Blood Glucose Regulation
BIOE 4200
Glucose Regulation Revisited
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input: desired blood glucose
output: actual blood glucose
error: desired minus
measured blood glucose
disturbance: eating, fasting,
etc.
desired
glucose
a&b
cells
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controller: a and b cells
actuator: glucose storing or
releasing tissues
plant: glucose metabolism
sensor: a and b cells (again)
eating,
fasting
glucose
tissues
a&b
cells
glucose
metabol.
actual
glucose
Insulin/Glucagon Secretion
error signal =
desired – actual
(mg/dl)
a&b
cells
insulin (mg/sec)
glucagon (mg/sec)
C6 H12O6    ATP    insulin   glucagon 
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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
Insulin/Glucagon Secretion
insulin (mg/sec)
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Attempt to model process empirically from
experimental data
Data shows how hormone secretion rate changes
when constant glucose concentration is applied
 actual
~100 sec
glucagon (mg/sec)
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 error
~100 sec
Insulin/Glucagon Secretion
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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)
d
(insulin rate )  k r (insulin rate )  k f (error )
dt
 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)
d
(glucagon rate )  k r (glucagon rate )  k f (error )
dt
Insulin/Glucagon Secretion
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Can now formulate
state equations
– x1 = insulin (mg/sec)
– x2 = glucagon (mg/sec)
– u = error (mg/dl)
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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)
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Parameters kr and kf
have units 1/sec
Adjust kr and kf to get
hormone secretion rate
observed in laboratory
d
x1   k r x1  k f u
dt
d
x 2  k r x 2  k f u
dt
y1  x1
y2  x 2
Insulin/Glucagon Diffusion
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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)
glucagon (mg/sec)
hormone
diffusion
insulin (mg/dl)
glucagon (mg/dl)
Insulin/Glucagon Diffusion
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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
m m t dm dt
concentrat ion 


Q Q t dQ dt
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This assumes that the hormones are uniformly and
rapidly mixed within the entire blood supply as it
passes through
Insulin/Glucagon Diffusion
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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)
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Parameter kv is inverse
of blood flow (sec/dl)
Obtain kv from known
values
Blood flow is 8 – 10
l/min in normal adults
y1  k v u1
y2  k vu 2
Insulin/Glucagon Diffusion
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Model spread of hormones between pancreas and
target tissues with diffusion equation
d
C tissue  k d (C pancreas  C tissue)
dt
 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
Insulin/Glucagon Diffusion
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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)
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kd = diffusion coefficient
(1/sec)
Determine value of kd
from laboratory or clinic
d
x1   k d x1  k d u1
dt
d
x 2  k d x 2  k d u 2
dt
y1  x1
y2  x 2
Glucose Uptake/Release
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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)
glucagon (mg/dl)
target
tissues
glucose (mg/sec)
Glucose Uptake/Release
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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)
glucagon (mg/dl)
Liver and
Adipose
glucose (mg/sec)
insulin (mg/dl)
Kidneys
glucagon (mg/dl)
Glucose Uptake/Release
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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)
C6 H12O6  insulin  ...  glycogen / adipose  glucagon
d
(C6 H12O6 rate )  k b (C6 H12O6 rate )  k h (glucagon )  k h (insulin )
dt
Glucose Uptake/Release
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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
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Parameter kb has units
1/sec
Parameter kh has units
dl/sec
Set parameters to
match time course of
glucose release
d
x   k b x  k h u1  k h u 2
dt
yx
Glucose Uptake/Release
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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)
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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
y  k n u1  k n u 2
Glucose Diffusion
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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 (mg/sec)
glucose
diffusion
glucose (mg/dl)
Glucose Diffusion
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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)
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Parameter kv is inverse
of blood flow (sec/dl)
Obtain kv from known
values
Blood flow is 8 – 10
l/min in normal adults
y  k vu
Glucose Diffusion
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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)
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ke = diffusion coefficient
(1/sec)
Do not assume same
value for hormone
diffusion
Smaller molecule and
different direction
d
x  k e x  k e u
dt
yx
Final Notes
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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)
desired
blood
glucose
hormone
secretion
(6, 9, 11)
liver and
adipose
(15)
glucose
diffusion
(18, 19)
actual
blood
glucose
kidneys
(16)
Slide numbers with relevant state equations are indicated for each process