE � RANDOMLY GENERATED FROM MUA ��float ls� �� SCATTERING DISTANCE � RANDOMLY GENERATED FROM MUS ��float sz� dz� �� SOURCE AND DETECTOR Z POSITION� THE SOURCE IS

AT X�Y� ��float hgpart�hgpart��hgpart��hgpart��hgpart �hgpart��

�� USED IN HENYEY�GREENSTEIN CALCULATION ��FILE �fp� �� FILE POINTER FOR SAVING THE DATA ��double fluxout�MAX�RHOBIN��MAX�TBIN�� �� SCORE THE PHOTON FLUX ��double fluxin�MAX�RHOBIN��MAX�TBIN��

double DeltaYout�MAX�RHOBIN��MAX�DeltaYBIN�� �� SCORE THE TOTAL MOMENTUM ��double DeltaYin�MAX�RHOBIN��MAX�DeltaYBIN�� �� TRANSFER ��float rnm� �� RANDOM NUMBER ��float ran� int �idum� int �ncall �� �� FUNCTION DECLARATION ��float B�C�L�T�DELL�T�T��LSP�LAP�LSP��LAP��B��

Page 22: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

�� USED FOR SPHERE INTERSECTION ROUTINE ���� B�C to be used in B����AC calculation� ���� L� T are also for cross passing� entering sphere etc checks ���� B��B����AC��

�� T�T��T are for parameterizations ���� DELL�delta L���� LSP� ls temp ���� LAP � la temp ��

char filenm����� �� FILE NAME FOR DATA FILE ��

�� GET THE COMMAND LINE ARGUMENTS ��if� argc��� � �printf� �usage� monte nphotons musp mua musp� mua� radius sz dz DeltaYmax DeltaYstep output�file�n� ��exit���

�� GET THE NUMBER OF PHOTONS TO MIGRATE ��sscanf� argv��� �!f�� "foo �� b � �int�foo�

�� GET THE OPTICAL PROPERTIES ���� mua " musp are for background medium ���� mua� " musp� are for the spherical tumor� all variables with

#�# as the last character are for the tumor ��sscanf� argv���� �!f�� "musp �� mus � musp����g�� musinv � ��mus�

sscanf� argv���� �!f�� "mua �� muainv � � � mua�sscanf� argv���� �!f�� "musp� �� mus�� musp�����g�� musinv�� ��mus��sscanf� argv� �� �!f�� "mua� �� muainv� � � � mua��

�� GET THE SPHERE RADIUS ���� radi� is the radius of the spherical tumor ���� radisq is for cross pass calculation ��sscanf� argv���� �!f�� "radi� ��radisq�radi��radi��

�� SOURCE AND DETECTOR Z POSITION ��sscanf� argv�$�� �!f�� "sz ��sscanf� argv���� �!f�� "dz ��

�� RANGE OF MOMENTUM TRANSFERS AND BIN STEP ��sscanf� argv���� �!f�� "DeltaYmax ��sscanf� argv��� �!f�� "DeltaYstep ��

�� INITIALIZE OTHER PARAMETERS ��velocity � ����$�e � ����� �� PHOTON VELOCITY IN WATER ��g � �� �� SCATTERING ANISOTROPY ��tbin��e�� � tgate����e��� �� TEMPORAL BIN WIDTH AND GATE ��rhostep��� �� DETECTOR WIDTH ��

�� CALCULATE THE NUMBER OF TEMPORAL AND MOMENTUM BINS THAT ARE NEEDED ��nt � tgate�tbin�nDeltaY � DeltaYmax�DeltaYstep��

�� MAKE SURE THAT ENOUGH MEMORY HAS BEEN ALLOCATED FOR THE NEEDED BINS ��if� nt�MAX�TBIN � �

printf� �Maximum number of time bins exceeded�� ��exit���

if� nDeltaY�MAX�DeltaYBIN � �printf� �Maximum number of DeltaY bins exceeded�� ��

exit���

�� INITIALIZE THE FLUX TO BE ZERO ��for� i�� i�MAX�RHOBIN� i�� �

for� j�� j�nt� j�� � �fluxin�i��j� � ��fluxout�i��j� � ��

�� INITIALIZE DeltaY TO ZERO ��for� i�� i�MAX�RHOBIN� i�� �

for� j�� j�nDeltaY� j�� � �DeltaYin�i��j� � ��DeltaYout�i��j� � ��

�� NUMBER PHOTONS EXECUTED SO FAR ��p��

�� SEED THE RANDOM NUMBER GENERATOR � A LARGE NEGATIVE NUMBER IS REQUIRED ��idum � �������ncall � � �� NUMBER OF CALLS MADE TO THE GENERATOR ��

Page 23: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix C� Monte Carlo Code ��

�� START MIGRATING THE PHOTONS ���� GENERATING PHOTONS UNTIL NUMBER OF PHOTONS EXECUTED �p� IS EQUAL TO THE

NUMBER TO BE GENERATED �b� ��

while �p�b��OUTSIDE�FALSE� �� START OUTSIDE THE OBJECT ��ABSORBED�FALSE� �� PHOTON NOT ABSORBED YET ��

t��� �� START PHOTON AT TIME � ��DeltaY � �� �� NO SCATTERING FROM A MOVING PARTICLE YET�

ZERO MOMENTUM TRANSFER ����p� �� INCREMENT THE PHOTONS EXECUTED COUNTER ��

�� INITIAL SOURCE POSITION ��x��� y��� z�sz� �� CURRENT POSITION ��xold�x� yold�y� zold�z� �� PREVIOUS POSITION ��

�� RADIAL POSITION OF PHOTONUSED FOR CHECKING FOR INTERSECTIONS WITH SPHERICAL DETECTORSAND THE SPHERICAL OBJECT ��

r � sqrt�x�x�y�y�z�z�� �� RADIAL POSITION OF PHOTON ��rold � r� �� OLD RADIAL POSITION ��

