- Email: [email protected]
Fine control in the human temperature regulation system
J. Theoret. Biol. (1%7) 16,406-426
Fine Control in the Human Temperature Regulation System RONALD W. CORNEWt
Department of Electrical Engineering and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. JAMESC.HOUK~
Department of Physiology, Harvard University, Boston, Massachusetts, U.S.A. LAWRENCESTARK~
Department of Bioengineering, University of Illinois at Chicago Circle and Presbyterian-St. Luke’s Hospital, Chicago, Illinois, U.S.A. (Received 11 October 1966,and in revisedform 4 April 1967) A model is developed for the regulation of temperature in man. The thermal properties of the body are presented in circuit form and simplifkd to eliminate all features not essential to an understanding of control. In this modeling, perspiration is treated as a sourcewhich can pump heat
against a temperature gradient while lung heat loss appearsas a highfrequency alteration in effective heat production. To this simplified thermal circuit areappendedthe principal mechanisms which are available to the nervous system for controlling body heat balance--increased metabolism
as in shivering,
increased
heat loss by perspiration,
and
alteration of thermal conductancethrough changesin blood flow-and a block diagram of the resultingcontrol systemis given. The descriptionof thesemechanismsfollows that of Benzinger and co-workers (Benzinger, 1959; 1961a,b;Benzinger,Pratt & Kitzinger, 1961). To formulate the problem on a feedbackcontrol basis,an error signal representingdeviationsof hypothalamictemperaturefrom a referencevalue t Supported in part by the Joint Services Electronics Program (Contract DA2l3-043AMC-O2536(E)), the National Science Foundation (G&t GK-835), the National Institutes of Health (Grant 2 PO1 MH-04737-06): , and the National Aeronautics and Space Admiuistration‘(Grant NsG-496). $ Supported by a predoctoral fellowship from the National Institutes of Health. $ Supported by National Institutes of Health Grants NE3055, NE3090, MH-06175; Office of Naval Research Nonr-609(39), Nonr-1841(70); Air Force (AF-33(616)-7282 and 7588), AFOSR 49(638)1313; and Army Medical Corps (DA-18-108~405-Cml-942) at the Massachusetts Institute of Technology. 406
FINE
CONTROL
OF
HUMAN
TEMPERATURE
407
or set point of 37°C is de&d. The reference temperature may have no precise natural correlate and its choice seems arbitrary. Since man normally spends most of his time neither shivering or perspiring, particular emphasis is placed upon an analysis of the vascular mechanism by which line control is achieved. A non-linear differential equation governing the system under this mode of control is presented and solved for transient disturbances in metabolic heat production and environmental temperature. The analysis indicates a quantitydependent on level of metabolism and degree of control over peripheral blood flow-which is a measure of the strength or effectivenessof vascular control. Both response time and steadystate error are reduced by this factor, which is estimated to have a numerical value of about 10. From this analysis it is clear that regardless of how additional assumptions might complicate a description of the system, the data on vascular control reported by Benzinger (1959) lead to a model of human temperature regulation exhibiting powerful multiplicative control over heat flow out of the body. 1. A Model of the Human Thermoregdatory System A model for the distribution of temperature in the nude human is the first of four topics to be discussed in this part of the paper. The emphasis will be placed on developing a model that (i) contains enough of the features of the real system to enable one to envision the way in which control is exercised and which (ii) yields workable expressions for the system dynamics. In this it is necessary to achieve a balance between inclusiveness and complexity, a balance which is facilitated by eventual restriction to the fine-control region, where signals are small and response mechanisms limited. 1.1.
THERMAL
CIRCUIT
Problems associated with the complicated geometry of the body are avoided by regarding man as a cylinder of shoulder height and waist-like girth as shown in Fig. 1. Following this simplification attention will be restricted to thermal gradients in the radial direction with leakage from the ends of the cylinder neglected. The best form to assume for this radial dependence of
FIO. 1. Cylinder model of man. The complicated geometry of the human body has been avoided by regarding man 89 a cylinder of shoulder height and waist-like girth.
408
R. W. CORNEW,
J. C. HOUK
AND
L. STARK
temperature has been a matter of long-term speculation. An early thought by Burton (1934) was to treat this cylinder as a distributed parameter system possessing uniform thermal capacity, conductivity, and heat generation. This entailed solution of a nonhomogeneous thermal diffusion equation subject to the boundary condition imposed by skin temperature. Such an approachthough neglecting non-uniformity in the active mechanism of heat transport via blood flow and in the generation of heat throughout the body-has been successful in describing qualitatively the parabolic shape of the static thermal gradient. While distributed parameter models may be extended to predict with sufficient accuracy dynamic changes in temperature as well as the static distribution (the latter has been provided by Wissler, 1961), the effects of control in any such model are difficult to visualize because of the complexity of the ensuing solution. Instead, another approach has evolved-based on division of the cylinder into a number of concentric layers each at constant temperature-which offers considerable advantage in conceptualizing the way in which control is exerted and in facilitating the computation of responses. The history of such lumped-parameter representations of the thermal system goes back to about 1950 (MacDonald & Wyndham, 1950; Wyndham, Bouwer, Devine, Paterson & MacDonald, 1952) and has subsequently been tied to several analog computer simulations of human thermoregulation (Crosbie, Hardy dz Fessenden, 1963; Brown, 1963; Smith & Jones, 1964, and, most recently, Stolwijk & Hardy, 1966u). We will begin with a threelayer model, since it affords considerable freedom in obtaining a good fit to the dynamics of the system and has the advantage of being the least complicated to contain explicitly all of the essential, involuntary control features of the system.? In Fig. 2 this model is shown with the layers selected so that their tempera-
FIG. 2. Three-layer model of man. To simplify analysis, three thermal layers are distinguished within the cylinder model of man. t Crosbie, Hardy & Fessenden-in the first analog computer study referred to abovealso used a three-layer model to simulate the temperature distribution of the body. In this case,computer capacity dictated the number of layers. In another of these studies, Brown used a three-layer model with good success over a liited range of simulation.
FINE
CONTROL
OF
HUMAN
TEMPERATURE
409
tures correspond to the n?ean temperatures of the inner core, the musculature, and the skin layer. Following this the resulting three-layer thermal circuit of a nude human is given in Fig. 3. In this circuit each layer has been given the
+fjtqJjjrgG Inner
core I I I
Muscle
layer
; I
Skin
loyer
, ;
arr%ient
Fro. 3. Thermal circuit of a nude subject. A thermal resistance, capacitance, and heat source has been associated with each of the three thermal layers of man. Their interconnection to form the thermal circuit of a nude subject is completed by the inclusion of a temperature source representing the thermal constraint of the environment and a resistance from core to environment accounting for the flow of heat through the lungs.
ability to store heat through the inclusion of thermal capacitance. Further, thermal gradients between layers are allowed for by the presence of thermal resistance.t The heat source in the inner core represents the heat generated by the metabolic processes in the internal organs ; that in the muscle layer results from muscle tone, shivering, and movement; and the source between the skin layer and the outside represents the perspiration mechanism which, through evaporation, pumps heat out of the body.$ The environment is represented by a temperature source. The description of Fig. 3 is completed by observations on two parameters appearing in it, Rs and RL. The former is a resistance combining the effects of heat loss from the body by conduction to the air enclosing it and radiation to the walls and objects surrounding it. Specifically included is the insulating property of the surface layer of air surrounding the body (Hendler, Crosbie & Hardy, 1958). In using a single, linear resistance it has been assumed that 7 The reader who is unfamiliar with thermal resistance and capacitance and the laws governing their behavior in thermal circuits is referred first to the Appendix and then to any good book on elementary circuit theory. A circuit theory approach will be used extensively here. $ The use, instead, of a sweat resistance (in parallel with CS) as suggested by MacDonald & Wyndham (1950) and Wyndham et al. (1952) strikes us as artificial. With that modeling (i) evaporative heat loss does not reappear in the environment as it should, (ii) the active nature of the evaporative process is lost needkssly and (iii) important experimental data cannot be explained. Under their assumption these authors were forced to conclude “that heat transfer by sweat is unimportant in comparison with that by radiation and convection” which was “an evident source of disagreement with theory”. These diiculties disappear in this modeling.
410
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
the air and wall temperatures are equal and change by an amount small compared with the normal skin temperature of about 300°K. This is a reasonable assumption for the range of temperatures that a nude subject could regulate against (Stolwijk & Hardy, 19663). The resistance R, should also be considered in more detail. It represents the number that relates heat loss through the lungs to the difference in temperature between the inside of the body and the surrounding air. It is apparent that this loss is intermittent, since it depends on the breathing cycle. Thus, modeling the nature of RL to fine detail would require a periodically time-varying resistance. Now one could-since the breathing cycle is short compared to times characterizing thermal responses-immediately replace this with an average value determined over a few breathing cycles. An alternate approach which preserves this time-varying character of R, and allows as well for an evaporative component of lung heat loss will be discussed next. 1.2. SIMPLIFICATION OF THE THERMAL CIRCUIT It is convenient to eliminate the explicit appearance of R, shown in Fig. 3 by allowing its heat flow to be absorbed into Q,, the heat source of the inner core. Noticing that the net body heat contributed by the inner core consists of that generated there less that lost through the lungs prompts definition of a new variable
Qs = QI-CT,- TOY& and alteration of the thermal circuit to the form shown in Fig. 4, in which the path through R, to the outside air has been eliminated. Any evaporative
FIG. 4. Modification of thermal circuit of a nude subject. The thermal circuit of a nude subject has been mod&d here by altering the source of heat within the core to include the flow of heat through the lungs. In addition, the environmental temperature source now includes the drop in skin temperature accompanying perspiration.
component of heat loss through breathing would have appeared in the original circuit as a current source in parallel with RL and would now show up as an additional term subtracted from QP An approximate form,
QB= QrG%-
ToYR,,
FINE
CONTROL
OF
HUMAN
TEMPERATURE
411
is possible since T,, the temperature of the inner core, is stable enough under most conditions to be replaced by T,,, a constant. (This constant will eventually coincide with the reference of the temperature regulation system but no such interpretation is presently necessary.) Thus Q, is independent of any body temperaure which shows appreciable variation; this further justifies its treatment as a single source. The previously noted time variation in R,, if included, would now show up as a ripple in this new heat source. This effect-high in frequency and small in amplitude-is certainly responsible for some of the noise observed in the system. Other sources of noise include diurnal variation in Q, as well as effects resulting from eating, drinking, digestion, and other independent changes in metabolism. Taken as a whole, QB therefore contains long-term drift as well as high frequency noise which together constitute a broad spectrum of internal disturbances against which the system must regulate. Figure 4 includes a second change in which the thermal effect of the perspiration source Q, in Fig. 3 has been absorbed into the temperature source To by the construction of a Thevenin equivalent of everything remaining to the right of C,. This equivalence is not affected by non-linearities which may lie to the left of the dashed bracket in Fig. 4. Incidentally, the new source
T; = To-Q,R, appearing in Fig. 4 may be regarded as an effective environmental temperature; it is the originally defined environmental temperature diminished by the temperature drop of the skin due to perspiration. If an imbalance of wall and air temperatures were allowed in the original modeling, it would appear here only as a change in T& Two assertions about the dynamics of temperature change in the human body will prove valuable: (i) changes in average core and muscle temperature closely parallel one another owing to a continuous exchange of large quantities of blood, at least for small disturbances about thermal neutrality; (ii) changes in skin temperature occur much more rapidly than changes in other temperatures under comparable conditions owing to the greatly reduced thermal mass of this layer. From the first of these it is assumed that RI, the thermal resistance between core and muscle, should be small compared to other resistances in the system. From the second a similar assumption is made concerning C, and the other capacitances of the system. This situation is idealized by taking
RI=&=0
412
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
in the model of Fig. 4. The ultimate test of validity for such assumptions lies in comparison of predicted and experimental transients; the modeling here leads to first-order transients throughout the system.
