Feedback and Control

, K$$^{+}$$, and Ca$$^{2+}$$, (f) volume, (g) osmolality (water/dissolved particle ratio), and (h) the organic nutrient concentrations, such as glucose.


A114622_2_En_13_Fig1_HTML.gif


Fig. 13.1
Control of the respiratory (ventilation) system during exercise. The respiratory area in the brain gets signals from the brain motor region, central chemoreceptors, and other receptors, including carotid artery and aortic chemoreceptors, lung intercostal muscle and diaphragm receptors, and skeletal muscle mechanical and metabolic receptors. During exercise this results in increased neural activity to the intercostal muscles and diaphragm through the phrenic and intercostal nerves. This increases the rate and depth of breathing, which regulates arterial oxygen, carbon dioxide, and pH (From [4]. Used with permission)


We have many other important control systems. Our eye irises open and close in response to light levels. The ciliary muscles control the focal length of the crystalline lenses in our eyes to focus on objects (accommodation, Fig. 11.​22). The force we use to grasp objects depends on the normal force feedback we get from it (and if the object “gives” we apply less force). During exercise the intercostal and abdominal muscles that control breathing are regulated by the brain, which receives input from chemical receptors in the blood, mechanical and metabolic receptors in skeletal muscles, as well as from receptors in the lungs (Fig. 13.1). Similarly, during exercise the brain receives input from the heart and muscles and then sends signals to control the heart and blood vessels to increase cardiac output and blood pressure (Fig. 13.6 below). Body heat balance and temperature are also controlled during exercise (Fig. 13.4 below). Of course, these control systems are also essential during resting. For general discussions about such control systems and control in the body see [1, 4, 6, 7, 10, 12, 15, 19, 21].


13.1 Basics of Feedback and Control


Figure 13.2 is a block diagram depicting feedback and control. There is a sensor or receptor that measures a quantity (stimulus), such as blood pressure by baroreceptors. The signal from this receptor is transmitted to an integration center, such as the transmission of signals from sensory centers to the brain by nerves. The integration center induces an effect. This response is transmitted by nerves (motor pathways) to an effector control center. There is some effect or response of interest, such as changing the heart rate or stroke volume to change the blood pressure. The success of this control is determined by measuring the response, the new blood pressure, which provides feedback for the control.

A114622_2_En_13_Fig2_HTML.gif


Fig. 13.2
Feedback and control in the body, with an example of controlling high blood pressure in parenthesis (Based on [19])


A114622_2_En_13_Fig3_HTML.gif


Fig. 13.3
a Negative and b positive feedback (Based on [19])

Negative feedback reverses the direction of the change of a variation, to keep the measured parameter near the desired set point. This type of feedback is very common in the body. There is usually an operating range centered about this set point bounded by allowable values. Figure 13.3a shows that the effector is activated to correct the parameter when it wanders above the highest value allowed or below the lowest value allowed. Room thermostats usually operate in the same manner, with a several degree operating range about the set point; our bodies also have thermostats.

Positive feedback causes the effector to produce more of a change in the same direction that the parameter is already changing, as in Fig. 13.3b. This type of feedback is rare in the body. One example is suckling which leads to the production of more milk in mothers. Another occurs during the final stages of death by chronic congestive heart failure [16]. With this condition, the cardiac muscle is not able to contract enough to attain the needed stroke volume. Fluid backs up in the veins, causing increased pressure on the lungs and other organs and not enough blood is circulated to the organs. Instead of returning the system to a better set point, positive feedback worsens heart performance and, as damaging, causes the blood volume to increase steadily much beyond the optimal value. Poor heart performance leads to the thickening of the cardiac muscle, but the new muscle is of low quality and decreases heart performance even further. The already limited blood flow to the cardiac muscle becomes increasingly insufficient to keep the increased amount of muscle healthy, which deteriorates performance even more. Concurrently, blood volume increases, increasing the load on the damaged heart, because of the response of the kidneys. The kidneys filter out some salt and water and normally remove them from circulation. For a healthy person this maintains the overall blood volume. For someone with congestive heart failure, such removal would decrease blood volume a bit and lessen the load on the heart, which would be good. However, the kidneys sense the lower blood flow and instead return to circulation the salt and water it has filtered out in an effort to increase the low blood volume to normal levels. This undesirably increases the blood volume, which increases the load on the heart, which causes the heart to deteriorate further and pump less blood, which causes the kidneys to increase total blood volume, and so on, accelerating the death process.


