Basement
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oi
MIT
Sloan School of
Management
Working Paper 4228-01
December 2001
A MATHEMATICAL MODEL OF THE IMPACT OF
NOVEL TREATMENTS ON THE Afi BURDEN IN THE
ALZHEIMER'S BRAIN, CSF AND PLASMA
Lawrence M. Wein, David
© 2001
by Lawrence M. Wein, David
Short sections of text, not
to
permission provided that
L. Craft,
L. Craft,
Dennis
exceed two paragraphs,
full credit
including
Dennis
J.
J.
Selkoe
Selkoe. All rights reserved.
may be quoted
© notice
is
without explicit
given to the source.
This paper also can be downloaded without charge from the
Social Science Research
Network Electronic Paper Collection:
http://papers.ssm. com/abstract_id=295076
A
Mathematical Model of the Impact of Novel Treatments
on the A(3 Burden
David
L.
CSF and Plasma
in the Alzheimer's Brain,
Craft
1
,
Lawrence M. Wein 2 *, Dennis
J.
Operations Research Center, MIT, Cambridge,
2
3
Sloan School of Management, MIT, Cambridge,
Selkoe 3
MA 02139
MA 02139
Center for Neurologic Diseases, Harvard Medical School, Brigham and Women's Hospital,
Boston,
"Person to
whom
MA
02115
correspondence should be addressed; email: lwein@mit.edu
1
Abstract
With the advent
AD,
of novel therapies for
there
is
a pressing need for
biomarkers that are easy to monitor, such as the amyloid-beta (A/3) levels in
the cerebrospinal fluid (CSF) and plasma.
the explanatory power of
To gain a better understanding
these biomarkers, we formulate and analyze a
partmental mathematical model
for the A/3
accumulation
reveals that the total A/3 burden in the brain
tity called the
polymerization ratio, which
is
is
CSF
in the brain,
and plasma throughout the course of Alzheimer's treatment.
Our
analysis
dictated by a unitless quan-
the product of the production
and elongation rates divided by the product of the fragmentation and
rates.
of
cora-
loss
In this ratio, the production rate and loss rate include a source and
sink term, respectively, related to the inter-compartmental transport.
Our
results suggest that production inhibitors are likely to reduce the A/3 levels
in all three
compartments. In contrast, agents that ingest monomers
polymers, or that increase fragmentation or block elongation,
A/3 burden in the brain, but
increase -
CSF and plasma
may produce
little
change
in
may
- or even transiently
A/3 levels. Hence, great care must be taken
interpreting these biomarkers.
off of
also reduce
when
Introduction
Amyloid-beta (A/3) polymerization and plaque deposition are central to the
1
pathogenesis of Alzheimer's disease (AD)
.
Several novel treatments, such as the
enhancement of A/3 clearance by an A/3 vaccine
tion
by a 7-secretase inhibitor
6
5,
,
2-4
and the reduction of A/3 produc-
have shown promise
in preclinical studies.
agents
7
,
it
will still
be important to estimate the A/3 burden
therapies enter clinical
While
and other
cognitive tests will provide the ultimate assessment of the efficacy of these
in the brain.
As these
a key challenge will be to infer treatment-induced
trials,
changes in the A/3 burden in the brain from the A/3
compartments, such as the cerebrospinal
fluid
more
levels in
easily
monitored
(CSF) and the plasma. These
infer-
ences need to be based on a solid understanding of the A/3 kinetics throughout the
three interrelated compartments. Although the transport of A/3 between the various
compartments has been studied
in recent years
8-12
,
the impact of treatment on the
A/3 kinetics in these three compartments has not been elucidated.
In previous
work 13
,
we formulated and analyzed a mathematical model that
tracks the dynamics of A/3 production
mentation
sults
in the
and
loss,
and polymer elongation and
by considering a three-compartment model, where A/3
the brain,
CSF and
that provide
we
brain during the course of treatment. Here
some
plasma.
is
frag-
generalize these re-
transported between
Simple formulas and numerical results are presented
insights into system behavior,
and that can be used
key transport, production and clearance parameters as
new
clinical
to estimate
data becomes
available.
