a mathematical biothermal model of the california sea lion

J Thermal Btoloy', 1975 Vol 1 pp 35 to 45 Peryamon Press Printed m Great Britain

A MATHEMATICAL BIOTHERMAL MODEL OF THE CALIFORNIA SEA LION*

RICHARD H LUECKE, V NATARAJAN AND FRANK E SOUTH

Department of Chemical Engineering, Dalton Research Center and Department of Veterinary Anatomy and Physiology, Umverstty of Mtssoun, Columbm, M~ssourL U S A

(Recetted 1 April 1975)

Abstract--The thermal behavior of the California sea hon ts modeled by a set of seventeen simultaneous parttal dlfferentml equations Numerical solution of these equations y~elds temperature profiles m the sea hon that closely reproduce experimental data obtained with the animals at rest m a~r The model predicts that exercise ~s necessary to mamtam body temperature whde ~mmersed m very cold water (0°C) but that very efficient heat conservatton occurs during dwmg because of bradycardm, hm~ted c~rculat~on, and absence of resp~ratton

INTRODUCTION

MARINE mammals face particularly severe thermal challenges from their environment An ammal dwmg m the ocean encounters not only the lamudmal and seasonal temperature changes in oceanic waters, but also passes through a vertical stratfficat|on of w~de- ranging temperatures on a very short term bas~s. Whde hauled out on land, seals and sea hons also encounter broad seasonal and dmrnal changes m temperature The study of the thermoregulatory system of these mammals during both surface act~wty and diwng could be of considerable importance to the un- derstanding of thermoregulat~on of mammals m general

Most physiological control systems are very complex and are difficult to describe with an intumve approach Prediction or interpretation of experimental results often revolves separat|on of many interact- mg factors. Although thermoregulatlon has this characteristic m common w~th most phys|ological control systems, ~t has the d~stmct~on that parts of the overall system obey relatwely simple physical laws which can be readdy hnked together m quantltatwe form. Thus, thermoregulatory systems lend them- selves to the development of mathematical models which are useful m the mterpretation of experimental data, the design of experiments and prediction of phy- stologlcal responses m situations where it would be d~fficult to make experimental measurements

Modeling has found extensive use in the physical and biological sciences. Apphcations of thermal modehng to humans preceded apphcattons to other animals m the development of space suits (Chambers et al , 1973, Morgan et at., 1970; Stolwljk & Hardy, 1966; Wissler, 1961, 1964), and for research and analysis of heat tolerance m deep mining envtron- ments (Wyndham & Atkms, 1968) Other.problems have caused a recent surge of interest m this field; a summary by Shltzer (1973) of blothermal mammal models hsts 87 pubhcatlons, most of which appeared between 1970 and 1972

* Supported by NSF Grant GB-42761 and the John M Dalton Research Center

Only recently have modehng techmques recewed apphcat~on m other ammal systems Mathematical models were used to study thermoregulatory responses in hibernating marmots (Luecke et al., 1970, Luecke & South, 1972). Models have also been used m evaluation of the effect of heat exchange on domes- tic anmaals such as dairy cattle (Hahn et al., 1971); a recent study has more exotic lmphcaUons (Spottla et al , 1973)

In the present study, a dlstnbuted parameter un- steady state mathematical model is developed for the thermal behavior of the Califorma sea hon (Zalophus cahformanus) whde it is swtmmmg, dwing, and hauled out on land The form of the model consists of four cyhnders which simulate the head, trunk, fore-flippers, and hind-flippers Each cyhndncal section is dwided into four concentric layers core, muscle, blubber, and skin Both conductwe and convectwe heat exchange take place m the model; conductive heat exchange radially through the layers and convective heat exchange via the blood flow The coupled set of seventeen simultaneous partml dlfferentml equations generated by thls arrangement ts solved numerically on the digital computer Many of the needed physiological parameters, such as local metabolic heat generation rates, local blood flow rates, etc., are not avatl- able so that approximations for many of these system characterist~cs are taken from analogous structures m other mammals The major source of these approximations is the human model for which relatively abundant data ~s available