Q;1= QB+QMH, CB = c,+c&f, R, = R,+R,, r---Skin
cl;,
temperature
$-+-~g
I
1
I
FIG. 5. Simplified thermal circuit of a nude human. This model emerges after placing RI = Cs = 0 and defining the total body heat Qk = QB + &, the total body capacitance easily approxiC, = Cl + CM, and the total body resistance RB = Rr + Rs. It pos~esas mated parameters with, for a 150 lb man, 100 w < @a < 200 w or more, depending upon the degree of muscular activity, C, = 200,000 Joules/T, and RB = 0*12”C/w. The value of RB varies greatly, depending upon the vascular state of the body. The specik heat of the body has been taken as 70% that of water in the capacitance calculation.
Figure 5 shows this considerably simplified model. Here are, respectively, the total effective body heat and the total remaining capacitance and resistance. T,, the inner core temperature, is the only body temperature remaining explicitly in this model although T,, the skin temperature, can be simply recovered if desired by noticing T, = ( TI - T&)R,/R,.
From Fig. 5, the differential equation governing the thermal system in the absence of control can be written by balancing the flows of heat at the upper node as Q~-cB!$-(EgT2=(-). B
This can be rewritten as R&B
‘2
+ TI = RBQ;P+ Th.
In the absence of shivering and perspiration which really lie beyond the range for which this model has been constructed, the primes on the variables of the right side of this equation may be dropped.
FINE
CONTROL
OF
HUMAN
TEMPERATURE
413
1.3. MECHANISMS OF CONTROL The simplified model of Fig. 5, describing the flow of heat through an idealized body, resembles a well-stirred tank possessing a source of heat and a large thermal mass, both interior to an insulating wall in contact with an external environment. At this point the elements of control will be added. Recent work by Benxinger and others (Benxinger, 1959; 1961a,b; Benxinger, Pratt & Kitzinger, 1961) demonstrates quantitatively that a key reference of the system is located in the hypothalamus, a neural structure within the underside of the brain. The controlled variable of the system is the mean inner core temperature as attained from the hypothalamic blood supply which is derived largely from the Circle of Willis, an arterial ring Iocated beneath the hypothalamus. The difference between the reference temperature (called T,, here and typically taken to be 98~6°F = 37.O”C) and the controlled temperature (TI) is the error signal of the system (AT,); that is, AT, = T,, - TI. Depending upon the sign and magnitude of this quantity, combinations of at least the following three control mechanisms can be brought into play: (i) Low fenrperutute re@un. The body reacts to low temperatures by creating more heat. This is accomplished by increased muscular tone, shivering, and voluntary activity (grouped as QM earlier). The heat generated by the first two of these is related to the error signal in Fig. 6(a) (Benzinger et al., 1961). The heat generated by the third, voluntary muscular activity, will receive no further attention here, since it is not predictably related to the error. It will later be argued that this component-while sometimes purposely employed to fight cold-is better treated as a disturbance to the system. The data of Fig. 6(a) is presented as a set of curves parameterixed in T,, the skin temperature. A major eEect is the downward shift with increasing skin temperature of the point at which shivering is initiated. An additional effect-variation of gain from one skin temperature to another-is also present. (ii) High temperature region. In this region the body loses heat by perspiration (called Qs earlier). The relation between this heat loss and error is given in Fig. 6(b) (Benzinger, 1961b). The data assumes that humidity and air velocity about the body are not subject to variation. Once again the curves exhibit a shift in set point with skin temperature-a downward change in skin temperature being accompanied by an upward change in the hypothalamic temperature at which perspiration begins. In this case gain variation is less evident. T.B. 2s
414
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
(iii) iVeurru2 temperature region. Since substantial heat is carried from core to skin by blood flow, fine control can be achieved by regulating the size of the blood vessels in the periphery of the body. Detection of small errors to the cold side of the reference (i.e. a positive AT,) results in constriction with its accompanying conservation of heat. Small errors to the warm side results in dilation with subsequent loss of heat. This vasomotor action is modeled by allowing variations in R,, the total body resistance. In Fig. 6(c) (taken from Benzinger, 1959) this variation has been approximated by a piecewise-linear curve. Since this alteration in resistance introduces a non-linearity which is purposefully employed to reduce the magnitude of error, the thermoregulatory system is seen to exhibit adaptive control. Unlike the control mechanisms at work for high and low temperatures, Benzinger has found no important dependence of this mechanism on skin temperature.? A more detailed discussion of the error-reducing power of this mechanism will be the topic of the second part of this paper. In presenting the data given in Fig. 6 and described above, Benzinger has been careful to point out that all measurements were made in the steady state and may not be valid dynamically. Nonetheless, it is often assumed that these control mechanisms are inherently faster than thermal responses in the remainder of the system. Evidence exists to indicate that this is true for the vascular mechanism which we are particularly interested in studying here1 and allows application of the steady-state measurements to the dynamic control situation in this case. Failure of this assumption for the other mechanisms would necessitate inclusion of appropriate dynamic elements into their control pathways. 1.4.
BLOCK
DIAGRAM
OF THE THERMORBGULATORY VASCULAR
SYSTEM
UNDER
CONTROL
The discussion above has re-presented Benzinger’s data which quantify the concept that involuntary control of body temperature results principally 7 This importance of skin temperature in determming vascular responses has been a matter of considerable discussion. Rawson & Randall (1961) make it abundantly clear that vascular responses to regional heating of the body surface exist but others, such as Wyndham (1965), find these effects several times less important than core variations at normal skin temperatures. The lack of quantitative data to the contrary and our self-imposed restriction to the region of thermal neutrality where skin temperature variations are not great leads us to exclude such effects totally. $ Investigation with an unanesthetized dog subjected to periodic step-changes in hypothalamic temperature reveals that vasomotor activity as measured by temperature changes in the ear pinna is at least several times more rapid than changes in deep body temperature at the rectum or back brain (Hammel, Stromme & Comew, 1963). This is no doubt more pronounced in man where thermal masses are larger and passive system response times longer.
FINE
CONTROL
OF
HUMAN
415
TEMPERATURE
from the action of three mechanisms: increased metabolic activity, increased evaporative loss, and vasomotor activity. Of these, the first two are class&cl here as gross control mechanisms functioning for large errors. The last, vasomotor activity, works to achieve fine control in the presence of small errors. The graph of the vascular control mechanism given in Fig. 6 can be coupled with the earlier differential equation governing heat flow in the body
-AT,* Hypothalamic
-Ar, temperature
PC)
--I
Hypothalamic
(0)
g,z
I
oc
PC1
W
3 40 % +=f 30 8 s s.f gE 8 2
temperature
l L-. ’
20
.
IO
lL
\
t_l
2.2
37.0
37.6
-AT,-+ Hypothalamic
temperature
PC)
(Cl
Fro. 6. Human themoxegulatory responses. These graphs, adapted from Benzinger (1959, 19616; Benzinger et al., 1961). show the particular way by which skin and hypothalamic temperature determine (a) muscular heat production (largely shivering), (b) rate of sweating, and (c) body resistance. In the first two of these, the curves given by Buuinger have been wed to approximate the behavior of the actual data. The last graph contains his or&al data after inversion and has been approximated by a straight lie having a fractional change of resistance of 2.4/“C. It is emphasized that only the internal, or variable component of Rs (roughly 33 yO of the total) is shown here. AT,, the deviation from 37°C = 98.6”F is included at the bottom of all graphs and will later be identified as the error of the system. Measurem~t of hypothalamic temperature was made at the tympanic membrane in all casea.
to yield a block diagram of the thermoregulatory system under vascular control (see Fig. 7). Here the symbol p has replaced d/d& the time derivative operator, to represent in compact form the dynamics of the system being controlled.