Mathematical model
We focus on A/342
in
(the 42 amino-acid form of A/3), which
parenchymal plaques
only
AJ4n
or
A542
.
14
.
If
is
the primary ingredient
we assumed that each brain polymer consisted
of
then the model could also be applied to A/34 o- which produces
8,
cerebrovascular deposits
10
;
however, we do not pursue this avenue here, and for
ease of notation refer to A/342 simply as A/3.
which
likely to
is
be
in a
number and
also only consider extracellular A/3,
dynamic equilibrium with
burden
primarily interested in the total A/3
to the
We
size of plaques,
we
intracellular A/3.
in the three
restrict
our attention to A/3 polymerization,
and do not attempt to capture the downstream processes
formation.
While
different parts of the brain
is
safely
we model the brain
much
faster
is
an
infinite
CSF and
,
system of nonlinear ordinary
and galaxy formation,
crystallization
the polymerization and depolymerization processes
and are
model
less
is
,
as a single
general than
many
,
plasma.
differential equations that
17
are related to a large class of models used to study actin formation
chemistry
16
15
than transport across the blood-brain barrier (BBB)
assume homogenous mixing within the
Our model
18
levels
compartment. Finally, because diffusion through extracellular spaces
homogeneous
we can
of fibrillization or plaque
have markedly different A/3
these levels tend to change proportionately and hence
in the brain
Because we are
compartments, as opposed
and cloud formation
we employ
considered in the literature,
19
,
polymer
" 20
.
While
are specialized to
the main novelty
AD
of our
the incorporation of multiple compartments with sources (production) and
sinks (loss).
For ease of understanding, nearly
For i=
1,2
we
let
all
of our mathematical notation
6,(0 be the concentration of A/3 i-mers at time
very few oligomers have been found in the
appears in the
is
CSF and plasma
CSF
21
or plasma,
mnemonic.
t.
Because
we assume that A/3
only as monomers, and denote their time-dependent
concentrations by c(t) and p(t), respectively.
The
differential equations specify the
time rate of change of these concentrations.
denoted by
b,(t), c(t)
and p(0, and are given by
+
h(t)
Vb
brain A/?
production
-c6i(*)(26 1 (t)
/f>(*)
vJ-^
monomers
'
={p
'
r
c}
^
=
bi(t)
{p c}
of
(i)-mers
^
degradation)
to
CSF and plasma
csf
+
c6i(t)6(_i(t)
brain A/?
elongations
loss^T! by
<
monomers
monomers from
plasma and
all
'
^
„
& \\ fragmentations
?
+ 53&i(t))
/bi+ i(t)
fragmentation
(i + l)-mers
elongation
(i - l)-mers
of
-/(l-^I{i=2})M*).
e6i(t)6((t)
(
2)
'
elongation
of 7-mers
=
c(t)
CSF
+
Pc
Yl
A/?
r
pp
plasma A/5
E
r
^ meis
-cW
,
monomers
to
brain and plasma
+
-
production
r xp x(t)
Yl
-
Yl VP(0
s
_^
jrijW
monomers
brain and
•
loss
>
v
monomers from
(3)
loss
brain and plasma
J~
'
fe^S
»={M^
monomers from
p{t)
of
-x W -
f={M^
production
fragmentation
to
brain and csf
CSF
(4)
The model
is
also depicted in Figure
1.
In equation (1), A/?
monomers
in the brain
are produced by cleavage of the amyloid precursor protein (APP) at the constant
rate
pb
.
These monomers
live for l^
1
time units on average before being
via cell internalization or protease degradation. Similar production
appear
only by
in equations (3)
monomer
and
(4).
and
lost, e.g.,
Equations (l)-(2) assume that elongation occurs
addition with elongation constant e
(i.e..
e&i
(f
)6,_i(f)
is
that (i-l)-mers elongate to i-mers), and that fragmentation occurs only by
break-offs at rate
/
(i.e.,
terms
loss
i-mers fragment into an
(i
-
l)-mer and a
the rate
monomer
monomer
at rate
The
fbi(t)).
factor 2 in front of b\(t) in equation (1) arises because the elongation of a
dimer requires 2 monomers. The indicator variable
otherwise; the factor \ in front of I{i=2}
due to the
is
1 if
i
fact that a
=
2
and equals
dimer can only
two potential fragmentation
location, whereas larger i-mers possess
fragment in one
equals
1 {,=2}
sites.
