Louisiana Tech University Ruston, LA 71272 Slide 1 Review Steven A. Jones BIEN 501 Friday, May 14,...
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Transcript of Louisiana Tech University Ruston, LA 71272 Slide 1 Review Steven A. Jones BIEN 501 Friday, May 14,...
Louisiana Tech UniversityRuston, LA 71272
Slide 1
Review
Steven A. Jones
BIEN 501
Friday, May 14, 2007
Louisiana Tech UniversityRuston, LA 71272
Slide 2
Simple Flow Field
What is the pathline?
Louisiana Tech UniversityRuston, LA 71272
Slide 3
Simple Flow Field
Louisiana Tech UniversityRuston, LA 71272
Slide 4
Simple Flow Field
Pathline follows the particle
Louisiana Tech UniversityRuston, LA 71272
Slide 5
Simple Flow Field
What is the streakline?
Louisiana Tech UniversityRuston, LA 71272
Slide 6
What is the Differential Equation that Describes a Streamline?
w
dz
v
dy
u
dx
w
u
dz
dx
v
u
dy
dx ,
Assume we know that:
kzyxhjzyxgizyxfzyxu
,,,,,,,,
Answer: Since
zyxh
zyxf
z
x
zyxg
zyxf
y
x
.,
,,,
,,
,,
So
Louisiana Tech UniversityRuston, LA 71272
Slide 7
Continuity
For a two-dimensional flow:
yexu cos
Use the equation of continuity to determine v.
Louisiana Tech UniversityRuston, LA 71272
Slide 8
Answer
yfdxx
uv
y
v
x
u
y
v
x
u0
yexx
u
sin
yfdxexv y sin
yfexv y sin
Louisiana Tech UniversityRuston, LA 71272
Slide 9
What is the equation for a pathline?
A pathline follows a fluid particle. Assume that you know the entire velocity field: and that the particle passes through the point
at time 0.
Answer:
dttutxdttx
txu ,
000 ,,0 zyxx
txudt
xd,
Louisiana Tech UniversityRuston, LA 71272
Slide 10
Example
Assume that:
Answer:
Yes
jxyixtxu
2, 2
Is continuity satisfied?
Louisiana Tech UniversityRuston, LA 71272
Slide 11
What is the equation for a pathline?
Answer: xv
dt
dyxu
dt
dx ,
Assume that:
jxyixxu
22 What is the equation for the pathline through (1,2)?
xy
dydtxy
dt
dyx
dt
dx
22,2
y
x
dy
dxx
dy
dxxy
22 2
Louisiana Tech UniversityRuston, LA 71272
Slide 12
What is the equation for a pathline?
Cyxy
dy
x
dx
y
x
dy
dx 2lnln
22
Cyex 2ln
Write: lnC
yx2
Louisiana Tech UniversityRuston, LA 71272
Slide 13
What is the equation for a pathline?
yx2
422
1
so
yx2
4
Louisiana Tech UniversityRuston, LA 71272
Slide 14
Answer (Continued)
xgxytty
yftxtx
22
1 2
Louisiana Tech UniversityRuston, LA 71272
Slide 15
Two Compartment Model
111 CVkm
Peripheral
Central
2221112
2
112221111
1
CkVCkVdt
dCV
CkVCkVCkVdt
dCV e
Conservation of Mass
dq
C1 C2
222 CVkm
Clearance
Central Compartment Peripheral Compartment
1V 2V
Louisiana Tech UniversityRuston, LA 71272
Slide 16
Two Compartment Model
Peripheral
Central
2221112
21
12221111
CkCkdt
dC
CkCkCkdt
dCe
Conservation of Mass
In terms of the volume ratio1
221 V
V
00,0 201
1 CCV
DCInitial Conditions
Solve the two ODEs for C1
0121
32121
2
Ckkdt
dCkkk
dt
Cde
Louisiana Tech UniversityRuston, LA 71272
Slide 17
ICs in terms of C1
0001
0,0 111
21201
Ckkdt
dCCCC e
011
01
0
0
Ckkdt
dC
CC
e
Louisiana Tech UniversityRuston, LA 71272
Slide 18
Solution
The solution to:
With
0121
32121
2
Ckkdt
dCkkk
dt
Cde
011
01
1
0,0 Ckk
dt
dCC
V
DC e
Is tt eeCC 21 101
Where: 2
4 22
21212,1
eee kkkkkkkk
Louisiana Tech UniversityRuston, LA 71272
Slide 19
Two Compartment Model
tt eeCC 21 101
Rapid Release Slow Release
One Compartment
Louisiana Tech UniversityRuston, LA 71272
Slide 20
Two Compartment Model
The two-compartment model obeys the same differential equations as the simple RLC circuit.
It is useful to compare the individual components to the RLC circuit:
0121
2121
2
Ckkdt
dCkkk
dt
Cdee
Damping
Transfer from L to C
Louisiana Tech UniversityRuston, LA 71272
Slide 21
Two Compartment Model
One might expect that overshoot (ringing) could happen. However, ringing will only happen for imaginary values of . In our case:
2
4 22
21212,1
eee kkkkkkkk
And for the RLC Circuit:
2
4 2
2
2
2,1CLR
LR
LR
Can make the square root imaginary with small R or large C.
As you increase k2 or ke, you must also increase (k1+k2+k3).
Louisiana Tech UniversityRuston, LA 71272
Slide 22
Two Compartment Model
To see if the square root can become imaginary, minimize it’s argument w.r.t. ke and see if it can be less than 0.
2
4 22
21212,1
eee kkkkkkkk
04 22
21 eee
kkkkkdk
d
21
21
221
0222
042
kkk
kkk
kkkk
e
e
e
Louisiana Tech UniversityRuston, LA 71272
Slide 23
Two Compartment Model
What value does the argument of the square root take on at the minimum?
ee kkkkk 22
21 4
21 kkk e
12212
2222
222
22
21
444
444
4
kkkkkkk
kkkkkkk
kkkkk
ee
ee
ee
Since k2 and k1 cannot be negative, the argument of the square root can never be negative. I.e. no ringing.
Louisiana Tech UniversityRuston, LA 71272
Slide 24
Pharmacokinetic Models
Vascular
Interstitial
Cellular
vC
LQ
pC
Q
iC
LsJ
iC
q
PBPK: Physiologically-Based Pharmocokinetic Model
Q: Plasma Flow
L: Lymph Flow
Js, q: Exchange rates
Louisiana Tech UniversityRuston, LA 71272
Slide 25
Pharmacokinetic Models
cc
c
ii
si
i
vsvpv
v
Rqdt
dCV
RqZ
LCJ
dt
dCV
RJCLQQCdt
dCV
Z: Equilibrium concentration ratio between interstitium and lymph.
Li ZCC
Louisiana Tech UniversityRuston, LA 71272
Slide 26
More Complicated Models
Plasma
Liver
Kidney
Muscle
G.I. Track
Louisiana Tech UniversityRuston, LA 71272
Slide 27
Note on Complexity
• While the equations become more complicated as more components are added, the basic concepts remain the same, and the systems can be analyzed with the same tools you would use to analyze a linear system in electrical engineering (e.g. transfer functions, Laplace transforms, Mason’s rule).
Louisiana Tech UniversityRuston, LA 71272
Slide 28
Louisiana Tech UniversityRuston, LA 71272
Slide 29
What is the Differential Equation that Describes a Streamline?
ybdz
zyxh
zyxfx
zadyzyxg
zyxfx
.,
,,
,,
,,