416
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
It is noted from Fig. 7 that there are three inputs to the thermoregulatory system. These would be the same regardless of the control mechanism under study. Two, net body heat contributed by the core (QJ and environmental temperature (To), take the form of inputs to the system from inside and 0
Summation
0
Multiplicotlon
Neutral
temperature
region
7. Block diagram of temperature regulation system in man under vascular control. The graph of RB shown for the neutral temperature region includes both variable and nonvariable components of that resistance and gives rise to multiplicative control. Qa, net body heat, and TO, environmental temperature, are inputs to the system from inside and outside the body. T,, (delined as 37°C = 98.6”F) is the reference of the system. The letter p symbolizes the time derivative operator, d/dt. FIG.
outside the body, respectively. Changes in these inputs from their equilibrium values constitute disturbances to the system. Specifically included are digestive heat and diurnal variation in basal metabolism, voluntary muscular activity, and changes in environmental temperature arising from alteration of air or wall temperature. The third input is the reference of the system (T’,) and can be disturbed by, among other things, fever inducing agents. 2. Fine Control Resulting from the Vasomotor Non-linearity
Because of the complicating effect of skin temperature and the possible interaction of the regulatory mechanisms the best approach to understanding the entire thermoregulatory system would probably be computer simulation guided by experiment and analysis. Crosbie, Hardy & Fessenden (1963) and -more recently-Stolwijk and others have tried this approach using an analog computer. The final word concerning their success has not been heard.7 However, by restricting attention to the neutral region where the t After submission of this manuscript the comprehensive article of Stolwijk & Hardy (1966u) which presents a much more detailed model than that used here (together with its analog simulation) was called to our attention. It seems appropriate to comment on that moddinrclationtoonrown: (a) The region of solution of the thermoregulatory problem is very different, it being our intention to present an analysis of line control where neither increased metabolic activity nor evaporative loss are important. In contrast, Stolwijk and Hardy’s simulation is validated against the latter responses; i.e. Benzinger’s ice ingeatlon
FINE
CONTROL
OF HUMAN
TEMPERATURE
417
vasomotor mechanism may act alone, it is possible to go into the analysis of some salient features of fine control. From common experience we know that, with the aid of clothing, most of our time is spent in this region neither shivering or perspiring. Specifically, Benzinger’s data shows that when the inner core temperature, T,, is at or near 37°C (98~6°F) there is a range of skin temperature, T,, from about 27 to 32°C (roughly 80 to 90°F) for which the vasomotor mechanism acts alone and is, furthermore, independent of skin temperature. While this formally restricts the analysis to a narrow range of skin and hypothalamic temperature (the latter no larger than, say, *OSl’C), it should be pointed out that such vascular responses underlie the entire region between 36.7 and 374T and an analysis of them provides a lower bound to regulatory success throughout this region. Moreover, whatever &lays exist in the shivering and perspiration mechanisms further widen the range of initial errors that can be accommodated solely by the vascular mechanism. In short, we seek an analysis of thermoregulation which is valid when the core-temperature dependent vascular mechanism acts alone. experiment (1959) resulting in evaporative loss and Park and Palme’s typhoid vaccine experiments resulting in increased metabolism are the offered comparisons of the model with experimental evidence. (b) The description of vascular control employed by these authors differs signiticantly from that used here. Citing results of a recent paper, where it was concluded that evaporative losses can be correlated with a product of deviations in skin and central temperatures from reference values (Hardy & Stolwijk, 1966), these authors genemlize to a similar control law for all thermoregulatory responses under study, including the vascular. No direct experimental verification of this multiplicative dependence on skin temperature deviation is given for these other responses and is, of course, absent from the Benzinger data and hence our model. The models agree in assuming that vascular responses, however they may be controlled, are essentially instantaneous and proportional. (c) Smce they were utiliig analog techniques, Stolwijk and Hardy were able to study a more complicated passive model involving three distinct cylinders and a separate compartment for blood. It would he valuable to have a quantitative estimate of how much thls added complexity influences the accuracy of control, particularly since the system is known to contain much smoothing. It is particularly important that such simulation studies do not discourage simpler models which seek to describe the system in more limited regions. Though outside the vascular response range for which it was prepared, the model used here can, for instance, yield the oppositely moving skin and tympanic temperatures of the Benzinger ice ingestion experiment if 1 AQs IiF’---Rg - Rs’ I a condition which can be met if central control of sweating is strong enough. For that matter, reintroducing Cs in our model (or placing delay in the evaporative loas echanism) will produce the observeddelay in onset of the rise in Ts: its shortness %ally argues for the deletion of both such effects in pesxil and paper studies.
418
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
2.1. DJFFJZRENTIAL EQUATION OF VASOMOTOR CONTROL By reference to Fig. 6(c), it is seen that the variation of body resistance, R,, with error (ATr or henceforth, simply E) may be described by R B = R&l +a&); -0*4”C < E < 0*3”C, (2) where R,, is the value of RB in the absence of error and a, the strength of control, is a constant determined by the product of two ratios: (i) the fractional change of the resistance with temperature as determined from a straight-line approximation to the data of Fig. 6(c); (ii) the fraction of body resistance undergoing such change. The second fraction is not unity, since RB has been defined to include the constant resistance of the layer of surface air about the body which was not included in the previous experimental data. If desired, one thermal effect of clothing could be incorporated at this point by adding a constant resistance to RB in equation 2, thereby increasing RBo and decreasing a. Clothing therefore reduces the variability of RB and, while giving more insulation, limits control.? The differential equation for body temperature which was obtained in the first part of the paper [equation (I)] may be rewritten in terms of the error e as
&G$ + E= Go-G+R,Q,), since the reference T,, is assumed constant. [The primes on To and QB have been dropped since perspiration (Q,) or shivering (QJ have been eliminated by the definition of the region under study.] Substituting RB from equation (2) into this relation and bringing all terms in E to the left yields a differential equation of vasomotor control valid for any properly bounded initial error:
R,oC,(1+4 2 + Cl+aR,oQ&
= T,, - ( To + b,Q,h.
(3)
2.2. STEADY-STATE SOLUTION The steady-state error in the presence of control may be determined from ds equation(3) by placing - = 0 and solving for E = sSS, dt
%s=
T,o - (To + ReoQd 1+QR,oQ, -
This equation gives the steady-state error for each set of values for To and t It is well known that one may wear too much clothing on a cold day to effectively control body temperature with the uncomfortable result of placing the body in a limit cycle between shivering and perspiration. However, purposeful clothing changes can be looked upon as yielding even more control of Rg. Clothing, of course, also inWoduces an additional thermal capacity that helps to smooth out brief changes in environmental temperature.
FINE
CONTROL
OF
HUMAN
TEMPERATURE
419
Q,, the environmental temperature and net body heat, which as previously noted constitute the inputs of the system. In the absence of control (a = 0) the corresponding error would be ~so = Txo -(To+R,o&)Thus
%SO Qs = 1+aReoQe' from which it is seen that the error-reducing power of vascular control is 1 +aR,,Q,, a number necessarily greater than 1. The range of disturbances which can be accommodated without necessity of shivering or perspiration is thereby extended by this same factor. A numerical estimate of the error-reducing power of vascular control can be obtained. From Benzinger’s data which yield a value of 2.4 for the fractional change of resistance with temperature and from Burton’s (1934) measurements of thermal circulation index, which show that as little as 33 % of RB may be variable, a is determined as 0.8. A value of RBoQB can be estimated by noticing from equation (4) that R,oQ, = T,o- To, where the term on the right is the temperature gradient from core to environment when the steady-state error is zero. Rough estimation suggests thatfor a nude subject-this would correspond to an environment temperature of about 25°C (77°F) for which the thermal stress from core to environment is 12°C. From these considerations an error-reducing power results which is slightly greater than 10. While this is only an estimate, a value of this magnitude is to be anticipated in view of the ability of the system to render environmental changes of several degrees into core errors of only a few tenths of a degree (Hardy & DuBois, 1938). From equation (4), the steady-state error is linearly related to To; however, its dependence on QB is not so simple. In any case, the transient after a disturbance in either of these quantities is nonlinear and will be described below. 2.3. TRANSIENT SOLUTION Despite its non-linearity it is possible by separation of variables to obtain an exact solution to equation (3) for the error transient resulting from any initial error .ao. The only provision-as discussed in the next paragraph-is that QB and To undergo no further change beyond the time when so is determined. To accomplish the solution this equation is rewritten as (1 +a&) z-de+dt=O, (8 - %s)
420
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
where RBOCB
Z= 1+
@I&B
is a characteristic time for which an interpretation will be given in the next section. Solving for dt and integrating from 0 to t as Egoes from e. to Eyields t&) = (1 + ass) In 80 - - %S + ho - 4, (5) E--ESS which is explicit in the normalized time tN = t/T. The second term on the right side of this equation destroys the possibility of obtaining a useful solution explicit in s.
Controlled
w 7‘
(a)
f
Controlled
(b)
FIG. 8. Typical error transients with and without vascular control. The solid curves of (a) and (b) represent oppositely-directed responses of the model in the presence of vascular control while the dotted curves would result if such control were absent. In each case, the initial error is taken as zero. The disturbances, which could be caused by previous changes in either QB or To, are equal and opposite and have ken chosen large enough to produce shivering or perspiration in the absence of control. Comparison shows a s&able reduction in steady-state error and response time in both cases, Since the strength of control 1 + ccRBoQB is independent of To, but not QB, a pair of responses to equal but opposite alterations in To will possess equal but opposite steady-state errors (reduced about 10 to 1 from the uncontrolled level-see text), while responses to changes in Q0 will be unequal. The time courses of the responses under vascular control are always unsymmetric.
FINE
CONTROL
OF
HUMAN
TEMPERATURE
421
A remark about how transient errors arise is appropriate. It has already been noted that changes in the inputs to the system-particularly environmental temperature and net body heat-constitute disturbances. Such disturbances may be gauged by the effect they have in changing core temperature from equilibrium; i.e. in producing steady-state errors. Should either of these inputs change, the system will undergo a transient in passing from one steady state to another. We have sacrificed description of the system during the course of the input change focusing instead on the lingering time course of the error after the input reattains a constant value. In the special case of sudden changes in an input, the transient is simply a step response of the system, and the previous steady-state error becomes the initial error of this transient. That the initial error E,, is sufficient to summarize the input history of the system is apparent from the solution obtained above and results from its assumed first-order dynamics. Figure 8(a) contains a plot of such a transient (solid line). Regrettably, an experimental response suitable for comparison (i.e. resulting only from vascular action) is not available in the literature. We predict a response which is altered from the time course of a simple first-order, linear system by the generally smaller second term of equation (5), which can speed up or slow down the response depending on its sign [as in, respectively, the solid lines of Fig. 8(a) and (b)]. To illustrate the importance of control, we have included the transients which would accompany the same two disturbances in the absence of control (dotted lines). Comparison evidences both a reduction in steady-state error, as determined in the previous section, as well as a reduction in response time, to be discussed next. 2.4. REDUCTION IN RESPONSE TIME In the absence of control (a = 0) equation 5 can be solved explicitly for the error 40 = %so- (%s, - 4 exp ( - WdAJ~ which is recognized as the transient of a first-order, linear system having a time constant r0 = RBo C,, and was plotted in Fig. 8. (The same result could have been derived directly from the single-layer model of Fig. 5 with RB held constant at RB,,.) The characteristic time of the system in the presence of vascular control (defined in Section 2.3) may now be expressed as R,BC, ‘=
l+(rRBOQB
70
=1+aRmQB9
a form which demonstrates the ability of this mechanism of control to hasten the reattainment of a steady state. Thus, 1 +aR,Q, becomes the factor governing both response time and error reduction in the fine control region.
