Supplementary Material to:
Restricted Diffusion in Annular Geometrical Pores
Bahman Ghadirian, Allan M. Torres, Nirbhay N. Yadav and William S. Price
This supplementary material contains the derivations of the important equations that constitute the
results of this paper. It also contains the cross checks with the limiting cases that can be found in the
literature.
Contents
Concentric Cylinder
2
1.
Diffusion Propagator
2
2.
Pulse Gradient Spin-Echo Attenuation in a Concentric Cylinder
8
3.
Mean Square Displacement in a Concentric Cylinder
17
Concentric Sphere
21
1.
Diffusion Propagator
21
2.
Pulse Gradient Spin-Echo Attenuation in a Concentric Sphere
28
3.
Mean Square Displacement in a Concentric Sphere
33
The limiting cases
36
1.
Concentric Cylinder
36
2.
Concentric Sphere
39
REFERENCES
40
A.
B.
C.
D.
1
A. Concentric Cylinder
1. Diffusion Propagator
(a) Zero eigenvalue solution
For the case of λ = 0 (i.e., zero eigenvalue),
2 0  0 .
(S1)
Consequently, the general form of the spatial part of the propagator is also given by a constant
(arbitrarily set to 1)
  0  BRZ   C  1,
(S2)
where B and C are constants and that cannot simultaneously be equal to zero (i.e., a trivial solution).
Thus,
P(r , , z , t ) 0    0T 0  A0 .
(S3)
The normalisation constant is determined by
N 0    0   0 dr   dr   L  a 2  b 2  .
V
V
Therefore
P 0  r0 , 0 , z0 , r1 ,1 , z1 , t  
  0  r0    0  r1 
N 0 N 0
2

1
.
 L  a 2  b2 
(S4)
(b) Positive eigenvalue solution
When    2   2   0, where the spatial eigenvalues (separation constants) ,  and n (a
non-negative integer) are defined by
2
 2 n2 
1 2 Z
1   2 R R 
1 1 2
n
2



,
,
and
r







  2  .


Z z 2
r 2   2
Rr  r 2 r 
r 
r

The solution for the Z component is
Z  A1 cos(  z )  A2 sin(  z ).
(S5)
The solution for the Φ component is
    B1 cos(n )  B2 sin(n )
(S6)
where B1 and B2 are constants. According to the periodic boundary condition B2 = 0, therefore
    B1 cos(n ).
From
 2 R 1 R  2 n2

   2
r 2 r r 
r
(S7)

 R  0, the solution is given by

R  r   En,m Cn ( nm r ),
3
(S8)
where En,m are constants corresponding to each eigenvalue. Cn ( nm r )  F1 J n ( nm r )  F2Yn ( nm r ), where
the index m is a non-negative integer, F1 and F2 are constants and Jn and Yn are Bessel functions of
the first and second kind respectively.
The boundary condition on the inner surface of the top of the cylinder is
 

 D  M L /2  Z
 z

0
(S9)
z  L /2
where ML/2 is the relaxivity on the inner surface of the cylinder, when applied to the cos function in
Eq. (S5) gives
 L M
 s tan  s   L / 2 .
D
 2 
(S10)
This root function also holds for the boundary condition at z = -L/2 (but changing M-L/2).
The boundary condition on the inner surface of the bottom of the cylinder is



  D  M  L /2  Z
z


0
(S11)
z  L /2
where M-L/2 is the relaxivity. Applied to the sin function in Eq. (S5) gives the root function
M
 u L 
   L/2 .

D
 2 
 u cot 
4
(S12)
This root function also holds for the boundary condition at z = L/2 (but changing ML/2).
Thus the z-component eigenfunction is


s 0
u 0
Z   As cos( s z )   Au sin( u z ).
(S13)
The normalisation constants are
L /2
 z sin(2 s z ) 
L  sin( s L) 
N s  A   cos( s z )cos(v z )dz   
 1 

.
 L /2
4 s   L /2 2 
s L 
2
2
s
L /2
and
L /2
 z sin(2 u z ) 
L  sin( u L) 
Nu  A   sin( u z )sin( w z )dz   
 1 

.
 L /2
4 u   L /2 2 
uL 
2
2
u
L /2
The inner radial boundary condition is given by
 

0
 D  Ma  R
 r
 r a
(S14)
where Ma is the relaxivity on the outer annular surface at r = a, when applied to the J solution the
following root function is obtained
 nl
J n ( nl a)
M
 a.
J n ( nl a)
D
From the Bessel function relation1
5
(S15)
Yn  x  
J n  x  cos  n   J  n  x 
,
sin  n 
(S16)
Yn  x  
J n  x  cos  n   J  n  x 
,
sin  n 
(S17)
and thus
also the following property J  n  x    1 J n  x  , this boundary condition for the Y solution give the
n
same eigenvalues as for the J solution.
The inner radial boundary condition is given by



 D  M b  R  0
r

 r b
(S18)
where Mb is the relaxivity on the outer annular surface at r = b. Again applying this operator to the
Y solution gives the following root function for computing the corresponding eigenvalues
nk
Yn nk b 
Yn nk b 

Mb
.
D
(S19)
Regardless of whether the J or Y solution is used on this boundary the eigenvalues will be the same.
Therefore the radial solution is given by
6




R   Fl J n   nl r    Fk Yn nk r .
n 0 l 0
(S20)
n 0 k 0
Thus the diffusion propagator for the annular cylinder will be of the form



P(r ,  , z, t )   H nls J n   nl r  cos  n  cos  s z  exp     nl2   s2  Dt 
n 0 l 0 s 0



  H nlu J n   nl r  cos  n  sin  u z  exp    nl2   u2  Dt 
n 0 l 0 u 0






(S21)
  H nks Yn nk r  cos  n  cos  s z  exp   nk2   s2  Dt 
n 0 k 0 s 0
   H nku Yn nk r  cos  n  sin  u z  exp   nk2   u2  Dt  .
n 0 k 0 u 0
The above propagator is not complete yet as it should be normalised. For this purpose we use the
definition of normalisation for each term in the above expression and boundary conditions of the
radial eigenfunctions at r = a and r = b, thus

N1    J n   nl r  cos  n  cos  s z    J m   mu r  cos  m  cos v z   dv

 2 n2
 L  sin  s L    2 
2
1

a
J

a



a  2

  n nl
4 
 s L  
 nl


 2 n2
2
2
2

J

a

b
J

b





 n nl
b  2
n
nl
 nl


(S22)

2
J

b
.


 n nl 


Note since we put F2 = 0 corresponding to the boundary condition on r = a, the second part of the
radial eigenfunction has no contribution to the normalisation factor. Similarly, the following
normalisation factors are obtained for the second, third and last part of the propagator,
N2 
N3 
L
sin  u L    2
 2 n2
2

a
J

a

  a  2

 u L   n nl
 nl


 2 n2
2
2
2

J

a

b
J

b

 n  nl 
b  2
n  nl 
 nl



2
 J n   nl b   , (S23)


L 
sin  s L    2
 2 n2
2

a
Y

a



a  2

 s L   n nk
nk


 2 n2
2
2
2 
Y

a

b
Y

b





 n nk
b  2
n
nk
nk



2
 Yn nk b   , (S24)


1 
4 
1 
4 
and
7
N4 
L 
1 
4 
sin  u L    2
 2 n2
2

a
Y

a




a  2

 u L   n nk
nk


 2 n2
2
2
2 
Y

a

b
Y

b





 n nk
b  2
n
nk
nk



2
 Yn nk b   .


(S25)
Finally the normalised diffusion propagator in an annular cylinder with inner and outer radius equal
to b and a, respectively and length L is given by
P  r0 , 0 , z0 , r1 ,1 , z1 , t  