�� INITIAL DIRECTION OF PHOTON ��

�ifdef COLLIMATED�� COLLIMATED ALONG Z�AXIS ��c��� co�c�c���� c�o�c��c���� c�o�c��

�endif�ifndef COLLIMATED

�� ISOTROPIC ��rnm � ran�"idum�"ncall��phi���pi�rnm�cphi�cos�phi��sphi�sin�phi��rnm � ran�"idum�"ncall��theta�acos������rnm��ctheta�cos�theta��stheta�sin�theta��c � stheta�cphi�

c� � stheta�sphi�c� � ctheta�co � c�c�o � c��c�o � c��

�endif

�� LOOP UNTIL TIME EXCEEDS GATE� PHOTON IS ABSORBED� OR PHOTON ESCAPES ���ifndef INFINITE �� SEMI�INFINITE MEDIUM ��

while � t�tgate "" ABSORBED��FALSE "" z��sz � ��endif�ifdef INFINITE �� INFINITE MEDIUM ��

while � t�tgate "" ABSORBED��FALSE � ��endif

�� CALCULATE SCATTERING AND ABSORPTION LENGTHS ��

�ifdef HOMOGENEOUS �� HOMOGENEOUS MEDIUM ��rnm � ran�"idum�"ncall��ls � �musinv � log�rnm��rnm � ran�"idum�"ncall��la � �muainv � log�rnm��OUTSIDE�TRUE�if�ls�la� �

L�ls� else �

L�la�ABSORBED�TRUE�

�� ACCUMULATE THE MOMENTUM TRANSFER FROM A MOVING PARTICLE ���ifndef PARTIAL�DYNAMIC

DeltaY �� ��c�co�c��c�o�c��c�o��endif

�ifdef PARTIAL�DYNAMICrnm � ran�"idum�"ncall��if� rnm�VOL�FRAC � DeltaY �� ��c�co�c��c�o�c��c�o�

�endif�endif

Page 24: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

�ifndef HOMOGENEOUS �� HETEROGENEOUS MEDIUM� I�E� A SPHERE IS PRESENT ���� CHECK IF WE ARE OUTSIDE THE SPHERE� OR INSIDE THE SPHERE ��if�radi��r��

rnm � ran�"idum�"ncall��ls��musinv �log�rnm��rnm � ran�"idum�"ncall��la��muainv �log�rnm��OUTSIDE�TRUE�

else �

��INCREMENT TOTAL WAVEVECTOR TRANSFER IF INSIDE SPHERE��DeltaY �� ��c�co�c��c�o�c��c�o�

rnm � ran�"idum�"ncall��ls��musinv��log�rnm��rnm � ran�"idum�"ncall��la��muainv��log�rnm��OUTSIDE�FALSE�

�� CHECK FOR INTERSECTIONS WITH THE SURFACE OF THE SPHERE ��

�� CALCULATE A�B�C ��

rold � r�B��xold�c�yold�c��zold�c���C�rold�rold�radisq�B��B�B�C�

�� CALCULATE MINIMUM DISTANCE ��if�ls�la� �

L�ls� else �

L�la�ABSORBED�TRUE�

�� FINE TUNE ��if �OUTSIDE��

if �B����

T���B�sqrt�B����T����B�sqrt�B����

�� FIND IF WE INTERSECTED THE TUMOR ���� I�E ENTERED IT ��if��T� "" T�L� %% �T�� "" T��L�� �

�� FIND MINIMUM OF T� T���if �T�T���

DELL�T�T�T��

else�

DELL�T��T�T�

L�DELL�

LSP��ls�DELL��musinv�muainv��LAP��la�DELL��muainv�muainv��

�� CHECK IF WE CROSSED OUT THE TUMOR ��if�LSP��T�DELL� "" LAP��T�DELL�� �

L�T�LSP���LSP�T�DELL��musinv��musinv�LAP���LAP�T�DELL��muainv��muainv�if�LSP��LAP���

L��LSP��ABSORBED�FALSE�

else �

L��LAP��ABSORBED�TRUE�

��STILL INSIDE THE TUMOR ��else �

if�LSP�LAP��L��LSP�ABSORBED�FALSE�

Page 25: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix C� Monte Carlo Code ���

else�

L��LAP�ABSORBED�TRUE�

�� END OF INTERSECTION WITH TUMOR ��

�� INSIDE TUMOR ��else �

T���B�sqrt�B����T����B�sqrt�B����

�� FIND IF WE INTERSECTED THE TUMOR ���� I�E EXITED IT ��if��T� "" T�L� %% �T�� "" T��L�� �

�� FIND MAXIMUM OF T� T���if �T�T���

DELL�T�

else�DELL�T��

L�DELL�LSP��ls�DELL��musinv�musinv��LAP��la�DELL��muainv�muainv��

if�LSP�LAP��L��LSP�ABSORBED�FALSE�

else �

L��LAP�ABSORBED�TRUE�

��END OF INSIDE TUMOR ���endif

�� CALCULATE THE NEW POSITION ��xold � x�yold � y�zold � z�rold � r�

x�xold�L�c�y�yold�L�c��z�zold�L�c��r�sqrt�x�x�y�y�z�z��

�� SCORE THE PHOTON ��

�� INCREMENT THE TIME ��t��L�velocity�

�� SCORE THE PHOTON WITH A RING DETECTOR ���ifndef SPHERICAL�DETECTOR

if� z�dz "" zold�dz � �lm��zold�dz���zold�z��L�xm�xold�lm�c�ym�yold�lm�c��rho�sqrt�xm�xm�ym�ym��rhoindex��int�floor�rho�rhostep���� tindex � �t�lm�velocity��tbin���tindex � t�tbin�DeltaYindex��int�ceil�DeltaY�DeltaYstep��if� rhoindex�MAX�RHOBIN "" tindex�MAX�TBIN � �

fluxout�rhoindex��tindex������c�� �� THE EFFECTIVE WIDTH OF THEDETECTOR VARIES WITH THEANGLE OF INCIDENCE� NORMALIZEOUT THIS DEPENDENCE ON ANGLE��

�� fluxout�rhoindex��tindex������if� DeltaYindex�MAX�DeltaYBIN �

DeltaYout�rhoindex��DeltaYindex����

Page 26: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

�ifdef INFINITEif� z�dz "" zold�dz � �

lm��zold�dz���zold�z��L�xm�xold�lm�c�ym�yold�lm�c��rho�sqrt�xm�xm�ym�ym��rhoindex��int�floor�rho�rhostep���� tindex � �t�lm�velocity��tbin���tindex � t�tbin�DeltaYindex��int�ceil�DeltaY�DeltaYstep��if� rhoindex�MAX�RHOBIN "" tindex�MAX�TBIN � �

fluxin�rhoindex��tindex�����c�� �� THE EFFECTIVE WIDTH OF THEDETECTOR VARIES WITH THEANGLE OF INCIDENCE� NORMALIZEOUT THIS DEPENDENCE ON ANGLE��

�� fluxin�rhoindex��tindex������if� DeltaYindex�MAX�DeltaYBIN �

DeltaYin�rhoindex��DeltaYindex����

�endif�endif

�� SCORE THE PHOTON WITH SPHERICAL DETECTORS ��

�ifdef SPHERICAL�DETECTORtindex � t�tbin�DeltaYindex��int�ceil�DeltaY�DeltaYstep��if� r�rold � �

rhoindex � �int�ceil�r�rhostep��while� �float�rhoindex�rhostep�rold � �

if� rhoindex�MAX�RHOBIN � �fluxin�rhoindex��tindex����if� DeltaYindex�MAX�DeltaYBIN �

DeltaYin�rhoindex��DeltaYindex���� rhoindex���

if� r�rold � �

rhoindex � �int�ceil�rold�rhostep��while� �float�rhoindex�rhostep�r � �

if� rhoindex�MAX�RHOBIN � �fluxout�rhoindex��tindex����if� DeltaYindex�MAX�DeltaYBIN �

DeltaYout�rhoindex��DeltaYindex���� rhoindex���

�endif

�� CALCULATE THE NEW SCATTERING ANGLE ASSUMING ISOTROPIC SCATTERING ��rnm � ran�"idum�"ncall��phi���pi�rnm�cphi�cos�phi��sphi�sin�phi��rnm � ran�"idum�"ncall��

theta�acos�����rnm��ctheta�cos�theta��stheta�sin�theta��

�� CALCULATE THE NEW SCATTERING ANGLE USING HENYEY�GREENSTEIN ���� phi���pi�rnm�

cphi�cos�phi��sphi�sin�phi��rnm � ran�"idum�"ncall��hgpart�����g�g���hgpart������g�g���hgpart�����g�����g�rnm���hgpart��hgpart��hgpart��hgpart ������g��hgpart���hgpart����hgpart���foo � �hgpart��hgpart��hgpart�����hgpart ��theta�acos�foo��stheta�sin�theta��

ctheta�cos�theta����

co � c�c�o � c��c�o � c��

Page 27: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix C� Monte Carlo Code ���

c � stheta�cphi�c� � stheta�sphi�c� � ctheta�

�� LOOP UNTIL END OF SINGLE PHOTON �� �� LOOP UNTIL ALL PHOTONS EXHAUSTED ��

�� SAVE TIME DOMAIN DATA USING A FORMAT UNDERSTOOD BY PMI ��for� q�� q�MAX�RHOBIN� q�� � �

rho � rhostep � ��float�q��sprintf� filenm� �!s�! ��f�tdd�� argv��� rho ��fp�fopen� filenm� �w���fprintf� fp� �!d !e ��n�� nt� tgate ��fprintf� fp� � !f !d �n��sz� b ��

�ifndef SPHERICAL�DETECTORfprintf� fp� �!f !f�n�� rho�rhostep��� dz��

�endif�ifdef SPHERICAL�DETECTOR

fprintf� fp� �!f �n�� rho���endif

for� tindex�� tindex�nt� tindex�� ��ifndef SPHERICAL�DETECTOR

fprintf�fp��!lf�n������float��fluxin�q��tindex��fluxout�q��tindex����pi�rhostep�rhostep���pi�rhostep�rho����endif�ifdef SPHERICAL�DETECTOR

fprintf�fp��!lf�n������float��fluxin�q��tindex��fluxout�q��tindex�������pi�rho�rho����endif

fclose�fp��

�� SAVE MOMENTUM TRANSFER DATA ��for� q�� q�MAX�RHOBIN� q�� � �

rho � rhostep � ��float�q��sprintf� filenm� �!s�! ��f�DelY�� argv��� rho ��fp�fopen� filenm� �w���fprintf� fp� �!d !e !e�n�� nDeltaY� DeltaYmax� DeltaYstep ��fprintf� fp� � !f !d �n��sz� b ��

�ifndef SPHERICAL�DETECTORfprintf� fp� �!f !f�n�� rho�rhostep��� dz��

�endif�ifdef SPHERICAL�DETECTOR

fprintf� fp� �!f �n�� rho���endif

for� DeltaYindex�� DeltaYindex�nDeltaY� DeltaYindex�� ��ifndef SPHERICAL�DETECTOR

fprintf�fp��!lf�n������float��DeltaYin�q��DeltaYindex��DeltaYout�q��DeltaYindex����pi�rhostep�rhostep���pi�rhostep�rho���

�endif�ifdef SPHERICAL�DETECTOR

fprintf�fp��!lf�n������float��DeltaYin�q��DeltaYindex��DeltaYout�q��DeltaYindex�������pi�rho�rho����endif

fclose�fp��

C�� ran�c

I use a combination of two random number generators from Numerical Recipes in

C ����� This particular routine was strongly suggested by Dr� Murray Penney at

GE�CRD in Schenectady� NY� Dr� Penney has thoroughly tested the randomness of

this routine�

�include �stdio�h�

�include �math�h�

Page 28: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

static long idum��

float ran� int �idum� int �ncall ��

float rn�float ran�� int �idum ��float ran�� long �idum� ��

�ncall � �ncall � �if� �ncall�� � idum� � �idum��$��if� fmod��ncall�e���� � �idum���������ran��"idum���rn � ran��idum��

if� rn�e�� � rn��e���ran��idum��if� rn�� � rn�e���

return rn�

C�� ran��c

This routine is from Numerical Recipes in C �����

�include �math�h�

�define M $�� �define IA ����define IC ���

float ran��idum�long �idum��static long iy�ir�����static int iff��int j�void nrerror���

if ��idum � %% iff �� � �iff��if ���idum��IC���idum�� ! M� � � �idum � ���idum��for �j��j���$�j��� ��idum��IA���idum��IC� ! M�ir�j����idum�� �idum��IA���idum��IC� ! M�iy���idum�� j� � �$��iy�M�if �j � �$ %% j � � nrerror��RAN�� This cannot happen����iy�ir�j���idum��IA���idum��IC� ! M�ir�j����idum��return �float� iy�M�

�undef M�undef IA�undef IC

C�� ran��c

This routine is from Numerical Recipes in C �����

�define MBIG �define MSEED �������define MZ �define FAC ���MBIG�

Page 29: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Appendix C� Monte Carlo Code ���

float ran��idum�int �idum��static int inext�inextp�static long ma� ���static int iff��long mj�mk�int i�ii�k�

if ��idum � %% iff �� � �iff��mj�MSEED���idum � & ��idum � �idum��mj !� MBIG�ma� ��mj�mk��for �i��i�� ��i��� �ii����i� ! �ma�ii��mk�mk�mj�mk�if �mk � MZ� mk �� MBIG�mj�ma�ii�� for �k��k����k���for �i��i�� �i��� �

ma�i� �� ma���i��� ! ��if �ma�i� � MZ� ma�i� �� MBIG� inext��inextp����idum�� if ���inext �� �� inext��if ���inextp �� �� inextp��mj�ma�inext��ma�inextp��if �mj � MZ� mj �� MBIG�ma�inext��mj�return mj�FAC�

�undef MBIG�undef MSEED

�undef MZ�undef FAC

Page 30: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

Page 31: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Bibliography

�� B� Chance� ed�� Photon Migration in Tissues �Plenum Press� �����

��� A� Yodh and B� Chance� Spectroscopy and imaging with di�using light� Physics

Today ��� �� ������

��� B� J� Tromberg� L� O� Svaasand� T� Tsay� and R� C� Haskell� Properties of photon

density waves in multiple�scattering media� Applied Optics ��� ��� ������

��� M� S� Patterson� B� Chance� and B� C� Wilson� Time resolved re�ectance and

transmittance for the non�invasive measurement of tissue optical properties� Ap�

plied Optics ��� ��� ������

��� G� A� Millikan� Experiments on muscle haemoglobin in vivo� The instantaneous

measurement of muscle metabolism� Proc� Roy� Soc� London B ��� �� ������

��� G� A� Millikan� The oximeter� an instrument for measuring continuously the

oxygen saturation of arterial blood in man� Rev� Sci� Instrum� ��� ��� ������

��� F� F� Jobsis� Nonivasive� infrared monitoring of cerebral and myocardial oxygen

su�ciency and circulatory parameters� Science ��� ��� ������

��� J� M� Schmitt� Simple photon di�usion analysis of the e�ects of multiple scatter�

ing on pulse oximetry� IEEE Transaction on Biomedical Engineering ��� ��

�����

��� M� R� Neuman� Pulse oximetry� physical principles technical realization and

present limitations� Adv� Exp� Med� Biol� ��� �� ������

���

Page 32: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

��� J� W� Severinghaus� History and recent developments in pulse oximetry� Scand�

inavian J� of Clinical and Laboratory Investigations ��� �� ������

�� W� F� Cheong� S� A� Prahl� and A� J� Welch� A review of the optical propeties

of biological tissues� IEEE J� Quantum Electronics ��� ������� ������

��� B� C� Wilson� E� M� Sevick� M� S� Patterson� and B� Chance� Time�dependent

optical spectroscopy and imaging for biomedical applications� Proceedings of the

IEEE �� �� ������

��� See related works by S� R� Arridge et al�� J� P� Kaltenbach et al�� and R� L�

Barbour et al�� in G� Muller et al�� eds��Medical Optical Tomography� Functional

Imaging and Monitoring� SPIE Optical Engineering Press� Bellingham� WA�

Vol� IS� pp� ���� ������

��� S� R� Arridge� P� ven der Zee� M� Cope� and D� T� Delpy�Reconstruction methods

for infra�red absorption imaging� SPIE ����� ��� �����

��� S� P� Gopinath� C� S� Robertson� R� G� Grossman� and B� Chance� Near�infrared

spectroscopic localization of intracranial hematomas� J� Neurosurg � �� ������

��� A� P� Shepherd and P� A� Oberg� eds�� Laser�Doppler Blood Flowmetry �Kluwer

Academic Press� Boston� �����

��� L� E� Drain� in The Laser Doppler Technique �J� Wiley� New York� �����

��� G� V� Belcaro� U� Ho�mann� A� Bollinger� and A� N� Nicolaides� eds�� Laser

Doppler �Med�Orion Publishing Company� London� �����

��� A� D� Edwards� C� Richardson� P� van der Zee� M� Cope� and D� T� Delpy�

Measurement of hemoglobin �ow and blood �ow by near�infrared spectroscopy�

Journal of Applied Physiology �� ��� ������

Page 33: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Bibliography ��

���� E� M� Sevick� B� Chance� J� Leigh� S� Nioka� and M� Maris� Quantitation of

time� and frequency�resolved optical spectra for the determination of tissue oxy�

genation� Analytical Biochemistry ��� ��� �����

��� J� B� Fishkin and E� Gratton� Propagation of photon density waves in strongly

