5. [Total: 8 pts] Muscle stretch reflex The figure below shows a simplified model of muscle stretch reflex in an experimental anesthetized animal. The model describes how the muscle stretch reflex regulates the muscle length. All variables shown have been linearized and represent fluctuations around its mean value under normal resting condition. Z represents external neural frequency stimulating the spinal cord. fa represents the total change in afferent neural frequency. The spinal cord, represented by the gain Gc, converts the incoming neural input to the neural output fe- The changes in the efferent neural frequency, fe, are relayed through motor neurons to the muscle. The muscle converts the efferent neural frequency to changes in muscle length L. The gain of this conversion is-G. Note that the gain is negative because increases in efferent neural frequency lead to decreases in muscle length. The change in the muscle length is sensed by the muscle spindle. The muscle spindle relays the neural output, fs, to the spinal cord through sensory neurons. This process is represented by the gain Gg. fa fe Gc Z -GM Gs fs L a) [4 pts] Suppose that animal A is injected with a drug that disrupts neurotransmission, blocking the sensory neurons. Under such condition, is the system operating as a negative feedback system? Explain your answer. Please also state which gain(s) is affected by the sensory nerve blockade. b) [4 pts] Under which condition between normal and sensory nerve blockade would result in stronger muscle contraction for the same level of the external stimulation Z? Explain your answer.

2. [15 pts] Assume the metabolic hyperbola for CO₂ given by the following equation: 863 - Vcoz V₁ PACO2 Pico2+ where the normal steady-state CO₂ production is 235 mL/min and the inspired CO₂ concentration (or volumetric fraction) is 0. Also, assume a dead-space ventilation rate of 1 L/min. Also, suppose the steady-state ventilatory response to CO₂ is given by the following equation: 32 Vc = Drive external +(1.46 + (Paco2-37) and Vc 20 Pa02-38.6) Note that the above equation is slightly modified from what we derived in class in that a constant term, Drive external is added to the equation. This drive represents an additional source of control that affects ventilatory response e.g., sleep-wake state, alertness, etc. At normal state (resting and awake), let Drive external = 0. a) [5 pts] Use MATLAB to plot the metabolic hyperbola for CO₂ and the ventilatory response to CO₂ under normal condition. Show both graphs on the same figure (see hold function). Label both axes with appropriate units. Also indicate which graph represents the me the metabolic hyperbola for CO₂ and which graph represents the ventilatory response to CO₂ (see legend function). The plot should show the range of ventilation from 0-20 L/min and the range of PACO₂ from 30-60 mmHg (see xlim and ylim functions). Determine the normal steady-state values of ventilation and Pacoz from the plot (e.g., click on the plot where you want to obtain the x-y coordinates). Note: For the metabolic hyperbola for CO₂, please plot V against PACO₂ i.e., you would take the dead-space ventilation into consideration. b) [4 pts] The onset of sleep causes the external drive to breathe to drop by -5 L/min. State which parameter changes and to what value. Show the graphs representing this situation and report the steady-state values of ventilation and Pacoz. c) [4 pts] Follows from b), how would inhalation of a gas mixture containing 3% CO₂ in air (i.e., volumetric fraction of 0.03) during sleep affect the steady-state values of ventilation and Pacoz? State which parameter changes and to what value. Show the graphs representing this situation and report the steady-state values of ventilation and Pacoz. You may assume that the subject is at sea level. d) [2pt] Copy and paste the MATLAB code you used in this question (all parts). You may take a picture of your MATLAB script, but please make sure that the code is clearly readable from the image.

