Introductory Modeling Exercises with the Shivering Model

Six exercises are provided for students with an introductory interest in physiology. These exercises may be completed from the information on the website and in Modeling the Environment.

1. Build and Verify

Build the second model and verify that it performs as indicated in Figure 9.
The model equations are listed below:

energy_in_core(t) = energy_in_core(t - dt) + (basal_heat_production +exercise_heat_production + shivering_heat_production - lung_heat_loss - eat_transfer) * dt INIT energy_in_core = 2146 basal_heat_production = GRAPH(core_temp) (37.0, 70.0), (38.0, 77.0), (39.0, 84.0), (40.0, 91.0), (41.0, 98.0), (42.0, 105) exercise_heat_production = 22 shivering_heat_production = std_heat_from_shivering*first_factor*second_factor lung_heat_loss = 6 heat_transfer = core_gradient/core_to_skin_resistance energy_in_skin(t) = energy_in_skin(t - dt) + (heat_transfer - heat_loss_from_insensible_perspiration - skin_heat_loss_from_radiation) * dt INIT energy_in_skin = 1831 heat_transfer = core_gradient/core_to_skin_resistance heat_loss_from_insensible_perspiration = 6 skin_heat_loss_from_radiation = skin_gradient/skin_to_air_resistance ambient_temp = if (time <12) then start_temp else start_temp-temp_drop Ambient_Temp_in_F = 32+1.8*ambient_temp Body_Temp_in_F = 32+core_temp*1.8 core_capcitance = 58 core_gradient = core_temp-skin_temp core_temp = energy_in_core/core_capcitance core_to_skin_resistance = .0617 critical_skin_temp_for_extra_shivering = 21 first_factor = shivering_initiation_mult_from_cooler_core normalized_skin_temp_difference = skin_temp_difference/normalizing_factor_K second_factor = shivering_mult_from_normalized_temp_diff skin_capcitance = 58 skin_gradient = skin_temp-ambient_temp skin_temp = energy_in_skin/skin_capcitance skin_temp_difference = max(0,critical_skin_temp_for_extra_shivering-skin_temp) Skin_Temp_in_F = 32+1.8*skin_temp skin_to_air_resistance = .05 start_temp = 27.5 std_heat_from_shivering = 252 temp_drop = 5 normalizing_factor_K = GRAPH(core_temp) (33.0, 47.0), (33.5, 42.0), (34.0, 36.0), (34.5, 31.0), (35.0, 25.0), (35.5, 19.0), (36.0, 14.0), (36.5, 8.00), (37.0, 1.00) shivering_initiation_mult_from_cooler_core = GRAPH(core_temp) (34.8, 1.00), (35.0, 0.99), (35.2, 0.98), (35.5, 0.97), (35.8, 0.95), (36.0, 0.91), (36.2, 0.83), (36.5, 0.69), (36.8, 0.45), (37.0, 0.00) shivering_mult_from_normalized_temp_diff = GRAPH(normalized_skin_temp_difference) (-1.00, 4.00), (-0.8, 3.20), (-0.6, 2.60), (-0.4, 2.00), (-0.2, 1.40), (0, 1.00)

2. Find the "Span of Control" Under Cold Conditions

Repeat the cold step tests with larger and larger values of the temperature drop. Does the model show homeostatic control of the core temperature with ambient temperatures as low as 5 degrees? Does the homeostatic control fail with ambient temperatures below 5 degrees?

3. Sensitivity of the "Span of Control"

One of the input parameters in Riggs' model is the "std heat from shivering." He recommends a value of 252 kcal/hr. Suppose the uncertainty on this input is plus-or-minus 20%. How does the uncertainty in this single parameter change your estimate of the span of control from the previous exercise?

4. Experiment with a Hot Step Test

Rerun the model with a 5 degree increase in the ambient temperature in the 12th month of the simulation. Does the model provide homeostatic control under hot conditions?

5 What if there is a delay in the Onset of Shivering?

The second model assumes that shivering will commence immediately whenever the core temperature falls below 37 degrees C. Figure 9 shows that that the body would find the new equilibrium without any oscillations, and shivering would remain constant over time. Milsum (1966, p. 76) describes work by Benzinger (1963) in which "sudden changes in skin temperature, either up or down, produced transient overshoots in the metabolic heat rates before these settled down to suitable new steady states." These observations might lead as to ask whether the shivering model could be expanded to include the possibility for overshoot and oscillations.

Expand the shivering model to represent a delay in the physiological mechanism which triggers shivering. (Perhaps there is a time lag before the hypothalamus sends the signal down the spinal cord to actuate shivering.) Follow the approach explained in Chapter 17. Define a new variable called the lagged core temperature using Stella's SMTH3 function. Set the lag time to 0.5 hours and assume that the lagged value of the core temperature triggers shivering as shown in Figure 6. Simulate the new model to verify the results shown in Figure 10.

Figure 10. Simulation of hypothetical oscillations in shivering.

6. Impact of Shorter Delay in the "Lagged Core Temperature"
After verifying the hypothetical results in Figure 10, experiment with lag times much shorter than the 0.5 hours in the previous expercise. (Make sure to check that your value of DT is sufficiently small to achieve accurate, numerical results.) Figure 10 shows around 6 pulses of shivering in a 12 hour span --- there would be one pulse every half hour. Experiment with the model to learn if the pulses will appear more and more frequently.

7. Discussion of Hypothetical Results in Figure 10

The results in Figure 10 are strictly hypothetical. They show what would happen if the signal down the spinal column to actuate shivering were delayed. With shorter and shorter values assigned to the delay, the pulses of shivering heat production would appear more and more frequently. Looking at this pattern, you might suspect that the we have explained through computer simulation the rhythmtic pattern of shivering. But the reason for the rhythmtic pattern we feel when we are shivering arises from different mechanisms, as described by Arthur Guyton 's (1971, p. 839) textbook of Mecical Physiology as follows: .

Hypothalamic Stimulation of Shivering (by Guyton)

Located in the dorsomedial portion of the posterior hypothalamus near the wall of the third ventricle is an area called the primary motor center for shivering. This area is normally inhibited by signals from the preoptic thermostatic area. However, when the preoptic thermostat becomes cooled and thereby loses its excitation, the normal inhibition of the primary motor center no longer exists. The self-excitation property of this center then causes it to transmit impulses through bilateral tracts down the brain stem, into the lateral columns of the spinal cord, and, finally, to the anterior motoneurons. These impulses are nonrhythmic and do not cause the actual muscle shaking. Instead, they progressively increase the tone of the skeletal muscles throughout the body. The resulting increase in muscle metabolism increases the rate of heat production, often raising the total body heat production as much as 50 per cent even before shivering occurs. Then, once the tone of the muscles rises above a certain critical level, shivering begins. This probably results from feedback oscillation of the muscle spindle stretch reflex mechanism. When shivering begins, body heat production can rise to as high as 200 to 400 percent of normal.