Blood Glucose Regulation

1
Blood Glucose Regulation

• BIOE 4200

2
Glucose Regulation Revisited

• 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)

eating, fasting
desired glucose
actual glucose
glucose tissues
a b cells
glucose metabol.
a b cells
3
Insulin/Glucagon Secretion
insulin (mg/sec)
error signal desired actual (mg/dl)
a b cells
glucagon (mg/sec)

• Complex chemical reaction
• Not all details have been worked out
• Need to simplify our analysis
• Suppose error gt 0 (actual lt desired), then
  glucagon will be secreted
• Suppose error lt 0 (actual gt desired), then
  insulin will be secreted

4
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
5
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)

6
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

7
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

insulin (mg/dl)
insulin (mg/sec)
hormone diffusion
glucagon (mg/dl)
glucagon (mg/sec)
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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

9
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

10
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

11
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

12
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

insulin (mg/dl)
target tissues
glucose (mg/sec)
glucagon (mg/dl)
13
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)
14
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)

15
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

16
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

17
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)
18
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

19
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

20
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

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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