Blood Glucose Regulation

BIOE 4200

Glucose Regulation Revisited

input: desired blood glucose output: actual blood glucose error: desired minus

measured blood glucose disturbance: eating, fasting,

etc.

controller: and cells actuator: glucose storing or

releasing tissues plant: glucose metabolism sensor: and cells

(again)

glucose tissues

& cells

desired glucose

actual glucose

& cells

glucose metabol.

eating, fasting

Insulin/Glucagon Secretion

Complex chemical reaction Not all details have been worked out Need to simplify our analysis Suppose error > 0 (actual < desired), then glucagon

will be secreted Suppose error < 0 (actual > desired), then insulin will

be secreted

& cells

error signal =desired – actual

(mg/dl)

insulin (mg/sec)

glucagon (mg/sec)

glucagoninsulinATPOHC 6126

Insulin/Glucagon Secretion

Attempt to model process empirically from experimental data

Data shows how hormone secretion rate changes when constant glucose concentration is applied

insu

lin (

mg

/se

c)

~100 sec

actual

glu

cag

on

(mg/

sec)

~100 sec

error

Insulin/Glucagon Secretion

Rate of insulin secretion decreases with error (increases with actual blood glucose)

Rate of insulin secretion decreases as more insulin is released (chemical equilibrium drives reaction back)

Rate of glucagon secretion increases with error (decreases with actual blood glucose)

Rate of glucagon secretion decreases as more glucagon is released (chemical equilibrium again)

)error(k)rateinsulin(k)rateinsulin(dt

dfr

)error(k)rateglucagon(k)rateglucagon(dt

dfr

Insulin/Glucagon Secretion

Can now formulate state equations– x1 = insulin (mg/sec)– x2 = glucagon (mg/sec)– u = error (mg/dl)

Note dx1/dt and dx2/dt represent the change in hormone secretion rate

Output equations are written to get states– y1 = insulin (mg/sec)– y2 = glucagon (mg/sec)

Parameters kr and kf have units 1/sec

Adjust kr and kf to get hormone secretion rate observed in laboratory

22

11

f2r2

f1r1

xy

xy

ukxkxdt

d

ukxkxdt

d

Insulin/Glucagon Diffusion

We have modeled the rate of insulin and glucagon secretion at the pancreas

How does this translate to insulin and glucagon concentration at target tissues?

First calculate concentration of insulin and glucagon in pancreas given hormone secretion rates

Then use diffusion equation to estimate hormone concentration in target tissues

hormone diffusion

insulin (mg/dl)

glucagon (mg/dl)

insulin (mg/sec)

glucagon (mg/sec)

Insulin/Glucagon Diffusion

Hormone is added to the bloodstream at a rate of dm/dt (mg/sec)

Blood is flowing through the body at a rate of dQ/dt (dl/sec)

The concentration of hormone (mg/dl) is

This assumes that the hormones are uniformly and rapidly mixed within the entire blood supply as it passes through

dtdQ

dtdm

tQ

tm

Q

mionconcentrat

Insulin/Glucagon Diffusion

This is a simple gain process (no states)

Input u1 = insulin secretion rate (mg/sec)

Input u2 = glucagon secretion rate (mg/sec)

Output y1 = insulin concentration in pancreatic blood (mg/dl)

Output y2 = glucagon concentration in pancreatic blood (mg/dl)

Parameter kv is inverse of blood flow (sec/dl)

Obtain kv from known values

Blood flow is 8 – 10 l/min in normal adults

2v2

1v1

uky

uky

Insulin/Glucagon Diffusion

Model spread of hormones between pancreas and target tissues with diffusion equation

Assumes diffusion is uniform across entire volume of blood between pancreas and target tissues

Assumes all target tissues in same location This models diffusion across static volume and

neglects spread due to blood flow The diffusion coefficient can be increased to partially

account for effects of blood flow

)CC(kCdt

dtissuepancreasdtissue

Insulin/Glucagon Diffusion

Input u1 = insulin concentration in pancreatic blood (mg/dl)

Input u2 = glucagon concentration in pancreatic blood (mg/dl)

State x1 and output y1 = insulin concentration in target tissues (mg/dl)