13.1.1 Control Theory (Advanced Topic)


Feedback is needed to control a parameter P, such as blood pressure or temperature, to keep it near a set point $$P_{0}$$. The level of feedback is often proportional to the difference between the current parameter value and the desired set point [15]. The gain g is the proportionality factor and this determines the magnitude of the response. This response causes a change $$\mathrm {d}P$$ in time $$\mathrm {d}t$$ and so


$$\begin{aligned} \mathrm {d}P=-g(P-P_{0})\mathrm {d}t. \end{aligned}$$

(13.1)
This feedback is not instantaneous. It is delayed by a time $$\tau _{\mathrm {d}}$$ due to delays associated with sensing, neuronal transmission to the brain, processing in the brain, neuronal transmission to the effectors, and how fast the effectors can cause a change. Consequently, a change at time t is due to feedback provided from sensors at time $$t-\tau _{\mathrm {d}}$$, and so (13.1) changes to


$$\begin{aligned} \mathrm {d}P(t)=-g(P(t-\tau _{\mathrm {d}})-P_{0})\mathrm {d}t, \end{aligned}$$

(13.2)
or defining $$p=P-P_{0}$$


$$\begin{aligned} \frac{\mathrm {d}p(t)}{\mathrm {d}t}=-gp(t-\tau _{\mathrm {d}}). \end{aligned}$$

(13.3)
The desired response is a monotonic decrease of the magnitude of p, a stable oscillation of p about 0 (and therefore of P about $$P_{0}$$), or some combination of the two. What values of g and $$\tau _{\mathrm {d}}$$ can produce this desired stable negative feedback? Obviously g must be positive and it may seem that the larger the gain the better, but we will see that large values of the gain are not always desirable.

Let us try to solve (13.3) by substituting the possible solution $$p(t)=p\,\exp (zt)$$ into it. In general, z can be a complex number, with $$z=x+\mathrm{i}y$$. The imaginary term y gives oscillatory behavior about the set point, while $$x<0$$ gives exponential decay to the set point and $$x>0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_13_Chapter_IEq15.gif”></SPAN> gives an exponential increase away from the set point, and this last case is undesirable. Equation (<SPAN class=InternalRef><A href=13.3) then becomes


$$\begin{aligned} z=-g\,\exp (-z\tau _{\mathrm {d}}). \end{aligned}$$

(13.4)
Reasonable solutions can exist for $$x\le 0$$. Because $$\exp (\mathrm{i}q)=\cos q+\mathrm{i}\sin q$$, (13.4) becomes separate equations for the real and imaginary parts.


$$\begin{aligned} x= & {} -g\,\exp (-x\tau _{\mathrm {d}})\cos y\tau _{\mathrm {d}}, \end{aligned}$$

(13.5)



$$\begin{aligned} y= & {} g\,\exp (-x\tau _{\mathrm {d}})\sin y\tau _{\mathrm {d}}. \end{aligned}$$

(13.6)
These are transcendental equations that in general must be solved numerically. We will, however, be able to solve them analytically in special regimes.

The solution with $$x=0$$ is one that oscillates about the set point. Using $$x=0$$ in (13.5) leads to $$0=-g\cos y\tau _{\mathrm {d}}$$, which gives $$y=\pi /2\tau _{\mathrm {d}}$$. Using $$x=0$$ in (13.6) leads to $$ y=g\sin y\tau _{\mathrm {d}}$$ and with $$y=\pi /2\tau _{\mathrm {d}}$$, this becomes $$\pi /2\tau _{\mathrm {d}}=g\sin ((\pi /2\tau _{\mathrm {d}})\tau _{\mathrm {d}})=g$$. This means


$$\begin{aligned} g\tau _{\mathrm {d}}=\frac{\pi }{2}, \end{aligned}$$

(13.7)
and so if the delay time is $$\tau _{\mathrm {d}}$$, there is an oscillatory response when the gain $$g=\pi /2\tau _{\mathrm {d}}$$. This is an acceptable solution.

Now let us say that x has a small magnitude but is not necessarily 0. Equation (13.5) still gives $$y\simeq \pi /2\tau _{\mathrm {d}}$$ and so $$\sin y\tau _{\mathrm {d}} \simeq 1$$. Because $$\exp (q)\approx 1\,+\,q$$ for $$ \mid q\mid \; \ll 1$$, (13.6) becomes $$y\simeq g(1-x\tau _{\mathrm {d} })\sin y\tau _{\mathrm {d}}$$ $$\simeq g(1-x\tau _{\mathrm {d}})$$ and so


$$\begin{aligned} x\simeq \frac{1}{\tau _{\mathrm {d}}}\left( 1-\frac{\pi }{2g\tau _{\mathrm {d}}} \right) . \end{aligned}$$

