<s>
Diffusion-weighted	B-Algorithm
magnetic	I-Algorithm
resonance	I-Algorithm
imaging	I-Algorithm
(	O
DWI	O
or	O
DW-MRI	B-Algorithm
)	O
is	O
the	O
use	O
of	O
specific	O
MRI	B-Algorithm
sequences	I-Algorithm
as	O
well	O
as	O
software	O
that	O
generates	O
images	O
from	O
the	O
resulting	O
data	O
that	O
uses	O
the	O
diffusion	O
of	O
water	O
molecules	O
to	O
generate	O
contrast	O
in	O
MR	O
images	O
.	O
</s>
<s>
A	O
special	O
kind	O
of	O
DWI	O
,	O
diffusion	B-Algorithm
tensor	I-Algorithm
imaging	I-Algorithm
(	O
DTI	B-Algorithm
)	O
,	O
has	O
been	O
used	O
extensively	O
to	O
map	O
white	O
matter	O
tractography	B-Algorithm
in	O
the	O
brain	O
.	O
</s>
<s>
In	O
diffusion	B-Algorithm
weighted	I-Algorithm
imaging	I-Algorithm
(	O
DWI	O
)	O
,	O
the	O
intensity	O
of	O
each	O
image	O
element	O
(	O
voxel	B-Algorithm
)	O
reflects	O
the	O
best	O
estimate	O
of	O
the	O
rate	O
of	O
water	O
diffusion	O
at	O
that	O
location	O
.	O
</s>
<s>
For	O
instance	O
,	O
DWI	O
is	O
more	O
sensitive	O
to	O
early	O
changes	O
after	O
a	O
stroke	O
than	O
more	O
traditional	O
MRI	B-Algorithm
measurements	O
such	O
as	O
T1	B-Algorithm
or	O
T2	B-Algorithm
relaxation	I-Algorithm
rates	O
.	O
</s>
<s>
A	O
variant	O
of	O
diffusion	B-Algorithm
weighted	I-Algorithm
imaging	I-Algorithm
,	O
diffusion	O
spectrum	O
imaging	O
(	O
DSI	O
)	O
,	O
was	O
used	O
in	O
deriving	O
the	O
Connectome	O
data	O
sets	O
;	O
DSI	O
is	O
a	O
variant	O
of	O
diffusion-weighted	O
imaging	O
that	O
is	O
sensitive	O
to	O
intra-voxel	O
heterogeneities	O
in	O
diffusion	O
directions	O
caused	O
by	O
crossing	O
fiber	O
tracts	O
and	O
thus	O
allows	O
more	O
accurate	O
mapping	O
of	O
axonal	O
trajectories	O
than	O
other	O
diffusion	B-Algorithm
imaging	I-Algorithm
approaches	O
.	O
</s>
<s>
It	O
is	O
also	O
used	O
more	O
and	O
more	O
in	O
the	O
staging	O
of	O
non-small-cell	O
lung	O
cancer	O
,	O
where	O
it	O
is	O
a	O
serious	O
candidate	O
to	O
replace	O
positron	B-Application
emission	I-Application
tomography	I-Application
as	O
the	O
'	O
gold	O
standard	O
 '	O
for	O
this	O
type	O
of	O
disease	O
.	O
</s>
<s>
Diffusion	B-Algorithm
tensor	I-Algorithm
imaging	I-Algorithm
is	O
being	O
developed	O
for	O
studying	O
the	O
diseases	O
of	O
the	O
white	O
matter	O
of	O
the	O
brain	O
as	O
well	O
as	O
for	O
studies	O
of	O
other	O
body	O
tissues	O
(	O
see	O
below	O
)	O
.	O
</s>
<s>
However	O
,	O
DWI	O
also	O
remains	O
sensitive	O
to	O
T1	B-Algorithm
and	O
T2	B-Algorithm
relaxation	I-Algorithm
.	O
</s>
<s>
Diffusion	B-Algorithm
tensor	I-Algorithm
imaging	I-Algorithm
(	O
DTI	B-Algorithm
)	O
is	O
important	O
when	O
a	O
tissuesuch	O
as	O
the	O
neural	O
axons	B-Algorithm
of	O
white	O
matter	O
in	O
the	O
brain	O
or	O
muscle	O
fibers	O
in	O
the	O
hearthas	O
an	O
internal	O
fibrous	O
structure	O
analogous	O
to	O
the	O
anisotropy	O
of	O
some	O
crystals	O
.	O
</s>
<s>
Diffusion	O
Basis	O
Spectrum	O
Imaging	O
(	O
DBSI	O
)	O
further	O
separates	O
DTI	B-Algorithm
signals	O
into	O
discrete	O
anisotropic	O
diffusion	B-Algorithm
tensors	I-Algorithm
and	O
a	O
spectrum	O
of	O
isotropic	O
diffusion	B-Algorithm
tensors	I-Algorithm
to	O
better	O
differentiate	O
sub-voxel	O
cellular	O
structures	O
.	O
</s>
<s>
For	O
example	O
,	O
anisotropic	O
diffusion	B-Algorithm
tensors	I-Algorithm
correlate	O
to	O
axonal	O
fibers	O
,	O
while	O
low	O
isotropic	O
diffusion	B-Algorithm
tensors	I-Algorithm
correlate	O
to	O
cells	O
and	O
high	O
isotropic	O
diffusion	B-Algorithm
tensors	I-Algorithm
correlate	O
to	O
larger	O
structures	O
(	O
such	O
as	O
the	O
lumen	O
or	O
brain	O
ventricles	O
)	O
.	O
</s>
<s>
Traditionally	O
,	O
in	O
diffusion-weighted	O
imaging	O
(	O
DWI	O
)	O
,	O
three	O