Much general biothermal reformat|on has been pubhshed about marine mammals (Andersen, 1969; Irwng et al., 1935, Matsuura & Whittow, 1972; Pierce, 1970, Rldgeway, 1972, South et al., 1973, 1975, and Whlttow et al., 1972), but no comprehensive mathematical model mcorporating th~s reformation mto a smgle representation of a marine mammal has appeared The model described here, to be used m a general study of thermoregulat|on and sleep m sea hons, has already proved useful m predicting and m- terpretmg experimental data. It also lends itself to extrapolation to other marine mammals Structural variations can be incorporated and, because of the

35

36 RICHARD H LUECKE. V NATARAJAN AND FRANK E SOUTH

OVERALL MODEL CONSIDERATIONS

The head and trunk of the Cahfornm sea lion form an elongated ovate elhpsold (Fig 1) The flippers extend so that the flattened parts divide naturally into trmngles con- nected by short cyhndrlcal hmbs to the body Elhpsoldal and triangular shaped models, however, generate excessi- vely complicated mathemaucs, a good representauon of thermal behavior can be obtained from cyhndrlcal approximations (Stolwtjk, 1972, Wtssler, 1964)

Separate cylindrical secttons were spectfied for the head, trunk, fore-flippers and hmd-fltppers with each of these sections bemg &wded into four concentric layers (Fig 2) (1) a core containing the organs, (2) a layer of muscle, (3) the blubber, and (4) the skin and fur The weight frac- tions of the four cyhndrtcal sections are computed on the basts of the measurements taken from two sea lions used as experimental subjects (Table 1) A surface area-weight relationship for seals and sea hons are given by Irving et al (1935)

A = 800 W 2/3 (1)

where A is surface area in cm 2 and W Is weight m kg The predtcted surface area was within 1% of the measured values for two experimental animals

The skm layer was taken as 2 mm thick as measured on a sea hon carcass Since weight distribution data for other sections of the animal are not available, values were adapted from other mammals, for the most part from the human model Sea lions have proporttonately more blubber than the average human has fat From measurements taken from anatomical drawings (Rldgeway, 1972), and on part of a carcass, it was estimated that 20°//0 of the ammal weight is blubber as compared to 15% fat m the human The

Fig 2 Cross-sectional view of a section Layer I, central core, Layer II, muscle, Layer III, blubber, Layer IV, skin, RI, R2, R3, R4-radn of core, muscle, blubber and skin,

respectively

mass and physical dimensions of the various compartments are given in Table 2

A central blood compartment must be deducted from the core weight of the trunk, head, fore, and hmd fltppers Although such a blood compartment has no clearly defined anatomical counterpart, the mass of blood in the large artertes and veins can be taken as a homogeneous mass at a single weighted average temperature because almost all of the blood heat exchange occurs m captllary beds For the human model, Stoiwtjk & Hardy (1966) assumed a volume of 1350ml for the central pool; 300ml for the heart, 375 ml for the venae cavae, and 100ml each for the carottd arteries, jugular veins and bronchtal arteries and vems The total blood volume of the sea hon ts 50°,/0 greater than that of an eqmvalent human (Andersen, 1969) but it is rattoed among sections m the model on the same basis In humans

INTERNAL HEAT TRANSPORT AND GENERATION UNDER BASAL CONDITIONS

The two impor tan t mechamsms for internal heat dis tr ibut ion are &rect conducuon through the Ussue and convect ion via the blood flow. Conductive heat flow Is assumed to occur only m the ra&al direction, axial temperature gra&ents are neglected with httle heat passing between different secUons by this mechamsm Physical properties are taken to be

Name Sex Weight Length

Trunk Head Fore-flippers Hmd-fhppers

Circumference Trunk (ave) Head (ave)

Average width Fore-flippers Hind-flippers Surface area Trunk Head Fore-flippers Hind-flippers Total