4 J n   nl r1  cos  n1  cos  s z1  J n   nl r0  cos  n 0  cos  s z0  exp     nl2   s2  Dt 


n0 l 0 s 0

 L 1 



4 J n   nl r1  cos  n1  cos  u z1  J n   nl r0  cos  n 0  cos  u z0  exp     nl2   u2  Dt 

 
n 0 l 0 u 0
sin  s L    2
 2 n2 
 2 n2 
2
2
2
2
2
  a J n   nl a    a  2  J n   nl a   b J n   nl b    b  2  J n   nl b  
s L  
 nl 
 nl 




 L 1 



n 0 k 0 s 0

 L 1 




 
n 0 k 0 u 0

 2 n2
2
2
2

J

a

b
J

b





 n nl
b  2
n
nl
 nl


sin  s L    2
 2 n2
2
  a Yn  nk a    a  2
s L  
nk


 2 n2
2
2
2
 Yn  nk a   b Yn  nk b    b  2
 nk



2
 J n   nl b  


4Yn  nk r1  cos  n1  cos  s z1  Yn  nk r0  cos  n 0  cos  s z0  exp    nk2   s2  Dt 

 
sin  u L    2
 2 n2
2

a
J

a




a  2
 u L   n nl
 nl


2
 Yn  nk b  


4Yn  nk r1  cos  n1  cos  u z1  Yn  nk r0  cos  n 0  cos  u z0  exp    nk2   u2  Dt 
 sin  u L    2
 2 n2 
 2 n2 
2
2
2
2
2
 L 1 
  a Yn nk a    a  2  Yn  nk a   b Yn  nk b    b  2  Yn  nk b  
uL  
nk 
nk 




.
(S26)
2. Pulse Gradient Spin-Echo Attenuation in a Concentric Cylinder
(a) Zero eigenvalue contribution
Substituting the propagator for  = 0 for the concentric cylinder (Eq. (11)) in to the SGP
equation (Eq. (1)), where ρ(r0) is the equilibrium spin density (equal to 1/(volume) of the restricting
pore) at the point r0 and initial time zero gives,
8
E  q,    0     r0  P  0  r0 , r1 ,   e
 4
a
b

L/2
a

0
 L/2 b
0
  
i 2 q  r1  r0 
dr0 dr1


 i 2 q  r  r
1
1





 L / 2   L  a 2  b2     L  a 2  b2  e 1 0 r0 r1dr0 d0 dz0 dr1d1dz1



L/2


4
I I ,

  2 L2  a 2  b 2 2  r0 r1


where
I r0  
a
b

L/2
0
 L/2
 
exp i 2  r0 q sin  cos 0  z0 q cos    r0 dr0 d 0 dz0 .
I r1 is the same as I r0 with r1, θ1 and z1 instead of r0, θ0 and z0 and negative exponential term. Thus,
I r0  
a
b


0
r0 exp i 2 qr0 sin  cos  0 
L/2
L/2
a

b
0
exp  i 2 z0 q cos   dz0 d 0 dr0

1
i 2 q cos 

 exp  2i qL cos    1  a 
1
r0 exp  i 2 qr sin  cos  0 d 0 dr0


i 2 q cos   exp  i qL cos    b 0

1
i 2 q sin 2 
2
 
r0 exp  i 2 qr0 sin  cos  0  exp  i qL cos    exp  i qL cos    d 0 dr0
 exp  2i qL cos    1 

  aJ1  2 qa sin    bJ1  2 qb sin    .
 exp  i qL cos   
Similarly for I r the following expression is obtained
1
I r1 
 exp  2i qL cos    1 
1

  aJ1  2 qa sin    bJ1  2 qb sin    .
2
2
i 2 q sin   exp  i qL cos    
Then the product of these two integrals will be
1  cos  2 qL cos  
2
I r0 I r1  
 aJ1  2 qa sin    bJ1  2 qb sin  
2 4
2
2 q sin  2 
9
(S27)
The echo attenuation corresponding to the zero-eigenvalue becomes
E  q,    0 
2 1  cos  2 qL cos    aJ1  2 qa sin    bJ1  2 qb sin  
 L q sin  2   a  b
4 2
4
2
2
2

2 2
.
(S28)
(b) Positive eigenvalue contribution
Substituting the propagator for  > 0 for the concentric cylinder (Eq. (12)) in to the SGP
equation (Eq. (1)) gives,
E  q,    4 
a
b

L/2
a

L/2
0
 L/2 b
0
 L/2
   
  r0  P  r0 , r1 ,  ei 2 qr r  r0 r1dr1d1dz1dr0 d0 dz0 .
1
0
(S29)
This multiple integrals are treated as follows,



E  q,    
n0 l 0 s 0



 
n 0 l 0 u 0



 
n 0 k 0 s 0



 
n 0 k 0 u 0
16
exp     nl2   s2  D  I r0 I r1
  a  b 2  L2
2
2
 sin  s L    2
 2 n2 
 2 n2
2
2
2
2


1

a
J

a

a

J

a

b
J

b










b  2

n
nl
 s L   n nl
 nl2  n nl
 nl



16
exp     nl2   u2  D  I r0 I r1
2
2
2
2
 a  b  L

2
 J n   nl b  


 sin  u L    2
 2 n2 
 2 n2
2
2
2
2


1

a
J

a

a

J

a

b
J

b

  


 
b  2

n  nl 
 u L   n nl
 nl2  n nl
 nl



16
exp    nk2   s2  D  I r0 I r1
2
2
  a  b 2  L2


 J n   nl b  


 sin  s L    2
 2 n2 
 2 n2
2
2
2
2 

1

a
Y

a

a

Y

a

b
Y

b










b  2

n
nk
 s L   n nk
 nk2  n nk
 nk



16
exp    nk2   u2  D  I r0 I r1
2
2
2
2
 a  b  L

2
 Yn  nk b  


 sin  u L    2
 2 n2
2
1 
  a Yn nk a    a  2
uL  
nk



2
 Yn  nk b  



 2 n2
2
2
2
 Yn  nk a   b Yn  nk b    b  2
nk


(S30)
10
,
where
 

L/2
0
L/ 2 0
a

L/2
b
0
L/ 2 0
I r0  
a
I r0  
b
 

I     
a
r0
L/2
r J n   nl r0  cos  n 0  cos  s z0  exp i 2  r0 q sin  cos 0  zq cos    dr0 d 0 dz0 ,
r J n   nl r0  cos  n 0  cos  u z0  exp i 2  r0 q sin  cos 0  zq cos    dr0 d 0 dz0 ,
r0Yn nl r0  cos  n 0  cos  s z0  exp i 2  r0 q sin  cos 0  zq cos    dr0 d 0 dz0 ,
b
0
L/ 2
a

L/2
0
L/ 2 0 n
I r0  
b
 
(S31)
r Y nl r0  cos  n 0  cos  u z0  exp i 2  r0 q sin  cos 0  zq cos    dr0 d 0 dz0 .
The corresponding integrals with r1 are the same as Eq. (S31) but with r1, θ1 and z1 instead of r0, θ0
and z0 and negative exponential terms. Next calculating each integral by parts, the first integral will
be subdivided as
 ei 2 q cos  z  i 2 q cos   cos  s z0    s sin  s z0   
cos

z
exp
i
2

q
cos

z
dz







s 0
0
 L / 2
 s2  4 2 q 2 cos 2 

  L / 2
L/2
L/2


 L
  L 
 L
  L 
ei q cos  L  i 2 q cos   cos  s    s sin  s    e i q cos  L  i 2 q cos   cos  s    s sin  s  
 2 
 2 
 2 
 2 



2
2 2
2
 s  4 q cos 
 L
 L
2 s cos  qL cos   sin  s    4 q cos   sin  qL cos   cos  s 
 2 
 2 .