gradient-directions	O
are	O
applied	O
,	O
sufficient	O
to	O
estimate	O
the	O
trace	O
of	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
or	O
'	O
average	O
diffusivity	O
 '	O
,	O
a	O
putative	O
measure	O
of	O
edema	O
.	O
</s>
<s>
More	O
extended	O
DTI	B-Algorithm
scans	O
derive	O
neural	O
tract	O
directional	O
information	O
from	O
the	O
data	O
using	O
3D	O
or	O
multidimensional	O
vector	O
algorithms	O
based	O
on	O
six	O
or	O
more	O
gradient	O
directions	O
,	O
sufficient	O
to	O
compute	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
.	O
</s>
<s>
The	O
diffusion	B-Algorithm
tensor	I-Algorithm
model	O
is	O
a	O
rather	O
simple	O
model	O
of	O
the	O
diffusion	O
process	O
,	O
assuming	O
homogeneity	O
and	O
linearity	O
of	O
the	O
diffusion	O
within	O
each	O
image	O
voxel	B-Algorithm
.	O
</s>
<s>
From	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
,	O
diffusion	O
anisotropy	O
measures	O
such	O
as	O
the	O
fractional	B-Algorithm
anisotropy	I-Algorithm
(	O
FA	O
)	O
,	O
can	O
be	O
computed	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
principal	O
direction	O
of	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
can	O
be	O
used	O
to	O
infer	O
the	O
white-matter	O
connectivity	O
of	O
the	O
brain	O
(	O
i.e.	O
</s>
<s>
tractography	B-Algorithm
;	O
trying	O
to	O
see	O
which	O
part	O
of	O
the	O
brain	O
is	O
connected	O
to	O
which	O
other	O
part	O
)	O
.	O
</s>
<s>
Recently	O
,	O
more	O
advanced	O
models	O
of	O
the	O
diffusion	O
process	O
have	O
been	O
proposed	O
that	O
aim	O
to	O
overcome	O
the	O
weaknesses	O
of	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
model	O
.	O
</s>
<s>
Amongst	O
others	O
,	O
these	O
include	O
q-space	O
imaging	O
and	O
generalized	O
diffusion	B-Algorithm
tensor	I-Algorithm
imaging	I-Algorithm
.	O
</s>
<s>
Diffusion	B-Algorithm
imaging	I-Algorithm
is	O
an	O
MRI	B-Algorithm
method	O
that	O
produces	O
in	O
vivo	O
magnetic	O
resonance	O
images	O
of	O
biological	O
tissues	O
sensitized	O
with	O
the	O
local	O
characteristics	O
of	O
molecular	O
diffusion	O
,	O
generally	O
water	O
(	O
but	O
other	O
moieties	O
can	O
also	O
be	O
investigated	O
using	O
MR	O
spectroscopic	O
approaches	O
)	O
.	O
</s>
<s>
MRI	B-Algorithm
can	O
be	O
made	O
sensitive	O
to	O
the	O
motion	O
of	O
molecules	O
.	O
</s>
<s>
Regular	O
MRI	B-Algorithm
acquisition	O
utilizes	O
the	O
behavior	O
of	O
protons	O
in	O
water	O
to	O
generate	O
contrast	O
between	O
clinically	O
relevant	O
features	O
of	O
a	O
particular	O
subject	O
.	O
</s>
<s>
The	O
versatile	O
nature	O
of	O
MRI	B-Algorithm
is	O
due	O
to	O
this	O
capability	O
of	O
producing	O
contrast	O
related	O
to	O
the	O
structure	O
of	O
tissues	O
at	O
the	O
microscopic	O
level	O
.	O
</s>
<s>
This	O
causes	O
many	O
of	O
the	O
protons	O
in	O
water	O
molecules	O
to	O
precess	O
simultaneously	O
,	O
producing	O
signals	O
in	O
MRI	B-Algorithm
.	O
</s>
<s>
To	O
sensitize	O
MRI	B-Algorithm
images	O
to	O
diffusion	O
,	O
the	O
magnetic	O
field	O
strength	O
(	O
B1	O
)	O
is	O
varied	O
linearly	O
by	O
a	O
pulsed	O
field	O
gradient	O
.	O
</s>
<s>
The	O
refocusing	O
will	O
not	O
be	O
perfect	O
for	O
protons	O
that	O
have	O
moved	O
during	O
the	O
time	O
interval	O
between	O
the	O
pulses	O
,	O
and	O
the	O
signal	O
measured	O
by	O
the	O
MRI	B-Algorithm
machine	O
is	O
reduced	O
.	O
</s>
<s>
In	O
order	O
to	O
localize	O
this	O
signal	O
attenuation	O
to	O
get	O
images	O
of	O
diffusion	O
one	O
has	O
to	O
combine	O
the	O
pulsed	O
magnetic	O
field	O
gradient	O
pulses	O
used	O
for	O
MRI	B-Algorithm
(	O
aimed	O
at	O
localization	O
of	O
the	O
signal	O
,	O
but	O
those	O
gradient	O
pulses	O
are	O
too	O
weak	O
to	O
produce	O
a	O
diffusion	O
