Closed-loop Regulation of Blood Glucose based on Subcutaneous Measurements

B. Wayne Bequette

• Background• Model Predictive Control (MPC), State Estimation

• Model and Controller Design• Unique Challenges of Diabetes

• Simulation Results• Focus on Meal Disturbance Rejection• Sensor Degradation and Compensation

Automated Feedback Control

controller

Sensor (Therasense)

pump patient

glucose setpoint

Model Predictive Control

Find current and future insulin infusion rates that best meet a desired future glucose trajectory. Implement first “move.”

Correct for model mismatch (estimate states), then perform new optimization.

tkcurrent step

setpoint (desired glucose)y

actual glucose (past)

PPredictionHorizon

past controlmoves

u max

min

MControl Horizon

past future

model prediction

tk+1current step

setpoint (glucose)y

PPredictionHorizon

past controlmoves

u max

min

MControl Horizon

model predictionfrom k

new model prediction

insulin

insulin

MPC Issues

Type of Model Linear differential equations

Model Update Additive “correction”? Explicit disturbance (meal) or parameter estimation?

Error Due to Disturbance or Noise? Future Prediction?

Classical MPC - assume constant for future

Sensors & Estimation Measure subcutaneous, control blood glucose

MPC Literature Review

Dogs, Venous Blood, Glucose+Insulin Delivery Kan et al. (2000)

Simulation, s.c. Sensor + Delivery, ANN Trajanoski and Wach (1998)

Simulation, i.v.-i.v., EFK-based MPC Parker et al. (1999)

Simulation, i.v.-s.c., EKF-based MPC Lynch and Bequette (2002)

Simulation, s.c.-s.c., EKF-based MPC Lynch (2002)

Estimation - Basic Idea

Blood glucose

Measured subcutaneous glucose

Sensor

Insulin infusionrate

Meal disturbance

IDDMPatient

+_

PatientModel

ModelFeedback

SensorModel

Predicted subcutaneousglucose

Estimates: Blood glucose Subcutaneous glucose Glucose meal disturbance

Estimator

Optimal Estimation - Kalman Filter

Measurement noise vs. process noise (disturbances)

If little measurement noise Trust measurement more than model

If much measurement noise Trust model more than measurement

Estimate unmeasured states Blood glucose based on s.c. measurement, for

example

Simulation Study Simulated Type I Diabetic

Minimal Model - Bergman (3-state) Lehman and Deutsch (1992) Meal Model

Gastric

emptying

Absorption into circulation

Constant Disturbance Assumption (Classical) Additive step

output

Additive step

input

Glucose

conc.

Insulin

infusion

Improved Meal Effect Prediction (ramp)

Simulation Study Using s.c. Sensor

Simulated Type I Diabetic 19 Differential Equations (Sorenson, 1985) - Extended

Model for Estimator/Controller Modified Bergman “minimal model” Parameters fit to Sorenson response Augmented equation for meal disturbance

Simulation Results - s.c. Sensor Degradation

50% sensor sensitivity decrease over 3 days

Motivates use of additional blood capillary measurement for s.c. sensor calibration

Simulation results: Sensor compensation

• Sensor degradation (50% over 3 days)

• Sensitivity estimate

5% s.c. noise (s.d. = 3.8 mg/dl) 2% capillary blood noise (s.d. =1.6 mg/dl)

Summary

Kalman Filter (estimation)-based MPC Disturbance (meal) estimation

Improved disturbance prediction

Low-order linear model, high-order patient State estimation: measure s.c., estimate blood

glucose Estimate sensor sensitivity with capillary blood

measurement Dual rate Kalman Filter

Future Multiple Models

Acknowledgment

Sandra Lynch MS Thesis, RPI (2002)

Kalman Filter w/Augmented States

xk+1

dk+1

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

xk+1a

1 2 3 =

Φ Γd

0 1

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Φ a1 2 4 3 4

xk

dk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

xka

{+

Γ

0

⎡

⎣ ⎢

⎤

⎦ ⎥

Γ a{

uk +0

1

⎡

⎣ ⎢

⎤

⎦ ⎥

Γ a,w{

wk

yk = C 0[ ]Ca

1 2 3 xk

dk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

xka

{+vk

Predictor-corrector equations:

ˆ x k|k−1a =Φa ˆ x k−1|k−1

a +Γ auk−1

ˆ x k|ka =ˆ x k|k−1

a +Lk yk −Ca ˆ x k|k−1a

( )

Kalman gain

Augmented state (includes meal disturbance)

Measured s.c. glucose

Insulin infusion

Aug. state

estimate