Table 1 Phystcal measurements of sea hon

Joy Female 31 5kg

94 5 cm 22 0 cm 3if0 cm 300cm

500cm 200em

150cm 8 0cm

4725 0 cm 2 440 0 cm 2

1800-0 cm 2 960 0 cm 2

7925-0 cm 2

59 62% 5 55%

22 72% 1211%

1o~oo%

Noel Female 31 6kg

95 5 cm 22 0 cm 300cm 300cm

490cm 20 0 cm

15 5cm 80cm

4679 5 cm 2 440 0 cm 2

18600cm 2 960 0 cm 2

7939 5 cm 2

58 94% 5 54%

23 43% 1209%

10000%

Mathematical biothermal model of Cahforma sea hon

Table 2 Anatomical &menslons of the ammal of body weight 31 413 kg

37

No of &fferentmt Weight Radms--thtckness layers used m

(kg) (era) computation

Head 2 033 5 560 80 Core 1 479 4 695 25 Muscle 0 264 0 368 25 Blubber 0-209 0-297 25 Skin 0 081 0 200 5

Trunk 24 430 8 742 80 Core 7 900 5 293 25 Muscle 9 580 2 339 25 Blubber 5 320 1 110 25 Skin 1 630 0 200 5

Fore-fhppers 3 300 5 827 80 Core 1 05 3 424 25 Muscle 1 32 1 566 25 Blubber 0 62 0 637 25 Skin 0 31 0 200 5

Hmd-fllppers 1 650 4 179 80 Core 0 525 2 421 25 Muscle 0-660 1 108 25 Blubber 0 310 0 450 25 Skm 0 155 0 200 5

Total 31 413

homogeneous w~thm each layer; a summary of these physical properties and their source is given m Table 3.

In addmon to its other important physiological functions, blood orculat~on plays a wtal role m heat exchange Most of the heat transport throughout the animal occurs by this mechamsm Blood is assumed to arrive at t~ssue s~tes w~th ~ts temperature unchanged from the central pool. Because of the h~gh surface area in tissue capdlanes, complete heat transfer is assumed there with the blood leaving every micro- region at the msue temperature

The possibihty of countercurrent heat exchange between blood flow to and from sea lion flippers has been suggested Such heat exchange has, however, been found of ltmlted effectiveness in larger mammals such as humans where heat exchange surface is small compared with the heat flux due to bulk blood flow. In any case, th~s mode of heat conservation was unne- cessary for the model since the controller permitted

Table 3 Physlcal parameters for sea hon model

Name Value Source

Thermal conductwlty (Ussue) cal

3 6 cm hr °C 32

Thermal conductlwty (blubber) cal 061

em hr °C

Density

Specific heat (tissue)

Spectfic heat (blood)

0-98 g 32 CITI 3

cal 0-83 gO--~

cal 0-92

g°C

32

32

almost complete cessation of blood flow to outer layers of all sections Under such condmons, heat loss from the flippers was neghglble.

Unfortunately, very htfle &rect mformauon Is available to quantify either total blood flow or Its &stribuUon m the sea hon. For the model, the basal flow and flow &stnbution were adapted from values measured in human beings. Some &fferences were assessed on the basis of presumed metabohc requirements for anatomical structures of sea hons as compared to humans.

Basal metabolic rates for mammals have been correlated with body weight by Kleiber (1947) "from mouse to elephant":

Q = 2-92 w 3/4 (2)

where Q Is the heat generation rate (kcal/hr). Irving & Hart (1957), however, reported that metabohc rates m harbor seals exceeded 'standard' values by factors of 1.5-2 Recent experimental measurements on the sea hon by South et al. (1973, 1975) showed that the constant in equation (2) had to be increased by a factor of 1-8. In both cases these h~gher values poss- ibly may be due to the youth of the antmals under consideration, about 2 yr-old m our ease.

The &stnbutlon of heat generation was based on that presented for humans (Stolwijk & Hardy, 1966), for eqmvalent muscle and organ weights. The overall distribution was much &fferent due to anatomical variations. The values for partmoned basal metabo- hsm are hsted in Table 4

EXERCISE M E T A B O L I S M

During exercise, heat is generated m the muscle layers of the trunk and fore-flippers. When the ammal ts swimming or &ving, :t must develop enough work to overcome the resistance due to acceleration and fired friction.