2
2 2
2
 s  4 q cos 
Similarly the same expression is obtained for the integral corresponding to r0,

L/2
 L/2
cos  s z0  exp  i 2 q cos  z0  dz0 
L/2
 L/2
cos  s z0  exp  i 2 q cos  z0  dz0
2

 s L 
 L 
  4 q cos   sin  qL cos   cos  s  
 2 s cos  qL cos   sin 

 2 
 2  .

2
2 2
2


 s  4 q cos 




Also the second and third parts of the integral can be evaluated together as
11
a

b
0

 i n
r0 J n   nl r0  cos  n 0  exp  i 2 qr sin  cos  0 d 0 dr0
 2 qa sin   J n   nl a  J n  2 qa sin      nl a  J n   nl a  J n  2 qa sin   


   2 qb sin   J n   nl b  J n  2 qb sin      nl b  J n   nl b  J n  2 qb sin   
 nl2  4 2 q 2 sin 2 
.
The integral corresponding to r0, which contains a negative exponential instead, is given by
a

b
0

r0 J n   nl r0  cos  n 0  exp  i 2 qr0 sin  cos 0 d 0 dr0
 2 qa sin   J n   nl a  J n  2 qa sin      nl a  J n   nl a  J n  2 qa sin   


   2 qb sin   J n   nl b  J n  2 qb sin      nl b  J n   nl b  J n  2 qb sin   
n

i 
.
 nl2  4 2 q 2 sin 2 
Then
I r0 I r1   i 2   1  2
n
n
 2 qa sin   J n   nl a  J n  2 qa sin      nl a  J n   nl a  J n  2 qa sin   


   2 qb sin   J n   nl b  J n  2 qb sin      nl b  J n   nl b  J n  2 qb sin   
 nl2   2 q sin  2 


2
2
2

 s L 
 s L 
 2 s cos  qL cos   sin  2    4 q cos   sin  qL cos   cos  2  





,
2 2
2
 s   2 q cos   


which after simplification gives
  2 qa sin   J n   nl a  J n  2 qa sin      nl a  J n   nl a  J n  2 qa sin    
 

    2 qb sin   J n   nl b  J n  2 qb sin     nl b  J n  nl b  J n  2 qb sin    


   2 cos  qL cos   sin   s L    4 q cos   sin  qL cos   cos   s L   
s




 
 2 
 2   
I r0 I r1   

 nl2   2 q sin  2   s2   2 q cos  2 















12
2
(S32)
The second integral I r , is again subdivided to be determined, viz.
0
 ei 2 q cos  z  i 2 q cos   sin  u z0    u cos  u z0   

 L / 2 sin  u z0  exp  i 2 q cos  z0  dz0  
 u2  4 2 q 2 cos 2 
  L / 2

L/2
L/2


 L
  L 
 L
  L 
ei q cos  L  i 2 q cos   sin  u    u sin  u    e i q cos  L    i 2 q cos   sin  u    u cos  u  
 2 
 2 
 2 
 2 



2
2 2
2
 u  4 q cos 

 uL 
 L
 i 2 u sin  qL cos   cos  u 

 2 
 2 .
2
2 2
2
 u  4 q cos 
 i 4 q cos   cos  qL cos   sin 
Similarly the above expression for the integral corresponding to r0 is given by

L /2
 L /2
sin  u z0  exp  i 2 q cos  z0  dz0
 L
 L
  i 4 q cos   cos  qL cos   sin  u   i 2 u sin  qL cos   cos  u 
 2 
 2 .

2
2 2
2
 u  4 q cos 
Then

L/2
L/2
sin  u z0  exp  i 2 q cos  z0  dz 
L/2
L/2
sin  u z0  exp  i 2 q cos  z0  dz0
2

 uL 
 L 
  4 q cos   cos  qL cos   sin  u  
 2 u sin  qL cos   cos 

 2 
 2  .

2
2 2
2


 u  4 q cos 




Again the following product of two multiple integrals after simplification gives
13
(S33)
  2 qa sin   J n   nl a  J n  2 qa sin      nl a  J n   nl a  J n  2 qa sin    
 

    2 qb sin   J n   nl b  J n  2 qb sin      nl b  J n   nl b  J n  2 qb sin    


   2 sin  qL cos   cos   u L    4 q cos   cos  qL cos   sin   u L   
u





 

 2 
 2  
I r0 I r1   

2
2
2
2
 nl   2 q sin     u   2 q cos   

















2
(S34)
The third integral I r , is determined similarly,
0

L /2
 L /2
cos  s z0  exp  i 2 q cos  z0  dz0
 L
 L
2 s cos  qL cos   sin  s    4 q cos   sin  qL cos   cos  s 
 2 
 2 .

2
2 2
2
 s  4 q cos 
Similarly for the integral corresponding to r0 we get the same expression as above, thus the product
of these two integrals is given by

L/2
 L/2
cos  s z  exp  i 2 q cos  z  dz 
L/2
 L/2
cos  s z0  exp  i 2 q cos  z0  dz0
2

 s L 
 L 
  4 q cos   sin  qL cos   cos  s  
 2 s cos  qL cos   sin 

 2 
 2  .

2
2 2
2


 s  4 q cos 




Also
a

b
0

r0 Yn  nk r0  cos  n 0  exp  i 2 qr0 sin  cos  0 d 0 dr0
 2 qa sin   Yn  nk a  J n  2 qa sin     nk a  Yn nk a  J n  2 qa sin   


   2 qb sin   Yn  nk b  J n  2 qb sin     nk b  Yn  nk b  J n  2 qb sin   
n
i 
,
nk2  4 2 q 2 sin 2 
14
and
a

b
0

r1Yn nk r1  cos  n1  exp  i 2 qr1 sin  cos 1 d1 dr1
 2 qa sin   Yn nk a  J n  2 qa sin     nk a  Yn nk a  J n  2 qa sin   


   2 qb sin   Yn nk b  J n  2 qb sin     nk b  Yn  nk b  J n  2 qb sin   
 i n
nk2  4 2 q 2 sin 2 
.
Therefore the product of two integrals is
  2 qa sin   Yn  nk a  J n  2 qa sin     nk a  Yn  nk a  J n  2 qa sin    
 

    2 qb sin   Yn  nk b  J n  2 qb sin     nk b  Yn  nk b  J n  2 qb sin    


   2 cos  qL cos  sin   s L   4 q cos  sin  qL cos  cos   s L  

   
 
    
  s
2


 2 

I r0 I r1   

2
2
2
2
nk   2 q sin     s   2 q cos   















Similarly the fourth integral in Eq. (S31) can be calculated as

L/2
L/2
sin  u z0  exp  i 2 q cos  z0  dz0 
L/2
L/2
sin  u z1  exp  i 2 q cos  z1  dz1
2

 uL 
 L 
  4 q cos   cos  qL cos   sin  u  
 2 u sin  qL cos   cos 

 2 
 2  .