related	O
attenuation	O
)	O
with	O
additional	O
"	O
motion-probing	O
"	O
gradient	O
pulses	O
,	O
according	O
to	O
the	O
Stejskal	O
and	O
Tanner	O
method	O
.	O
</s>
<s>
The	O
equation	O
set	O
by	O
Stejskal	O
and	O
Tanner	O
then	O
becomes	O
inaccurate	O
and	O
the	O
signal	O
attenuation	O
must	O
be	O
calculated	O
,	O
either	O
analytically	O
or	O
numerically	O
,	O
integrating	O
all	O
gradient	O
pulses	O
present	O
in	O
the	O
MRI	B-Algorithm
sequence	I-Algorithm
and	O
their	O
interactions	O
.	O
</s>
<s>
The	O
result	O
quickly	O
becomes	O
very	O
complex	O
given	O
the	O
many	O
pulses	O
present	O
in	O
the	O
MRI	B-Algorithm
sequence	I-Algorithm
,	O
and	O
as	O
a	O
simplification	O
,	O
Le	O
Bihan	O
suggested	O
gathering	O
all	O
the	O
gradient	O
terms	O
in	O
a	O
"	O
b	O
factor	O
"	O
(	O
which	O
depends	O
only	O
on	O
the	O
acquisition	O
parameters	O
)	O
so	O
that	O
the	O
signal	O
attenuation	O
simply	O
becomes	O
:	O
</s>
<s>
However	O
,	O
those	O
diffusion-weighted	O
images	O
are	O
still	O
also	O
sensitive	O
to	O
T1	B-Algorithm
and	O
T2	O
relaxivity	O
contrast	O
,	O
which	O
can	O
sometimes	O
be	O
confusing	O
.	O
</s>
<s>
This	O
deviation	O
from	O
a	O
free	O
diffusion	O
behavior	O
is	O
what	O
makes	O
diffusion	B-Algorithm
MRI	I-Algorithm
so	O
successful	O
,	O
as	O
the	O
ADC	O
is	O
very	O
sensitive	O
to	O
changes	O
in	O
tissue	O
microstructure	O
.	O
</s>
<s>
Then	O
,	O
given	O
conservation	O
of	O
mass	O
,	O
the	O
continuity	O
equation	O
relates	O
the	O
time	O
derivative	O
of	O
the	O
concentration	O
with	O
the	O
divergence	B-Application
of	O
the	O
flux	O
:	O
</s>
<s>
which	O
has	O
terms	O
for	O
precession	O
,	O
T2	B-Algorithm
relaxation	I-Algorithm
,	O
and	O
T1	B-Algorithm
relaxation	I-Algorithm
.	O
</s>
<s>
In	O
1956	O
,	O
H.C.	O
Torrey	O
mathematically	O
showed	O
how	O
the	O
Bloch	B-Algorithm
equations	I-Algorithm
for	O
magnetization	O
would	O
change	O
with	O
the	O
addition	O
of	O
diffusion	O
.	O
</s>
<s>
The	O
Bloch-Torrey	B-Algorithm
equation	I-Algorithm
is	O
:	O
</s>
<s>
where	O
is	O
now	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
.	O
</s>
<s>
For	O
the	O
simplest	O
case	O
where	O
the	O
diffusion	O
is	O
isotropic	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
is	O
a	O
multiple	O
of	O
the	O
identity	O
:	O
</s>
<s>
Anisotropic	O
diffusion	O
will	O
have	O
a	O
similar	O
solution	O
for	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
,	O
except	O
that	O
what	O
will	O
be	O
measured	O
is	O
the	O
apparent	O
diffusion	O
coefficient	O
(	O
ADC	O
)	O
.	O
</s>
<s>
An	O
apparent	O
diffusion	O
coefficient	O
(	O
ADC	O
)	O
image	O
,	O
or	O
an	O
ADC	O
map	O
,	O
is	O
an	O
MRI	B-Algorithm
image	O
that	O
more	O
specifically	O
shows	O
diffusion	O
than	O
conventional	O
DWI	O
,	O
by	O
eliminating	O
the	O
T2	O
weighting	O
that	O
is	O
otherwise	O
inherent	O
to	O
conventional	O
DWI	O
.	O
</s>
<s>
Diffusion	B-Algorithm
tensor	I-Algorithm
imaging	I-Algorithm
(	O
DTI	B-Algorithm
)	O
is	O
a	O
magnetic	O
resonance	O
imaging	O
technique	O
that	O
enables	O
the	O
measurement	O
of	O
the	O
restricted	O
diffusion	O
of	O
water	O
in	O
tissue	O
in	O
order	O
to	O
produce	O
neural	O
tract	O
images	O
instead	O
of	O
using	O
this	O
data	O
solely	O
for	O
the	O
purpose	O
of	O
assigning	O
contrast	O
or	O
colors	O
to	O
pixels	O
in	O
a	O
cross-sectional	O
image	O
.	O
</s>
<s>
In	O
DTI	B-Algorithm
,	O
each	O
voxel	B-Algorithm
has	O
one	O
or	O
more	O
pairs	O
of	O
parameters	O
:	O
a	O
rate	O
of	O
diffusion	O
and	O
a	O
preferred	O
direction	O
of	O
diffusiondescribed	O
in	O
terms	O
of	O
three-dimensional	O
spacefor	O
which	O
that	O
parameter	O
is	O
valid	O
.	O
</s>
<s>
The	O
properties	O