38 RICHARD H LLECKE, V NATARAJAN AND FRANK E SOUTH

Table 4 Distribution of basal blood flow and metabolism

Basal metabolism Basal blood flow Compartment (kcal/hr) (I/mln)

Core 53 020 1 763 Muscle 12 049 0 481 Blubber 3 838 0 084 Skin 0 897 0 073 Total 69 804 2 401 Head

Core 11 191 0 350 Muscle 0 268 0 011 Blubber 0 128 0 003 Skin 0 070 0 004 Total 11 657 0 368

Trunk Core 40 677 1 382 Muscle 9 884 0 389 Blubber 3 150 0 069 Skin 0 597 0 043 Total 54 308 1 883

Fore-flippers Core 0 775 0 023 Muscle 1 276 0 054 Blubber 0 377 0 008 Skin 0 155 0017 Total 2 583 0 102

Hind-flippers Core 0 377 0 008 Muscle 0 621 0 027 Blubber 0 183 0004 Skin 0 175 0 009 Total 1 256 0 048

Animal total 69 804 2 401

The amount of resistance offered by the fired around the ammars body can be evaluated ff the pattern of flow ,s known. Lang (1966) reported that whales maintain a laminar flow over a high propor- uon of the body whde m water If the animal ,s swunmmg In deep water at a submers,on depth of the body ax,s greater than three tunes ,ts effectwe dmmeter, the flow around the animal is laminar (Andersen, 1969) If the anmaal swims closer to the surface, waves are created and the consequent change m flow pattern mcreases the remtance. Th,s does not quite create a turbulent mot,on of the fluid around the anunal and ,t ,s said to be m the crmcal region, a transmon between laminar to turbulent flow.

Drag coefficients (Cr) for these regions have been reported from experunental data taken from rig,d, smooth, streamhned bodies (Hoerner, 1957). For the laminar region,

Cr = 1"33 Re- 1/2 (3)

for the crmcal regmn,

Cr = 0-074 Re- t/s (4)

and for the turbulent region,

CF = (4 15 log Re) -2/3 (5)

where Re is the Reynold's number

Lv Re = - - (6) %,

where v is the speed at which the anunal is swmumng and v ~s the kinematic viscosity of seawater

The power reqmred to overcome the frictional drag (PF) can be calculated using the Lang (1966) equation:

CrpSV 3 Pe = 2 (7)

where p is the fluid density The internal heat generated Q,.t m the muscle is then computed

(1 - ~/M) P r Q,., = (8) r/.~t r/E

where r/~t is the efficiency of conversion of muscle energy to the power and qE is the efficiency with which fhppers convert muscle power to swimming motion. Internal muscle efficiency was taken as 25%, a commonly accepted value (Andersen, 1969, Sea- grave, 1971; Wlssler, 1964). The flipper efficiency was taken as 50%; Lang (1966) estmaates the flukes of dol- phins at 80--85% efficient whde Wood (1973) reported the efficiency of seals at a much lower level

HEAT TRANSFER TO ENVIRONMENT

The sea hon has an outer layer of short hair that becomes completely wetted and hes so fiat that its thickness is less than I mm This mat can contribute very little insulating value and has been entirely neglected m this model

39 Mathematical biothermal model of Cahforma sea hon

Heat exchange w~th the enwronment can occur by conduction, convecuon, radiation and evaporation The rate of heat transfer by each mode depends on the state of the anLmal and the physical properties of the surrounding medium. When the sea hon ts on land, heat ts lost to the environment by convection, radmtion and evaporative coohng via respiration and insensible moisture losses When the ammal is sw~m- rmng in water, smce most of its body ~s mmmersed ~t loses no heat by skin evaporation, and since ~ts skin temperature ~s very close to that of the water ~t also loses very httle heat by radmtlon which vanes as the &fference of T 4. Evaporative heat loss to the environment does occur via respiration During diving, essentially all of the heat loss to the environment must occur by external convection

(a) Radiatzon and convection Radiant and convectwe heat transfer can be est~-

mated using well known correlations Radmnt heat transfer ~s computed from the Boltzmann equation taking the effective emlsswlty as 0.9 (Seagrave, 1971, Wlssler, 1961):

QRA~ = 4-4 x 10-12(T~ - T~) Keal (9) cm2hr

where T, is the temperature of the skin and To Is the ambtent temperature m °K. The ambient temperature for ra&ant exchange depends on the actual temperature of environmental objects which may or may not correspond to ambient atr temperature Ra&ant exchange on a clear night or m a bright sun would have very low or very high effective mean ra&ant temperatures, respectwely (multtplled by appropriate view factors). In the work c~ted here to cahbrate the model (South et al., 1975), the radmnt exchange was with calorimeter walls maintained at ambient air temperature

The convectwe heat transfer coefficaent (he) was taken to be that for forced convecUon across a flat plate (Knudsen & Katz, 1958),

hc --- 0-324 L k-- Re 1/~ Pr 1/3 (10)

where k = thermal conductwlty of the fluid (keals/ cm°C hr).

