The Power of Optimal Control in Biological Systems Suzanne Lenhart

Size: px

Start display at page:

Download "The Power of Optimal Control in Biological Systems Suzanne Lenhart"

Sharyl Chapman
6 years ago
Views:

1 The Power of Optimal Control in Biological Systems Suzanne Lenhart August 13, 2012

2 Outline Background, some about UT and some about me Viewpoint on Models Cardiopulmonary Resuscitation Rabies in Raccoons More current work

3 How our Math Biology program got started Started with one faculty member (Tom Hallam) in late 1970s and 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 years with faculty from many different disciplines Built interdisciplinary institute (The Institute for Environmental 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

4 Fellowship Grants! In the same summer, Cynthia Peterson was awarded the SCALE-IT and PEER grants. Mentoring Team for PEER students - Lenhart leading the team

5 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 Oak Ridge National Lab comfortable with a service role interested in outreach and REUs for many years found another great collaborator, Lou Gross, math ecologist at UT new phase of my life, NIMBioS work hard, be flexible, willing to work on many things

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

7 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

8 Mathematical Models Inputs to a system of equations are adjusted until the desired goal output is obtained. Optimal control theory is a tool to choose optimal inputs. Equations involve rates of change and interaction and movement terms among the components of the system.

9 Improving Cardiopulmonary Resuscitation Each year, more than 250,000 people die from cardiac arrest in the USA alone. Despite widespread use of cardiopulmonary resuscitation, the survival 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 rate of survival is 10-15%.

10 Goal The goal is to improve traditional CPR technique by using optimal control methods. The standard and various alternative CPR techniques such as interposed abdominal compression IAC, and Lifestick CPR have been represented in various models. We consider a model for CPR allowing chest and abdomen compression and decompression. Design optimal PATTERN of compression/decompression!

11 Model by Babbs We apply the optimal control strategy for improving resuscitation rates to a circulation model developed by Babbs. (model -discrete in time, with seven compartments) In his model, heart and blood vessels are represented as resistance-capacitive networks, pressures in the chest and in the vascular components as voltages, blood flow as electric current, and valves. Reference: Babbs, Circulation 1999.

Copyright 2005 Lippincott Williams & Wilkins.

Concepts of Altered Health States, Seventh Edition.

valve Pulmonic valve Tricuspid valve Papillary muscle Inferior vena

The atrioventricular valves are in an open position, and the semilunar

12 Copyright 2005 Lippincott Williams & Wilkins. Instructor's Resource CD-ROM to Accompany Porth's Pathophysiology: Concepts of Altered Health States, Seventh Edition. Heart Diagram Superior vena cava Aortic valve Pulmonary veins Mitral valve Pulmonic valve Tricuspid valve Papillary muscle Inferior vena cava Figure 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.

13 Diagram of Circulation Model Thoracic aorta Abdominal aorta Carotid Artery Thoracic pump Jugular vein Right heart Inferior vena cava

14 Seven Components in the Model P 1 P 2 P 3 P 4 P 5 P 6 P 7 pressure in abdominal aorta pressure in inferior vena aorta pressure in carotid artery pressure in jugular vein pressure in thoracic aorta pressure in rt. heart, superior vena cava pressure in thoracic pump and left heart.

15 Goal for this model Design compression/depression patterns for chest and abdomen pressures To increase pressure differences across thoracic aorta and right heart SPP - Systemic Prefussion Pressure

16 The chosen CPR model consists of seven difference equations, with time as the discrete underlying variable. At the step n, when time is n t, the pressure vector is denoted by: P(n) = (P 1 (n),p 2 (n),...,p 7 (n)). We assume that the initial pressure values are known, when n = 0. To make the chest pressure profiles medically reasonable, assume i.e., u i (0) = u i (N 1). u 1 = (u 1 (0),u 1 (1),...,u 1 (N 2),u 1 (0)), u 2 = (u 2 (0),u 2 (1),...,u 2 (N 2),u 2 (0)),

17 Difference Equations Model for n = 1,2,...,N 1 (in vector notation) P(1) = P(0) + T 1 (u 1 (0)) + T 2 (u 2 (0)) + tf(p(0)), (1) P(n + 1) = P(n) + T 1 (u 1 (n) u 1 (n 1)) (2) +T 2 (u 2 (n) u 2 (n 1)) + tf(p(n)), (3) T 1 (u 1 (n)) = (0,0,0,0,t p u 1 (n),t p u 1 (n),u 1 (n)), T 2 (u 2 (n)) = (u 2 (n),u 2 (n),0,0,0,0,0).