State x2 and output y2 = glucagon concentration in target tissues (mg/dl)

kd = diffusion coefficient (1/sec)

Determine value of kd from laboratory or clinic

22

11

2d2d2

1d1d1

xy

xy

ukxkxdt

d

ukxkxdt

d

Glucose Uptake/Release

Target tissues include kidney, liver, adipose tissue Can model this as separate processes in parallel Each process has two inputs - insulin and glucagon

concentration in mg/dl Each process has single output for glucose release

rate (mg/sec) Negative output value indicates glucose uptake or

excretion

target tissues

glucose (mg/sec)insulin (mg/dl)

glucagon (mg/dl)

Glucose Uptake/Release

Liver and adipose tissues incorporate glucose into larger molecules (glycogen and fat) as storage

Kidney controls flow of glucose between blood and urine

Consider liver and adipose tissues together Consider kidney separately

Liver and Adipose

glucose (mg/sec)

insulin (mg/dl)

glucagon (mg/dl)

Kidneysinsulin (mg/dl)

glucagon (mg/dl)

Glucose Uptake/Release

Similar to model for secretion of insulin and glucagon driven by glucose

Complex chemical reaction that we will simplify Rate of glucose secretion decreases with insulin Rate of glucose secretion increases with glucagon Rate of glucose secretion decreases as more

glucose is released (chemical equilibrium drives reaction back)

)insulin(k)glucagon(k)rateOHC(k)rateOHC(dt

d

glucagonadipose/glycogen...insulinOHC

hh6126b6126

6126

Glucose Uptake/Release

Input u1 = insulin concentration at target tissues (mg/dl)

Input u2 = glucagon concentration at target tissues (mg/dl)

State x and output y = glucose release rate (mg/sec)

Note dx/dt represents the change in glucose secretion rate

Parameter kb has units 1/sec

Parameter kh has units dl/sec

Set parameters to match time course of glucose release

xy

ukukxkxdt

d2h1hb

Glucose Uptake/Release

Model kidney function as a simple gain process (no states)

Assumes response of glucose uptake or excretion rate changes rapidly

Uptake increases with glucagon, excretion increases with insulin

Output y = glucose release rate (mg/sec)

Input u1 = insulin concentration at target tissues (mg/dl)

Input u2 = glucagon concentration at target tissues (mg/dl)

Parameter kn has units of dl/sec

2n1n ukuky

Glucose Diffusion

Must translate glucose release/uptake from target tissues into blood glucose concentration

Blood glucose concentration will be measured at pancreas, so this will serve as convenient output

Like we did earlier, calculate concentration of glucose at target tissues given glucose secretion rates

Then use diffusion equation to estimate blood glucose concentration at pancreas

glucose diffusion glucose (mg/dl)glucose (mg/sec)

Glucose Diffusion

First convert from glucose release rate to concentration at target tissues

Input u = glucose secretion rate (mg/sec)

Output y = glucose concentration in blood around target tissues (mg/dl)

Parameter kv is inverse of blood flow (sec/dl)

Obtain kv from known values

Blood flow is 8 – 10 l/min in normal adults

uky v

Glucose Diffusion

Then use diffusion equation to model spread of glucose from target tissues back to pancreas

Input u = glucose concentration in target tissues (mg/dl)

State x and output y = glucose concentration in pancreas (mg/dl)

ke = diffusion coefficient (1/sec)

Do not assume same value for hormone diffusion

Smaller molecule and different direction

xy

ukxkxdt

dee

Final Notes

We are now ready to assemble the individual processes and simulate the system in MATLAB

Desired blood glucose is system input (constant) Disturbance input is glucose intake and metabolism Disturbance input will generally be negative to

indicate basal glucose metabolism with positive periods to indicate glucose intake

Model feedback as unity gain process Assumes measured glucose equals glucose

concentration in pancreas

Model Summary

desired blood

glucose

actual blood

glucose

hormone secretion(6, 9, 11)

glucose diffusion(18, 19)

glucose intake and metabolism (20)

liver and adipose

(15)

kidneys (16)

Slide numbers with relevant state equations are indicated for each process