of	O
each	O
voxel	B-Algorithm
of	O
a	O
single	O
DTI	B-Algorithm
image	O
are	O
usually	O
calculated	O
by	O
vector	O
or	O
tensor	B-Device
math	O
from	O
six	O
or	O
more	O
different	O
diffusion	O
weighted	O
acquisitions	O
,	O
each	O
obtained	O
with	O
a	O
different	O
orientation	O
of	O
the	O
diffusion	O
sensitizing	O
gradients	O
.	O
</s>
<s>
The	O
higher	O
information	O
content	O
of	O
a	O
DTI	B-Algorithm
voxel	B-Algorithm
makes	O
it	O
extremely	O
sensitive	O
to	O
subtle	O
pathology	O
in	O
the	O
brain	O
.	O
</s>
<s>
In	O
addition	O
the	O
directional	O
information	O
can	O
be	O
exploited	O
at	O
a	O
higher	O
level	O
of	O
structure	O
to	O
select	O
and	O
follow	O
neural	O
tracts	O
through	O
the	O
brain	O
—	O
a	O
process	O
called	O
tractography	B-Algorithm
.	O
</s>
<s>
Diffusion	B-Algorithm
MRI	I-Algorithm
relies	O
on	O
the	O
mathematics	O
and	O
physical	O
interpretations	O
of	O
the	O
geometric	O
quantities	O
known	O
as	O
tensors	B-Device
.	O
</s>
<s>
Only	O
a	O
special	O
case	O
of	O
the	O
general	O
mathematical	O
notion	O
is	O
relevant	O
to	O
imaging	O
,	O
which	O
is	O
based	O
on	O
the	O
concept	O
of	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Diffusion	O
itself	O
is	O
tensorial	O
,	O
but	O
in	O
many	O
cases	O
the	O
objective	O
is	O
not	O
really	O
about	O
trying	O
to	O
study	O
brain	O
diffusion	O
per	O
se	O
,	O
but	O
rather	O
just	O
trying	O
to	O
take	O
advantage	O
of	O
diffusion	O
anisotropy	O
in	O
white	O
matter	O
for	O
the	O
purpose	O
of	O
finding	O
the	O
orientation	O
of	O
the	O
axons	B-Algorithm
and	O
the	O
magnitude	O
or	O
degree	O
of	O
anisotropy	O
.	O
</s>
<s>
Tensors	B-Device
have	O
a	O
real	O
,	O
physical	O
existence	O
in	O
a	O
material	O
or	O
tissue	O
so	O
that	O
they	O
do	O
not	O
move	O
when	O
the	O
coordinate	O
system	O
used	O
to	O
describe	O
them	O
is	O
rotated	O
.	O
</s>
<s>
There	O
are	O
numerous	O
different	O
possible	O
representations	O
of	O
a	O
tensor	B-Device
(	O
of	O
rank	O
2	O
)	O
,	O
but	O
among	O
these	O
,	O
this	O
discussion	O
focuses	O
on	O
the	O
ellipsoid	O
because	O
of	O
its	O
physical	O
relevance	O
to	O
diffusion	O
and	O
because	O
of	O
its	O
historical	O
significance	O
in	O
the	O
development	O
of	O
diffusion	O
anisotropy	O
imaging	O
in	O
MRI	B-Algorithm
.	O
</s>
<s>
The	O
following	O
matrix	B-Architecture
displays	O
the	O
components	O
of	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
:	O
</s>
<s>
The	O
same	O
matrix	B-Architecture
of	O
numbers	O
can	O
have	O
a	O
simultaneous	O
second	O
use	O
to	O
describe	O
the	O
shape	O
and	O
orientation	O
of	O
an	O
ellipse	O
and	O
the	O
same	O
matrix	B-Architecture
of	O
numbers	O
can	O
be	O
used	O
simultaneously	O
in	O
a	O
third	O
way	O
for	O
matrix	B-Architecture
mathematics	O
to	O
sort	O
out	O
eigenvectors	O
and	O
eigenvalues	O
as	O
explained	O
below	O
.	O
</s>
<s>
The	O
idea	O
of	O
a	O
tensor	B-Device
in	O
physical	O
science	O
evolved	O
from	O
attempts	O
to	O
describe	O
the	O
quantity	O
of	O
physical	O
properties	O
.	O
</s>
<s>
Properties	O
that	O
can	O
be	O
described	O
this	O
way	O
are	O
called	O
scalars	O
;	O
these	O
can	O
be	O
considered	O
tensors	B-Device
of	O
rank	O
0	O
,	O
or	O
0th-order	O
tensors	B-Device
.	O
</s>
<s>
Tensors	B-Device
can	O
also	O
be	O
used	O
to	O
describe	O
quantities	O
that	O
have	O
directionality	O
,	O
such	O
as	O
mechanical	O
force	O
.	O
</s>
<s>
Vectors	O
of	O
this	O
sort	O
can	O
be	O
considered	O
tensors	B-Device
of	O
rank	O
1	O
,	O
or	O
1st-order	O
tensors	B-Device
.	O
</s>
<s>
A	O
tensor	B-Device
is	O
often	O
a	O
physical	O
or	O
biophysical	O
property	O
that	O
determines	O
the	O
relationship	O
between	O
two	O
vectors	O
.	O
</s>
<s>
If	O
the	O