2
2 2
2


 u  4 q cos 




Also we obtain the following integral as
a

b
0

r0 Yn  nk r0  cos  n 0  exp  i 2 qr0 sin  cos  0 d 0 dr0
 2 qa sin   Yn  nk a  J n  2 qa sin     nk a  Yn nk a  J n  2 qa sin   


   2 qb sin   Yn  nk b  J n  2 qb sin     nk b  Yn  nk b  J n  2 qb sin   

n
i 
,
nk2  4 2 q 2 sin 2 
15
2
(S35)
and also
a

b
0

 i n
r1Yn nk r1  cos  n1  exp  i 2 qr1 sin  cos 1 d1 dr1
 2 qa sin   Y  a  J   2 qa sin     a  Y   a  J  2 qa sin   
n
nk
n
nk
n
nk
n


   2 qb sin   Y  b  J   2 qb sin     b  Y   b  J  2 qb sin   
n
nk
n
nk
n
nk
n


nk2  4 2 q 2 sin 2 
.
Thus the product of the two integrals becomes,
  2 qa sin   Yn  nk a  J n  2 qa sin     nk a  Yn  nk a  J n  2 qa sin    
 

    2 qb sin   Yn  nk b  J n  2 qb sin     nk b  Yn  nk b  J n  2 qb sin    


   2 sin  qL cos  cos   u L   4 q cos  cos  qL cos  sin   u L   

 
 
 
 

  u
 2 
 2   

I r0 I r1  

 nk2   2 q sin  2   u2   2 q cos  2 

 (S36)














2
Substituting the integral products (i.e., Eqs. (S32), (S34), (S35) and (S36)) into the expression for
the signal attenuation (Eq. (1)), gives the final result for the spin-echo attenuation for diffusion in a
concentric cylinder of finite length with arbitrary angle between the longitudinal axis and the
direction of the applied magnetic field, viz.
E  q,    0 
16
2
 a  b2  L2


n0
2
  O  q, a   O  q , b   2  
 
 L

  M  q, 

 l  0 H  q, a   H  q, b   s  0  2 
2


 L
 L  L
A  q,  C  q,     N  q,  B  q,  F  q,   
 2
 2  2
u 0

2
 P  q, a   P  q, b    
 L
 
  M  q, 
k  0 K  q, a   K  q, b  
 s 0  2 
2

 
 L
 L  L
A  q,  D  q,     N  q,  B  q,  G  q,     ,
 2
 2  2
u 0
 

2
16
(S37)
where
1
 sin  2 s z  
A  q, z   1 
 ,
2 s z 

1
 sin  2 u z  
B  q, z   1 
 ,
2 u z 

exp     nl2   s2  Dt 
C  q, t  
,
2
2
 nl2   2 q sin  2   s2   2 q cos  2 

 

F  q, t  
exp     nl2   u3  Dt 
,
2
 nl2   2 q sin  2   u2   2 q cos  2 



2
2
exp    nk   s  Dt 
D  q, t  
,
2
2
nk2   2 q sin  2   s2   2 q cos  2 

 

2
2
exp    nk   u  Dt 
G  q, t  
,
2
2
 nk2   2 q sin  2   u2   2 q cos  2 

 


n2 
2
H  q, r   r 2 J n   nl r    r 2  2  J n2   nl r  ,
 nl 


n2 
2
K  q, r   r 2Yn nk r    r 2  2  Yn2 nk r  ,
 nk 

M  q, z   2 s cos  2 qz cos   sin  s z    4 q cos   sin  2 qz cos   cos  s z  ,
(S38)
N  q, z   2 u sin  2 qz cos   cos  u z    4 q cos   cos  2 qz cos   sin  u z  ,
O  q, r    2 qr sin   J n   nl r  J n  2 qr sin      nl r  J n   nl r  J n  2 qr sin   ,
P  q, r    2 qr sin   Yn  nk r  J nk  2 qr sin     nk r  Yn  nk r  J n  2 qr sin   .
3. Mean Square Displacement in a Concentric Cylinder
The MSD in a concentric cylinder is calculated from Eq. (20) and the expressions for the diffusion
propagator (Eqs. (11) and (12)  Eqs. (S4) and (S26)). It is possible to redefine the expression for
the MSD in terms of the mean propagator2,3
P  R, t      r0  P  r0 , r0  R, t  dr0 ,
(S39)
where R is the vector of displacement during t, that is R = r1 – r0. An equivalent expression for the
MSD to Eq. (20) but using the mean propagator is3
17
r1  r0 
2
 R t 
2
  P  R, t  R 2 dR .
(S40)
Using Eqs. (S39) and (S40), the MSD component corresponding to the zero-eigenvalue component
(Eq. (11)) of the propagator will be
 r1  r0 
2
 0
2
1
  2 L2  a 2  b2  .
3
(S41)
Using the cylindrical coordinate system the MSD component for the non-zero eigenvalues is
calculated by substituting Eq. (12) into Eq. (20). The integral over r0 is over the three elements of
the coordinate system (i.e., z0, 0, and r0). Integration over the z0 coordinate gives
 r1  r0 2  1   0 2   z1  z0 2  cos  s z0  dz0 


2
2
 L 
 L
4 L s cos  s    8  L2  4  r0  r1    0  1   z12  s2  sin  s 




 2 
 2 .
2 s2

L/2
 L/2



(S42)
Also, the integral over 0 is


0
 L
4 L s cos  s
 2



2
2
 
 s L 
2
2
2
   8  L  4  r0  r1    0  1   z1  s  sin  2 


 cos n d 
 0 0
2 s2
(S43)
 L 
8n1 s2  8n    1  s2 cos  n   sin  s  .
 2 
Note that in the above integration the terms containing sin(nπ) are eliminated as they are zero for
different values of n. The integration over r0 gives
18
 L 
8n1 s2  8n    1  s2 cos  n   sin  s  J n  nl r0  r0 dr0 
 2 
 n  2i  1 n
 L  1 
8n1 s2  8n    1  s2 cos  n   sin  s  
 aJ n  2i 1  nl a   bJ n  2i 1  nl b .
 2   nl i 0 2  1 n  i  1 1 n  i 



2
 2


a
b
(S44)
By using the above integration results (i.e., Eqs. (S42)–(S44)) in the above equation and repeating
the integration for r1 (i.e., z1, θ1 and r1) the final form of the MSD expression the for non-zero
eigenvalues is obtained,
2

 L 
16 s  1  cos2  n  sin 2  s   

n

2
i

1
n


 2 

 aJ n  2i 1  nl a   bJ n  2i 1  nl b   .

n nl
 i 0 2  1 n  i  1 1 n  i 






2
 2

2
(S45)
This calculation is then repeated for the second term in the propagator. The third and last terms that
are related to Bessel function of the second kind are calculated in a similar way and the final form
of the MSD for diffusion in a concentric cylinder (i.e., including zero and non-zero eigenvalues
components) is
19
 r1  r0 
2
2
1
1
  2 L2  a 2  b 2  

3
 L  a2  b2 
2









2
n  2i  1 n

2
2  s L 


64 s  1  cos  n   sin 
 aJ n  2i 1   nl a   bJ n  2i 1   nl b   
 


 2   i  0 2  1 n  i  1 1 n  i 

  






2
 2



 n0 l 0 s 0

 sin  s L    2
 2 n2 
 2 n2 
2
2
2
2
2
n nl Ln 1 

  a J n   nl a    a  2  J n   nl a   b J n   nl b    b  2  J n   nl b   
s L  
 nl 
 nl 




 


2
2
 exp     nl   s  Dt 



2


 

 
2
 n  2i  1 n
 L  

64 u  1  cos 2  n   sin 2  u   
 aJ n  2i 1   nl a   bJ n  2i 1   nl b    

 2   i  0 2  1 n  i  1 1 n  i 
 



   

