The Power of Optimal Control in Biological Systems …bioquest.org/peer2012/files/2012/08/peertalk12.pdf · The Power of Optimal Control in Biological Systems Suzanne ... Control

The Power of Optimal Control in Biological Systems Suzanne Lenhart

The Power of Optimal Control in Biological

Systems

Suzanne Lenhart

August 13, 2012


Outline

Background, some about UT and some about me

Viewpoint on Models

Cardiopulmonary Resuscitation

Rabies in Raccoons

More current work


How our Math Biology program got started

Started with one faculty member (Tom Hallam) in late 1970sand added more over 20 years

Developed 100-level math for biology courses

Developed 2-year grad sequence and degree concentration

Organized interdisciplinary seminar every semester for 30 yearswith faculty from many different disciplines

Built interdisciplinary institute (The Institute forEnvironmental Modeling) with links to ORNL

Used TIEM to foster collaborations and attract external funds(no UT funds involved)

Moved-in projects from other units as appropriate

Won the NIMBioS award! summer 2008


Fellowship Grants!

In the same summer, Cynthia Peterson was awarded the SCALE-ITand PEER grants.

Mentoring Team for PEER students - Lenhart leading the team


teachers, mentors make a big impact

born to be a teacher

attended Bellarmine College, graduate work at U of Kentucky

went to UT straight from graduate school, spouse support

found great collaborator, Vladimir Protopopescu, at OakRidge National Lab

comfortable with a service role

interested in outreach and REUs for many years

found another great collaborator, Lou Gross, math ecologistat UT

new phase of my life, NIMBioS

work hard, be flexible, willing to work on many things


Models

WHAT is a MODEL?A model is like a map — it represents part of reality but not all ofit!


Tools?

MODELS!!Use mathematical models for research work

Drug treatment strategies for HIV/AIDS

Control practices for tuberculosis epidemics

Drug treatments for leukemia

West Nile virus

Cholera management

Fishery models

Invasive species


Mathematical Models

Inputs to a system of equations are adjusted until the desired goaloutput is obtained.

Optimal control theory is a tool to choose optimal inputs.

Equations involve rates of change and interaction and movementterms among the components of the system.


Improving Cardiopulmonary Resuscitation

Each year, more than 250,000 people die from cardiac arrest in theUSA alone.

Despite widespread use of cardiopulmonary resuscitation, thesurvival of patients recovering from cardiac arrest remains poor.

The rate of survival for CPR performed out of the hospital is 3%,while for patients who have cardiac arrest in the hospital, the rateof survival is 10-15%.


Goal

The goal is to improve traditional CPR technique by using optimalcontrol methods.

The standard and various alternative CPR techniques such asinterposed abdominal compression IAC, and Lifestick CPR havebeen represented in various models.

We consider a model for CPR allowing chest and abdomencompression and decompression.

Design optimal PATTERN of compression/decompression!


Model by Babbs

We apply the optimal control strategy for improving resuscitationrates to a circulation model developed by Babbs. (model -discretein time, with seven compartments)

In his model, heart and blood vessels are represented asresistance-capacitive networks, pressures in the chest and in thevascular components as voltages, blood flow as electric current,and valves .

Reference: Babbs, Circulation 1999.


Heart Diagram

Figure 23-11 Valvular structures of the heart. The atrioventricular valves are in an open position, and the semilunar

valves are closed. There are no valves to control the flow of blood at the inflow channels (i.e., vena cava and

pulmonary veins) to the heart.

Copyright © 2005 Lippincott Williams & Wilkins. Instructor's Resource CD-ROM to Accompany Porth's Pathophysiology: Concepts of Altered Health States, Seventh Edition.

Superior

vena cavaAortic valve

Pulmonary

veins

Inferior

vena cava

Mitral

valve

Pulmonic

valve

Tricuspid

valvePapillary

muscle


Diagram of Circulation Model

← Thoracic aorta → Abdominal aorta↓

Carotid Artery ↑ ↓↓

Thoracic pump ↓

↓

Jugular vein ↑ ↓

↓→ Right heart → Inferior vena cava


Seven Components in the Model

P1 pressure in abdominal aortaP2 pressure in inferior vena aortaP3 pressure in carotid arteryP4 pressure in jugular veinP5 pressure in thoracic aortaP6 pressure in rt. heart, superior vena cavaP7 pressure in thoracic pump and left heart.