1. [5 pts] Construct the glucose-insulin regulation model in Simulink as outlined in Fig. 3. The parameter values shown in Fig. 3 are the values of a normal adult. Set the Simulink model to run simulations for a total duration of 24 hours. Although Simulink generally assumes time to be measured in seconds, the units of the model parameters give time in hours-so, you can treat the durations you enter into Simulink as hours (instead of seconds). The initial conditions of normal blood glucose and insulin concentrations are 0.81 mg/mL and 0.055 mU/mL, respectively. These initial values of blood glucose and insulin concentrations should be set in the respective Integrator blocks (shown as in Simulink). To set the initial values in the integrator block, double click the block to open the Block Parameters window. Enter the initial value (e.g., 0.81 for glucose) in the Initial Condition field, and click OK. Run a simulation for a normal adult (Subject N) over 24 hours without any external glucose input (i.e., U(t) = 0). Show the graph of glucose and insulin concentrations as a function of time as 2-by-1 subplots i.e., the glucose concentration is the top subplot and the insulin concentration is the bottom subplot. Label all axes. You may plot using the Scope in Simulink or plot in MATLAB using the simulation results that are sent to MATLAB Workspace. Save this Simulink model as "glucose_N.slx".

3. [5 pts] Simulate the oral glucose tolerance test (OGTT) in N and D1 subjects. During the OGTT, a large amount of glucose is infused rapidly and the corresponding blood glucose and insulin trajectories are tracked over a period of up to 5 hours. Assume the total amount of glucose infused is 25,000 mg and the infusion is carried out in the form of a bolus lasting 15 minutes (0.25 h), starting from t = 1 h. Plot the trajectories of blood glucose and insulin predicted by both models. To aid comparison across the 2 cases (N and D1), plot the glucose responses of both N and D1 cases using the same time- and glucose concentration axes i.e., there are two graphs in the top subplot for glucose concentration. Similarly, plot the insulin responses of both cases on the same axes. Label all axes and add a legend indicating which graph belongs to which case. You may plot these simulation results in MATLAB using the simulation results that are sent to the MATLAB Workspace. Report the values of blood glucose and insulin from your simulations: just before you impose the glucose challenge, and at times 1, 2 and 3 hours following the start of the challenge. Glucose concentration (mg/mL) Insulin concentration (mU/mL) D1 N D1 N Time Before OGTT Time = 1 hr Time = 2 hr Time = 3 hr

Submit your solutions on Gradescope. 1. [Total: 16 pts] The figure below shows a schematic control block diagram of the control of respiration, with the respiratory controller representing the respiratory chemoreflexes, brain respiratory neural centers and the respiratory muscles, and the lungs representing the CO₂ exchange in the lungs. In amyotrophic lateral sclerosis (ALS, also known as Lou Gehrig's disease), motor neuron degeneration results in weakness and eventual paralysis of the respiratory muscles. Note that the regulation of ventilation model here is slightly different (more simplified) from the one we discussed in class. V/₁ Respiratory controller Lungs Pacoz a) [2 pts] Suppose that, in a particular ALS patient, the (steady-state) respiratory controller equation (which characterizes the chemoreflex response) is given by the following equation: V₁ = Paco2 - 37 Where V represents the alveolar ventilation (in L/min), and Pacoz is the partial pressure for CO₂ in the arterial blood (in units of mmHg). Draw (as accurately as possible on the provided graph) the line representing the steady-state controller response to Paco2- Note that we are neglecting the effects of O₂ here. Label this line as "(a)". b) [2 pts] Assume that gas exchange in this patient is normal and can be characterized by the following plant equation: Paco2 = 200/V Draw (as accurately as possible on the provided graph) the relationship between Pacoz and V. Label the curve as "(b)"./nVdotA (L/min) 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 40 41 (mm Hg) c) [2 pts] Estimate the steady-state operating values of Pacoz and V₁ in this patient. d) [4 pts] In a normal healthy person, the controller response is given by V₁ = 2(Paco2-37). Draw this controller response on the graph below. Estimate the steady-state operating values of Paco2 and V in the normal subjects. Label the curve as "(d)". 35 36 37 38 39 P aCO2 42 43 44 45 e) [4 pts] A pressure support ventilator can be used to provide assistance to the ALS patient - such a ventilator would produce an increase of x L/min on top of the subject's own natural ventilatory output, independent of the Pacoz level. However, the ventilator has to be triggered by a minimal inspiratory effort by the patient himself. If the patient does not generate any effort, the ventilator does not provide any ventilatory assistance. Draw on the graph below the total controller response (patient+ ventilator) required to restore the steady-state V and Pacoz values to the levels seen in the normal subject. Label the total controller response as "(e)". f) [2 pts] What is the value of x (L/min)?