The "flate plate" equation describes the heat transfer rate through the boundary layer formed as the solid surface moves with respect to the fired It would be necessary to use an equation developed for cylinders only if free convection were an important factor or ff fluid flow were toward the side of the animals producing flow vortices.

The effectwe area for heat transfer whde the antmal is swimming or diving is the same as the total surface area. The effective area for radiative and convectwe heat transfer when the anmaal is hauled out on land cannot be easdy determined because of the complex geometry of the sea lion The area of heat transfer vanes markedly as the posture of the ammal changes. As the body temperature rises, the amrnal changes ~ts posture to obtain the maximum possible coohng surface area At maxLmal exposure, the radmnt vt.ew factor was taken as 0-90. Posture change is assumed

to occur when the ambient air temperature decreases At 20°C the ammal is assumed to expose about 80% of ~ts body As the ambient air temperature falls further, the ammal folds Its forefllppers reside the hmd-fl~ppers to conserve heat. An exposure ratio of 655/o Is assumed for temperatures at or below 5°C

(b) Evaporatice cooh~

The actual sites and mechanisms of evaporatwe cooling revolve active sweat secretion from the fl~p- pers (Matsuura & Whlttow, 1972, South et al., 1973, 1975) as well as from excessive sahvatlon which soaks the fur of the ventrum of the neck Since urination has been observed to have occurred rather frequently dunng exposure to the higher tempeatures (South et al, 1973, 1975), wetting of the clrcumanal area may also be a source of evaporatwe coohng

Evaporatwe cooling depends on the amount of surface area wetted. Since the sweat glands are found only m the flippers, it is easy to evaluate the surface area wetted by persplratmn But the determination of surface area m the trunk and the head where the evaporatwe cooling occurs due to sahvation and urination ~s less certain and. based on observations m metabolic tests (South et al, 1975), it is assumed here that the ammal wets about 100/o of its surface area of the trunk by urination and sahvatlon. The total evaporative coohng Is distributed over different sections according to the surface area available for the evaporative coohng Sweat and sahvat~on rates were taken to be high enough to saturate the wetted regions

(c) Respiration

The total rate of heat loss through the respiratory tract depends on the resp,rat~on rate, the temperature of the core, and on the temperature and humi&ty of the ambient a~r The respiration rate m turn is dependent on the total heat production m the body at an RQ of 0 83, every kflocalone of heat hberated requires approximately 0-2071 of oxygen It was estimated that 24~ of the oxygen m inspired atr Is con- sumed Thus, the respiratory volume (RV) m 1/hr can be calculated from the following equation:

RV = 4 145 x Q (11)

where Q is the total heat production (kcals/hr) Some of the heat loss of the respiratory tract occurs

in the core of the head and some m the trunk but there is htfle avadable data on the actual &stnbutlon. Based on approximate evaporative surface area it is assumed that 10°/0 of the heat loss through the respiratory tract Is from the central head core and 90°/0 from the central trunk core. Further, it ~s assumed that the expired air is saturated with water vapor at a mean temperature Tin:

7", = 0-1 Tnc + 09 Trc (12)

where the head core temperature (Tnc) and trunk core temperature (Trc) are measured in °C. Thus, the rate of heat loss through the respiratory tract (Q~) can be calculated from the ambient parttal pressure of water (Pw) and the relatwe huml&ty of mr (rh) (Wlssler, 1964)

Q~ = 15-12 x IO-4RV{Tm - IT a = 0-37(1 - rh)pw]} keals/hr. (13)

40 RICHARD H LUECKE, V NATARAJAN AND FRANK E SOUTH

Note that this heat loss was deducted directly from the inner trunk and head cores