 
2
2



 

 sin  u L    2
 2 n2 
 2 n2 
 n 0 l 0 u 0
2
2
2
2 
2
n nl Ln 1 
  a J n   nl a    a  2  J n   nl a   b J n   nl b    b  2  J n   nl b   


L 
 nl 
 nl 
u



 



2
2


 exp     nl   u  Dt 


2 

 



 

2
n

2
i

1
n


 L

64 s  1  cos 2  n   sin 2  s   
 aYn  2i 1  nk a   bYn  2i 1  nk b    
 2   i  0 2  1 n  i  1 1 n  i 
 





 
   
2
 2

 

 sin  s L    2
 2 n2 
 2 n2 
2
2
2
2 
 n 0 k 0 s 0
2
n nk Ln 1 
 a Y   a    a  2  Yn  nk a   b Yn  nk b    b  2  Yn  nk b   

 s L   n nk
nk 
 nk 





 exp    2   2 Dt 

s 
  nk




2 

 



 

2
n

2
i

1
n



L

2
2 u



64


1

cos
n

sin

aY

a

bY

b








u 
n  2 i 1
nk
 2   1
 n  2i 1 nk
 


  i  0 2  n  i  1 1 n  i 
 





 
   
2
 2

 

2
2
 n  0 k  0 s  0 n Ln 1  sin  u L    a 2Y   a 2   a 2  n  Y  a 2  b 2Y   b 2   b 2  n  Y  b 2  







nk
n
nk

 u L   n nk
nk2  n nk
nk2  n nk  





 exp    nk2   u2  Dt 





(S46)
This can be simplified to
 r1  r0 
2
2
2
1
1

  2 L2  a 2  b 2  
1  cos  n 
2
2  
3
 L  a  b   n0
  V 1 a   V 1 b   2  


L  L
L  L

 
S
1
A
q
,
R
1
t

U 1  B  q ,  R 2  t  




  


 l  0  H  q, a   H  q, b    s  0  2   2 
2  2
u 0


2


W 1 a   W 1 b      L   L 
  
L  L

  S 1  A  q ,  R 3  t    U 1  2  B  q , 2  R 4  t     ,
  

k 0 
u 0
 
 H  q, a   H  q, b    s  0  2   2 


20
(S47)
where
 L
 L 
S1   64 s sin 2  s  ,
2
 2 
 L
 L 
U 1   64 u sin 2  u  ,
2
 2 
n 1  2i  n 
r J 2i  n 1  nl r  ,
 n  n 
i 0
2  i   1 i  
 2  2 

n 1  2i  n 
W 1 r   
rY2i  n 1 nk r  ,
 n  n 
i 0
2  i   1 i  
 2  2 

V 1 r   
R1 t  
R2  t  
R3  t  
R4  t  
exp    nl2   s2  Dt 
 nL nl
,
exp    nl2   u2  Dt 
 nL nl
,
exp   nk2   s2  Dt 
 nLnk
,
exp   nk2   u2  Dt 
 nLnk
.
(S48)
B. Concentric Sphere
1. Diffusion Propagator
(a)Zero eigenvalue solution
For the case of λ = 0 (i.e., zero eigenvalue), the time variable is equal to a constant
T 0  A0
(S49)
and similarly for the spatial eigenfunction
2 0  0 .
21
(S50)
Consequently, the general form of the spatial part of the propagator is also given by a constant
(arbitrarily set to 1)
   0  BR  C  1,
(S51)
where B and C can be any constants that do not simultaneously. Therefore the diffusion propagator
corresponding to the zero-eigenvalue will be given by
P  r ,  , t   0     0T  0  A0 .
(S52)
The normalised propagator of zero-eigenvalue (i.e., Nλ=0) is obtained as
P 0  r0 , 0 , r1 ,1 , t  
  0  r0    0  r1 
N  0 N  0

3
.
4  a 3  b3 
(S53)
(b) Positive eigenvalue solution
For non-zero eigenvalues, it is convenient to set λ = -α2, the Eq. (23) becomes
1 T 2  1   2 R 2 R  1 1  
 

  2 
1   2     2 ,

 2

DT t

R  r
r r  r   
 
(S54)
 n  n  1 
1 1  
 
1 2     

,

2
2
r   
 
 r

(S55)
setting
22
and rearranging gives a Legendre’s differential equation of the form

 
 n  n  1   0.
1   2  
 
 
(S56)
The solutions of this equation are given in terms of Legendre’s polynomials of the first Pn and
second Qn kind,
     C1Pn     C2Qn   
(S57)
where C1 and C1 are constants. Since Qn(μ) have poles at μ = ±1, this solution has been excluded.
The spatial part of Eq. (S54) can be written as
 2 R 2 R  2 n  n  1 

  
 R  0.
r 2 r r 
r2 
(S58)
The solution for this equation is
R  r   En,m nmr 
1/2
Cn1/2 (nmr )
(S59)
where Cn 1/ 2 ( nm r )  F1 J n 1/ 2 ( nm r )  F2Yn 1/ 2 ( nm r ) .
The temporal part of the diffusion equation (Eq. (S54)) is
1 T
  2 ,
DT t
23
(S60)
with solution
T  t   exp   D 2t  .
(S61)
Thus, combining Eqs. (S57), (S59), and (S61) the solution for this propagator is given by


P  r ,  , t     r ,   T  t    Anm  nm r 
n 0 m0
1/ 2
2
Cn 1/ 2  nm r  Pn    exp   D nm
t .
(S62)
From boundary condition on the outer surface (r = a) given by
 

 D  M a  R  0,
 r
 r a
(S63)
where Ma is the relaxivity on the outer surface, gives the root-function,
 nl
J n1/2   nl a   M a 1

 .
J n1/2   nl a 
D
2a
(S64)
Similar to the case of the concentric cylinder, by using the definition of Y function in terms of J, it
can be shown that both solutions on this surface results in the same eigenvalues. Similarly for the
boundary condition on r = b the boundary condition



  D  M b  R  0,
r

 r b
gives the equation for the eigenvalues to be
24
(S65)
nk
Yn1/2 nk b 
Yn 1/2 nk b 

Mb
1
 .
D 2b
(S66)
This boundary condition applied on either Y or J solutions gives the same eigenvalues. Thus,


R   Bnl   nl r 
1/2
n 0 l 0


J n1/2   nl r    Bnk nk r 
n 0 k 0
1/2
Yn1/2 nk r .
(S67)
Therefore the non-normalised diffusion propagator for diffusion within a concentric sphere is


P  r ,  , t    Bnl   nl r 
1/ 2
n 0 m 0


  Bnk  nk r 
n 0 k 0
1/ 2
J n 1/ 2   nl r  Pn    exp   D nl2 t 
(S68)
Yn 1/ 2  nk r  Pn    exp   D t  .
2
nk
The above propagator must be normalised by determining the normalisation factor for each part in
keeping with the boundary conditions of the radial eigenfunctions at r = a and r = b. The first
normalisation factor is,
N1     nl r 


1/ 2
J n 1/ 2   nl r  Pn       mu r  J m 1/ 2   mu r  Pm     dV


 a2 
 
  n  1 / 2 2  2
  J n1/ 2   nl a 2  1 
 J n 1/ 2   nl a   
2

2 

 nl a  

4
 
 


.
 2n  1  nl  b2 

  n  1 / 2 2  2
2
 J n 1/ 2   nl b   
  J n1/ 2   nl b   1 
2

 

b


 2 
nl



1/ 2
25
(S69)
Since F2 = 0 corresponds to the boundary condition on r = a, the second part of the radial
eigenfunction has no contribution to the normalisation factor. The normalisation factor for the
second part is,
N 2   nk r 