Goal for this model

Design compression/depression patterns for chest and abdomenpressures

To increase pressure differences across thoracic aorta and rightheart

SPP - Systemic Prefussion Pressure


The chosen CPR model consists of seven difference equations, withtime as the discrete underlying variable.At the step n, when time is n∆t , the pressure vector is denoted by:

P(n) = (P1(n),P2(n), ...,P7(n)).

We assume that the initial pressure values are known, when n = 0.To make the chest pressure profiles medically reasonable, assumei.e., ui (0) = ui (N − 1).

u1 = (u1(0), u1(1), ..., u1(N − 2), u1(0)),

u2 = (u2(0), u2(1), ..., u2(N − 2), u2(0)),


Difference Equations Model

for n = 1, 2, ...,N − 1 (in vector notation)

P(1) = P(0) + T1(u1(0)) + T2(u2(0)) + ∆tF (P(0)), (1)

P(n + 1) = P(n) + T1(u1(n)− u1(n − 1)) (2)

+T2(u2(n)− u2(n − 1)) + ∆tF (P(n)), (3)

T1(u1(n)) = (0, 0, 0, 0, tpu1(n), tpu1(n), u1(n)),

T2(u2(n)) = (u2(n), u2(n), 0, 0, 0, 0, 0).


Note that the pressure vector depends on the control, P = P(u),and the calculation of the pressures at the next time step requiresthe values of the controls at the current and previous time steps.We use extension of the discrete version of Pontryagin’s MaximumPrinciple.


Show function F (P(n)) by some of its seven components:

1

cjug

[

1

Rh

(P3(n)− P4(n))−1

Rj

V (P4(n)− P6(n))

]

1

cao

[

1

Ro

V (P7(n)− P5(n))−1

Rc

(P5(n)− P3(n))

]

+1

Ra

(P5(n)− P1(n))−1

Rht

V (P5(n)− P6(n))

]

where the valve function is defined byV (s) = s if s ≥ 0V (s) = 0 if s ≤ 0. Three valves: between compartments 4 - 6

AND 5 - 7 AND 5 - 6.


Goal

Choose the control set U ⊂ ℜ2N , defined as:

U = (u1, u2)|ui (0) = ui(N − 1)

−Ki ≤ ui (n) ≤ Li , i = 1, 2, n = 0, 1, . . . ,N − 2.

We define the objective functional J(u1, u2) to be maximized

N∑

n=1

[P5(n)− P6(n)]−

N−2∑

n=0

[B1

2u21(n) +

B2

2u22(n)] (4)

Use OPTIMAL CONTROL THEORY to solve this problem.


Pressure Profiles

0 0.2 0.4 0.6 0.8 1−20

0

20

40

60IAC−CPR for chest

0 0.2 0.4 0.6 0.8 1−20

0

20

40

60

80

100

120IAC−CPR for abdomen

0 0.2 0.4 0.6 0.8 1−20

0

20

40

60Lifestick−CPR for chest

0 0.2 0.4 0.6 0.8 1−20

0

20

40

60

80

100

120Lifestick−CPR for abdomen

Figure: Each waveform represents one cycle.


Optimal Control on Chest only and Standard Profiles

0 2 4 6 8 10 12

0

10

20

30

40

50

60(a

)

STD−CPR: SPP=24.7630

0 2 4 6 8 10 12

−20

0

20

40

60

Time (second)

(b)

Optimal Control: SPP=36.1164


OC Profiles

0 2 4 6 8 10 12

−20

0

20

40

60

Time (s)

Che

st C

ontr

ol (

mm

Hg)

0 2 4 6 8 10 12

−20

0

20

40

60

80

100

120

Time (s)

Abd

omin

al C

ontr

ol (

mm

Hg)

Figure: The controlled chest and abdominal pressure using Lifestick


Concluding Remarks about CPR

This procedure with RAPID compression and decompression cycleshas recently been recommended by several medical groups.

We can increase the pressure difference across the thoracic aortaand the right heart by about 25 percent.

We received a US patent for this idea!through Oak Ridge National Labwith Protopopescu and Jung


Rabies in Raccoons

Rabies is a common viral disease.Transmission is through the bite of an infected animal.Raccoons are the primary terrestrial vector for rabies in theeastern US.Vaccine is distributed through food baits. (preventative)Medical and Economic Problem -death to humans andlivestock and COSTS


Costs and Treatment associated with Rabies in USA

30,000 persons/year given rabies post exposure prophylaxis at acost of $30 million

Treatment - one dose of rabies immune globulin (injected near thesite of the bite)

and- five doses of vaccine over 28 days (injected into upper arm)

Symptoms - flu-like at first, about 10-60 days after exposure, laterdelirium, disruption of nervous system


Goal

Develop models and numerical results to investigate otherdistribution patterns for vaccine baits, as it impacts the spread ofrabies among raccoons.