An
alternate modeling approach, which
direct clearance of
monomers
i-mers
off of
is
pursued later
(e.g.,
the
monomer
The remaining terms
is
typically
in the
CSF—»plasma
an
than obeying Michaelis-Menten kinetics
j
is
it
denoted by r l3
for i,j
.
diffusion
and active or carrier-mediated
and plasma—>brain transport are
nated by active transport, we assume that
that
monomer fragmentation
model describe the intercompartmental transport,
some combination of passive
Although
transport.
is
to allow
This alternative would result in the omission of
clearance.
13
ingestion rate, and gives qualitatively similar results
ment
is
fragmentations" term in equation (1) and the re-interpretation of /as
"all
which
the paper,
to represent microglial ingestion of
polymers), rather than requiring a two-step procedure of
followed by
in
=b,c
22
;
all
domi-
transport rates are first-order, rather
the rate from compartment
or p.
likely
Our main
i
to compart-
reason for this simplification
eases the parameter estimation task. However, the A/3 levels in these three
compartments do not undergo many orders-of-magnitude changes during the course
of treatment; hence, the saturation effect
may be minor
in the practically relevant
range.
Finally,
Similarly,
bound
we note that most
CSF
plasma A/3
A,J binds to gelsolin
25
.
is
23
bound, primarily to albumin
Moreover,
many
only free A/3
(i.e.,
'
24
.
not yet understood whether
it is
or free A/3, or both, gets transported across the
further,
occurs
of
BBB 11 To
.
confuse matters
laboratory and clinical studies that report plasma A/3 values quantify
23
.
free
Here,
and
we
implicitly
assume that
linear
total A/3 are in direct proportion
relationship between the total A/3
and
its
(i.e.,
22
)
unsaturated) binding
and that there
is
a linear
clearance rate out of the plasma. There-
when we model
fore
monomers
free
entering the plasma,
is
it
understood that a
proportion of them bind to plasma proteins but that this binding does not affect
the linear degradation and transport laws applied to the total A/3.
Results
Steady-state solution
The
steady-state analysis of equations (l)-(4) consists of setting the
sides of these
left
equations to zero and solving for the steady-state A(3 levels b t c and p; the details
,
of the derivation are omitted.
first
To present our
determine the effective production and
augment the actual
manner, we
results in a transparent
loss rates for
each compartment, which
by incorporating monomer exchange with the other two
rates
compartments. Because only monomers are transported between compartments, by
symmetry
effective
it
index the three compartments by
suffices to
production and
survival probability that a
compartment
h2
=
r 2i
+
r 23
We
1.
compartment
loss rates for
monomer
entering
h3
=
r 31
+
+
r 32
I3,
production and loss rates for compartment
probabilities s 2
and
By
S3.
Figure
1,
For
compartment
also define the total exit rate
+ U and
1.
2
1,
i
i
and
=
3,
2, 3,
and derive the
we
let s t
eventually makes
from compartments
respectively.
we need
To
r 31
to derive the
to
compartment
1
it
compartment
to
r x
(
^
)
as
unknown
r23
survival
f<\
h2
,
r 32
/ fi \
"3
«3
partment 2 makes
to
calculate the effective
,
h2
(5) states that
2
it
and 3
follows that
1, it
r21
For example, equation
be the
the probability that a
1
equals the probability that
plus the probability that
(^p) and from there eventually makes
monomer
it
to
it
first
compartment
it
entering com-
goes directly
goes to compartment 3
1
(S3).
The
solution to
equations (5)-(6)
is
2
''32
'"31
h3
V
With the
'
H3
/
H 2 /t3
\
r 21 \ /,
h2 h3
^23^32
yi--i^)
h h3
)
(8)
.
2
survival probabilities in hand, the effective production
and
loss rates are
given by
Pi
i1
Note that the
of
1
monomers
is
increased by the
compartment
We
1
1
compartments, whereas the
1
-
s terms,
to the other
1,
c for 2
).