1/ 2
Yn 1/ 2 nk r  Pn     mv r  Ym 1/ 2 mv r  Pm     dV


2
 a2 
 
  n  1 / 2  2
 Yn1/ 2 nk a 2  1 
 

Y

a


2

 n 1/ 2 nk  
2 


a


nk


4
 
 


.
2
2
n

1


 nk  b2 



 n  1 / 2 Y 2  b  
2
 Yn1/ 2 nk b   1 
n 1/ 2  nk  
2
2 
 
nk b  




1/ 2
(S70)
Thus, the normalised diffusion propagator for diffusion in a concentric sphere is
P  r0 , 0 , r1 , 1 ; t  



n0 l 0
 2n  1  nl  nl r1 
2
1/ 2
J n 1/ 2   nl r1  Pn  1   nl r0 
1/ 2
J n 1/ 2   nl r0  Pn  0 
 
 
  n  1 / 2  2
a 2  J n1/ 2   nl a 2  1 
 
J

a



n

1/
2
nl
2 2



a
 
nl


 


2



 2

n

1
/
2


2
 J n21/ 2   nl b   
b  J n1/ 2   nl b   1 
2 2


b

 
nl



2
exp   nl2 Dt 
1/ 2
1/ 2
 2n  1nk nk r1  Yn1/ 2 nk r1  Pn  1 nk r0  Yn1/ 2 nk r0  Pn  0  exp  2 Dt .
 nk 
2
 
 
n0 k 0
  n  1 / 2 2  2
2
2
a Yn1/ 2  nk a   1 
Y
 a  
(S71)

 2 a 2  n 1/ 2 nk  
 


 
nk


 
 


2

  n  1 / 2  2
 2
2
Y
 b  
b Yn1/ 2  nk b   1 
nk2 b 2  n 1/ 2 nk  




The half-integer order Bessel functions of the first kind can be converted to spherical Bessel using
the following definitions1
26
J n1/2  z  
2z
J n 1/2  z  
1
2z
jn  z  
j  z  ,
 n
2 z
Yn1/2  z  
2z
Yn1/2  z  
1
2z
yn  z  
y  z  .
 n
2 z
zJ n1/2  z  
z 1

jn  z   zjn  z   ,

2  2



jn  z  ,
(S72)
yn  z  ,
Therefore
z
zYn1/2  z  
2
1

 2 yn  z   zyn  z   .
(S73)
And thus Eq. (S71) by using Equations (S72) and (S73) becomes
P  r0 , 0 , r1 , 1 , t  



 2n  1  nl exp   nl2 Dt 
n0 l 0


jn   nl r1  Pn  1  jn   nl r0  Pn  0 


5
1 
2
2
4

   nl a     nl a   2  n    jn   nl a   2   nl a  jn   nl a   jn   nl a     nl a  jn   nl a    

2  


 2



2

5
 
1 
2
2
4

    nl b     nl b   2  n    jn   nl b   2   nl b  jn   nl b   jn   nl b     nl b  jn   nl b    
2  

 2
 
 
2


 
n 0 k 0
 2n  1nk exp  nk2 Dt 


yn  nk r1  Pn  1  yn  nk r0  Pn  0 
2


5
1 
2
2
4

  nk a    nk a   2  n    yn  nk a   2  nk a  yn  nk a   yn  nk a    nk a  yn  nk a    

2  


 2



2

5
 
1 
2
2
4

  nk b   nk b   2  n    yn  nk b   2  nk b  yn  nk b   yn  nk b    nk b  yn  nk b    
2  

 2
 
 
(S74)
27
.
2. Pulse Gradient Spin-Echo Attenuation in a Concentric Sphere
(a) Zero eigenvalue contribution
Substituting the propagator for  = 0 for the concentric sphere (Eq. (24)) in to the SGP
equation (Eq. (1)) gives, the signal attenuation corresponding to this part of the propagator to be
E  q,   0  4 2 
a
b

     r P  r , r ,   e
1
a
1 b
1
0
1
9
a
3
b
0
0
i 2 q r1 r0 
1
d 1r12 dr1d 0 r0 2 dr0
sin  2 qa    2 qa  cos  2 qa   sin  2 qb    2 ab  cos  2 qb   .
2
  2 q 
3 2
6
(S75)
(b) Positive eigenvalue contribution
Substituting the propagator for  > 0 for the concentric sphere (Eq. (S71)) and (r0) that is
the equilibrium spin density (equal to 1/(volume) of the restricting pore) at the point r0 and initial
time zero in to the SGP equation (Eq. (1)) gives,
E  q,    0  4 2 
a
b
     r P  r , r ,   e
1
a
1 b
1
1
0
0
i 2 q  r1 r0 
1
d 1r12 dr1d 0 r0 2 dr0
 
 2n  1  nl exp   nl2 D  I r0 I r1
3


2  a 3  b3  n 0 l 0  
 
  n  1 / 2 2  2
a 2  J n1/ 2   nl a 2  1 
 
J

a



n

1/
2
nl
2 2



a
 
 
nl




2



 2

n

1
/
2


2
 J n21/ 2   nl b   
b  J n1/ 2   nl b   1 
2 2

 nl b

 



 
 2n  1nk exp  nk2 D  I r0 I r1
3

,

2
2  a 3  b3  n 0 k  0  
 


n

1
/
2


2
a 2 Yn1/ 2 nk a   1 
 Yn21/ 2 nk a   
2 2



a
 
nk


 



  n  1 / 2 2  2
 2
2
 Yn 1/ 2 nk b   
b Yn1/ 2 nk b   1 
2 2


b

 
nk



where μ = cos θ, (θ is inclination angle and 0     ) and also
28
(S76)
  r 
    r 
a
1
b
1
I r0  
I r1
a
1
b
1
1/ 2
J n 1/ 2   nl r0  Pn  0  exp  i 2 qr0 0  d 0 r0 2 dr0
1/ 2
J n 1/ 2   nl r1  Pn  1  exp  i 2 qr1 1  d 1 r12 dr1 ,
nl 0
nl 1
and
  r 
I      r 
a
I r0  
r1
1
b
1
a
1
b
1
1/ 2
Yn 1/ 2 nk r0  Pn  0  exp  i 2 qr0 0  d 0 r0 2 dr0
1/ 2
Yn 1/ 2 nk r1  Pn  1  exp  i 2 qr1 1  d 1 r12 dr1 .
nk 0
nk 1
The first integral becomes
I r0     nl r0 
a
1/2
    nl r0 
J n1/2   nl r0  i n  qr0 

1/2
b

i n   nl q 
1/2
 nl2   2 q 
1
1
b
a
J n1/2   nl r0   Pn  0  exp  i 2 qr0 0  d 0 r0 2 dr0
2
1/2
J n1/2  2 qr0   r0 2 dr0

 2 qa  J n1/2   nl a  J n1/2  2 qa     nl a  J n1/2  2 qa  J n1/2   nl a 
(S77)
  2 qb  J n1/2   nl b  J n1/2  2 qb     nl b  J n1/2  2 qb  J n1/2   nl b   .
The second integral becomes
I r1     nl r1 
a
1/2
b
    nl r1 
a
b
1/2
J n1/2   nl r1   Pn  1  exp  i 2 qr11  d 1r12 dr1
1
1
J n1/2   nl r1  i n  qr1 