422
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
However, in contrast to the linear case where t = r0 implies 63 % completion of the transient independent of its initial and final values, no fixed percentage of the non-linear transient is complete when t = z; i.e. fN = 1. Nonetheless, a sizeable reduction in response time is indicated-typically from several hours down to a fraction of a single hour. Computation of the response time of the body based on the thermal properties of tissue in a nominal, but uncontrolled, vascular state yields a figure of 7 hr for a 150 lb human subject. (See the caption of Fig. 5 for values of R, and C,.) In the presence of control, the response time should be reduced to about 2/3 of an hour. 2.5. FURTHER DISCUSSION: APPARENT NON-EXISTENCE OF A UNIQUE REFERENCE The reader should observe that the discussions of this part of the paper did not invoke the concept of a unique, physiologically delved reference. All that could be noticed under hypothalamic vascular control was an ability of the system to minimize the effects of imposed changes. In fact, since R, as a function of TI is known [Fig. 6(c)], equation (1) could be solved for the final TI that prevails for each set of values of QB and To. This would have been one solution to the steady-state aspect of the vascular control problem and would not have involved the concept of feedback. The fact that the system does evidence tight regulation in this region tempted us to look for an operational definition of a reference so that it could be put into feedback control form in the first part of the paper and so that a quantitative measure of the changes in core temperature resulting from disturbances could be given in the second. Apparently the condition used, that R, = RBo when s = 0, could serve to defme the reference of the system as any core temperature within the region of vasomotor response. That is, T,, can be taken anywhere in the vasomotor region as long as RB is taken to be Rso there. Convention prompted selection of 37°C = 98.6”F as the reference, roughly the mid-point of the vasomotor response region. Now this raises several questions. (i) Does not the intersection of the heat generation (Q,) and heat dissipation curves (QJ yield a unique reference as claimed by Benzinger et al. (1961)? Evidently it does not, since these curves possess no unique intersection for a common T,! Instead a band of temperatures of 0.5”C or more is enclosed. As mentioned above, man spends most of his time in this region where changes in vascular resistance provide the only control of core temperature. Now it may be acceptable to look upon this band of temperatures as the target of control in a gross sense, but it is certainly not a unique reference.
PINE
CONTdOL
OF
HUMAN
TEMPERATURE
423
(ii) How can a reference which is operationally defined be shifted by, among other causes, pyrogens? In fever, the entire band of temperatures defined above (with TIo operationally defined somewhere in it) moves up and down. Incidentally, the time course of this motion itself contains little information about natural temperature regulation, although, if quick enough, can serve as a useful input in studies of the regulatory system as shown by Stolwijk & Hardy (1966u). (iii) Could a natural reference exist but be unmeasurable? One answer to this is another question: By what attributes would such a reference be distinguished? The feedback approach to the analysis of biological systems has many advantages, but a consequence of taking this point of view toward a system is the introduction of concepts, such as reference, which may have no natural correlate. 3. Conclusion In the application of mathematical techniques to the study of biological systems there is always need of a model which is at once simple enough to support analysis and yet not m&e. By selecting a very simple model for the human temperature control system, one in which anatomical features such as the neural pathways from hypothalamus to effector mechanisms are not considered and in which the passive thermal properties and active heat sources of the body are treated as lumped, it has been possible to formulate the system on a feedback control basis and go some distance into the analysis of the salient features of vascular control. More exacting assumptionstreating the system as distributed, or reintroducing its geometric complexity as by increasing the number of cylinders, etc.-would lead to greater depth of description at the expense of increased complexity of analysis. Regardless of how these assumptions might complicate a description of the passive system, the data on vascular control reported by Benzinger (1959) leads to a model of human temperature regulation exhibiting powerful multiplicative control over heat flow out of the body. This contrasts with the situation for larger errors where his data on increased metabolism (Benzinger et al., 1961) or evaporative loss (1961b) shows roughly proportional control with a parametric dependence on skin temperature. In consequence, the response to an initially large error should depart from that of a linear system as error decreases-a behavior which would not be well described by, say, small-signal frequency analysis using sinusoidal excitation. Step responses, which are easily analyzed and more physiological, expose the properties of this system better. A non-linear analysis which is valid for such steps and
424
R. W. CORNEW,
I. C. HOUK
AND
L. STARK
other transient disturbances in metabolic heat production and environmental temperature has been presented here; this analysis demonstrates that vascular control diminishes the ensuing perturbation in core temperature by about 10 to 1.7 In spite of the apparent absence of a precise reference, the feedback approach remains useful because it focuses attention on incremental deviations of the system about a steady state. REFERENCES BENZINGER, T. H. (1959). Proc. natn Acad. Sci. U.S.A. 45, 645. BENZINGER, T. H. (1961a). Scient. Am. 204,134. BENZINGER, T. H. (19616). Proc. natn Acud. Sci. U.S.A. 47, 1683. BENZINGER, T. H. (1964). Symp. Sot. exp. Biol. 18,49. B~N~~NGER, T. H., PRATT, A. W. & KITZINGER, C. (1961). Proc. natn Acud. Sci. U.S.A. 47, 730. BORN, M. (1948). “Natural Philosophy of Cause and Chance”. Oxford: Clarendon Press. BROWN, A. C. (1963). ASTIA, AD428144 BURTON, A. C. (1934). J. Nutr. 7,497. CROSBIE,R. J., HARDY, J. D. & FESSENDEN,E. (1963). “Temperature, Its Measurement and Control in Science and Industry”. New York: Reinhold. HAMMEL, H. T., ANDERSON,H. T., JACKSON.D. C. &HARDY, J. D. (1961). ASD TR 61489. HAMMEL, H. T., STRBMME, S. & &RNEW, R. W. (1963). Life Sckces, i, 933. HARDY. J. D. & DuBors. E. F. (1938). J. Nutr. 15. 477. HARDY; J. D. & %OLWIiK (1966). J. ;?ppl. Physiol: 31, 1799. &NDLRR, E., Cnosme, R. & HARDY, 3. D. (1958). J. uppI. Physiol. 12, 177. bfACL)oN.UD, D. K. C. & WYNDHAM, C. H. (1950). J. appl. Pkysiof. 3,352. RAWSON, R. 0. & RANDALL, W. C. (1961). J. upid. Physiol. 16, 1006. SMITH, P. E., JR. & JONES,E. W., END (1964). Archs e&r. H&h, 9, 332. STOLWIJK, 3. A. J. & HARDY, J. D. (1966~). Ffliigers Arch. ges. Physiol. 291, 129. STOLWUK, J. A. J. & HARDY, J. D. (1966&. J. up&. Phydol. 21,967. WISSLER, E. H. (1961). J. appl. Physiol. 16, 734. WYNDHAM, C. H. (1965). J. uppl. Physiol. 20, 31. WYNDHAM, C. H., B~UWER, W. v. D. M., DEVINE, M. G., PATERSON,H. E. dc MACDONALD, D. K. C. (1952). J. appl. Physiol. 5, 299.
Appendix : Thermal Andogies This Appendix will discuss a useful analogy which is employed throughout this paper. Because of similarities in the form of thermal and electrical equations, temperature (2”) and voltage (V), as well as heat flow (Q) and current (I), have been taken as analogous quantities. This selection allows t Certain other advantages result from possession of a vasomotor response, rather than dimct regulation of body heat production, as a principal mode of control. The Srst is a matter of economics. In the cold, vasoconstriction increases the insulation of the body making batter use of the heat which is available. The second is a matter of priority. In warmer environments the processes which generate core heat could not reasonaMy be turned down at the beck and call of the temperature control system-they are at least as vitalasitis.Athirdisspeed,vascularchangeabeingmorerapidthanchangesincoreheat production. All other things being equal, the system with vasomotor control would be expeded to operate with a smaller mean variation in core temperature than if slower metabolic changea were relied upon for t?ne control.
FINE
CONTROL
OF
HUMAN
TEMPERATURE
425
thermal resistance (Rr) and thermal capacitance (C,) to be deflned in the same way as the corresponding electrical quantities. (A subscript “T” will be used in this Appendix to denote thermal quantities which might be confused with their electrical counterparts.) First consider the Fourier Law of Heat Flow which, in one dimension, is Q = k,Adq, where = heat flow (in direction T(x) = temperature distribution = cross-sectional area of A kT = thermal conductivity. Applying this to a homogeneous uniform thermal resistivity pT (=
Q
of temperature gradient) as a function of x structure in question slab of cross-sectional area A, length Z, and l/k,) yields
Q=-$, where T is the temperature difference between the ends of the slab. Solving for T yields the thermal form of Ohm’s Law: T = RTQ, where PT1 R r=- A defines thermal resistance in a manner analogous to electrical resistance. In practice R, often has the dimensions of “C/w, since T is usually in “C and Q is frequently given in watts. Thermal capacitance (C,) is defined by the relation
where H denotes heat accumulation of a body over time (the time integral of heat flow Q) and T is the accompanying difference in temperature. This is analogous to the electrical definition of capacitance as the charge accumulated per unit voltage. Thermal capacitance has the dimensions of Joules/T in the system of units established in the previous paragraph and may be considered constant throughout the range of temperatures for which this modeling is intended. Recalling that resistance, capacitance, and inductance constitute the set of commonly encountered, two-terminal passive electrical elements prompts a
426
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
question about the def%tition of thermal inductance. However, to the best of anyone’s knowledge, nature has provided no phenomena which require the existence of thermal inductance for satisfactory explanation. A “thought” experiment reveals that its absence is as fundamental as the Second Law of Thermodynamics which assures, in the words of Born (1948), “in no way can heat be entirely converted into work or raised to a level of higher temperature”. If thermal inductance did exist it would be possible to do the latter by placing the inductance across two reservoirs at different temperatures, waiting until a measurable heat flow was set up, and then instantaneously interchanging the reservoirs. Heat would then momentarily flow back into the higher temperature bath against a temperature gradient in violation of the Second Law. Some further similarity between electrical and thermal quantities can be uncovered. For instance ,rT = R,C,
has dimensions of time and can be properly identified as a thermal time constant. However, the discussion of the Appendix must end on a note of caution. Observe that, for electrical circuits, Power = VI but, for thermal circuits, Power # TQ, since Q is itself power. The analogy between electrical and thermal quantities is then a formal one growing out of the fact that two sets of basically different quantities satisfy equations of the same form.