scattering media containing an absorbing �semi�innite plane bounded by a

straight edge� J� Opt� Soc� of Am� A �� �� ������

���� M� A� O�Leary� D� A� Boas� B� Chance� and A� G� Yodh� Refraction of di�use

photon density waves� Phys� Rev� Lett� �� ���� ������

���� B� W� Pogue and M� S� Patterson� Frequency�domain optical�absorption spectro�

scopy of nite tissue volumes using di�usion�theory� Phys� Med� Biol� �� ��

������

���� S� J� Matcher� C� E� Elwell� C� E� Cooper� M� Cope� and D� T� Delpy� Per�

formance comparison of several published tissue near�infrared spectroscopy al�

gorithms� Analytical Biochemistry �� � �� ������

���� H� Liu� D� A� Boas� Y� Zhang� A� G� Yodh� and B� Chance� Determination of

optical properties and blood oxygenation using continuous NIR light� Phys� Med�

Biol� �� ��� ������

���� V� G� Peters� D� R� Wyman� M� S� Patterson� and G� L� Frank� Optical properties

of normal and diseased human breast tissues in the visible and near infrared�

Phys� Med� Biol� ��� �� ������

���� A table of published optical properties compiled by Lihong Wang and Steve

Jacques is available at

ftp�""laser�mda�uth�tmc�edu"pub"tissue"Cheong����msw or

ftp�""laser�mda�uth�tmc�edu"pub"tissue"Cheong����ps�

Page 34: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

���� D� A� Boas� M� A� O�Leary� B� Chance� and A� G� Yodh� Scattering and

wavelength transduction of di�use photon density waves� Phys� Rev� E � �

R���� ������

���� D� A� Boas� M� A� O�Leary� B� Chance� and A� G� Yodh� Scattering of di�use

photon density waves by spherical inhomogeneties within turbid media� analytic

solution and applications� Proc� Natl� Acad� Sci� USA �� ���� ������

���� P� N� den Outer� T� M� Nieuwenhuizen� and A� Lagendijk� Location of objects

in multiple�scattering media� J� Opt� Soc� Am� A �� ��� ������

��� S� Feng� F� Zeng� and B� Chance� Photon migration in the presence of a single

defect� a perturbation analysis� Appl� Opt� ��� ���� ������

���� J� M� Schmitt� A� Knuttel� and J� R� Knutson� Interference of di�usive light

waves� J� Opt� Soc� Am� A � ��� ������

���� B� Chance� K� Kang� L� He� J� Weng� and E� Sevick� Highly sensitive object loc�

ation in tissue models with linear in�phase and anti�phase multi�element optical

arrays in one and two dimensions� Proc� Natl� Acad� Sci� USA � ���� ������

���� J� Maier and E� Gratton� Frequency domain methods in optical tomography�

Detection of localized absorbers and a backscattering reconstruction scheme� in

B� Chance and R� R� Alfano� eds�� Proceedings of Photon Migration and Imaging

in Random Media and Tissues� SPIE Optical Engineering Press� Bellingham�

WA� Vol� ���� pp� ������ ������

���� A� Knuttle� J� M� Schmitt� and J� R� Knutson� Spatial Localization of Absorbing

Bodies by Interfering Di�usive Photon Density Waves� Applied Optics ��� ��

������

���� N� A� Clark� J� H� Lunacek� and G� B� Benedek� A study of Brownian motion

using light scattering� American Journal of Physics ��� ��� ������

Page 35: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Bibliography ���

���� P� J� Berne and R� Pecora� in Dynamic Light Scattering �Wiley� New York�

�����

���� B� J� Berne and R� Pecora� in Dynamic Light Scattering with Applications

to Chemistry� Biology� and Physics �Robert Krieger Publishing Co�� Florida�

�����

���� W� Brown� ed�� Dynamic Light Scattering � The Method and Some Applications

�Clarendon Press� New York� �����

���� G� G� Fuller� J� M� Rallison� R� L� Schmidt� and L� G� Leal� The measurement

of velocity gradients in laminar �ow by homodyne light�scattering spectroscopy�

J� Fluid Mech� �� ��� ������

��� P� Tong� W� I� Goldburg� C� K� Chan� and A� Sirivat� Turbulent transition by

photon�correlation spectroscopy� Phys� Rev� A � � ��� ������

���� M� Bertolotti� B� Crosignani� P� Di� Porto� and D� Sette� Light scattering by

particles suspended in a turbulent �uid� J� Phys� A �� �� ������

���� P� J� Bourke� J� Butterworth� L� E� Drain� P� A� Egelsta�� E� Jakeman� and E� R�

Pike� A study of the spatial structure of turbulent �ow by intensity��uctuation

spectroscopy� J� Phys� A �� �� ������

���� T� Tanaka� C� Riva� and I� Ben�Sira� Blood velocity measurements in human

retinal vessels� Science ���� ��� ������

���� M� Stern� In vivo evaluation of microcirculation by coherent light scattering�

Nature ���� �� ������

���� R� Bonner and R� Nossal� Model for laser Doppler measurements of blood �ow

in tissue� Applied Optics �� ���� �����

Page 36: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

���� H� Z� Cummings and E� R� Pike� eds�� Photon Correlation and Light�Beating

Spectroscopy� Vol� � of NATO Advanced Study Institute Series B� Physics

�Plenum� New York� �����

���� D� J� Pine� D� A� Weitz� P� M� Chaikin� and Herbolzheimer� Di�using�wave

spectroscopy� Phys� Rev� Lett� �� �� ������

���� F� C� MacKintosh and S� John� Di�using�wave spectroscopy and multiple scat�

tering of light in correlated random media� Phys� Rev� B �� ���� ������

���� G� Maret and P� E� Wolf� Multiple light scattering from disordered media� The

e�ect of brownian motion of scatterers� Z� Phys� B ��� ��� ������

��� A� Y� Val�kov and V� P� Romanov� Sov� Phys� JETP ��� ��� ������ �Zh� Eksp�

Teor� Fiz� � ������� �������

���� J� D� Briers� Laser Doppler and time�varying speckle� A reconciliation� J� Opt�

Soc� of Am� A ��� ��� ������

���� M� A� O�Leary� D� A� Boas� B� Chance� and A� G� Yodh� Experimental images of

heterogeneous turbid media by frequency�domain di�using�photon tomography�

Optics Letters �� ��� ������

���� K� M� Case and P� F� Zweifel� in Linear Transport Theory �Addison�Wesley�

MA� �����

���� B� Davison and J� B� Sykes� in Neutron Transport Theory �Oxford Unversity

Press� London� �����

���� S� Glasstone and M� C� Edlund� in the Elements of Nuclear Reactor Theory �D�

van Nostrand Co�� Princeton N�J�� �����

���� B� J� Ackerson� R� L� Dougherty� N� M� Reguigui� and U� Nobbman� Correlation

transfer� Application of radiative transfer solution methods to photon correla�

tion problems� J� Thermophys� and Heat Trans� �� ��� ������

Page 37: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Bibliography ���

���� R� L� Dougherty� B� J� Ackerson� N� M� Reguigui� F� Dorri�Nowkoorani� and

U� Nobbmann� Correlation transfer� Development and application� J� Quant�

Spectrosc� Radiat� Transfer ��� �� ������

���� J� E� Fernandez and V� G� Molinari� Di�usion of polarized photons in the frame

of the transport theory� Nuclear Instruments and Methods in Physics Research

�� �� ������

���� S� Chandrasekhar� in Radiative Transfer �Dover� New York� �����

��� R� Aronson� Subcritical problems in spherical geometry� Nuclear Science and

Engineering ��� ��� ������

���� M� D� Alexandrov� V� S� Remizovich� and D� B� Rogozkin� Multiple light scat�

tering in a two�dimensional medium with large scatterers� J� Opt� Soc� Am� A

�� ���� ������

���� J� D� Jackson� in Classical Electrodynamics �Wiley and Sons Inc�� New York�

�����

���� K� Furutsu and Y� Yamada� Di�usion approximation for a dissipative random

medium and the applications� Phys� Rev� E �� ���� ������

���� A� H� Gandjbakhche� R� F� Bonner� and R� Nossal� Scaling relationships for

anisotropic random walks� J� Stat� Phys� �� �� ������

���� A� H� Gandjbakhche� R� Nossal� and R� F� Bonner� Scaling relationships for

theories of anisotropic random walks applied to tissue optics� Applied Optics

��� ��� ������

���� R� Graa�� M� H� Koelink� F� F� M� de Mul� W� G� Zijlstra� A� C� M� Dassel�

and J� G� Aarnoudse� Condensed Monte Carlo simulations for the description

of light transport� Applied Optics ��� ��� ������

Page 38: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

���� E� Gratton� W� Mantulin� M� J� van de Ven� J� Fishkin� M� Maris� and B�

Chance� in Proceedings of the Third International Conference� Peace through

Mind�Brain Science �Hamamatsu Photonics KK� Hamamatsu� ����� p� ���

���� For an interesting example of a di�usion wave in the context of heat conduc�

tion see A� Sommerfeld� in Partial Di�erential Equations in Physics �Academic

Press� NY� ���� p� ���

���� A� M� Zhabotinsky� M� D� Eager� and I� R� Epstein� Refraction and re�ection

of chemical waves� Phys� Rev� Lett� �� ��� ������

��� The Intralipid used here can be obtained from Kabi Pharmacia in Clayton�

North Carolina�

���� H� J� van Staveren� C� J� M� Moes� J� van Marle� S� A� Prahl� and M� J� C� van

Gemert� Light scattering in Intralipid�� � in the wavelength range of � ���

nm� Applied Optics �� ���� �����

���� S� T� Flock� S� L� Jacques� B� C� Wilson� W� M� Star� and M� J� C� van Ge�

mert� Optical properties of Intralipid� A phantom medium for light propagation

studies� Lasers in Surgery and Medicine ��� �� ������

���� C� J� M� Moes� M� J� C� van Gemert�W� M� Star� J� P� A� Marijnissen� and S� A�

Prahl� Measurements and calculations of the energy �uence rate in a scattering

and absorbing phantom at ��� nm� Applied Optics ��� ���� ������

���� C� M� Hale and M� R� Querry� Applied Optics ��� ��� ������

���� R� C� Haskell� L� O� Svaasand� T� Tsay� T� Feng� M� S� McAdams� and B� J�

Tromberg� Boundary conditions for the di�usion equation in radiative transfer�

J� Opt� Soc� of Am� A ��� ���� ������

���� R� Aronson� Boundary conditions for di�usion of light� J� Opt� Soc� of Am� A

��� ���� ������

Page 39: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Bibliography ���

���� S� Glasstone and M� C� Edlund� in The Elements of Nuclear Reactor Theory

�D� van Nostrand Co�� New York� ����� Chap� � # ��

���� K� M� Case and P� F� Zweifel� in Linear Transport Theory �Addison�Wesley�

MA� ����� Chap� ��

���� E� Hecht� in Optics �Addison�Wesley� Reading� MA� ���� Chapters �� �� ��

��� J� D� Jackson� in Classical Electrodynamics �Wiley and Sons Inc�� New York�

���� ch� ��

���� H� S� Carslaw and J� Jaeger� in Conduction of heat in solids �Oxford University

Press� New York� �����

���� A� Ishimaru� inWave Propagation and Scattering in Random Media �Academic

Press� New York� �����

���� J� D� Jackson� in Classical Electrodynamics �Wiley and Sons Inc�� New York�

���� Chap� � # ��

���� K� A� Kang� D� F� Bruley� J� M� Londono� and B� Chance� Highly scattering op�

tical system identication via frequency response analysis of NIR�TRS spectra�

Annals of Biomedical Engineering ��� ��� ������

���� H� Liu� M� Miwa� B� Beauvoit� N� G� Wang� and B� Chance� Characterization

of absorption and scattering properties of small�volume biological samples using

time�resolved spectroscopy� Analytical Biochemistry ���� ��� ������

���� J� M� Kaltenbach and M� Kaschke� Frequency and time domain modelling of

light transport in random media� Medical Optical Imaging� Functional Imaging

and Monitoring SPIE IS��� �� ������

���� W� M� Star� Comparing the P��approximation with di�usion theory and with

monte carlo calculations of light propagation in a slab geometry� SPIE Institute

Series SPIE IS�� �� ������

Page 40: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

���� D� Pattanayuk� General Electric Consumer Research and Development Report

on Resolution of optical images formed by di�usive waves in highly scattering

media�

���� J� A� Moon and J� Reintjes� Image resolution by use of multiply scattered light�

Optics Letters �� �� ������

��� A� H� Gandjbakhche� R� Nossal� and R� F� Bonner� Resolution limits for optical

transillumination of abnormalities deeply embedded in tissues� Med� Phys� ���

�� ������

���� M� Essenpreis� C� E� Elwell� M� Cope� v� P� S� R� Arridge� and D� T� Delpy�

Spectral dependence of temporal point spread functions in human tissues� Appl�

Opt� ��� �� ������

���� D� G� Papaioannou� G� W� Thooft� J� J� M� Baselmans� and M� J� C� Vange�

mert� Image quality in time resolved transillumination of highly scattering me�

dia� Appl� Opt� ��� ��� ������

���� D� A� Boas� L� E� Campbell� and A� G� Yodh� Scattering and imaging with

di�using temporal eld correlations� Phys� Rev� Lett� �� ��� ������

���� P� R� Bevington� in Data Reduction and Error Analysis for the Physical Sciences

�McGraw�Hill Book Company� NY� �����

���� Y� Yao� Y� Wang� R� L� Barbour� H� L� Graber� and J� Chang� Scattering

characteristics of photon density waves from an object embedded in a spherically

two�layer turbid medium� Proc� SPIE ���� �� ������

���� P� N� Pusey and J� M� Vaughan� Light Scattering and Intensity Fluctuation

Spectroscopy� in M� Davies� ed�� Dielectric and Related Molecular Processes �

vol� � Specialist Periodical Report �London� The Chemical Society� �����

Page 41: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Bibliography ���

���� H� Z� Cummings and H� L� Swinney� Light beating spectroscopy� Prog� Opt� ��

�� ������

���� W� B� Davenport and W� L� Root� in An Introduction to the Theory of Random

Signals and Noise �McGraw�Hill� New York� �����

���� S� O� Rice in N� Wax� ed�� Noise and Stochastic Processes �New York� Dover�

���� p� ���

��� P� N� Pusey� J� M� Vaughan� and D� V� Willets� E�ect of spatial incoherence of

the laser in photon�counting spectroscopy� J� Opt� Soc� Am� �� �� ������

���� T� Bellini� M� A� Glaser� N� A� Clark� and V� Degiorgio� E�ects of nite laser

coherence in quasielastic multiple scattering� Phys� Rev� A ��� ��� �����

���� R� Nossal� S� H� Chen� and C� C� Lai� Use of laser scattering for quantitative

determinations of bacterial motility� Optics Communications �� �� �����

���� X� L� Wu� D� J� Pine� P� M� Chaikin� J� S� Huang� and D� A� Weitz� Di�using�

wave spectroscopy in a shear �ow� J� Opt� Soc� Am� B � � ������

���� A� G� Yodh� P� D� Kaplan� and D� J� Pine� Pulsed di�using�wave spectroscopy�

High resolution through nonlinear optical gating� Phys� Rev� B ��� ���� ������

���� D� J� Pine� D� A� Weitz� J� X� Zhu� and E� Herbolzheimer� Di�using�wave

spectroscopy� Dynamic light scattering in the multiple scattering limit� J� Phys�

France ��� �� ������

���� K� Katayama� G� Nishimura� M� Kinjo� and M� Tamura� Absorbance measure�

ments in turbid media by the photon correlation method� Appl� Optics ��� ���

������

���� G� Nishimura� K� Katayama� M� Kinjo� and M� Tamura� Di�using�wave absorp�

tion spectroscopy in the homogeneous turbid media� to be published in Optics

Communications ������

Page 42: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

���� M� J� Stephen� Temporal �uctuations in wave propagation in random media�

Phys� Rev� B � � ������

��� J� Z� Xue� D� J� Pine� S� T� Milner� X� L� Wu� and P� M� Chaikin� Nonergodicity

and light scattering from polymer gels� Phys� Rev� A ��� ���� ������

�� K� Schatzel� Accuracy of photon correlation measurements on nonergodic

samples� Applied Optics ��� ���� ������

��� J� G� H� Joosten� E� T� F� Gelade� and P� N� Pusey� Dynamic light scattering

by nonergodic media� Brownian particles trapped in polyacrylamide gels� Phys�

Rev� A ��� �� ������

��� A� C� Kak and M� Slaney� in Principles of Computerized Tomographic Imaging

�IEEE Press� New York� �����

��� D� Bicout and R� Maynard� Di�using wave spectroscopy in inhomogeneous �ows�

Physica A �� ��� ������

��� D� Bicout and G� Maret� Multiple light scattering in Taylor�Couette �ow� Phys�

ica A ��� �� ������

��� D� J� Bicout and R� Maynard�Multiple light scattering in turbulent �ow� Physica

B ��� �� ������

��� D� Bicout� Non�newtonian behavior of colloidal suspensions from multiple light

scattering� Physics Letters A ��� ��� ������

��� H� C� van de Hulst� in Light Scattering by Small Particles �Dover Publications

