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 desired), then insulin will be secreted  &  cells error signal = desired – actual (mg/dl) insulin (mg/sec) 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 insulin (mg/sec) ~100 sec  actual glucagon (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)

Insulin/Glucagon Secretion Can now formulate state equations – x 1 = insulin (mg/sec) – x 2 = glucagon (mg/sec) – u = error (mg/dl) Note dx 1 /dt and dx 2 /dt represent the change in hormone secretion rate Output equations are written to get states – y 1 = insulin (mg/sec) – y 2 = glucagon (mg/sec) Parameters k r and k f have units 1/sec Adjust k r and k f to get hormone secretion rate observed in laboratory

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

Insulin/Glucagon Diffusion This is a simple gain process (no states) Input u 1 = insulin secretion rate (mg/sec) Input u 2 = glucagon secretion rate (mg/sec) Output y 1 = insulin concentration in pancreatic blood (mg/dl) Output y 2 = glucagon concentration in pancreatic blood (mg/dl) Parameter k v is inverse of blood flow (sec/dl) Obtain k v from known values Blood flow is 8 – 10 l/min in normal adults

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

Insulin/Glucagon Diffusion Input u 1 = insulin concentration in pancreatic blood (mg/dl) Input u 2 = glucagon concentration in pancreatic blood (mg/dl) State x 1 and output y 1 = insulin concentration in target tissues (mg/dl) State x 2 and output y 2 = glucagon concentration in target tissues (mg/dl) k d = diffusion coefficient (1/sec) Determine value of k d from laboratory or clinic

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) Kidneys insulin (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)

Glucose Uptake/Release Input u 1 = insulin concentration at target tissues (mg/dl) Input u 2 = 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 k b has units 1/sec Parameter k h has units dl/sec Set parameters to match time course of glucose release

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 u 1 = insulin concentration at target tissues (mg/dl) Input u 2 = glucagon concentration at target tissues (mg/dl) Parameter k n has units of dl/sec

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 k v is inverse of blood flow (sec/dl) Obtain k v from known values Blood flow is 8 – 10 l/min in normal adults

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) k e = diffusion coefficient (1/sec) Do not assume same value for hormone diffusion Smaller molecule and different direction

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