Fine Control in the Human Temperature Regulation System RONALD W. CORNEWt
Department of Electrical Engineering and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. JAMESC.HOUK~
Department of Physiology, Harvard University, Boston, Massachusetts, U.S.A. LAWRENCESTARK~
Department of Bioengineering, University of Illinois at Chicago Circle and Presbyterian-St. Luke’s Hospital, Chicago, Illinois, U.S.A. (Received 11 October 1966,and in revisedform 4 April 1967) A model is developed for the regulation of temperature in man. The thermal properties of the body are presented in circuit form and simplifkd to eliminate all features not essential to an understanding of control. In this modeling, perspiration is treated as a sourcewhich can pump heat
against a temperature gradient while lung heat loss appearsas a highfrequency alteration in effective heat production. To this simplified thermal circuit areappendedthe principal mechanisms which are available to the nervous system for controlling body heat balance--increased metabolism
as in shivering,
increased
heat loss by perspiration,
and
alteration of thermal conductancethrough changesin blood flow-and a block diagram of the resultingcontrol systemis given. The descriptionof thesemechanismsfollows that of Benzinger and co-workers (Benzinger, 1959; 1961a,b;Benzinger,Pratt & Kitzinger, 1961). To formulate the problem on a feedbackcontrol basis,an error signal representingdeviationsof hypothalamictemperaturefrom a referencevalue t Supported in part by the Joint Services Electronics Program (Contract DA2l3-043AMC-O2536(E)), the National Science Foundation (G&t GK-835), the National Institutes of Health (Grant 2 PO1 MH-04737-06): , and the National Aeronautics and Space Admiuistration‘(Grant NsG-496). $ Supported by a predoctoral fellowship from the National Institutes of Health. $ Supported by National Institutes of Health Grants NE3055, NE3090, MH-06175; Office of Naval Research Nonr-609(39), Nonr-1841(70); Air Force (AF-33(616)-7282 and 7588), AFOSR 49(638)1313; and Army Medical Corps (DA-18-108~405-Cml-942) at the Massachusetts Institute of Technology. 406
FINE
CONTROL
OF
HUMAN
TEMPERATURE
407
or set point of 37°C is de&d. The reference temperature may have no precise natural correlate and its choice seems arbitrary. Since man normally spends most of his time neither shivering or perspiring, particular emphasis is placed upon an analysis of the vascular mechanism by which line control is achieved. A non-linear differential equation governing the system under this mode of control is presented and solved for transient disturbances in metabolic heat production and environmental temperature. The analysis indicates a quantitydependent on level of metabolism and degree of control over peripheral blood flow-which is a measure of the strength or effectivenessof vascular control. Both response time and steadystate error are reduced by this factor, which is estimated to have a numerical value of about 10. From this analysis it is clear that regardless of how additional assumptions might complicate a description of the system, the data on vascular control reported by Benzinger (1959) lead to a model of human temperature regulation exhibiting powerful multiplicative control over heat flow out of the body. 1. A Model of the Human Thermoregdatory System A model for the distribution of temperature in the nude human is the first of four topics to be discussed in this part of the paper. The emphasis will be placed on developing a model that (i) contains enough of the features of the real system to enable one to envision the way in which control is exercised and which (ii) yields workable expressions for the system dynamics. In this it is necessary to achieve a balance between inclusiveness and complexity, a balance which is facilitated by eventual restriction to the fine-control region, where signals are small and response mechanisms limited. 1.1.
THERMAL
CIRCUIT
Problems associated with the complicated geometry of the body are avoided by regarding man as a cylinder of shoulder height and waist-like girth as shown in Fig. 1. Following this simplification attention will be restricted to thermal gradients in the radial direction with leakage from the ends of the cylinder neglected. The best form to assume for this radial dependence of
FIO. 1. Cylinder model of man. The complicated geometry of the human body has been avoided by regarding man 89 a cylinder of shoulder height and waist-like girth.
408
R. W. CORNEW,
J. C. HOUK
AND
L. STARK
temperature has been a matter of long-term speculation. An early thought by Burton (1934) was to treat this cylinder as a distributed parameter system possessing uniform thermal capacity, conductivity, and heat generation. This entailed solution of a nonhomogeneous thermal diffusion equation subject to the boundary condition imposed by skin temperature. Such an approachthough neglecting non-uniformity in the active mechanism of heat transport via blood flow and in the generation of heat throughout the body-has been successful in describing qualitatively the parabolic shape of the static thermal gradient. While distributed parameter models may be extended to predict with sufficient accuracy dynamic changes in temperature as well as the static distribution (the latter has been provided by Wissler, 1961), the effects of control in any such model are difficult to visualize because of the complexity of the ensuing solution. Instead, another approach has evolved-based on division of the cylinder into a number of concentric layers each at constant temperature-which offers considerable advantage in conceptualizing the way in which control is exerted and in facilitating the computation of responses. The history of such lumped-parameter representations of the thermal system goes back to about 1950 (MacDonald & Wyndham, 1950; Wyndham, Bouwer, Devine, Paterson & MacDonald, 1952) and has subsequently been tied to several analog computer simulations of human thermoregulation (Crosbie, Hardy dz Fessenden, 1963; Brown, 1963; Smith & Jones, 1964, and, most recently, Stolwijk & Hardy, 1966u). We will begin with a threelayer model, since it affords considerable freedom in obtaining a good fit to the dynamics of the system and has the advantage of being the least complicated to contain explicitly all of the essential, involuntary control features of the system.? In Fig. 2 this model is shown with the layers selected so that their tempera-
FIG. 2. Three-layer model of man. To simplify analysis, three thermal layers are distinguished within the cylinder model of man. t Crosbie, Hardy & Fessenden-in the first analog computer study referred to abovealso used a three-layer model to simulate the temperature distribution of the body. In this case,computer capacity dictated the number of layers. In another of these studies, Brown used a three-layer model with good success over a liited range of simulation.
FINE
CONTROL
OF
HUMAN
TEMPERATURE
409
tures correspond to the n?ean temperatures of the inner core, the musculature, and the skin layer. Following this the resulting three-layer thermal circuit of a nude human is given in Fig. 3. In this circuit each layer has been given the
+fjtqJjjrgG Inner
core I I I
Muscle
layer
; I
Skin
loyer
, ;
arr%ient
Fro. 3. Thermal circuit of a nude subject. A thermal resistance, capacitance, and heat source has been associated with each of the three thermal layers of man. Their interconnection to form the thermal circuit of a nude subject is completed by the inclusion of a temperature source representing the thermal constraint of the environment and a resistance from core to environment accounting for the flow of heat through the lungs.
ability to store heat through the inclusion of thermal capacitance. Further, thermal gradients between layers are allowed for by the presence of thermal resistance.t The heat source in the inner core represents the heat generated by the metabolic processes in the internal organs ; that in the muscle layer results from muscle tone, shivering, and movement; and the source between the skin layer and the outside represents the perspiration mechanism which, through evaporation, pumps heat out of the body.$ The environment is represented by a temperature source. The description of Fig. 3 is completed by observations on two parameters appearing in it, Rs and RL. The former is a resistance combining the effects of heat loss from the body by conduction to the air enclosing it and radiation to the walls and objects surrounding it. Specifically included is the insulating property of the surface layer of air surrounding the body (Hendler, Crosbie & Hardy, 1958). In using a single, linear resistance it has been assumed that 7 The reader who is unfamiliar with thermal resistance and capacitance and the laws governing their behavior in thermal circuits is referred first to the Appendix and then to any good book on elementary circuit theory. A circuit theory approach will be used extensively here. $ The use, instead, of a sweat resistance (in parallel with CS) as suggested by MacDonald & Wyndham (1950) and Wyndham et al. (1952) strikes us as artificial. With that modeling (i) evaporative heat loss does not reappear in the environment as it should, (ii) the active nature of the evaporative process is lost needkssly and (iii) important experimental data cannot be explained. Under their assumption these authors were forced to conclude “that heat transfer by sweat is unimportant in comparison with that by radiation and convection” which was “an evident source of disagreement with theory”. These diiculties disappear in this modeling.
410
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
the air and wall temperatures are equal and change by an amount small compared with the normal skin temperature of about 300°K. This is a reasonable assumption for the range of temperatures that a nude subject could regulate against (Stolwijk & Hardy, 19663). The resistance R, should also be considered in more detail. It represents the number that relates heat loss through the lungs to the difference in temperature between the inside of the body and the surrounding air. It is apparent that this loss is intermittent, since it depends on the breathing cycle. Thus, modeling the nature of RL to fine detail would require a periodically time-varying resistance. Now one could-since the breathing cycle is short compared to times characterizing thermal responses-immediately replace this with an average value determined over a few breathing cycles. An alternate approach which preserves this time-varying character of R, and allows as well for an evaporative component of lung heat loss will be discussed next. 1.2. SIMPLIFICATION OF THE THERMAL CIRCUIT It is convenient to eliminate the explicit appearance of R, shown in Fig. 3 by allowing its heat flow to be absorbed into Q,, the heat source of the inner core. Noticing that the net body heat contributed by the inner core consists of that generated there less that lost through the lungs prompts definition of a new variable
Qs = QI-CT,- TOY& and alteration of the thermal circuit to the form shown in Fig. 4, in which the path through R, to the outside air has been eliminated. Any evaporative
FIG. 4. Modification of thermal circuit of a nude subject. The thermal circuit of a nude subject has been mod&d here by altering the source of heat within the core to include the flow of heat through the lungs. In addition, the environmental temperature source now includes the drop in skin temperature accompanying perspiration.
component of heat loss through breathing would have appeared in the original circuit as a current source in parallel with RL and would now show up as an additional term subtracted from QP An approximate form,
QB= QrG%-
ToYR,,
FINE
CONTROL
OF
HUMAN
TEMPERATURE
411
is possible since T,, the temperature of the inner core, is stable enough under most conditions to be replaced by T,,, a constant. (This constant will eventually coincide with the reference of the temperature regulation system but no such interpretation is presently necessary.) Thus Q, is independent of any body temperaure which shows appreciable variation; this further justifies its treatment as a single source. The previously noted time variation in R,, if included, would now show up as a ripple in this new heat source. This effect-high in frequency and small in amplitude-is certainly responsible for some of the noise observed in the system. Other sources of noise include diurnal variation in Q, as well as effects resulting from eating, drinking, digestion, and other independent changes in metabolism. Taken as a whole, QB therefore contains long-term drift as well as high frequency noise which together constitute a broad spectrum of internal disturbances against which the system must regulate. Figure 4 includes a second change in which the thermal effect of the perspiration source Q, in Fig. 3 has been absorbed into the temperature source To by the construction of a Thevenin equivalent of everything remaining to the right of C,. This equivalence is not affected by non-linearities which may lie to the left of the dashed bracket in Fig. 4. Incidentally, the new source
T; = To-Q,R, appearing in Fig. 4 may be regarded as an effective environmental temperature; it is the originally defined environmental temperature diminished by the temperature drop of the skin due to perspiration. If an imbalance of wall and air temperatures were allowed in the original modeling, it would appear here only as a change in T& Two assertions about the dynamics of temperature change in the human body will prove valuable: (i) changes in average core and muscle temperature closely parallel one another owing to a continuous exchange of large quantities of blood, at least for small disturbances about thermal neutrality; (ii) changes in skin temperature occur much more rapidly than changes in other temperatures under comparable conditions owing to the greatly reduced thermal mass of this layer. From the first of these it is assumed that RI, the thermal resistance between core and muscle, should be small compared to other resistances in the system. From the second a similar assumption is made concerning C, and the other capacitances of the system. This situation is idealized by taking
RI=&=0
412
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
in the model of Fig. 4. The ultimate test of validity for such assumptions lies in comparison of predicted and experimental transients; the modeling here leads to first-order transients throughout the system.