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

19 Show function F(P(n)) by some of its seven components: [ 1 1 (P 3 (n) P 4 (n)) 1 ] V (P 4 (n) P 6 (n)) R h R j c jug 1 c ao [ 1 V (P 7 (n) P 5 (n)) 1 ] (P 5 (n) P 3 (n)) R o R c + 1 R a (P 5 (n) P 1 (n)) 1 R ht V (P 5 (n) P 6 (n)) where the valve function is defined by V (s) = s if s 0 V (s) = 0 if s 0. Three valves: between compartments 4-6 AND 5-7 AND 5-6. ]

20 Goal Choose the control set U R 2N, defined as: U = {(u 1,u 2 ) u i (0) = u i (N 1) K i u i (n) L i,i = 1,2,n = 0,1,...,N 2}. We define the objective functional J(u 1,u 2 ) to be maximized N n=1 N 2 [P 5 (n) P 6 (n)] [ B 1 2 u2 1 (n) + B 2 2 u2 2 (n)] (4) n=0 Use OPTIMAL CONTROL THEORY to solve this problem.

21 Pressure Profiles 60 IAC CPR for chest 60 Lifestick CPR for chest IAC CPR for abdomen 120 Lifestick CPR for abdomen Figure: Each waveform represents one cycle.

22 Optimal Control on Chest only and Standard Profiles STD CPR: SPP= (a) Optimal Control: SPP= (b) Time (second)

23 OC Profiles 60 Chest Control (mmhg) Time (s) 120 Abdominal Control (mmhg) Time (s) Figure: The controlled chest and abdominal pressure using Lifestick

24 Concluding Remarks about CPR This procedure with RAPID compression and decompression cycles has recently been recommended by several medical groups. We can increase the pressure difference across the thoracic aorta and the right heart by about 25 percent. We received a US patent for this idea! through Oak Ridge National Lab with Protopopescu and Jung

25 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 the eastern US. Vaccine is distributed through food baits. (preventative) Medical and Economic Problem -death to humans and livestock and COSTS

26 Costs and Treatment associated with Rabies in USA 30,000 persons/year given rabies post exposure prophylaxis at a cost of $30 million Treatment - one dose of rabies immune globulin (injected near the site of the bite) and- five doses of vaccine over 28 days (injected into upper arm) Symptoms - flu-like at first, about days after exposure, later delirium, disruption of nervous system

27 Goal Develop models and numerical results to investigate other distribution patterns for vaccine baits, as it impacts the spread of rabies among raccoons. Reduce the chance of rabies spread while keeping the costs of vaccine distribution as low as possible. More Precise Goal Minimize the number of infected raccoons while taking into account limited amount of funding for the distribution of vaccine baits.

28 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

29 Movement In one time step, if the box size was the size of a home range (about 4 km 2 ), then 95 percent of the raccoons would not leave their box. The 5 percent moving out would be distributed inversely proportional to distance. But a raccoon could not move farther than their home range distance (2 km) in one time step. If the box size is smaller, then the percentage moving is changed appropriately.

30 Order of events Within a time step (about a week to 10 days): First movement: using home range estimate to get range of movement. See sum S, sum I and sum R to reflect movement. Then: some susceptibles become immune by interacting with vaccine Lastly: new infecteds from the interaction of the non-immune susceptibles and infecteds NOTE that infecteds from time step n die and do not appear in time step n + 1.

31 Susceptibles and Infecteds Equations v(k,l,t) S(k,l,t + 1) = (1 e 1 )sum S(k,l,t) v(k,l,t) + K v(k,l,t) (1 e 1 )sum S(k,l,t)sum I(k,l,t) v(k,l,t) + K β sum S(k,l,t) + sum R(k,l,t) + sum I(k,l,t),

32 Susceptibles and Infecteds Equations v(k,l,t) S(k,l,t + 1) = (1 e 1 )sum S(k,l,t) v(k,l,t) + K v(k,l,t) (1 e 1 )sum S(k,l,t)sum I(k,l,t) v(k,l,t) + K β sum S(k,l,t) + sum R(k,l,t) + sum I(k,l,t), v(k,l,t) (1 e 1 )sum S(k,l,t)sum I(k,l,t) v(k,l,t) + K I(k,l,t + 1) = β sum S(k,l,t) + sum R(k,l,t) + sum I(k,l,t).

