1

The Application of Advanced Control to the Management of Type 1 Diabetes

Graham C. GoodwinUniversity of Newcastle

Australia

Presented at IEEE MSC September 21-23, 2015, Manly, Australia

2 Motivation

Type 1 Diabetes is a major health issue.

Approximately 8% of the world’s population have (Type 1 or Type 2) diabetes, about 10% of these have Type 1.

Current treatments are intrusive and often lead to poor outcomes.

Consequences of poor blood glucose regulation include: Cardiovascular disease, coma and even death.

Diabetes is the sixth highest cause of death in Australia.

The disease is particularly debilitating for children who need to regularly take blood glucose measurements and to inject insulin at multiple instants every day.

3 hi

Hi

4 Relevance to Control Engineering

Some of the control engineering aspects associated with diabetes treatment are:

System identification and parameter estimation

Nonlinear observers with nonstandard prior knowledge

Design of sampling strategies

Design of controllers for systems having significant nonlinearities and constraints on inputs and states

Enunciation of fundamental design trade-offs

Combining feedforward and feedback action

Accounting for the high uncertainty associated with future disturbances

5

The above concepts, though familiar to control engineers, represent major challenges in the context of diabetes. For example, since patient lives are at stake, there is little or no room for error.

It will be argued that Diabetes management is a quintessential example of how control engineers can contribute to the broader field of personalized chronic disease treatment.

6 How Tough is the Problem?

Diabetes treatment is extremely difficult!

Many control groups around the world are working on this problem.

Typically “text book” control algorithms are suggested.

Results are often worse, or at best, marginally better than the results obtained by manual treatment.

My honest belief is that we need to take a radically different approach!

7 Today’s Talk

I will focus on 3 aspects that link Diabetes treatment to contemporary research in Control Theory.

a) Fundamental Limitations for diabetes treatment

b) Multiple daily injections

c) Dealing with disturbance uncertainty

Link to positive systems

Link to sparse optimization

Link to stochastic programming

8 Outline

1. Context of Research

2. Modelling

3. Control Aspects

4. Conclusions

9 Outline

1. Context of Research

2. Modelling

3. Control Aspects

4. Conclusions

10

 Implementation

The AP system will incorporate an insulin pump, a continuous glucose

monitor (sensor), and a phone sized control unit.

Infusion Set and Sensor ‘in situ’

11 Typical BGL response patient #102

12 Outline

1. Context of Research

2. Modelling

3. Control Aspects

4. Conclusions

13 Human Regulatory System

14 Mathematical Modelling

15 COMPLICATIONS: 1. Nonlinear 2. Model structure?

16 Model Fitting for Patient #200

17 Model Validation for Patient #200

18 Outline

1. Context of Research

2. Modelling

3. Control Aspects

3.1 Fundamental Limitations

3.2 Multiple Daily Injections

3.3 Dealing with Disturbance Uncertainty

4. Conclusions

19 Outline

3. Control Aspects

3.1 Fundamental Limitations

3.2 Multiple Daily Injections

3.3 Dealing with Disturbance Uncertainty

20 Well-known Fundamental Limitation Results

Important in all Control Problems

Bode Sensitivity Integral

Implications

log 0S d

21

Blood Glucose Regulation is an example of a Positive System

This leads to novel fundamental limitations.

BGL 0

Insulin Flows 0

Disturbances 0

22

Theorem (Fundamental Limitations: Blood Glucose Regulation)

Let B1 be blood glucose at time T1.

B2 be blood glucose at time T2.

Then if we aim for B1, then

C1, C2, r* are functions of the pulse responses

, .u ft th h

22 1 1B C C F r B

23

A key aspect of the result is that equality can be achieved by a very special insulin injection policy!

24 Proof of the Theorem

Let

denote impulse response due to food disturbance

denote impulse response due to bolus injection

dth

uth

25

Using the principle of superposition, the

response at time t due to a disturbance

sequence and to an input sequence

is

; 0,1,...jd j

; 0,1,...ju j

26

Apply a single pulse of food at t = 0.

Constrain lower limit of BGL response to be ymin

.

27 Key Step

For a given ymin occurring at time T2

There exists a best time to apply insulin to avoid low BGL response

2

2

1

min

1 20, 1uT k

T uT k

hy k T c

h

28 Illustrate the key idea of the proof via pictures

time

insulinT1 T2

A1 A2

insulinfood Produces Undershoot

29

time

insulinT1 T2

B1

B2

delayedinsulin

food

delay

30 Implications

It is optimal to apply a Bolus with food and any other strategy leads to a poorer trade-off.

Hence feedback from BGL to insulin unlikely to achieve good results

Go early , go hard!

31 Nonlinear Version

Because the proof uses time-domain arguments, it can be extended to the case of nonlinear models.

Recent work with Christopher Townsend and Diego Carrasco.

32 Illustration of Fundamental Limitations: Patient #101

33 Outline

3. Control Aspects

3.1 Fundamental Limitations

3.2 Multiple Daily Injections

3.3 Dealing with Disturbance Uncertainty

34

The fundamental limitation result suggests that it is optimal to inject once per meal

However, what happens if we have multiple meals?