6. [Total: 9pts] Chemical regulation of ventilation An investigator wants to investigate the effect of exercise on ventilation and Pacoz in a healthy volunteer. Assume that the baseline ventilatory response to CO₂ and the subject's metabolic hyperbola prior to participating in the study yields an operating point labeled "BL" in the figure below. In order to measure ventilation during the exercise study, when the volunteer has to be on a treadmill, she has to wear a face mask connected to a long breathing tube which is then connected to a device used to measure ventilation. This device does not provide any ventilation support, but it can provide different levels of inhaled oxygen. The graph below shows metabolic hyperbolas and ventilatory response to CO₂ under different conditions. в с VdotE or VdotC (l/min) BL A Measures ventilation PACO₂ or P or Pacoz (mmHg) O E F a) [3 pts] First, the subject's ventilation and Pacoz are measured while she is standing at rest on the treadmill and she breathes in air (21% O₂). Considering the long tube in this experimental setup, which curve (A, B, C, D, E or F) best represents this situation? Briefly explain your selection. Label on the figure "a", the new steady state equilibrium values of Pacoz and ventilation for the volunteer at rest.

2. [5 pts] Customize the model in Q.1 for a type-1 diabetic subject (Subject D1) by altering B, the model parameter that controls the production rate of insulin by the pancreas. Reduce B to 20% of the normal subject's value. Establish the equilibrium conditions for type-1 diabetic subject by running the simulation over 24 hours without any external glucose input (i.e., U (t) = 0). Show the graph of glucose and insulin concentrations as a function of time (similar to Q.1). Report the final values of blood glucose and insulin concentrations that appear at the end of your simulations. Assuming that both glucose and insulin concentrations have settled into a steady state, these final values can now be used as the initial values in the respective integrators for subsequent simulations of a type-1 diabetic subject. Save this Simulink model as "glucose_D1.slx".

6. [10 pts] Here, we assume that D1 uses a programmable insulin infusion pump, instead of the first one that was only able to generate a constant infusion rate. This new pump is capable of infusing insulin in rectangular pulses of different durations and insulin amounts at various times during the 24-hour period. Imagine you are subject D1, program your insulin pump to achieve the following goals: a) Bring your mean blood glucose level over 24 hours as close as possible to the corresponding level in Subject N; b) Reduce the fluctuation in blood glucose level (as measured by the standard deviation) to as low as possible over the 24 hours; and c) Minimize the total amount of insulin (in mU) infused over 24 hours, since insulin is an expensive drug. Explore a variety of insulin infusion patterns. After some trial and error, try to learn from the results what you think are the best strategies. Select the 3 infusion patterns that best satisfy the above criteria. For each pattern that you design, plot the trajectories of: i) glucose infusion, ii) insulin infusion, iii) blood glucose concentration and iv) blood insulin concentration over 24 hours of simulation. For each of these three selected cases, quantify the metrics that correspond to the 3 criteria (a, b and c) described above. Be sure you specify the formulas employed to arrive at each of the 3 criteria. Case N (from Q.4) I 24-h mean glucose conc. (mg/mL) 24-hr glucose conc. fluctuations (mg/mL) Total insulin infused over 24 hours (mU) Briefly justify why you chose these 3 patterns to be your "best 3", and what considerations you used to arrive at each pattern. Save the Simulink model, the best of the 3 cases, as glucose_D1_pump.slx.

