Biomedical Engineering Reference
In-Depth Information
⎧
⎨
x
=
h
(
x
)
,
t
budding
(9)
0
T
i
as
T
as
p
i
⎩
=
x
i
d
t
,
1
≤
i
≤
n
,
where we name the constant concentration of T lymphocytes of type
i
as
T
a
i
.
The infected cell clearance rate until budding can be computed by substituting
the solution of Eq. (
9
) into Eq. (
6
)toget
2
g
i
μ
(
)=
(
)
=
p
C
1
∑
i
A
i
p
i
where
A
i
and
t
budding
0
x
i
d
t
. We look for the set
p
∗
satisfying
=
g
i
p
∗
)=
∑
i
2
μ
(
min
A
i
(
p
i
)
.
(10)
p
i
≥
c
,
p
≥
0
∑
i
Solution of the Optimization Problem
For all
i
,
A
i
>
0 and the optimization problem is a constrained convex quadratic
optimization problem that can be solved using the KKT (Karush-Kuhn-Tucker)
theorem [
25
,
29
], leading to a unique positive minimum [
36
].
The inequality constraint
∑
i
p
i
≥
c
is always active (
∑
i
p
i
=
c
). We can thus write
−
∑
i
=
i
0
p
i
where
i
0
is the index of some nonzero element
p
i
and reformulate
the optimization problem (
10
) as follows:
p
i
0
=
c
⎛
−
i
=
i
0
p
i
2
⎞
c
⎝
2
⎠
.
i
=
i
0
A
i
(
p
i
)
+
min
p
A
i
0
(11)
≥
0
The Lagrangian is
⎛
−
i
=
i
0
p
i
2
⎞
c
⎝
i
=
i
0
A
i
(
p
i
)
2
⎠
−
i
=
i
0
η
i
p
i
.
L
(
p
,
η
)=
+
A
i
0
(12)
Up to a constant difference, this is equivalent to the maximization problem in Eq.
(
5
). The KKT equations are
c
−
i
=
i
0
p
i
∂
L
p
i
=
2
A
i
p
i
−
2
A
i
0
−
η
i
=
0
,
∂
(13a)
1
≤
i
≤
n
and
i
=
i
0
,
L
∂η
i
=
−
∂
p
i
≤
0
,
1
≤
i
≤
n
and
i
=
i
0
,
(13b)
η
i
p
i
=
0
,
1
≤
i
≤
n
and
i
=
i
0
,
(13c)
η
i
≥
0
,
1
≤
i
≤
n
and
i
=
i
0
.
(13d)
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