Multiple Daily Injections

35 Question

If we allow r injections in a day, then

When, and

How Much?

36 Example

Say we divide the period 7am to 11pm into 5 minute intervals and allow 4 injections.

192*191*190*189

4*3*2*1approximately 55 million discrete options!

2 years @ 1 second per option

37 Need to be Smarter!

Use recent research on sparse optimization

38 Sparse Optimzation

Common approach is add regularization to the cost function to “promote” sparsity.

39 Ridge Regression

Lasso

Elastic Net

A combination of Ridge and Lasso

2j

j

G u u

jj

G u u

40

What is the best choice?

Depends on prior knowledge or desired constraint.

For example, if we want a solution of a given complexity, then we need to count the number of entries in u i.e,

where

0

01 of 0

0 of 0

jj

j j

j

G u u

u u

u

41

Contours

1

1 1

1

2

22

2

>1

>1

42

Advantage of regularization: It is convex

Disadvantage of regularization: It doesn’t yield a solution of specified complexity.

We will adopt an alternative approach based on converting the complexity constraint into a bilinear constraint.

43

Theorem: Equivalent formulation of cardinality constrained optimization

is non-convex due to the bilinear constraint.

: min

cardinality

r uP f u

u r

,

1

1

: min min

0

0 1

n rbi u W

N

i ii

N

ii

i

P f u

u

N r

biP

44 Recall the Question

If we allow r injections in a day, then

When, and

How Much?

45 Patient Trials: No Bolus

46 Patient Trials: One Bolus

47 Patient Trials: Two Boluses

48 Patient Trials: Three Boluses

49 Patient Trials: Four Boluses

50 Performance Improvement with Number of Boluses

51 Why isn’t this the ultimate solution?

The above based on the premise of “Ground Hog” day i.e. the food and exercise disturbances repeat

In the real world there is considerable uncertainty about food and exercise patterns

To solve we need an entirely new approach that targets the uncertainty issue!

52 Outline

3. Control Aspects

3.1 Fundamental Limitations

3.2 Multiple Daily Injections

3.3 Dealing with Disturbance Uncertainty

53 Typical Food and Exercise Scenarios

54 Typical Robust Model Predictive Control Formulation

Single Sequence Optimization

1

arg min max , , ,N

optk k k k

U D k

U y u y d

0 1

0 1

,...,

,...,

N

N

U u u

D d d

55

This solution is not satisfactory since it is too conservative.

56 Standard MPC Controller

57

Need to take disturbances more seriously!

Use Rolling Horizon Stochastic Programming (Stochastic Dynamic Programming).

POLICY optimization rather than SEQUENCE optimization

In general computationally intractable

58 Dealing with computational complexity

Divide disturbances into a finite set of options (scenarios).

Place scenarios in a disturbance tree.

Associate a control sequence with each branch of the tree.

59 Simple Illustration

Say there are two possible disturbances at t = t* and the disturbance becomes known at t* + 1.

Control sequence for disturbance 1

Control sequence for disturbance 2

However, we only know the disturbance at t* + 1.

Hence add causality constraint

1 1 10 1 1, ,..., Nu u u

2 2 20 1 1, ,..., Nu u u

1 2 for 0,...,i iu u i t

60 Cost Function

Expectation over all possible disturbance scenarios

With a separate input sequence for each disturbance scenario and subject to causality constraint.

1

, , ,N

jk k k k

j

J y u y d

J E J

61

Note that this leads to a high dimensional but (a locally) convex optimization problem.

62 Recall Food and Exercise Scenarios

63 Standard MPC Controller

64 Stochastic Dynamic Programming Results

65

Control Aspects

3.1 Fundamental Limitations

3.2 Multiple Daily Injections

3.3 Dealing with Disturbance Uncertainty

66 Outline

1. Context of Research

2. Modelling

3. Control Aspects

4. Conclusions

67 Conclusions

Diabetes is a major health issue.

Half Billion suffers in the world.

Current treatment poor.

Advanced control offers genuine

prospects for improved patient outcomes.

However, we need to go beyond simple

“text book” strategies, eg MPC

68

Our proposed strategy uses rolling horizon stochastic dynamic programming which, amongst other things, accounts for future disturbance uncertainty.

Finally, I see benefit in all medical and allied health professionals being required, as part of their training, to study Systems and Control!

69 Acknowledgements

DART Millennium grant, Uni. Newcastle, NCIG

Hunter Medical Research Institute: Dr Bruce King,

Dr Prudence Lopez, Dr Carmel Smart, Dr Megan

Paterson, Tenele Smith, Dr Kirstine Bell.

Engineering Team: Dr Adrian Medioli, Dr Diego

Carrasco, Carly Stephen.

Students: Phan Vinh Hieu, Aaron Matthews, Natalie

Gouind.

Admin Support: Jayne Disney, Amy Crawford.

70

Our Team

Others: Dr Kirstine Bell, Aaron Matthews, Chris Townsend, Vinh Hieu Phan, Tenele Smith, Natalie Govind

71

Thank you!