(10)
1 is
enhanced by the survival
effective loss rate of
compartments but never return.
and
and p
for 3 (or,
loss rates in the brain, pb
by symmetry, p
for 2
and
subscripts of equations (9) and (10). Similarly, the effective rates for
respectively) are given
by substituting
compartment subscripts
We
are
now
in
compartment
by the monomers that are transported from
i.e.,
define the effective production
substituting b for
(9)
=l +r 12 (l-s 2 )+r 13 {l-s3
production rate of compartment
effective
in other
=Pi +P2S2 +P3S3,
for 2
and 3
c (p, respectively) for
1
and
lb
,
by
c for 3) in the
CSF
(plasma,
and the other two
in equations (9)-(10).
a position to present the steady-state solution. To do
so,
we
define
the polymerization ratio
r
=
M
(11)
hf
which
is
a unitless quantity that succinctly captures the four key processes of (effec-
tive) production, elongation, (effective) loss
of equations (l)-(4) to zero
and solving
for
and fragmentation. Setting the
finite
where r
>
1.
side
the steady-state concentrations reveal
that there are two regimes: a steady-state (or subcritical) regime where r
supercritical regime
left
<
1
and a
In the former case, the A/3 levels eventually attain
equilibrium values, where
&!
= &,
(12)
bi
=
26ir
i_1
1
,
c
=
2,3,...,
(13)
= £,
(14)
'c
(15)
P=fBy equations
(12)-(13), the total A/3 concentration in the brain
ber of A/3 molecules, whether they exist as
is
denoted by b
=
YlvLi
*&»>
'
monomers
the total
(i.e.,
or as part of polymers),
steady-state A/3 levels in the plasma (p), the
and increasing
in
which
s
There are several noteworthy features of our steady-state solution.
are linear
num-
CSF
(c),
First, the
and the brain monomers
(&i)
compartment production terms, and independent of the
elongation rate e and the fragmentation rate /; these latter two parameters only
z-mer concentration for
affect the brain
also decreasing in loss rates.
is
more
By
>
All steady-state concentrations are
2.
influence of transport parameters
on the A/?
levels
subtle, as explained in the Discussion.
the brain A/3 burden goes to infinity as the polymerization ratio r
(16),
approaches
i,j),
The
i
If
1.
we assume no transport
across
compartments
then the solution coincides with our one-compartment results
steady A/3 levels in
relevant one,
AD
patients
26 " 28
suggest that the r
and that the slow A/3 accumulation
<
1
13
regime
is less
increasing over the years. Consequently, the paper focuses on the r
>
1
regime
is
is
than
=
The
.
in the brain results
steady-state situation, where the polymerization ratio r
a brief discussion of the r
ru
(i.e.,
1
<
for all
relatively
the clinically
from a quasibut
1
is
slowly
regime, and
deferred until later.
Post-treatment kinetics
The impact
(i.e.,
of treatment can be assessed by changing the appropriate parameter
production,
loss,
transport, elongation or fragmentation rate) in the model
and substituting
To analyze the
assumptions.
simulations
13
it
into equations (14)-(16) to find the post-treatment steady state.
A/3 kinetics immediately after treatment,
The
first
assumption, which
that the total
number
that the total
is
simplifying
based on the observation from numerical
is
of brain oligomers changes
number
than the number of monomers,
we make two
much more
slowly
of oligomers in the brain,
Yylvbii remains constant immediately after treatment; we denote this quantity by
B2
,
which equals 2^(y^:) by equation
monomer
level after
much more
This new
level, 61, is
we
1
assumption, the brain
approximated by setting the
of equation (1) to zero and solving equations
values in Table
this
treatment quickly reaches a new level and thereafter changes
gradually.
presentation,
Under
(13).