1/2
J n1/2  2 qr1   r12 dr1

 i   nl q   2 qa J
 2
 n1/2  nl a  J n1/2  2 qa    nl a  J n1/2  2 qa  J n1/2  nl a 
2 
 nl   2 q 
  2 qb  J n1/2   nl b  J n1/2  2 qb     nl b  J n1/2  2 qb  J n1/2   nl b   .
n
1/2
The third integral becomes
29
(S78)
I r0   nk r0 
a
1/2

i  nk q 
1/2
nk2   2 q 
1
1
b
n
Yn1/2 nk r0   Pn  0  exp  i 2 qr0 0  d 0 r0 2 dr0
2
 2 qa  Yn1/2 nk a  J n1/2  2 qa    nk a  J n1/2  2 qa  Yn1/2  nk a 
(S79)
  2 qb  Yn1/2 nk b  J n1/2  2 qb   nk b  J n1/2  2 qb  Yn1/2 nk b   .
And the fourth integral becomes
 i  nk q   2 qa Y  a J  2 qa   a J
 2
 n1/2  nk  n1/2 
  nk  n1/2  2 qa Yn1/2 nk a 
2 
nk   2 q 
  2 qb  Yn1/2  nk b  J n1/2  2 qb    nk b  J n1/2  2 qb  Yn1/2  nk b   .
n
I r1
1/2
Combining Eqs. (S76)-(S80), the spin-echo attenuation can be written as
30
(S80)
E  q,    0 


 
n 0 l 0

3
 2n  1  nl exp   nl D 
2  a  b3 
3


 
  n  1 / 2 2  2
  n  1 / 2 2  2
2
2
 2
2





J

a

b
J

b

1

J

b









a  J n 1/ 2   nl a   1 
n

1/
2
nl
n

1/
2
nl
n

1/
2
nl

 nl2 a 2 
 nl2 b 2 


 
 


   nl q 1/ 2




2

qa
J

a
J
2

qa


a
J
2

qa
J

a
 2













n

1/
2
nl
n

1/
2
nl
n

1/
2
n

1/
2
nl
2 
  nl   2 q 



  2 qb  J n 1/ 2   nl b  J n1/ 2  2 qb     nl b  J n 1/ 2  2 qb  J n1/ 2  nl b  




1
3
 2n  1nk exp  nk D 

2  a 3  b3 
 
n 0 k 0
2
 


 
  n  1 / 2 2  2
  n  1 / 2 2  2
2
2
2
2


 Yn 1/ 2  nk a    b Yn 1/ 2  nk b   1 
 Yn 1/ 2 nk b   
a Yn 1/ 2 nk a   1 
2 2
2 2



nk a
nk b


 




 
2
 nk q 1/ 2




2

qa
Y

a
J
2

qa


a
J
2

qa
Y

a
 2












n

1/
2
nk
n

1/
2
nk
n

1/
2
n

1/
2
nk
2 

nk   2 q 



  2 qb  Yn 1/ 2  nk b  J n1/ 2  2 qb    nk b  J n 1/ 2  2 qb  Yn1/ 2  nk b  


 .

1
(S81)
Transforming from half-integer order Bessel functions of the first kind to spherical Bessel functions
using Eqs. (S72) and (S73), Eq. (S81) simplifies to
31
E  q,    0 
2


a3
 2 qa  jn   nl a  jn  2 qa     nl a  jn  2 qa  jn   nl a   

2
2 
   nl a    2 qa 



3
b

 2 qb  jn   nl b  jn  2 qb     nl b  jn  2 qb  jn   nl b   
   b 2   2 qb 2 

 
nl



2

5
n0 l 0 
1 
2
2
4

  nl a     nl a   2  n    jn   nl a   2   nl a  jn   nl a   jn   nl a     nl a  jn   nl a   
2  



 2


2
5

1 

2
2
4

   nl b   2   nl b   2  n  2   jn   nl b   2   nl b  jn   nl b   jn  nl b    nl b  jn  nl b   

 




6  2n  1  nl3 exp   nl2 D 
a
3
 b3 
2


a3
 2 qa  yn nk a  jn  2 qa    nk a  jn  2 qa  yn nk a   

2
2 
 nk a    2 qa 



3
b
 



2

qb
y

b
j
2

qb


b
j
2

qb
y

b

 n  nk  n 
  nk  n 
 n  nk   
2
2 

 
nk b    2 qb 


 
2




n0 k 0
5
1
2
2
4

nk a   nk a   2  n    yn nk a   2  nk a  yn  nk a   yn  nk a    nk a  yn  nk a   
2  

 2




2
5

1 

2
2
4

 nk b   2 nk b   2  n  2   yn nk b   2 nk b  yn nk b   yn nk b   nk b  yn nk b   

 




6  2n  1 nk3 exp   nk2 D 
a
3
 b3 
(S82)
In compact form we have for the annular sphere attenuation function
E  q,    0
2
   G  q, a   G  q , b   A   
6
 

 3


C  q, a   C  q, b 
n 0  l 0
 a  b3  

2
  H  q, a   H  q, b   B    




,
D  q, a   D  q, b 
k 0

where
32
(S83)
A  t    2n  1  nl2 exp   nl2 Dt  ,
B  t    2n  1nk2 exp   nk2 Dt  ,
2
5
1 
2
2
4

C  q, r     nl r     nl r   2  n    jn   nl r   2   nl r  jn   nl r   jn   nl r    nl r  jn  nl r   ,
2  

 2
2
5
1 
2
2
4

D  q, r   nk r   nk r   2  n    yn nk r   2 nk r  yn nk r   yn nk r   nk r  yn nk r   ,
2  

 2
r
G  q, r   2
 2 qr  jn   nl r  jn  2 qr     nl r  jn  2 qr  jn   nl r   ,
2 
 nl   2 q 
H  q, r  
r
   2 q 
2
nk
 2 qr  yn  nk r  jn  2 qr    nk r  jn  2 qr  yn  nk r   .
2
(S84)
3. Mean Square Displacement in a Concentric Sphere
The MSD for diffusion in a concentric sphere is determined similarly to the MSD for diffusion
within a concentric cylinder (see Section A.3). For the case of zero-eigenvalue the propagator (i.e.,
Eq. (24)  Eq. (S53)), the MSD component is
 r1  r0 
2  4 3 3 
 a  b  .
3  3
2
2

(S85)
Using the spherical coordinate system the MSD component corresponding to non-zero eigenvalues
(Eq. (25)  Eq. (S74)) can be obtained using Eq. (20). For this we start with the following integral,
  r  r         arccos
1
2
1
1
0
2
1
0
1
2
 arccos0   Pn  0  d 0 


 r1  r0 2  1  0 2  12  Pn  0  d 0  2arccos1  0 Pn  cos 0  sin  0d 0


 1
1

  02 Pn  cos0  sin 0d0 .

33
(S86)
It can be shown from standard tables of integrals (page 50, Eq. 1.14.1 Eq(1))4 that the first integral
in the above will be zero. Also the integrand in the third integral is an odd function and thus the
contribution will vanish when integrated from –π to π. Thus, the only remaining integral to be
evaluated is the second integral. Making the change of variable θ  π/2-θ, changing the limits of
integration to - π/2 to π/2 (and consequently multiplying by -2), the integral is transformed into the
standard integral (page 435, Eq. 2.17.7 Eq(15) in ref.4),
2


n
  n  2 !! 

1   1
2arccos 1   0 Pn  cos  0  sin  0 d 0  
4 arccos1 
 .