4. [5 pts] Modify glucose_N.slx and glucose_D1.slx to simulate glucose-insulin dynamics under more realistic conditions where each of these subjects eats 3 meals over 24 hours: - Breakfast starts at hour = 1, duration=30 minutes, glucose infusion rate = 15,000 mg/hr - Lunch starts at hour = 5, duration=45 minutes, glucose infusion rate = 10,000 mg/hr - Dinner starts at hour = 12, duration=1 hour, glucose infusion rate = 12,000 mg/hr You may assume that the glucose rate per meal is constant within the given duration (i.e., the external glucose infusion rate takes the form of rectangular pulses with different durations). Note that in this question, we are not incorporating any oral glucose tolerance test (as in Q.3). In this question, you would create 3 subplots: i) Plot the trajectories of the external glucose infusions (i.e., the meals). ii) Plot the trajectories of blood glucose predicted by both models (normal and type-1 diabetic) over 24 hours on the same axes. iii) Plot the trajectories of blood insulin predicted by both models (normal and type-1 diabetic) over 24 hours on the same axes. Label all axes. For the glucose and insulin subplots, add a legend indicating which graph belongs to which case (normal or type-1 diabetic). From the simulation results, calculate and report the average and standard deviation of blood glucose and insulin concentrations over 24-hour period. Glucose concentration (mg/ml) N D1 Average SD Insulin concentration (mU/mL) N D1

3. [Total: 19 pts] Blood glucose and insulin regulation The equations that characterize the steady state regulation of glucose and insulin can be derived by taking into account the main factors that affect the appearance and disappearance of each of these substances in the body. The mass balance for blood glucose (x) is given by: Q₁ = λx +vxy (Eqn. 1) where Q₁ represents the flow-rate (assume to be constant) at which glucose enters the blood through absorption from the gastrointestinal tract or through production from the liver, xx represents the rate at which the body tissues utilize glucose without help from insulin, and vxy is the rate at which glucose is metabolized with facilitation from insulin. The blood concentration of insulin is represented by y, and λ and v are constant parameters. Thus, Eqn. 1 represents the "glucose response to insulin". By applying similar mass-balance considerations to the production of insulin from the pancreas and the subsequent destruction of insulin, we can derive the following expression for the "insulin response to glucose": (Egn. 2a) (Eqn. 2b) where and are constant parameters. Insulin is not produced when glucose is lower than the threshold parameter (Eqn.2a). Note that the insulin production and destruction rates are combined into one parameter, (zeta). By using empirically derived values for the model parameters (Q₁, A, v, 3, 4) in Egns. 1 and 2a/b, these equations allow us to predict the blood concentrations of glucose (x) and insulin (y) under various conditions. The figure below displays the plots corresponding to these responses in normal and diabetic subjects. Point N is the operating point of a normal subject. B с D Insulin concentration (mu.mL-¹) 0.2 0.15 y = 0, x≤0 y = {(x-4), x> 0.1 0.05 0 0.2 0.4 0.6 0.8 N 1 1.2 1.4 1.6 1.8 2/na) [2 pts] Estimate from the figure the operating values of glucose and insulin in the blood of a normal subject. b) [3 pts] Suppose this person eats junk food everyday and does not exercise, and after many years, develops "insulin resistance" (i.e., the body requires more and more insulin to metabolize the same amount of glucose). Assuming his pancreas still functions normally. Which parameter(s) need to change to simulate this condition? Determine his new operating values of glucose and insulin. Label the new operating point "b". Briefly explain how you arrived at your answers. c) [3 pts] After many years of untreated insulin resistance, this subject's pancreas can now operate at half of its full capacity. Which parameter(s) need to change to simulate this condition? Determine his new operating values of glucose and insulin. Label the new operating point "c". Briefly explain how you arrived at your answers./nd) [5 pts] The subject does nothing to change his lifestyle and the pancreas eventually fails. Suppose that his pancreas is removed and replaced by a machine that continuously infuses insulin into his body independent of the glucose level. The machine is adjusted such that it infuses the appropriate amount of insulin, restoring the plasma glucose concentration to the normal level. Sketch as accurately as possible in the provided figure to reflect the described condition. Label the curve(s) "d". Briefly explain how you arrived at your answer. e) [3 pts] From the provided figure, is it possible for you to estimate the parameter of a normal healthy person? If so, provide your best estimate of pp. Briefly explain how you arrived at your answer. f) [3 pts] From the provided figure, is it possible for you to estimate the parameter < (Eqn. 2b) of a normal healthy person? If so, provide your best estimate of and from which graph. Briefly explain how you arrived at your answer.