(1), (3)
and
(4) for b x
.
left side
For ease of
present the solution for b x under the special case of the parameter
(i.e.,
61
=
=
r bp
-{l b
r cb
=
r pc
= p c = pp =
lc
=
0):
+ eB2 + J(lb + eB2 f + 8e(p + fB 2
b
)
)
(I'J
•
4e
In equation (17),
B2
is
number
the pre-treatment
of oligomers
and the four poly-
merization parameters represent the post-treatment values. Because the rate of
tercompartmental exchange (Table
1)
faster
is
than the changes
in-
in the brain A/?
polymers after treatment, we make the second simplifying assumption that the
steady-state relationships in equations (14)-(15) actually hold for
treatment. Therefore,
(again,
we
predict that the
assuming the parameters
in
Table
CSF and plasma
all
times after
levels rapidly
change to
1)
rtJ>i
-Mc
'
respectively,
r cv+ l
t'p
pb
and then gradually approach
(18)
r cp rbcbi
+
PP
'
*
their post-treatment steady states
10
(19)
Parameter estimation
10 parameters, which are listed along with their values in Table
Our model has
The
value of the A/?
minutes, which
group
is
monomer
loss rate
/
in
Table
1
corresponds to a
close to the crude experimental value of 38
in the brains of
APP
transgenic mice
6
.
Protofibrils
29
,
1.
half-life of 41.6
minutes reported by one
fibrils
30, 31
and plaques 32
appear to grow primarily via A/3 monomer addition, and we use an elongation rate
Table
e in
1
taken from a synthetic
of other estimates for fibrils
we
set the
CSF
We
and
protofibrils
,
CSF
31
29
.
which
,
is
within a factor of two
For lack of data to the contrary,
loss rate
lc
and the plasma production
zero.
assume that only three
and assume there
a circular
is
of the six possible exchanges
have a positive trans-
CSF—>brain,
we ignore brain—>plasma, plasma^CSF and
port rate. In particular,
flow given by brain—>CSF—>plasma^brain. To
we note that
these inclusions and omissions,
tify
analysis
fibril
production rate p c the
pp eq ual to
rate
30
CSF—>brain
jus-
and brain—^plasma
transport are likely to be by passive diffusion and these transport rates are probably
much
BBB
ics,
smaller than the
consistent with a specific transport
port
is
CSF—>plasma
difficult
the brain—>CSF flow
is
which passes through the
transport
is
quite rapid
is
is
10
,
diffusion
included in the model.
CSF
study of rats
9
8,
,
8 9
'
.
whereas plasma—>CSF trans-
drained/transported into the
taken from studies of guinea pigs
based on a
mechanism that dwarfs passive
33
because proteins do not readily pass through ultrafiltration
neuronally-produced A/3
is
rate,
>
according to a saturable mechanism that follows the Michaelis-Menten kinet-
Similarly,
rpb
plasma— brain transport
CSF
Our estimate
and the value
of mice
34
,
.
Also,
and hence
of the transport rate
for the transport rate r cp
10
.
This leaves four unknown parameter values, which are determined by solving four
equations (using equations (12) and (14)-(16)): the total A/?42 level in the brain b
is
1975 xlO 3
pM
(averaged over 5 different cortical regions of wet brain tissue of
patients with a clinical dementia rating
(CDR)
11
score of 5.0 (severe dementia)
35
,
assuming that the density of wet cortical
A/342 level c
Ad
is
115
pM
that consists of
Table
1,
36
;
tissue
the plasma A/342 level
monomers, ^,
is
1.3%
38,
39
.
is
is
equal to that of water); the
29
;
and the
CSF
fraction of brain
These estimates, which are given
lead to the polymerization ratio estimate r
Parameter
pM
37
=
0.84.
in
set equal to zero (details not
shown), the
monomer
directly proportional to each other in Figures
the
CSF and plasma
levels in the brain
2b and
drop to the values predicted
The
hours after
Note that
in this case the values in equations (18)
CSF
are
A/3 concentrations in
and
in equations (18)
treatment, and then slowly approach
(19) within
their post-treatment steady states.
the post-treatment steady-state values. Hence, the
approach their post-treatment steady-state A/3
2c.
and
and
(19) are very similar to
CSF and plasma compartments
levels
much
faster
than the brain
compartment. The post-treatment steady states represent an 18-fold reduction
the brain, a 1.67-fold reduction in the
Upon
the
CSF
the discontinuation of treatment, similar kinetics occur:
monomer
levels in all three
in
and a 1.67- fold reduction in the plasma.
a rapid change in
compartments, followed by a slow return to the
pre-treatment steady state.