(13.8)
For $$g\tau _{\mathrm {d}}<\pi /2$$, we see that $$x<0$$ and the parameter P approaches the set point as an oscillation with a magnitude that decreases with time. For $$g\tau _{\mathrm {d}}=\pi /2$$, it is clear that $$x=0$$ and the parameter P oscillates about the set point, as shown above. For $$g\tau _{\mathrm {d}}>\pi /2$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_13_Chapter_IEq38.gif”></SPAN>, we see that <SPAN id=IEq39 class=InlineEquation><IMG alt=0$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_13_Chapter_IEq39.gif”> and the parameter P oscillates about the set point with a magnitude that increases with time and this is unstable. Therefore, this stability criterion sets an upper limit to the possible gain for a given delay time, given by


$$\begin{aligned} g\tau _{\mathrm {d}}\le \frac{\pi }{2}. \end{aligned}$$

(13.9)


A114622_2_En_13_Fig4_HTML.gif


Fig. 13.4
The thermoregulatory system, including input to the brain from receptors and output from the brain to effectors. The hypothalamic center relays information to the effector organs to stimulate the heat production and conservation processes shown when the core temperature receptors differ from the $$37\,^{\circ }$$C set point (From [4]. Used with permission)


13.2 Regulation of the Body



13.2.1 Regulation of Temperature


The regulation of body temperature is closely tied to the discussion in Chap. 6 of the production of heat by the metabolism and the modes of heat loss from the body. The normal core body temperature is $$37\, \pm \, 2\,^{\circ }$$C ($$98.6\, \pm \, 3.6\,^{\circ }$$F). Large increases and decreases mean trouble. Far above normal body temperatures, at $$41\,^{\circ }$$C ($$106\,^{\circ }$$F) the central nervous system begins to deteriorate and convulsions occur. At $$45 \,^{\circ }$$C ($$115\,^{\circ }$$F), proteins denature, followed by death. For lower than normal temperatures, at $$33\,^{\circ }$$C ($$91\,^{\circ }$$F), nervous functions are depressed to the point that consciousness is lost. At $$30\,^{\circ }$$C ($$86\,^{\circ }$$F), the temperature regulation system fails. At $$28\,^{\circ }$$C ($$82\,^{\circ }$$F), there is cardiac fibrillation, leading to death. The body is quite good at regulating core temperature, as is clear from Fig. 6.​20, but it has its limits and can fail, as is clear from Figs. 6.​21, 6.​22 and 6.​23. Figure 13.4 illustrates how the receptors in the skin and in the core sense temperature and send information to the brain. Temperatures outside the range of normality lead to conscious acts and automatic activity by the body to correct the temperatures.

Each person has a set point temperature, which varies by $$\simeq \pm 0.3\,^{\circ }$$C ($$\pm 0.5\,^{\circ }$$F) among people [3]. The mean set point for skin temperature is approximately $$33\,^{\circ }$$C ($$91.4\,^{\circ }$$F). (In Chap. 6 we used a skin temperature of $$34\,^{\circ }$$C.) Skin temperatures above $$34.5\,^{\circ }$$C ($$94\,^{\circ }$$F) cause active sweating, while those below $$30\,^{\circ }$$C ($$86\,^{\circ }$$F) lead to the increased metabolic activity associated with shivering.

Recalling our discussion in Chap. 6, the conservation of energy implies a conservation of heat creation and loss


$$\begin{aligned} \left( \frac{\mathrm {d}Q}{\mathrm {d}t}\right) _{\mathrm {net}}=\left( \frac{\mathrm {d}Q}{\mathrm {d}t}\right) _{\mathrm {metabolism}}+\left( \frac{\mathrm {d}Q}{\mathrm {d}t}\right) _{\mathrm {passive \;loss} }+\left( \frac{\mathrm {d}Q}{\mathrm {d}t}\right) _{\mathrm {body \;controlled}}. \nonumber \\ \end{aligned}$$

(13.10)
The left side is the net heating rate due to all modes of heat production and heat loss. The first term on the right-hand side is the heat production from metabolic activity and physical activity; it is $$>$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_13_Chapter_IEq62.gif”></SPAN> <SPAN id=IEq63 class=InlineEquation><IMG alt=$$0$$ src=. The second term is heat loss due to passive modes, such as from the skin, radiation, convection, etc., it is $$<$$ $$0$$. The third term is that due to body-controlled heating ($$>$$” src=”http://basicmedicalkey.com/wp-content/uploads/2017/06/A114622_2_En_13_Chapter_IEq66.gif”></SPAN> <SPAN id=IEq67 class=InlineEquation><IMG alt=$$0$$ src=) and/or heat removal ($$<$$ $$0$$) [1].

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Jun 11, 2017 | Posted by in GENERAL & FAMILY MEDICINE | Comments Off on Feedback and Control

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