Blood Glucose Regulation BIOE 4200. Glucose Regulation Revisited input: desired blood glucose...
Transcript of Blood Glucose Regulation BIOE 4200. Glucose Regulation Revisited input: desired blood glucose...
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
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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