movement	O
is	O
in	O
a	O
single	O
direction	O
,	O
the	O
transformation	O
can	O
be	O
described	O
using	O
a	O
vector	O
—	O
a	O
tensor	B-Device
of	O
rank	O
1	O
.	O
</s>
<s>
The	O
relationship	O
between	O
the	O
properties	O
of	O
driving	O
force	O
that	O
generate	O
diffusion	O
of	O
the	O
water	O
molecules	O
and	O
the	O
resulting	O
pattern	O
of	O
their	O
movement	O
in	O
the	O
tissue	O
can	O
be	O
described	O
by	O
a	O
tensor	B-Device
.	O
</s>
<s>
These	O
can	O
be	O
written	O
as	O
a	O
matrix	B-Architecture
similar	O
to	O
the	O
one	O
at	O
the	O
start	O
of	O
this	O
section	O
.	O
</s>
<s>
Six	O
or	O
more	O
gradients	O
are	O
summed	O
to	O
get	O
all	O
the	O
measurements	O
needed	O
to	O
fill	O
in	O
the	O
matrix	B-Architecture
,	O
assuming	O
it	O
is	O
symmetric	B-Algorithm
above	O
and	O
below	O
the	O
diagonal	O
(	O
red	O
subscripts	O
)	O
.	O
</s>
<s>
These	O
six	O
variables	O
can	O
be	O
represented	O
by	O
a	O
matrix	B-Architecture
similar	O
to	O
the	O
tensor	B-Device
matrix	B-Architecture
defined	O
at	O
the	O
start	O
of	O
this	O
section	O
(	O
since	O
diffusion	O
is	O
symmetric	B-Algorithm
,	O
then	O
we	O
only	O
need	O
six	O
instead	O
of	O
nine	O
components	O
—	O
the	O
components	O
below	O
the	O
diagonal	O
elements	O
of	O
the	O
matrix	B-Architecture
are	O
the	O
same	O
as	O
the	O
components	O
above	O
the	O
diagonal	O
)	O
.	O
</s>
<s>
This	O
is	O
what	O
is	O
meant	O
when	O
it	O
is	O
stated	O
that	O
the	O
components	O
of	O
a	O
matrix	B-Architecture
of	O
a	O
second	O
order	O
tensor	B-Device
can	O
be	O
represented	O
by	O
an	O
ellipsoid	O
—	O
if	O
the	O
diffusion	O
values	O
of	O
the	O
six	O
terms	O
of	O
the	O
quadric	O
ellipsoid	O
are	O
placed	O
into	O
the	O
matrix	B-Architecture
,	O
this	O
generates	O
an	O
ellipsoid	O
angled	O
off	O
the	O
orthogonal	O
grid	O
.	O
</s>
<s>
When	O
the	O
ellipsoid/tensor	O
is	O
represented	O
by	O
a	O
matrix	B-Architecture
,	O
we	O
can	O
apply	O
a	O
useful	O
technique	O
from	O
standard	O
matrix	B-Architecture
mathematics	O
and	O
linear	B-Language
algebra	I-Language
—	O
that	O
is	O
to	O
"	O
diagonalize	O
"	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
first	O
one	O
is	O
the	O
measured	O
diffusion	O
ellipsoid	O
sitting	O
at	O
an	O
angle	O
determined	O
by	O
the	O
axons	B-Algorithm
,	O
and	O
the	O
second	O
one	O
is	O
perfectly	O
aligned	O
with	O
the	O
three	O
Cartesian	O
axes	O
.	O
</s>
<s>
The	O
term	O
"	O
diagonalize	O
"	O
refers	O
to	O
the	O
three	O
components	O
of	O
the	O
matrix	B-Architecture
along	O
a	O
diagonal	O
from	O
upper	O
left	O
to	O
lower	O
right	O
(	O
the	O
components	O
with	O
red	O
subscripts	O
in	O
the	O
matrix	B-Architecture
at	O
the	O
start	O
of	O
this	O
section	O
)	O
.	O
</s>
<s>
It	O
then	O
becomes	O
possible	O
to	O
do	O
a	O
vector	O
processing	O
step	O
in	O
which	O
we	O
rewrite	O
our	O
matrix	B-Architecture
and	O
replace	O
it	O
with	O
a	O
new	O
matrix	B-Architecture
multiplied	O
by	O
three	O
different	O
vectors	O
of	O
unit	O
length	O
(	O
length	O
=	O
1.0	O
)	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
is	O
diagonalized	B-Algorithm
because	O
the	O
off-diagonal	O
components	O
are	O
all	O
now	O
zero	O
.	O
</s>
<s>
Diagonalization	B-Algorithm
of	I-Algorithm
a	I-Algorithm
matrix	I-Algorithm
is	O
done	O
by	O
finding	O
a	O
second	O
matrix	B-Architecture
that	O
it	O
can	O
be	O
multiplied	O
with	O
followed	O
by	O
multiplication	O
by	O
the	O
inverse	O
of	O
the	O
second	O
matrix	B-Architecture
—	O
wherein	O
the	O
result	O
is	O
a	O
new	O
matrix	B-Architecture
in	O
which	O
three	O
diagonal	O
(	O
xx	O
,	O
yy	O
,	O
zz	O
)	O
components	O
have	O
numbers	O
in	O
them	O
but	O
the	O
off-diagonal	O
components	O
(	O
xy	O
,	O
yz	O
,	O