Q;1= QB+QMH, CB = c,+c&f, R, = R,+R,, r---Skin
cl;,
temperature
$-+-~g
I
1
I
FIG. 5. Simplified thermal circuit of a nude human. This model emerges after placing RI = Cs = 0 and defining the total body heat Qk = QB + &, the total body capacitance easily approxiC, = Cl + CM, and the total body resistance RB = Rr + Rs. It pos~esas mated parameters with, for a 150 lb man, 100 w < @a < 200 w or more, depending upon the degree of muscular activity, C, = 200,000 Joules/T, and RB = 0*12”C/w. The value of RB varies greatly, depending upon the vascular state of the body. The specik heat of the body has been taken as 70% that of water in the capacitance calculation.
Figure 5 shows this considerably simplified model. Here are, respectively, the total effective body heat and the total remaining capacitance and resistance. T,, the inner core temperature, is the only body temperature remaining explicitly in this model although T,, the skin temperature, can be simply recovered if desired by noticing T, = ( TI - T&)R,/R,.
From Fig. 5, the differential equation governing the thermal system in the absence of control can be written by balancing the flows of heat at the upper node as Q~-cB!$-(EgT2=(-). B
This can be rewritten as R&B
‘2
+ TI = RBQ;P+ Th.
In the absence of shivering and perspiration which really lie beyond the range for which this model has been constructed, the primes on the variables of the right side of this equation may be dropped.
FINE
CONTROL
OF
HUMAN
TEMPERATURE
413
1.3. MECHANISMS OF CONTROL The simplified model of Fig. 5, describing the flow of heat through an idealized body, resembles a well-stirred tank possessing a source of heat and a large thermal mass, both interior to an insulating wall in contact with an external environment. At this point the elements of control will be added. Recent work by Benxinger and others (Benxinger, 1959; 1961a,b; Benxinger, Pratt & Kitzinger, 1961) demonstrates quantitatively that a key reference of the system is located in the hypothalamus, a neural structure within the underside of the brain. The controlled variable of the system is the mean inner core temperature as attained from the hypothalamic blood supply which is derived largely from the Circle of Willis, an arterial ring Iocated beneath the hypothalamus. The difference between the reference temperature (called T,, here and typically taken to be 98~6°F = 37.O”C) and the controlled temperature (TI) is the error signal of the system (AT,); that is, AT, = T,, - TI. Depending upon the sign and magnitude of this quantity, combinations of at least the following three control mechanisms can be brought into play: (i) Low fenrperutute re@un. The body reacts to low temperatures by creating more heat. This is accomplished by increased muscular tone, shivering, and voluntary activity (grouped as QM earlier). The heat generated by the first two of these is related to the error signal in Fig. 6(a) (Benzinger et al., 1961). The heat generated by the third, voluntary muscular activity, will receive no further attention here, since it is not predictably related to the error. It will later be argued that this component-while sometimes purposely employed to fight cold-is better treated as a disturbance to the system. The data of Fig. 6(a) is presented as a set of curves parameterixed in T,, the skin temperature. A major eEect is the downward shift with increasing skin temperature of the point at which shivering is initiated. An additional effect-variation of gain from one skin temperature to another-is also present. (ii) High temperature region. In this region the body loses heat by perspiration (called Qs earlier). The relation between this heat loss and error is given in Fig. 6(b) (Benzinger, 1961b). The data assumes that humidity and air velocity about the body are not subject to variation. Once again the curves exhibit a shift in set point with skin temperature-a downward change in skin temperature being accompanied by an upward change in the hypothalamic temperature at which perspiration begins. In this case gain variation is less evident. T.B. 2s
414
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
(iii) iVeurru2 temperature region. Since substantial heat is carried from core to skin by blood flow, fine control can be achieved by regulating the size of the blood vessels in the periphery of the body. Detection of small errors to the cold side of the reference (i.e. a positive AT,) results in constriction with its accompanying conservation of heat. Small errors to the warm side results in dilation with subsequent loss of heat. This vasomotor action is modeled by allowing variations in R,, the total body resistance. In Fig. 6(c) (taken from Benzinger, 1959) this variation has been approximated by a piecewise-linear curve. Since this alteration in resistance introduces a non-linearity which is purposefully employed to reduce the magnitude of error, the thermoregulatory system is seen to exhibit adaptive control. Unlike the control mechanisms at work for high and low temperatures, Benzinger has found no important dependence of this mechanism on skin temperature.? A more detailed discussion of the error-reducing power of this mechanism will be the topic of the second part of this paper. In presenting the data given in Fig. 6 and described above, Benzinger has been careful to point out that all measurements were made in the steady state and may not be valid dynamically. Nonetheless, it is often assumed that these control mechanisms are inherently faster than thermal responses in the remainder of the system. Evidence exists to indicate that this is true for the vascular mechanism which we are particularly interested in studying here1 and allows application of the steady-state measurements to the dynamic control situation in this case. Failure of this assumption for the other mechanisms would necessitate inclusion of appropriate dynamic elements into their control pathways. 1.4.
BLOCK
DIAGRAM
OF THE THERMORBGULATORY VASCULAR
SYSTEM
UNDER
CONTROL
The discussion above has re-presented Benzinger’s data which quantify the concept that involuntary control of body temperature results principally 7 This importance of skin temperature in determming vascular responses has been a matter of considerable discussion. Rawson & Randall (1961) make it abundantly clear that vascular responses to regional heating of the body surface exist but others, such as Wyndham (1965), find these effects several times less important than core variations at normal skin temperatures. The lack of quantitative data to the contrary and our self-imposed restriction to the region of thermal neutrality where skin temperature variations are not great leads us to exclude such effects totally. $ Investigation with an unanesthetized dog subjected to periodic step-changes in hypothalamic temperature reveals that vasomotor activity as measured by temperature changes in the ear pinna is at least several times more rapid than changes in deep body temperature at the rectum or back brain (Hammel, Stromme & Comew, 1963). This is no doubt more pronounced in man where thermal masses are larger and passive system response times longer.
FINE
CONTROL
OF
HUMAN
415
TEMPERATURE
from the action of three mechanisms: increased metabolic activity, increased evaporative loss, and vasomotor activity. Of these, the first two are class&cl here as gross control mechanisms functioning for large errors. The last, vasomotor activity, works to achieve fine control in the presence of small errors. The graph of the vascular control mechanism given in Fig. 6 can be coupled with the earlier differential equation governing heat flow in the body
-AT,* Hypothalamic
-Ar, temperature
PC)
--I
Hypothalamic
(0)
g,z
I
oc
PC1
W
3 40 % +=f 30 8 s s.f gE 8 2
temperature
l L-. ’
20
.
IO
lL
\
t_l
2.2
37.0
37.6
-AT,-+ Hypothalamic
temperature
PC)
(Cl
Fro. 6. Human themoxegulatory responses. These graphs, adapted from Benzinger (1959, 19616; Benzinger et al., 1961). show the particular way by which skin and hypothalamic temperature determine (a) muscular heat production (largely shivering), (b) rate of sweating, and (c) body resistance. In the first two of these, the curves given by Buuinger have been wed to approximate the behavior of the actual data. The last graph contains his or&al data after inversion and has been approximated by a straight lie having a fractional change of resistance of 2.4/“C. It is emphasized that only the internal, or variable component of Rs (roughly 33 yO of the total) is shown here. AT,, the deviation from 37°C = 98.6”F is included at the bottom of all graphs and will later be identified as the error of the system. Measurem~t of hypothalamic temperature was made at the tympanic membrane in all casea.
to yield a block diagram of the thermoregulatory system under vascular control (see Fig. 7). Here the symbol p has replaced d/d& the time derivative operator, to represent in compact form the dynamics of the system being controlled.