Inc�� NY� ����

��� D� J� Durian� Accuracy of di�using�wave spectroscopy theories� Phys� Rev� E

��� ���� ������

Page 43: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Bibliography ��

���� A� A� Middleton and D� S� Fisher� Discrete scatterers and autocorrelations of

multiply scattered light� Phys� Rev� B ��� ���� �����

��� C� E� Elwell� M� Cope� A� D� Edwards� J� S� Wyatt� D� T� Delpy� and E� O� R�

Reynolds� Quantication of adult cerebral hemodynamics by near�infrared spec�

troscopy� J� Applied Physiology � ���� ������

���� American National Standards Institute� American National Standard for Safe

Use of Lasers� �The Laser Institute of America� ���� see table � and g� �a�

���� B� Chance� M� T� Dait� C� Zhang� T� Hamaoka� and F� Hagerman� Recovery from

excercise�induced desaturation in the quadriceps muscles of elite competitive

rowers� Am� J� Physiol� ���� C��� ������

���� P� D� Line� K� K� Sorensen� and K� Kvernebo� Postocclusive hyperemia in skin

measured in pigs by laser Doppler � In�uence of state of arterial�stenosis in the

limb� European Journal of Surgery ���� �� ������

���� D� Colin and J� L� Saumet� In�uence of external�pressure on transcutaneous

oxygen�tension and laser�Doppler �owmetry on sacral skin� Clinical Physiology

��� � ������

���� N� C� Abbot and J� S� Beck� Biological zero in laser Doppler measurements in

normal ischemic and in�amed human skin� International Journal of Microcir�

culation � Clinical and Experimental ��� �� ������

���� R� Pecora� in Dynamic Light Scattering� Applications of Photon Correlation

Spectroscopy �Plenum Press� New York� �����

���� P� N� Pusey and R� J� A� Tough� in Dynamic Light Scattering �Plenum� New

York� �����

���� H� A� Green� E� E� Burd� N� S� Nishioka� and C� C� Compton� Skin�graft take

and healing following ����nm excimer� continuous�wave carbon dioxide �CO���

Page 44: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

pulsed CO�� or pulsed Holmium�YAG laser�ablation of the graft bed� Archives

of Dermatology ��� ��� ������

���� M� A� Afromowitz� G� S� Van� Liew� and D� Heimbach� Clinical evaluation of

burn injuries using an optical re�ectance technique� IEEE Trans� Biomed� Eng�

��� � ������

��� M� A� Afromowitz� J� B� Callis� D� M� Heimbach� L� A� DeSoto� and M� K�

Norton� Multispectral imaging of burn wounds� A new clinical instrument for

evaluating burn depth� IEEE Trans� Biomed� Eng� ��� ��� ������

���� J� E� Gatti� D� LaRoosa� D� G� Silverman� and C� E� Hartford� Evaluation of

the burn wound with perfusion �uorometry� Journal of Trauma ��� ��� ������

���� H� A� Green� D� Bua� R� R� Anderson� and N� S� Nishioka�Burn depth estimation

using indocyanine green �uorescence� Arch� Dermatol� ���� �� ������

���� J� Micheels� B� Alsbjorn� and B� Sorensen� Clinical use of laser Doppler �ow�

metry in a burns unit� Scand� J� Plast� Reconstr� Surg� ��� �� ������

���� K� Waxman� N� Lefcourt� and B� Achauer� Heated laser Doppler �ow measure�

ments to determine depth of burn injury� Am� J� Surg� �� � �� ������

���� A� Sadhwani� K� T� Schomacker� G� J� Tearney� and N� S� Nishioka� Determin�

ation of Burn Depth using Laser Speckle I� Experiments in Tissue Phantoms�

submitted to Applied Optics ������

���� M� Firbank and D� T� Delpy� A design for a stable and reproducible phantom

for use in near infra�red imaging and spectroscopy� Phys� Med� Biol� ��� ���

������

���� M� Firbank� M� Oda� and D� T� Delpy� An improved design for a stable and re�

producible phantom material for use in near�infrared spectroscopy and imaging�

Phys� Med� Biol� �� ��� ������

Page 45: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

Bibliography ���

���� S� L� Jacques and L� Wang� Monte Carlo modeling of light transport in tissues�

in A� J� Welch and M� J� C� van Gemert� eds�� Optical�Thermal Response of

Laser�Irradiated Tissue �Plenum Press� New York� ���� pp� ������

���� L� Wang� S� L� Jacques� and L� Zheng� MCML�Monte Carlo modeling of light

transport in multi�layered tissues� Computer Methods and Programs in Bio�

medicine � � � ������

��� L� G� Henyey and J� L� Greenstein� Di�use radiation in the galaxy� Astrophys�

J� �� ����� �����

���� J� Ricka� Dynamic light scattering with single�mode and multimode bers� Ap�

plied Optics ��� ���� ������

���� E� R� Van� Keuren� H� Wiese� and D� Horn� Di�using�wave spectroscopy in con�

centrated latex dispersions� An investigation using single�mode bers� Colloids

and Surfaces A � �� ������

���� M� H� Koelink� F� F� M� de Mul� J� Greve� R� Graa�� A� C� M� Dassel� and

J� G� Aarnoudse� Laser Doppler blood �owmetry using two wavelengths� Monte

Carlo simulations and measurements� Appl� Opt� ��� ���� ������

���� C� S� Robertson� S� P� Gopinath� and B� Chance� A new application for near�

infrared spectroscopy� Detection of delayed intracranial hematomas after head

injury� Journal of Neurotrauma ��� �� ������

���� S� P� Gopinath� C� S� Robertson� C� F� Contant� R� K� Narayan� R� G� Grossman�

and B� Chance� Early detection of delayed traumatic intracranial hematomas

using near�infrared spectroscopy� J� Neurosurg� ��� ��� ������

���� G� B� Arfken� in Mathematical Methods for Physicists �Academic Press� Or�

lando� �����

Page 46: p pen dix A - Center for Biomedical Imaginged as L r t N X l l m lm Y A an d S r t N X l l m q lm Y A Th eph ot on uence r t is giv en b y s ee eq A an dt h e comp on en sof t e ux

��� Boas� Di�use Photon Probes of Turbid Media� Theory and Applications

���� M� E� Rose� in Elementary Theory of Angular Momentum �Wiley and Sons Inc��

New York� �����

���� W� H� Press� S� A� Teukolsky� W� T� Vetterling� and B� P� Flannery� in Numerical

Recipes in C� The Art of Scientic Computing �Cambridge University Press�

New York� �����