zx	O
)	O
are	O
0	O
.	O
</s>
<s>
The	O
second	O
matrix	B-Architecture
provides	O
eigenvector	O
information	O
.	O
</s>
<s>
The	O
ellipsoid	O
formalism	O
functions	O
also	O
as	O
a	O
mathematical	O
method	O
of	O
organizing	O
tensor	B-Device
data	O
.	O
</s>
<s>
Measurement	O
of	O
an	O
ellipsoid	O
tensor	B-Device
further	O
permits	O
a	O
retrospective	O
analysis	O
,	O
to	O
gather	O
information	O
about	O
the	O
process	O
of	O
diffusion	O
in	O
each	O
voxel	B-Algorithm
of	O
the	O
tissue	O
.	O
</s>
<s>
By	O
knowing	O
the	O
detailed	O
effects	O
of	O
diffusion	O
gradients	O
we	O
can	O
generate	O
a	O
formula	O
that	O
allows	O
us	O
to	O
convert	O
the	O
signal	O
attenuation	O
of	O
an	O
MRI	B-Algorithm
voxel	B-Algorithm
into	O
a	O
numerical	O
measure	O
of	O
diffusionthe	O
diffusion	O
coefficient	O
D	O
.	O
When	O
various	O
barriers	O
and	O
restricting	O
factors	O
such	O
as	O
cell	O
membranes	O
and	O
microtubules	O
interfere	O
with	O
the	O
free	O
diffusion	O
,	O
we	O
are	O
measuring	O
an	O
"	O
apparent	O
diffusion	O
coefficient	O
"	O
,	O
or	O
ADC	O
,	O
because	O
the	O
measurement	O
misses	O
all	O
the	O
local	O
effects	O
and	O
treats	O
the	O
attenuation	O
as	O
if	O
all	O
the	O
movement	O
rates	O
were	O
solely	O
due	O
to	O
Brownian	O
motion	O
.	O
</s>
<s>
Diffusion	O
is	O
fast	O
along	O
the	O
length	O
of	O
(	O
parallel	O
to	O
)	O
an	O
axon	B-Algorithm
,	O
and	O
slower	O
perpendicularly	O
across	O
it	O
.	O
</s>
<s>
Once	O
we	O
have	O
measured	O
the	O
voxel	B-Algorithm
from	O
six	O
or	O
more	O
directions	O
and	O
corrected	O
for	O
attenuations	O
due	O
to	O
T2	O
and	O
T1	B-Algorithm
effects	O
,	O
we	O
can	O
use	O
information	O
from	O
our	O
calculated	O
ellipsoid	O
tensor	B-Device
to	O
describe	O
what	O
is	O
happening	O
in	O
the	O
voxel	B-Algorithm
.	O
</s>
<s>
The	O
lengths	O
are	O
symbolized	O
by	O
the	O
Greek	O
letter	O
λ	B-General_Concept
.	O
</s>
<s>
The	O
long	O
one	O
pointing	O
along	O
the	O
axon	B-Algorithm
direction	O
will	O
be	O
λ1	O
and	O
the	O
two	O
small	O
axes	O
will	O
have	O
lengths	O
λ2	O
and	O
λ3	O
.	O
</s>
<s>
In	O
the	O
setting	O
of	O
the	O
DTI	B-Algorithm
tensor	B-Device
ellipsoid	O
,	O
we	O
can	O
consider	O
each	O
of	O
these	O
as	O
a	O
measure	O
of	O
the	O
diffusivity	O
along	O
each	O
of	O
the	O
three	O
primary	O
axes	O
of	O
the	O
ellipsoid	O
.	O
</s>
<s>
This	O
is	O
a	O
little	O
different	O
from	O
the	O
ADC	O
since	O
that	O
was	O
a	O
projection	O
on	O
the	O
axis	O
,	O
while	O
λ	B-General_Concept
is	O
an	O
actual	O
measurement	O
of	O
the	O
ellipsoid	O
we	O
have	O
calculated	O
.	O
</s>
<s>
The	O
diffusivity	O
along	O
the	O
principal	O
axis	O
,	O
λ1	O
is	O
also	O
called	O
the	O
longitudinal	O
diffusivity	O
or	O
the	O
axial	O
diffusivity	O
or	O
even	O
the	O
parallel	O
diffusivity	O
λ∥	O
.	O
</s>
<s>
where	O
is	O
a	O
diagonal	O
matrix	B-Architecture
with	O
eigenvalues	O
,	O
and	O
on	O
its	O
diagonal	O
.	O
</s>
<s>
where	O
is	O
the	O
matrix	B-Architecture
of	O
eigenvectors	O
and	O
is	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
.	O
</s>
<s>
Aside	O
from	O
describing	O
the	O
amount	O
of	O
diffusion	O
,	O
it	O
is	O
often	O
important	O
to	O
describe	O
the	O
relative	O
degree	O
of	O
anisotropy	O
in	O
a	O
voxel	B-Algorithm
.	O
</s>
<s>
One	O
objective	O
is	O
to	O
keep	O
the	O
number	O
near	O
0	O
if	O
the	O
voxel	B-Algorithm
is	O
spherical	O
but	O
near	O
1	O
if	O
it	O
is	O
elongate	O
.	O
</s>
<s>
This	O
leads	O
to	O
the	O
fractional	B-Algorithm
anisotropy	I-Algorithm
or	O
FA	O
which	O
is	O
the	O
square	O
root	O
of	O
the	O
sum	O
of	O
squares	O
(	O
SRSS	O
)	O
of	O
the	O
diffusivity	O
differences	O
,	O
divided	O
by	O
the	O
SRSS	O
of	O
the	O
diffusivities	O
.	O