416
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
It is noted from Fig. 7 that there are three inputs to the thermoregulatory system. These would be the same regardless of the control mechanism under study. Two, net body heat contributed by the core (QJ and environmental temperature (To), take the form of inputs to the system from inside and 0
Summation
0
Multiplicotlon
Neutral
temperature
region
7. Block diagram of temperature regulation system in man under vascular control. The graph of RB shown for the neutral temperature region includes both variable and nonvariable components of that resistance and gives rise to multiplicative control. Qa, net body heat, and TO, environmental temperature, are inputs to the system from inside and outside the body. T,, (delined as 37°C = 98.6”F) is the reference of the system. The letter p symbolizes the time derivative operator, d/dt. FIG.
outside the body, respectively. Changes in these inputs from their equilibrium values constitute disturbances to the system. Specifically included are digestive heat and diurnal variation in basal metabolism, voluntary muscular activity, and changes in environmental temperature arising from alteration of air or wall temperature. The third input is the reference of the system (T’,) and can be disturbed by, among other things, fever inducing agents. 2. Fine Control Resulting from the Vasomotor Non-linearity
Because of the complicating effect of skin temperature and the possible interaction of the regulatory mechanisms the best approach to understanding the entire thermoregulatory system would probably be computer simulation guided by experiment and analysis. Crosbie, Hardy & Fessenden (1963) and -more recently-Stolwijk and others have tried this approach using an analog computer. The final word concerning their success has not been heard.7 However, by restricting attention to the neutral region where the t After submission of this manuscript the comprehensive article of Stolwijk & Hardy (1966u) which presents a much more detailed model than that used here (together with its analog simulation) was called to our attention. It seems appropriate to comment on that moddinrclationtoonrown: (a) The region of solution of the thermoregulatory problem is very different, it being our intention to present an analysis of line control where neither increased metabolic activity nor evaporative loss are important. In contrast, Stolwijk and Hardy’s simulation is validated against the latter responses; i.e. Benzinger’s ice ingeatlon
FINE
CONTROL
OF HUMAN
TEMPERATURE
417
vasomotor mechanism may act alone, it is possible to go into the analysis of some salient features of fine control. From common experience we know that, with the aid of clothing, most of our time is spent in this region neither shivering or perspiring. Specifically, Benzinger’s data shows that when the inner core temperature, T,, is at or near 37°C (98~6°F) there is a range of skin temperature, T,, from about 27 to 32°C (roughly 80 to 90°F) for which the vasomotor mechanism acts alone and is, furthermore, independent of skin temperature. While this formally restricts the analysis to a narrow range of skin and hypothalamic temperature (the latter no larger than, say, *OSl’C), it should be pointed out that such vascular responses underlie the entire region between 36.7 and 374T and an analysis of them provides a lower bound to regulatory success throughout this region. Moreover, whatever &lays exist in the shivering and perspiration mechanisms further widen the range of initial errors that can be accommodated solely by the vascular mechanism. In short, we seek an analysis of thermoregulation which is valid when the core-temperature dependent vascular mechanism acts alone. experiment (1959) resulting in evaporative loss and Park and Palme’s typhoid vaccine experiments resulting in increased metabolism are the offered comparisons of the model with experimental evidence. (b) The description of vascular control employed by these authors differs signiticantly from that used here. Citing results of a recent paper, where it was concluded that evaporative losses can be correlated with a product of deviations in skin and central temperatures from reference values (Hardy & Stolwijk, 1966), these authors genemlize to a similar control law for all thermoregulatory responses under study, including the vascular. No direct experimental verification of this multiplicative dependence on skin temperature deviation is given for these other responses and is, of course, absent from the Benzinger data and hence our model. The models agree in assuming that vascular responses, however they may be controlled, are essentially instantaneous and proportional. (c) Smce they were utiliig analog techniques, Stolwijk and Hardy were able to study a more complicated passive model involving three distinct cylinders and a separate compartment for blood. It would he valuable to have a quantitative estimate of how much thls added complexity influences the accuracy of control, particularly since the system is known to contain much smoothing. It is particularly important that such simulation studies do not discourage simpler models which seek to describe the system in more limited regions. Though outside the vascular response range for which it was prepared, the model used here can, for instance, yield the oppositely moving skin and tympanic temperatures of the Benzinger ice ingestion experiment if 1 AQs IiF’---Rg - Rs’ I a condition which can be met if central control of sweating is strong enough. For that matter, reintroducing Cs in our model (or placing delay in the evaporative loas echanism) will produce the observeddelay in onset of the rise in Ts: its shortness %ally argues for the deletion of both such effects in pesxil and paper studies.
418
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
2.1. DJFFJZRENTIAL EQUATION OF VASOMOTOR CONTROL By reference to Fig. 6(c), it is seen that the variation of body resistance, R,, with error (ATr or henceforth, simply E) may be described by R B = R&l +a&); -0*4”C < E < 0*3”C, (2) where R,, is the value of RB in the absence of error and a, the strength of control, is a constant determined by the product of two ratios: (i) the fractional change of the resistance with temperature as determined from a straight-line approximation to the data of Fig. 6(c); (ii) the fraction of body resistance undergoing such change. The second fraction is not unity, since RB has been defined to include the constant resistance of the layer of surface air about the body which was not included in the previous experimental data. If desired, one thermal effect of clothing could be incorporated at this point by adding a constant resistance to RB in equation 2, thereby increasing RBo and decreasing a. Clothing therefore reduces the variability of RB and, while giving more insulation, limits control.? The differential equation for body temperature which was obtained in the first part of the paper [equation (I)] may be rewritten in terms of the error e as
&G$ + E= Go-G+R,Q,), since the reference T,, is assumed constant. [The primes on To and QB have been dropped since perspiration (Q,) or shivering (QJ have been eliminated by the definition of the region under study.] Substituting RB from equation (2) into this relation and bringing all terms in E to the left yields a differential equation of vasomotor control valid for any properly bounded initial error:
R,oC,(1+4 2 + Cl+aR,oQ&
= T,, - ( To + b,Q,h.
(3)
2.2. STEADY-STATE SOLUTION The steady-state error in the presence of control may be determined from ds equation(3) by placing - = 0 and solving for E = sSS, dt
%s=
T,o - (To + ReoQd 1+QR,oQ, -
This equation gives the steady-state error for each set of values for To and t It is well known that one may wear too much clothing on a cold day to effectively control body temperature with the uncomfortable result of placing the body in a limit cycle between shivering and perspiration. However, purposeful clothing changes can be looked upon as yielding even more control of Rg. Clothing, of course, also inWoduces an additional thermal capacity that helps to smooth out brief changes in environmental temperature.
FINE
CONTROL
OF
HUMAN
TEMPERATURE
419
Q,, the environmental temperature and net body heat, which as previously noted constitute the inputs of the system. In the absence of control (a = 0) the corresponding error would be ~so = Txo -(To+R,o&)Thus
%SO Qs = 1+aReoQe' from which it is seen that the error-reducing power of vascular control is 1 +aR,,Q,, a number necessarily greater than 1. The range of disturbances which can be accommodated without necessity of shivering or perspiration is thereby extended by this same factor. A numerical estimate of the error-reducing power of vascular control can be obtained. From Benzinger’s data which yield a value of 2.4 for the fractional change of resistance with temperature and from Burton’s (1934) measurements of thermal circulation index, which show that as little as 33 % of RB may be variable, a is determined as 0.8. A value of RBoQB can be estimated by noticing from equation (4) that R,oQ, = T,o- To, where the term on the right is the temperature gradient from core to environment when the steady-state error is zero. Rough estimation suggests thatfor a nude subject-this would correspond to an environment temperature of about 25°C (77°F) for which the thermal stress from core to environment is 12°C. From these considerations an error-reducing power results which is slightly greater than 10. While this is only an estimate, a value of this magnitude is to be anticipated in view of the ability of the system to render environmental changes of several degrees into core errors of only a few tenths of a degree (Hardy & DuBois, 1938). From equation (4), the steady-state error is linearly related to To; however, its dependence on QB is not so simple. In any case, the transient after a disturbance in either of these quantities is nonlinear and will be described below. 2.3. TRANSIENT SOLUTION Despite its non-linearity it is possible by separation of variables to obtain an exact solution to equation (3) for the error transient resulting from any initial error .ao. The only provision-as discussed in the next paragraph-is that QB and To undergo no further change beyond the time when so is determined. To accomplish the solution this equation is rewritten as (1 +a&) z-de+dt=O, (8 - %s)
420
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
where RBOCB
Z= 1+
@I&B
is a characteristic time for which an interpretation will be given in the next section. Solving for dt and integrating from 0 to t as Egoes from e. to Eyields t&) = (1 + ass) In 80 - - %S + ho - 4, (5) E--ESS which is explicit in the normalized time tN = t/T. The second term on the right side of this equation destroys the possibility of obtaining a useful solution explicit in s.
Controlled
w 7‘
(a)
f
Controlled
(b)
FIG. 8. Typical error transients with and without vascular control. The solid curves of (a) and (b) represent oppositely-directed responses of the model in the presence of vascular control while the dotted curves would result if such control were absent. In each case, the initial error is taken as zero. The disturbances, which could be caused by previous changes in either QB or To, are equal and opposite and have ken chosen large enough to produce shivering or perspiration in the absence of control. Comparison shows a s&able reduction in steady-state error and response time in both cases, Since the strength of control 1 + ccRBoQB is independent of To, but not QB, a pair of responses to equal but opposite alterations in To will possess equal but opposite steady-state errors (reduced about 10 to 1 from the uncontrolled level-see text), while responses to changes in Q0 will be unequal. The time courses of the responses under vascular control are always unsymmetric.
FINE
CONTROL
OF
HUMAN
TEMPERATURE
421
A remark about how transient errors arise is appropriate. It has already been noted that changes in the inputs to the system-particularly environmental temperature and net body heat-constitute disturbances. Such disturbances may be gauged by the effect they have in changing core temperature from equilibrium; i.e. in producing steady-state errors. Should either of these inputs change, the system will undergo a transient in passing from one steady state to another. We have sacrificed description of the system during the course of the input change focusing instead on the lingering time course of the error after the input reattains a constant value. In the special case of sudden changes in an input, the transient is simply a step response of the system, and the previous steady-state error becomes the initial error of this transient. That the initial error E,, is sufficient to summarize the input history of the system is apparent from the solution obtained above and results from its assumed first-order dynamics. Figure 8(a) contains a plot of such a transient (solid line). Regrettably, an experimental response suitable for comparison (i.e. resulting only from vascular action) is not available in the literature. We predict a response which is altered from the time course of a simple first-order, linear system by the generally smaller second term of equation (5), which can speed up or slow down the response depending on its sign [as in, respectively, the solid lines of Fig. 8(a) and (b)]. To illustrate the importance of control, we have included the transients which would accompany the same two disturbances in the absence of control (dotted lines). Comparison evidences both a reduction in steady-state error, as determined in the previous section, as well as a reduction in response time, to be discussed next. 2.4. REDUCTION IN RESPONSE TIME In the absence of control (a = 0) equation 5 can be solved explicitly for the error 40 = %so- (%s, - 4 exp ( - WdAJ~ which is recognized as the transient of a first-order, linear system having a time constant r0 = RBo C,, and was plotted in Fig. 8. (The same result could have been derived directly from the single-layer model of Fig. 5 with RB held constant at RB,,.) The characteristic time of the system in the presence of vascular control (defined in Section 2.3) may now be expressed as R,BC, ‘=
l+(rRBOQB
70
=1+aRmQB9
a form which demonstrates the ability of this mechanism of control to hasten the reattainment of a steady state. Thus, 1 +aR,Q, becomes the factor governing both response time and error reduction in the fine control region.