33 Immune and Vaccine Equations v(k,l,t) R(k,l,t + 1) = sum R(k,l,t) + e 1 sum S(k,l,t), v(k,l,t) + K 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).

34 Objective Functional maximize the susceptible raccoons, minimize the infecteds and cost of distributing baits ( ) I(m,n,T) S(m,n,T) + B c(m,n,t) 2, m,n,t m,n where T is the final time and c(m,n,t) is the cost of distributing the packets at cell (m,n) and time t, B is the balancing coefficient, c is the control, t = 1,2,...,T 1. Use OPTIMAL CONTROL THEORY to solve.

35 Numerical Example Using a square grid with 25 boxes, we do the math analysis followed by the numerical solution. In each box, 8 equations are solved at each time step. Four equations for S, I, R, and V and four equations for the optimizing procedure. To get convergence to optimal bait distribution, about 100 iterations are completed.

36 Disease Starts From the Corner: Initial Distribution susceptibles t=1 infecteds t=

37 Susceptibles, with control susceptibles susceptibles t=2, B= t=4, B=0.5 susceptibles susceptibles t=3, B= t=5, B=

38 Infecteds, with control infecteds infecteds t=2, B= t=4, B=0.5 infecteds infecteds t=3, B= t=5, B=

39 Immune, with control immunes immunes t=2, B= t=4, B=0.5 immunes immunes t=3, B= t=5, B=

40 Opt. Control -number of baits in each box at each time control control t=1, B= t=3, B=0.5 control control t=2, B= t=4, B=

41 Fishery Problem: Motivation No-take marine reserves may be a part of optimal harvest strategy designed to maximize yield. Marine reserves can protect habitat and defend endangered stock from overexploitation. Marine reserves as a part of fishery management plan are controversial.

42 Work -parabolic case Can we show that when considering the maximization of revenue only, 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 harvesting strategy. We have completed the analysis for general semilinear parabolic PDE in a multidimensional domain but here we present a simpler case.

43 Parabolic Fishery Model Our fishery model in domain Q = Ω (0,T) is : u t = u + u(1 u) hu inq (5) with initial and boundary conditions: u(x,0) = u 0 (x) u(x,t) = 0 for x Ω on Ω (0,T) u represents fish population h represents the proportion to be harvested

44 Goal We seek to maximize the objective functional over h U: J(h) = T 0 Ω e δt hu dx dt (6) where U = {h L (Q) : 0 h(x,t) M 1} is class of admissible controls and e ṭ represents a discount factor with interest rate δ. (1 + δ)/2 < M This problem is linear in the control. Use OPTIMAL CONTROL again.

45 Optimal Control for Different Initial Conditions Figure: Left IC -unexploited stock, Right IC -overexploited stock

46 Conclusions about Fishery Models Spatial optimal control for harvesting problems are relevant as technology enables the enforcement of spatially structured harvest constraints. In the future, investigate more spatial heterogeneities in the dynamics and in the domain, and and more realistic boundary conditions Ding and Lenhart are investigating a fishery problem for a specific species with age structure and discrete time.

47 Modeling the Hog Population in GSMNP Team: Chuck Collins, Suzanne Lenhart, Bill Stiver, Marguerite Madden, Rene Salinas, Eric Carr, Joe Corn

48 Modeling Feral Hogs in GSMNP Cosby Gatlinburg Cades Cove Cataloochee Calderwood High Region Cherokee Fontana

49 Big Picture by Month MODEL - discrete space and time USE database with 10,000 entries of harvesting hogs in the park SUMMARY description of actions in months January: 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-low movement, possible low-to-high movement August: all hogs in high region, survival, proportional movement to low regions September, October, November, December: all hogs in low regions, survival, movement low-to-low

50 Collaborators Vladimir Protopopescu, Eunok Jung and Charles Babbs Lou Gross, Wandi Ding, Keith Langston Mike Neubert, Ta Herrera, H. R. Joshi

Optimal Control of Epidemic Models of Rabies in Raccoons

Optimal Control of Epidemic Models of Rabies in Raccoons Wandi Ding, Tim Clayton, Louis Gross, Keith Langston, Suzanne Lenhart, Scott Duke-Sylvester, Les Real Middle Tennessee State University Temple Temple