Figure 3 depicts the impact of an agent that enhances fragmentation by 100%.
While the steady-state A/? burden
in the brain
drops 15.6-fold, the
CSF and plasma
A/? levels experience a transient rise and then return to the pre-treatment levels.
The
return to pre-treatment levels follows from the fact that the fragmentation rate does
not alter the steady-state brain
only interaction with the
monomer
CSF and plasma
the production inhibitor in Figure
different
2,
We now
is
levels in the
CSF and
is
deleted from equation (1)
monomer
an A/3 vaccine
2
and the fragmentation rate
ingestion off of oligomers.
plasma. As a
"all
is
fragmen-
interpreted
Figure 4 shows the impact of
that increases the ingestion rate by 100%.
levels rise
the brain's
similar for all three compartments.
consider an alternative to equations (l)-(4), in which the
term
is
the values in equations (18) and (19) are quite
approach to steady state
as the rate of
and plasma
equation (12), which
compartments. In contrast to the case of
than the post-treatment steady-state
result, the rate of
tations"
level in
In this case, the
CSF
monotonically to post-treatment steady states that are 2.5%
higher than the pre-treatment steady states, even though the steady-state brain A/3
burden decreases
8.6-fold.
13
Finally,
sible
we
investigate the impact of changing the transport rates.
improvements
pos-
reducing the A/3 burden in the brain are to increase the
for
brain—>CSF transport
Two
rate, r bc
and to decrease the plasma—>brain
,
ure 6 shows that increasing r bc by
rate, r pb
100% causes a modest 17% decrease
in the brain
A/3 level after one year (the post-treatment steady state level represents a
duction), while almost doubling the A/3 levels in the
CSF and
a 100-fold reduction in r pb reduces the brain A/3 burden by
Fig-
.
24%
re-
plasma. In contrast,
less
then 0.2%.
Supercritical regime
We now
turn to the case where the polymerization ratio r satisfies r
mathematical analysis of
this case
shows that the A/? burden
without bound, eventually increasing linearly at rate p b (r
-
>
in the brain
1.
A
grows
l)/r (Figure 5a).
The
polymer concentrations in the brain are not geometrically distributed as in the
r
<
1
where each z-mer successively achieves
case, but are uniformly distributed,
the concentration
=
6j
f/e, b z
plasma concentrations attain
brain concentration
is
=
2//e
for
finite levels in
unbounded. This
is
i
>
the r
Interestingly, the
2.
>
1 case,
CSF and
even though the total
because the brain A/3 burden grows by
accumulating larger polymers, not more monomers.
Discussion
Despite the existence of some mathematical models for A/3 fibrillization and
plaque growth, this paper appears to represent the
ical
model to
of the body.
CSF and
assess the effect of
We
AD
first
treatment on A/3
attempt to use a mathemat-
levels in various
compartments
provide simple formulas for the steady-state A/3 levels in the brain.
plasma, both before and after treatment. Our solution reveals that there
are two possible regimes, depending
the brain, r
=
Pf which
,
is
upon the value
of the polymerization ratio in
the product of the effective production rate and elon-
14
gation rate divided by the product of the effective loss rate and the fragmentation
The
rate.
and
effective
production and loss rates account not just
but also
loss in the brain,
the plasma and CSF.
When
for actual
and sinks due to transport
for sources
the polymerization ratio
We
A/3 levels are achieved throughout the body.
is
less
than
believe that this
1,
is
production
and from
to
steady-state
the clinically
relevant regime, in light of the slow accumulation of A/3 throughout the body.
r
>
1,
If
then the A/3 burden in the brain grows indefinitely (by accumulating larger
polymers while maintaining a constant monomer
early at rate
pb (r -
whereas the A/3
l)/r,
level),
levels in the
a steady state. This unusual state of affairs in the r
tant assumptions
in
CSF and plasma
1
case
is
lin-
remain
in
due to two impor-
our model: only monomers pass between compartments and no
polymerization occurs in the
with the plasma and
CSF
CSF and
plasma. Consequently, the brain A/? interacts
only via A/3 monomers.