422
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
However, in contrast to the linear case where t = r0 implies 63 % completion of the transient independent of its initial and final values, no fixed percentage of the non-linear transient is complete when t = z; i.e. fN = 1. Nonetheless, a sizeable reduction in response time is indicated-typically from several hours down to a fraction of a single hour. Computation of the response time of the body based on the thermal properties of tissue in a nominal, but uncontrolled, vascular state yields a figure of 7 hr for a 150 lb human subject. (See the caption of Fig. 5 for values of R, and C,.) In the presence of control, the response time should be reduced to about 2/3 of an hour. 2.5. FURTHER DISCUSSION: APPARENT NON-EXISTENCE OF A UNIQUE REFERENCE The reader should observe that the discussions of this part of the paper did not invoke the concept of a unique, physiologically delved reference. All that could be noticed under hypothalamic vascular control was an ability of the system to minimize the effects of imposed changes. In fact, since R, as a function of TI is known [Fig. 6(c)], equation (1) could be solved for the final TI that prevails for each set of values of QB and To. This would have been one solution to the steady-state aspect of the vascular control problem and would not have involved the concept of feedback. The fact that the system does evidence tight regulation in this region tempted us to look for an operational definition of a reference so that it could be put into feedback control form in the first part of the paper and so that a quantitative measure of the changes in core temperature resulting from disturbances could be given in the second. Apparently the condition used, that R, = RBo when s = 0, could serve to defme the reference of the system as any core temperature within the region of vasomotor response. That is, T,, can be taken anywhere in the vasomotor region as long as RB is taken to be Rso there. Convention prompted selection of 37°C = 98.6”F as the reference, roughly the mid-point of the vasomotor response region. Now this raises several questions. (i) Does not the intersection of the heat generation (Q,) and heat dissipation curves (QJ yield a unique reference as claimed by Benzinger et al. (1961)? Evidently it does not, since these curves possess no unique intersection for a common T,! Instead a band of temperatures of 0.5”C or more is enclosed. As mentioned above, man spends most of his time in this region where changes in vascular resistance provide the only control of core temperature. Now it may be acceptable to look upon this band of temperatures as the target of control in a gross sense, but it is certainly not a unique reference.
PINE
CONTdOL
OF
HUMAN
TEMPERATURE
423
(ii) How can a reference which is operationally defined be shifted by, among other causes, pyrogens? In fever, the entire band of temperatures defined above (with TIo operationally defined somewhere in it) moves up and down. Incidentally, the time course of this motion itself contains little information about natural temperature regulation, although, if quick enough, can serve as a useful input in studies of the regulatory system as shown by Stolwijk & Hardy (1966u). (iii) Could a natural reference exist but be unmeasurable? One answer to this is another question: By what attributes would such a reference be distinguished? The feedback approach to the analysis of biological systems has many advantages, but a consequence of taking this point of view toward a system is the introduction of concepts, such as reference, which may have no natural correlate. 3. Conclusion In the application of mathematical techniques to the study of biological systems there is always need of a model which is at once simple enough to support analysis and yet not m&e. By selecting a very simple model for the human temperature control system, one in which anatomical features such as the neural pathways from hypothalamus to effector mechanisms are not considered and in which the passive thermal properties and active heat sources of the body are treated as lumped, it has been possible to formulate the system on a feedback control basis and go some distance into the analysis of the salient features of vascular control. More exacting assumptionstreating the system as distributed, or reintroducing its geometric complexity as by increasing the number of cylinders, etc.-would lead to greater depth of description at the expense of increased complexity of analysis. Regardless of how these assumptions might complicate a description of the passive system, the data on vascular control reported by Benzinger (1959) leads to a model of human temperature regulation exhibiting powerful multiplicative control over heat flow out of the body. This contrasts with the situation for larger errors where his data on increased metabolism (Benzinger et al., 1961) or evaporative loss (1961b) shows roughly proportional control with a parametric dependence on skin temperature. In consequence, the response to an initially large error should depart from that of a linear system as error decreases-a behavior which would not be well described by, say, small-signal frequency analysis using sinusoidal excitation. Step responses, which are easily analyzed and more physiological, expose the properties of this system better. A non-linear analysis which is valid for such steps and
424
R. W. CORNEW,
I. C. HOUK
AND
L. STARK
other transient disturbances in metabolic heat production and environmental temperature has been presented here; this analysis demonstrates that vascular control diminishes the ensuing perturbation in core temperature by about 10 to 1.7 In spite of the apparent absence of a precise reference, the feedback approach remains useful because it focuses attention on incremental deviations of the system about a steady state. REFERENCES BENZINGER, T. H. (1959). Proc. natn Acad. Sci. U.S.A. 45, 645. BENZINGER, T. H. (1961a). Scient. Am. 204,134. BENZINGER, T. H. (19616). Proc. natn Acud. Sci. U.S.A. 47, 1683. BENZINGER, T. H. (1964). Symp. Sot. exp. Biol. 18,49. B~N~~NGER, T. H., PRATT, A. W. & KITZINGER, C. (1961). Proc. natn Acud. Sci. U.S.A. 47, 730. BORN, M. (1948). “Natural Philosophy of Cause and Chance”. Oxford: Clarendon Press. BROWN, A. C. (1963). ASTIA, AD428144 BURTON, A. C. (1934). J. Nutr. 7,497. CROSBIE,R. J., HARDY, J. D. & FESSENDEN,E. (1963). “Temperature, Its Measurement and Control in Science and Industry”. New York: Reinhold. HAMMEL, H. T., ANDERSON,H. T., JACKSON.D. C. &HARDY, J. D. (1961). ASD TR 61489. HAMMEL, H. T., STRBMME, S. & &RNEW, R. W. (1963). Life Sckces, i, 933. HARDY. J. D. & DuBors. E. F. (1938). J. Nutr. 15. 477. HARDY; J. D. & %OLWIiK (1966). J. ;?ppl. Physiol: 31, 1799. &NDLRR, E., Cnosme, R. & HARDY, 3. D. (1958). J. uppI. Physiol. 12, 177. bfACL)oN.UD, D. K. C. & WYNDHAM, C. H. (1950). J. appl. Pkysiof. 3,352. RAWSON, R. 0. & RANDALL, W. C. (1961). J. upid. Physiol. 16, 1006. SMITH, P. E., JR. & JONES,E. W., END (1964). Archs e&r. H&h, 9, 332. STOLWIJK, 3. A. J. & HARDY, J. D. (1966~). Ffliigers Arch. ges. Physiol. 291, 129. STOLWUK, J. A. J. & HARDY, J. D. (1966&. J. up&. Phydol. 21,967. WISSLER, E. H. (1961). J. appl. Physiol. 16, 734. WYNDHAM, C. H. (1965). J. uppl. Physiol. 20, 31. WYNDHAM, C. H., B~UWER, W. v. D. M., DEVINE, M. G., PATERSON,H. E. dc MACDONALD, D. K. C. (1952). J. appl. Physiol. 5, 299.
Appendix : Thermal Andogies This Appendix will discuss a useful analogy which is employed throughout this paper. Because of similarities in the form of thermal and electrical equations, temperature (2”) and voltage (V), as well as heat flow (Q) and current (I), have been taken as analogous quantities. This selection allows t Certain other advantages result from possession of a vasomotor response, rather than dimct regulation of body heat production, as a principal mode of control. The Srst is a matter of economics. In the cold, vasoconstriction increases the insulation of the body making batter use of the heat which is available. The second is a matter of priority. In warmer environments the processes which generate core heat could not reasonaMy be turned down at the beck and call of the temperature control system-they are at least as vitalasitis.Athirdisspeed,vascularchangeabeingmorerapidthanchangesincoreheat production. All other things being equal, the system with vasomotor control would be expeded to operate with a smaller mean variation in core temperature than if slower metabolic changea were relied upon for t?ne control.
FINE
CONTROL
OF
HUMAN
TEMPERATURE
425
thermal resistance (Rr) and thermal capacitance (C,) to be deflned in the same way as the corresponding electrical quantities. (A subscript “T” will be used in this Appendix to denote thermal quantities which might be confused with their electrical counterparts.) First consider the Fourier Law of Heat Flow which, in one dimension, is Q = k,Adq, where = heat flow (in direction T(x) = temperature distribution = cross-sectional area of A kT = thermal conductivity. Applying this to a homogeneous uniform thermal resistivity pT (=
Q
of temperature gradient) as a function of x structure in question slab of cross-sectional area A, length Z, and l/k,) yields
Q=-$, where T is the temperature difference between the ends of the slab. Solving for T yields the thermal form of Ohm’s Law: T = RTQ, where PT1 R r=- A defines thermal resistance in a manner analogous to electrical resistance. In practice R, often has the dimensions of “C/w, since T is usually in “C and Q is frequently given in watts. Thermal capacitance (C,) is defined by the relation
where H denotes heat accumulation of a body over time (the time integral of heat flow Q) and T is the accompanying difference in temperature. This is analogous to the electrical definition of capacitance as the charge accumulated per unit voltage. Thermal capacitance has the dimensions of Joules/T in the system of units established in the previous paragraph and may be considered constant throughout the range of temperatures for which this modeling is intended. Recalling that resistance, capacitance, and inductance constitute the set of commonly encountered, two-terminal passive electrical elements prompts a
426
R.
W.
CORNEW,
J.
C.
HOUK
AND
L.
STARK
question about the def%tition of thermal inductance. However, to the best of anyone’s knowledge, nature has provided no phenomena which require the existence of thermal inductance for satisfactory explanation. A “thought” experiment reveals that its absence is as fundamental as the Second Law of Thermodynamics which assures, in the words of Born (1948), “in no way can heat be entirely converted into work or raised to a level of higher temperature”. If thermal inductance did exist it would be possible to do the latter by placing the inductance across two reservoirs at different temperatures, waiting until a measurable heat flow was set up, and then instantaneously interchanging the reservoirs. Heat would then momentarily flow back into the higher temperature bath against a temperature gradient in violation of the Second Law. Some further similarity between electrical and thermal quantities can be uncovered. For instance ,rT = R,C,
has dimensions of time and can be properly identified as a thermal time constant. However, the discussion of the Appendix must end on a note of caution. Observe that, for electrical circuits, Power = VI but, for thermal circuits, Power # TQ, since Q is itself power. The analogy between electrical and thermal quantities is then a formal one growing out of the fact that two sets of basically different quantities satisfy equations of the same form.