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


Slide 2

Simple Flow Field

What is the pathline?


Slide 3

Simple Flow Field


Slide 4

Simple Flow Field

Pathline follows the particle


Slide 5

Simple Flow Field

What is the streakline?


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


Slide 7

Continuity

For a two-dimensional flow:

yexu cos

Use the equation of continuity to determine v.


Slide 8

Answer

yfdxx

uv

y

v

x

u

y

v

x

u0

yexx

u

sin

yfdxexv y sin

yfexv y sin


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,


Slide 10

Example

Assume that:

Answer:

Yes

jxyixtxu

2, 2

Is continuity satisfied?


Slide 11


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


Slide 12


Cyxy

dy

x

dx

y

x

dy

dx 2lnln

22

Cyex 2ln

Write: lnC

yx2


Slide 13


yx2

422

1

so

yx2

4


Slide 14

Answer (Continued)

xgxytty

yftxtx

22

1 2


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


Slide 16


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


Slide 17

ICs in terms of C1

0001

0,0 111

21201

Ckkdt

dCCCC e

011

01

0

0

Ckkdt

dC

CC

e


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


Slide 19


tt eeCC 21 101

Rapid Release Slow Release

One Compartment


Slide 20


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


Slide 21


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).


Slide 22


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


Slide 23


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.


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


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


Slide 26

More Complicated Models

Plasma

Liver

Kidney

Muscle

G.I. Track


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).


Slide 28


Slide 29

What is the Differential Equation that Describes a Streamline?

ybdz

zyxh

zyxfx

zadyzyxg

zyxfx

.,

,,

,,

,,

louisiana tech university ruston, la 71272 slide 1 review steven a. jones bien 501 friday, may 14,...

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