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## Blood Glucose Regulation

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### Blood Glucose Regulation

BIOE 4200

input: desired blood glucose

output: actual blood glucose

error: desired minus measured blood glucose

disturbance: eating, fasting, etc.

controller: a and b cells

actuator: glucose storing or releasing tissues

plant: glucose metabolism

sensor: a and b cells (again)

Glucose Regulation Revisitedeating, fasting

desired glucose

actual glucose

a & b cells

glucose tissues

glucose metabol.

a & b cells

Insulin/Glucagon Secretion

insulin (mg/sec)

error signal =

desired – actual

(mg/dl)

a & b cells

- 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

glucagon (mg/sec)

Insulin/Glucagon Secretion

- Attempt to model process empirically from experimental data
- Data shows how hormone secretion rate changes when constant glucose concentration is applied

- actual

- error

insulin (mg/sec)

glucagon (mg/sec)

~100 sec

~100 sec

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)

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

Insulin/Glucagon SecretionInsulin/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

insulin (mg/sec)

insulin (mg/dl)

hormone diffusion

glucagon (mg/sec)

glucagon (mg/dl)

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

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

Insulin/Glucagon DiffusionInsulin/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

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

Insulin/Glucagon DiffusionGlucose 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

insulin (mg/dl)

target tissues

glucose (mg/sec)

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

insulin (mg/dl)

Liver and Adipose

glucagon (mg/dl)

glucose (mg/sec)

insulin (mg/dl)

Kidneys

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)

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

Glucose Uptake/ReleaseModel 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

Glucose Uptake/ReleaseGlucose 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/sec)

glucose (mg/dl)

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

Glucose DiffusionThen 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

Glucose DiffusionFinal 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

glucose intake and metabolism (20)

hormone secretion

(6, 9, 11)

liver and adipose

(15)

glucose diffusion

(18, 19)

desired blood glucose

actual blood glucose

kidneys (16)

Slide numbers with relevant state equations are indicated for each process

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