the application of advanced control to the management of type 1 diabetes graham c. goodwin...
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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
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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!