Reduce the chance of rabies spread while keeping the costs ofvaccine distribution as low as possible.

More Precise GoalMinimize the number of infected raccoons while taking intoaccount limited amount of funding for the distribution of vaccinebaits.


Variables

Model with (k,l) denoting spatial location, t time

susceptibles = S(k,l,t)

infecteds = I(k,l,t)

immune = R(k,l,t)

vaccine = v(k,l,t)

control c(k, l, t), input of vaccine baits


Movement

In one time step, if the box size was the size of a home range(about 4 km2 ), then 95 percent of the raccoons would not leavetheir box. The 5 percent moving out would be distributed inverselyproportional to distance. But a raccoon could not move fartherthan their home range distance (2 km) in one time step. If the boxsize is smaller, then the percentage moving is changedappropriately.


Order of events

Within a time step (about a week to 10 days):

First movement: using home range estimate to get range ofmovement. See sum S, sum I and sum R to reflect movement.

Then: some susceptibles become immune by interacting withvaccine

Lastly: new infecteds from the interaction of the non-immunesusceptibles and infecteds

NOTE that infecteds from time step n die and do not appear intime step n + 1.


Susceptibles and Infecteds Equations

S(k, l , t + 1) = (1− e1v(k, l , t)

v(k, l , t) + K)sum S(k, l , t)

− β

(1− e1v(k, l , t)

v(k, l , t) + K)sum S(k, l , t)sum I (k, l , t)

sum S(k, l , t) + sum R(k, l , t) + sum I (k, l , t),


Susceptibles and Infecteds Equations

S(k, l , t + 1) = (1− e1v(k, l , t)

v(k, l , t) + K)sum S(k, l , t)

− β

(1− e1v(k, l , t)


sum S(k, l , t) + sum R(k, l , t) + sum I (k, l , t),

I (k, l , t + 1) = β

(1− e1v(k, l , t)


sum S(k, l , t) + sum R(k, l , t) + sum I (k, l , t).


Immune and Vaccine Equations

R(k, l , t + 1) = sum R(k, l , t) + e1v(k, l , t)

v(k, l , t) + Ksum S(k, l , t),

v(k, l , t + 1) =

Dv(k, l , t) max [0, (1 − e(sum S(k, l , t) + sum R(k, l , t)))] + c(k, l , t).


Objective Functional

maximize the susceptible raccoons, minimize the infecteds and costof distributing baits

∑

m,n

(

I (m, n,T )− S(m, n,T ))

+ B∑

m,n,t

c(m, n, t)2,

where T is the final time and c(m, n, t) is the cost of distributingthe packets at cell (m, n) and time t, B is the balancingcoefficient, c is the control, t = 1, 2, ...,T − 1.

Use OPTIMAL CONTROL THEORY to solve.


Numerical Example

Using a square grid with 25 boxes, we do the math analysisfollowed by the numerical solution.

In each box, 8 equations are solved at each time step. Fourequations for S, I, R, and V and four equations for the optimizingprocedure.

To get convergence to optimal bait distribution, about 100iterations are completed.