</s>
<s>
The	O
fractional	B-Algorithm
anisotropy	I-Algorithm
can	O
also	O
be	O
separated	O
into	O
linear	O
,	O
planar	O
,	O
and	O
spherical	O
measures	O
depending	O
on	O
the	O
"	O
shape	O
"	O
of	O
the	O
diffusion	O
ellipsoid	O
.	O
</s>
<s>
The	O
most	O
common	O
application	O
of	O
conventional	O
DWI	O
(	O
without	O
DTI	B-Algorithm
)	O
is	O
in	O
acute	O
brain	O
ischemia	O
.	O
</s>
<s>
With	O
perfusion	B-Algorithm
MRI	I-Algorithm
detecting	O
both	O
the	O
infarcted	O
core	O
and	O
the	O
salvageable	O
penumbra	O
,	O
the	O
latter	O
can	O
be	O
quantified	O
by	O
DWI	O
and	O
perfusion	B-Algorithm
MRI	I-Algorithm
.	O
</s>
<s>
Some	O
more	O
specialized	O
diffusion	B-Algorithm
MRI	I-Algorithm
techniques	O
such	O
as	O
diffusion	O
kurtosis	O
imaging	O
(	O
DKI	O
)	O
have	O
also	O
been	O
shown	O
to	O
predict	O
the	O
response	O
of	O
cancer	O
patients	O
to	O
chemotherapy	O
treatment	O
.	O
</s>
<s>
The	O
architecture	O
of	O
the	O
axons	B-Algorithm
in	O
parallel	O
bundles	O
,	O
and	O
their	O
myelin	O
sheaths	O
,	O
facilitate	O
the	O
diffusion	O
of	O
the	O
water	O
molecules	O
preferentially	O
along	O
their	O
main	O
direction	O
.	O
</s>
<s>
The	O
imaging	O
of	O
this	O
property	O
is	O
an	O
extension	O
of	O
diffusion	B-Algorithm
MRI	I-Algorithm
.	O
</s>
<s>
magnetic	O
field	O
variations	O
in	O
the	O
MRI	B-Algorithm
magnet	O
)	O
are	O
applied	O
that	O
can	O
determine	O
at	O
least	O
3	O
directional	O
vectors	O
(	O
use	O
of	O
6	O
different	O
gradients	O
is	O
the	O
minimum	O
and	O
additional	O
gradients	O
improve	O
the	O
accuracy	O
for	O
"	O
off-diagonal	O
"	O
information	O
)	O
,	O
it	O
is	O
possible	O
to	O
calculate	O
,	O
for	O
each	O
voxel	B-Algorithm
,	O
a	O
tensor	B-Device
(	O
i.e.	O
</s>
<s>
a	O
symmetric	B-Algorithm
positive	O
definite	O
33	O
matrix	B-Architecture
)	O
that	O
describes	O
the	O
3-dimensional	O
shape	O
of	O
diffusion	O
.	O
</s>
<s>
The	O
fiber	O
direction	O
is	O
indicated	O
by	O
the	O
tensor	B-Device
's	O
main	O
eigenvector	O
.	O
</s>
<s>
The	O
brightness	O
is	O
weighted	O
by	O
the	O
fractional	B-Algorithm
anisotropy	I-Algorithm
which	O
is	O
a	O
scalar	O
measure	O
of	O
the	O
degree	O
of	O
anisotropy	O
in	O
a	O
given	O
voxel	B-Algorithm
.	O
</s>
<s>
Mean	O
diffusivity	O
(	O
MD	O
)	O
or	O
trace	O
is	O
a	O
scalar	O
measure	O
of	O
the	O
total	O
diffusion	O
within	O
a	O
voxel	B-Algorithm
.	O
</s>
<s>
These	O
measures	O
are	O
commonly	O
used	O
clinically	O
to	O
localize	O
white	O
matter	O
lesions	O
that	O
do	O
not	O
show	O
up	O
on	O
other	O
forms	O
of	O
clinical	O
MRI	B-Algorithm
.	O
</s>
<s>
Diffusion	B-Algorithm
tensor	I-Algorithm
imaging	I-Algorithm
data	O
can	O
be	O
used	O
to	O
perform	O
tractography	B-Algorithm
within	O
white	O
matter	O
.	O
</s>
<s>
Tractography	B-Algorithm
is	O
a	O
useful	O
tool	O
for	O
measuring	O
deficits	O
in	O
white	O
matter	O
,	O
such	O
as	O
in	O
aging	O
.	O
</s>
<s>
The	O
use	O
of	O
DTI	B-Algorithm
for	O
the	O
assessment	O
of	O
white	O
matter	O
in	O
development	O
,	O
pathology	O
and	O
degeneration	O
has	O
been	O
the	O
focus	O
of	O
over	O
2,500	O
research	O
publications	O
since	O
2005	O
.	O
</s>
<s>
Applications	O
in	O
brain	O
research	O
include	O
the	O
investigation	O
of	O
neural	B-Architecture
networks	I-Architecture
in	O
vivo	O
,	O
as	O
well	O
as	O
in	O
connectomics	O
.	O
</s>
<s>
Brachial	O
plexus	O
:	O
DTI	B-Algorithm
can	O
differentiate	O
normal	O
nerves	O
(	O
as	O
shown	O
in	O
the	O
tractogram	B-Algorithm
of	O
the	O
spinal	O
cord	O
and	O
brachial	O
plexus	O
and	O
3D	O
4k	O
reconstruction	O
)	O
from	O
traumatically	O
injured	O
nerve	O
roots	O
.	O
</s>
<s>
Early	O
in	O
the	O
development	O
of	O
DTI	B-Algorithm
based	O
tractography	B-Algorithm