CSF and plasma
A/3 levels provide a
level in the brain,
but not necessarily
Consequently, from a biomarker viewpoint,
reliable indirect estimate of the A/3
of the total A/3
>
eventually increasing
burden
monomer
in the brain.
An
extreme example of
CSF and plasma
above, where the easily-monitored
levels
this
is
the r
>
1
case
remain constant and do
not hint at the unbounded A/3 accumulation in the brain. However, our analysis also
has implications for the monitoring and assessment of treatment. Agents that inhibit
the production of A/3
monomers 5
6
'
monomer
or increase the
are likely to cause significant reductions in the A/3 levels in
with the
levels
CSF and plasma compartments
more
polymers 2
,
all
loss rate in the brain
three compartments,
attaining their post- treatment steady-state
quickly than the brain. In contrast, agents that ingest
monomers
off of
reduce the elongation rate, or increase the fragmentation rate of A/3
polymers in the brain,
the steady-state
may
cause only minor changes - including increases - in
CSF and plasma
on the total A/3 burden
primary toxic moiety
levels,
in the brain.
38 42
'
,
which are not indicative of their impact
However,
then the plasma A/3 and
15
soluble A/3
if
CSF
monomers
A/9 levels
are the
may be excellent
biomarkers for
efficacious
all
Alzheimer's treatments, and production inhibitors
than other agents that have
impairment
of cognitive
in
less
may be more
impact on A/3 monomers. The prevention
mice by an A/? vaccine 7 suggests that this
may
not be
the case.
Our
analysis also reveals
The
A/3 throughout the body.
more
how
the transport parameters affect the distribution of
relationship of A/3 levels to transport parameters
For example, a drug that increases removal of A/? from the brain by
subtle.
increasing the transport rate to the
will increase
is
the
CSF
CSF
mechanism
(a possible
which
chosen
in this
will fall,
and the
CSF
which
(cr cp ),
paper (based on A/342 ), the plasma
than the brain
2
)
A/3 level but the change in plasma A/3 level can be positive or
monomers by
negative, depending on the relative contributions to plasma
(^i r 6P ),
of the Elan vaccine
With the parameters
will rise.
level rises
the brain
because the
CSF
rather
the primary extra-compartmental source of plasma A/3. However,
is
in the case of A/34 o,
where
it is
estimated that r^
is
10 times larger than
43
r;, c
,
our
analysis indicates that plasma A/34 o levels would decrease as a consequence of such
a therapy.
While many of the model's parameter values are imprecisely known and the
model
rather simple
is
and bound
stem
CSF
in the brain
was done
in
transport, linear relationship between free
A/3), the qualitative nature of our results should
in large part
in the
(e.g., first-order
from the empirical observation
21
be robust because they
that very few A/3 polymers reside
or plasma, suggesting that A/3 polymerization occurs almost exclusively
and A/3 polymers are not
HIV
research
44, 45
,
easily transported out of the brain.
mathematical analysis such as
with data generated by perturbation of
to derive estimates of
human
some parameters that
this
However, as
can be combined
A/3 compartments by novel agents
are difficult to measure in vivo, thereby
uncovering the primary flow dynamics of A/3 in the body. More generally, our model
provides a systematic framework with which to interpret upcoming
trials of
novel agents for Alzheimer's disease.
16
human
clinical
Acknowledgment
This research was partially supported by the Singapore-MIT Alliance
by the Foundation
for
Neurologic Diseases (DJS).
17
(LMW) and
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\
t
1,
Figure
1:
A
pictorial depiction of the
25
mathematical model.
(a)
z
1.5
//M
1
0.5
Brain total
(c)
CSF
(a)
z
1.5
(JM
1
0.5
Brain total
(c)
CSF
(a)
0.6
(JM
0.4
0.2
Brain total
(c)
CSF
(a)
Brain total
(c)
CSF
250
12
10
200
8
150
(J.M
pM
6
100
4
50
2
40
20
60
80
100
(b)
Brain
monomer
50000
40000
pM 30000
20000
10000
40
60
time (days)
40
(d)
60000
20
20
60
time (days)
time (days)
100
Plasma
80
100
(a)
/UM
1.8
(c)
Brain total
pM
CSF
Z.Z- Aj&j
Date Due
Z/jq
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