Disease Starts From the Corner: Initial Distribution

1 2 3 4 5 61

2

3

4

5

6

t=1

susc

epti

ble

s

1 2 3 4 5 61

2

3

4

5

6

t=1

infe

cted

s

5

10

15


Susceptibles, with control

1 2 3 4 5 61

2

3

4

5

6

t=2, B=0.5

susc

epti

ble

s

1 2 3 4 5 61

2

3

4

5

6

t=3, B=0.5

susc

epti

ble

s

1 2 3 4 5 61

2

3

4

5

6

t=4, B=0.5

susc

epti

ble

s

1 2 3 4 5 61

2

3

4

5

6

t=5, B=0.5

susc

epti

ble

s

5

10

15

5

10

15


Infecteds, with control

1 2 3 4 5 61

2

3

4

5

6

t=2, B=0.5

infe

cted

s

1 2 3 4 5 61

2

3

4

5

6

t=3, B=0.5

infe

cted

s

1 2 3 4 5 61

2

3

4

5

6

t=4, B=0.5

infe

cted

s

1 2 3 4 5 61

2

3

4

5

6

t=5, B=0.5

infe

cted

s

0.5

1

1.5

2

2.5

0.5

1

1.5

2

2.5


Immune, with control

1 2 3 4 5 61

2

3

4

5

6

t=2, B=0.5

imm

un

es

1 2 3 4 5 61

2

3

4

5

6

t=3, B=0.5

imm

un

es

1 2 3 4 5 61

2

3

4

5

6

t=4, B=0.5

imm

un

es

1 2 3 4 5 61

2

3

4

5

6

t=5, B=0.5

imm

un

es

2

4

6

8

10

12

14

2

4

6

8

10

12

14


Opt. Control -number of baits in each box at each time

1 2 3 4 5 61

2

3

4

5

6

t=1, B=0.5

con

tro

l

1 2 3 4 5 61

2

3

4

5

6

t=2, B=0.5

con

tro

l

2

4

6

8

1 2 3 4 5 61

2

3

4

5

6

t=3, B=0.5

con

tro

l

1 2 3 4 5 61

2

3

4

5

6

t=4, B=0.5

con

tro

l

2

4

6

8


Fishery Problem: Motivation

No-take marine reserves may be a part of optimal harveststrategy designed to maximize yield.

Marine reserves can protect habitat and defend endangeredstock from overexploitation.

Marine reserves as a part of fishery management plan arecontroversial.


Work -parabolic case

Can we show that when considering the maximization of revenueonly, the marine reserves occur in the optimal harvesting strategy?

MODEL

includes both time and space

multi-dimensional spatial domain

Investigate the presence of marine reserves in optimal harvestingstrategy.

We have completed the analysis for general semilinear parabolicPDE in a multidimensional domain but here we present a simplercase.


Parabolic Fishery Model

Our fishery model in domain Q = Ω× (0,T ) is :

ut = ∆u + u(1− u)− hu inQ (5)

with initial and boundary conditions:

u(x , 0) = u0(x) for x ∈ Ω

u(x , t) = 0 on ∂Ω× (0,T )

u represents fish population

h represents the proportion to be harvested


Goal

We seek to maximize the objective functional over h ∈ U:

J(h) =

∫ T

0

∫

Ω

e−δthu dx dt (6)

where U = h ∈ L∞(Q) : 0 ≤ h(x , t) ≤ M ≤ 1 is class ofadmissible controls and e−t. represents a discount factor withinterest rate δ.(1 + δ)/2 < M

This problem is linear in the control.

Use OPTIMAL CONTROL again.


Optimal Control for Different Initial Conditions

Figure: Left IC -unexploited stock, Right IC -overexploited stock


Conclusions about Fishery Models

Spatial optimal control for harvesting problems are relevant astechnology enables the enforcement of spatially structured harvestconstraints.

In the future, investigate more spatial heterogeneities in thedynamics and in the domain, and and more realistic boundaryconditions

Ding and Lenhart are investigating a fishery problem for a specificspecies with age structure and discrete time.


Modeling the Hog Population in GSMNP

Team: Chuck Collins, Suzanne Lenhart, Bill Stiver, MargueriteMadden, Rene Salinas, Eric Carr, Joe Corn


Modeling Feral Hogs in GSMNP

High Region

Fontana

Gatlinburg

CherokeeCalderwood

CataloocheeCades Cove

Cosby


Big Picture by Month

MODEL - discrete space and timeUSE database with 10,000 entries of harvesting hogs in the park

SUMMARY description of actions in monthsJanuary: hogs in low regions, survival, births, low-to-low movement

February: hogs in low regions, survival, low-to-low movement

March, April, May, June, July:hogs may be in low and high regions, survival, low-to-lowmovement, possible low-to-high movement

August: all hogs in high region, survival, proportional movement tolow regions

September, October, November, December: all hogs in lowregions, survival, movement low-to-low


Collaborators

Vladimir Protopopescu, Eunok Jung and Charles Babbs

Lou Gross, Wandi Ding, Keith Langston

Mike Neubert, Ta Herrera, H. R. Joshi

Documents

The Power of Optimal Control in Biological Systems …bioquest.org/peer2012/files/2012/08/peertalk12.pdf · The Power of Optimal Control in Biological Systems Suzanne ... Control