2n 2
  n  1 ! 
  2  
(S87)
Also the integration over r0 which involves Bessel functions of the first kind gives
2
2




n
n



a 1   1
1   1
 n  2 !!  r 1/2 J
 n  2 !!  
2

4

arccos


r
r
dr


4

arccos









1
nl
0
n

1/2
nl
0
0
0
1
b 2n2
2n 2
  n  1 ! 
  n  1 ! 
  2  
  2  
3 
3  1
1

 n     n  2i     n  i 
1  3
2 
2 2
3/2
  a 3/2 J
J n  2i 3/2  nl b 

n  2 i  3/2   nl a   b

5
 nl3/2
 1  i 0
1
 n
 n  i  
2
2 
2
2


n
  n  2 !! 
1   1

arccos1 
 
2n 2
  n  1 ! 
  2  
3 
3  1
1

  n     n  2i     n  i 
8
3
2
2
2
2

 
 
 a 2 j

 n  2i 1  nl a   b jn  2i 1  nl b .
5
1
  nl   1 n  i 0
 n  i  


2
2 
2
(S88)
Doing the similar calculations for θ1 and r1 also gives the same results and the final MSD with
contribution from both zero and non-zero eigenvalues is
34
 r1  r0 
2  4
3

    a 3  b3   

3  3
 4  a 3  b3 
2
2
2


 1

3 
3 1

  n     n  2i     n  i 




2n  1  3
2 
2 2
2
 a2 j






 n  2i 1   nl a   b jn  2i 1   nl b  


5
 2 nl2   1  i 0
1

 n
 n  i  
 



2
2 
2


2
 n0 l 0

5
1 
2
2
4



  nl a    nl a   2  n    jn  nl a   2  nl a  jn  nl a   jn  nl a    nl a  jn  nl a 
2  

 2




2 2




 
n






1   1
n  2 !! 

2





8

exp   nl Dt 
 2n  2


  n  1 !  



  2   




2


5

    nl b     nl b 2  2  n  1   jn   nl b 2  2   nl b 4 jn   nl b   jn   nl b     nl b  jn   nl b  



2  



 2


2
 1


3 
3 1


  n     n  2i     n  i 




2n  1  3
2 
2 2
2
 a2 y






a

b
y

b





n

2
i

1
nl
n

2
i

1
nl
2 2


5
  nk   1  i 0
1



 n
 n  i  


  

2
2
2





 

2
5
1 
 n0 k 0

2
2
4


a

a

2
n






nk
nk

  yn  nk a   2  nk a  yn  nk a   yn  nk a    nk a  yn  nk a   

2  

 2




2 2


 


n






1   1
 n  2 !!  exp  2 Dt
8





nk
n2
 2
  n  1 !  



  2   






2


5
1 
2
2
4

   nk b    nk b   2  n    yn  nk b   2  nk b  yn  nk b   yn  nk b    nk b  yn  nk b  

2  

 2


(S89)
This can be simplified to
 r1  r0 
2
2

2
 1  1 n  n  2 !!  2 




2  4
3



  a 3  b3   

 
 8  2n  1 

3  3
2n  2   n  1 / 2  !  
 4  a 3  b3   n  0



2
  V 2  a   V 2  b   2

 W 2  a   W 2  b  






Q1 t   
Q2  t   ,
 l  0 C  q, a   C  q, b 

k  0 D  q, a   D  q, b 


(S90)
where
35
3 
3 1
1

  n     n  2i     n  i 
2
2
2
2
 
 
 r2 j
V 2r   

n  2 i 1   nl r  ,
5
 1  i 0
1
 n
 n  i  
2
2 
2
3 
3 1
1

  n     n  2i     n  i 
2
2 
2 2
 r2 y
W 2r   

n  2 i 1   nl r  ,
1
1
5

 i 0


 n
 n  i  
2
2 
2
Q1 t  
exp   nl2 Dt 
Q2 t  
 2 nl2
,
exp  nk2 Dt 
 2nk2
(S91)
.
C. The limiting cases
1. Concentric Cylinder
The new model for diffusive spin-echo attenuation in a concentric cylinder (i.e., Eqs. (7),
(17) and (18) was successfully tested against known solutions for diffusion between parallel planes5
and in a normal cylinder6,7 (although the literature solution is for an infinite cylinder) for the
reflecting boundary condition (i.e., M = 0 for all surfaces) by setting the inner radius to zero (i.e.
b0) in the concentric model and simulating the attenuation profiles for q oriented at  = 0 (i.e.,
parallel planes – see Figure 1),  = 90 (i.e., perpendicular), and tilted at  = 45° to the long axis of
the cylinder as illustrated in Figure S1. The corresponding simulations and comparisons against
known solutions for diffusion between parallel planes7-9 and in a normal cylinder7 under the
relaxing boundary condition (i.e., M  0 for all surfaces) are given in Figure S2.
36
Figure S1. Simulated spin-echo attenuation profiles for diffusion in a reflecting concentric
cylindrical pore in the limiting case b = 0 m, where the symmetrical axis of the cylinder is oriented
at (A)  = 0° (equivalent to parallel planes; ) and  = 90° (equivalent to cylinder; ) and (B)
 = 45° to the direction of the gradient q. The simulations were performed with a = 20 m, b = 0
m, L = 40 m, Δ = 0.87 s and D = 2.3  10-9 m2s-1 such that
D / ( a  b )  4 D / L
2
2
 5 . In (A) the
existing literature solutions for attenuation in a simple infinite cylinder () and between parallel
plane () is included for comparison and are observed to be in excellent agreement.
37
Figure S2. Simulated spin-echo attenuation profiles for diffusion in a concentric cylindrical pore
with relaxing boundary conditions in the limiting case b = 0 m, whose symmetrical axis is oriented
at (A)  = 0° (equivalent to parallel planes; ) and  = 90° (equivalent to cylinder; ) and (B)
 = 45° to the direction of q. In these simulations, a = 20 m, b = 0 m, L = 40 m, Δ = 0.87 s and
D = 2.3  10-9 m2s-1 such that
D / ( a  b )  4 D / L
determining the eigenvalues were M z
2
2
 5 . The values of the relaxivity used in
 2 D / L, M r  D / a,
o
38
and
M ri  D / b .
In (A) the existing literature
solution for attenuation in simple infinite cylinder () and parallel plane () is included for
comparison and are observed to be in excellent agreement.
2. Concentric Sphere
The new model for diffusive spin-echo attenuation in a concentric sphere (i.e., Eqs. (7), (28)
and (29)) was successfully tested against known solutions for a sphere under reflecting (i.e., M = 0
for all surfaces)10-12 and relaxing (i.e., M  0 for all surfaces)7 boundary conditions by setting the
inner radius to zero (i.e. b0) in the concentric model as illustrated in Figure S3.
Figure S3. A comparison of the PGSE attenuation for the reflecting (M = 0; ) and relaxing (
M ro  D / a ;
) limiting cases for the concentric sphere, when the inner radius is shifted to zero
(i.e., b = 0) with Δ = 0.87 s such that
D / ( a  b)
2
 5 . The analytical solution for attenuation in a
simple reflecting () and relaxing () sphere is included for comparison and are observed to be in
excellent agreement.
39
D. REFERENCES
1. Handbook of Mathematical Functions, 1st ed. edited by M. Abramowitz and I. A. Stegun (Dover, New
York, 1970).
2. J. Kärger and W. Heink, J. Magn. Reson. 51, 1 (1983).
3. W. S. Price, NMR Studies of Translational Motion: Principles and Applications, 1st ed. (Cambridge
University Press, Cambridge, 2009).
4. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, 1st ed. (Gordon and
Breach, The Netherlands, 1992), Vol. 2.
5. J. E. Tanner and E. O. Stejskal, J. Chem. Phys. 49, 1768 (1968).
6. O. Söderman and B. Jönsson, J. Magn. Reson. A 117, 94 (1995).
7. P. T. Callaghan, J. Magn. Reson. A 113, 53 (1995).
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40