,	O
a	O
number	O
of	O
researchers	O
pointed	O
out	O
a	O
flaw	O
in	O
the	O
diffusion	B-Algorithm
tensor	I-Algorithm
model	O
.	O
</s>
<s>
The	O
tensor	B-Device
analysis	O
assumes	O
that	O
there	O
is	O
a	O
single	O
ellipsoid	O
in	O
each	O
imaging	O
voxel	B-Algorithm
as	O
if	O
all	O
of	O
the	O
axons	B-Algorithm
traveling	O
through	O
a	O
voxel	B-Algorithm
traveled	O
in	O
exactly	O
the	O
same	O
direction	O
.	O
</s>
<s>
This	O
is	O
often	O
true	O
,	O
but	O
it	O
can	O
be	O
estimated	O
that	O
in	O
more	O
than	O
30%	O
of	O
the	O
voxels	B-Algorithm
in	O
a	O
standard	O
resolution	O
brain	O
image	O
,	O
there	O
are	O
at	O
least	O
two	O
different	O
neural	O
tracts	O
traveling	O
in	O
different	O
directions	O
that	O
pass	O
through	O
each	O
other	O
.	O
</s>
<s>
In	O
the	O
classic	O
diffusion	O
ellipsoid	O
tensor	B-Device
model	O
,	O
the	O
information	O
from	O
the	O
crossing	O
tract	O
just	O
appears	O
as	O
noise	O
or	O
unexplained	O
decreased	O
anisotropy	O
in	O
a	O
given	O
voxel	B-Algorithm
.	O
</s>
<s>
The	O
idea	O
is	O
best	O
understood	O
by	O
conceptually	O
placing	O
a	O
kind	O
of	O
geodesic	O
dome	O
around	O
each	O
image	O
voxel	B-Algorithm
.	O
</s>
<s>
Basically	O
,	O
we	O
are	O
now	O
going	O
to	O
look	O
into	O
the	O
voxel	B-Algorithm
from	O
a	O
large	O
number	O
of	O
different	O
directions	O
(	O
typically	O
40	O
or	O
more	O
)	O
.	O
</s>
<s>
We	O
use	O
"	O
n-tuple	O
"	O
tessellations	B-Algorithm
to	O
add	O
more	O
evenly	O
spaced	O
apices	O
to	O
the	O
original	O
icosahedron	O
(	O
20	O
faces	O
)	O
an	O
idea	O
that	O
also	O
had	O
its	O
precedents	O
in	O
paleomagnetism	O
research	O
several	O
decades	O
earlier	O
.	O
</s>
<s>
If	O
two	O
tracts	O
cross	O
in	O
the	O
voxel	B-Algorithm
,	O
there	O
will	O
be	O
two	O
pairs	O
of	O
maxima	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
We	O
can	O
still	O
use	O
tensor	B-Device
math	O
to	O
use	O
the	O
maxima	O
to	O
select	O
groups	O
of	O
gradients	O
to	O
package	O
into	O
several	O
different	O
tensor	B-Device
ellipsoids	O
in	O
the	O
same	O
voxel	B-Algorithm
,	O
or	O
use	O
more	O
complex	O
higher	O
rank	O
tensors	B-Device
analyses	O
,	O
or	O
we	O
can	O
do	O
a	O
true	O
"	O
model	O
free	O
"	O
analysis	O
that	O
just	O
picks	O
the	O
maxima	O
and	O
go	O
on	O
about	O
doing	O
the	O
tractography	B-Algorithm
.	O
</s>
<s>
The	O
Q-Ball	O
method	O
of	O
tractography	B-Algorithm
is	O
an	O
implementation	O
in	O
which	O
David	O
Tuch	O
provides	O
a	O
mathematical	O
alternative	O
to	O
the	O
tensor	B-Device
model	O
.	O
</s>
<s>
Instead	O
of	O
forcing	O
the	O
diffusion	O
anisotropy	O
data	O
into	O
a	O
group	O
of	O
tensors	B-Device
,	O
the	O
mathematics	O
used	O
deploys	O
both	O
probability	O
distributions	O
and	O
a	O
classic	O
bit	O
of	O
geometric	O
tomography	B-Algorithm
and	O
vector	O
math	O
developed	O
nearly	O
100	O
years	O
agothe	O
Funk	B-Algorithm
Radon	I-Algorithm
Transform	I-Algorithm
.	O
</s>
<s>
For	O
DTI	B-Algorithm
,	O
it	O
is	O
generally	O
possible	O
to	O
use	O
linear	B-Language
algebra	I-Language
,	O
matrix	B-Architecture
mathematics	O
and	O
vector	O
mathematics	O
to	O
process	O
the	O
analysis	O
of	O
the	O
tensor	B-Device
data	O
.	O
</s>
<s>
In	O
some	O
cases	O
,	O
the	O
full	O
set	O
of	O
tensor	B-Device
properties	O
is	O
of	O
interest	O
,	O
but	O
for	O
tractography	B-Algorithm
it	O
is	O
usually	O
necessary	O
to	O
know	O
only	O
the	O
magnitude	O
and	O
orientation	O
of	O
the	O
primary	O
axis	O
or	O
vector	O
.	O
</s>
<s>
Only	O
one	O
axis	O
is	O
needed	O
as	O
it	O
is	O
assumed	O
the	O
largest	O
eigenvalue	O
is	O
aligned	O
with	O
the	O
main	O
axon	B-Algorithm
direction	O
to	O
accomplish	O
tractography	B-Algorithm
.	O
</s>