THE CONTROL SYSTEM

The controller was formulated so as to allow good simulation of observed phenomena while at the same tmae being as flexible as possible so that the formulation can be modified without making extensive changes in the other parts of the model

The controller is divided into two parts, sensor mechanisms and effector action The sensor mechanisms are not accessible to observation and study but It is generally accepted that the temperature sensitive stuctures in the hypothalamus play an important role in thermoregulaUon (Hammel, 1967, Nakayama et al., 1961). Similarly, there is general agreement that thermal receptors m the skin play an important role (Stolwijk & Hardy, 1966) In addition there are Indications that temperature sensitive structures exist in the voluntary muscles (Robinson et al, 1965)

The effector actions are well known m a quahtatwe sense: variations in skin blood flow and muscle blood flow and sweating with consequent evaporative loss However, the regulator which will be discussed below is by no means to be taken as final. It is anticipated that this hypothetical regulator will be useful m sug- gesting additional expermaental work to challenge Its validity

The signals received from the sensors are taken to vary linearly with the local temperature (Stolwijk & Hardy, 1966) These sensors are assumed to be in the head core and in an even distribution over the skin Each of these sensor systems is taken to have zero output at a local temperature corresponding to a set point for the head core (Tnc) and mean skin (T,) temperatures. The head core temperature determines whether the major effector response is that for increasing heat loss by sweating and vasodilation or for increasing heat storage via vasoconstriction. Shi- vering as a thermoregulatory response is seldom observed in the sea lion and was not included m the model

(a) Vasomotor actzon

Blood performs two important functions. In addi- Uon to the funcUon of serving as a transport medium for oxygen, nutrients and wastes, it also transports heat to the surface. Variations m blood flow to the core and muscle can be estimated from the oxygen demand caused by metabolic requirements. Control of blood flow to the outer sections is a principal means of overall body temperature control.

Metabolic heat production in the brain is quite constant and in the core of the flippers the heat production is low under all circumstances. This is not strictly true for the trunk core where the heat pro- duced by the heart, especmlly, is varmble with work The resulting effect on overall heat production, however, is relatively small Since the oxygen and nutrient demand remains relatively constant, the blood flow to the core of all the sections is taken to remain constant at the basal level

In contrast, the total heat production m the muscles changes over a wide range due to exercise This variation m heat production requires different amounts of oxygen whmh, in turn, requires variation m blood flow to muscles. Since oxygen requirements are 0.2071 of oxygen/kcal and since every liter of blood flow makes 0-201 of oxygen potentially available for the muscle tissues (Andersen, 1969; Rid- geway, 1972), approximately one liter of blood must be circulated m the muscles for every kcal of energy liberated The minimal blood flow to the muscle (BFM) is evaluated as

BF u = BMB + 1000 Wry, (14) where BMB (cm3/hr) is the basal blood flow to the muscle (Table 3) and Wex (kcal/hr) is exercise work

The blubber plays an important role as a heat insulator with a very low thermal conductwlty compared to that of other tissue. The blood flow to the blubber serves the purpose of thermoregulation almost exclus- ively by varying the effective thermal conductivity through the blubber.

Skin blood flow is also highly dependent on the thermoregulatory controller. The basal blood flow to skin at thermal neutrahty can be diminished by vasoconstriction or Increased by vasodilatation. In the model, vasodilatation and vasoconstriction are treated separately with control gains adjusted to fit experimental high and low temperature data, respect- lvely

When the animal is diving, there is a pronounced reflexive bradycardla, a reduction m heart rate Bra- dycardia is not an isolated activity but is accom- panied by a strong generahzed peripheral arteriolar vasoconstriction. There is reduced blood flow to the skeletal muscle and skin, with pooling in the viscera Cerebral and coronary flow is less affected (Andersen, 1969) Eisner (1969) measured cardiac output by directly recording the blood flow in the pulmonary artery of Zalophus cahformanus Slowing of heart rate was clearly visible while the stroke volume remained unchanged during the Immersion The heart rate fell from about 140 beats/mm in air to 30--40 in water

Clearly blood circulation during diving is slmdarly reduced A problem arises however in evaluating the amount of reduction of blood flow to different parts of the body. Scholander (1940) reports flow practically stops in the muscles during diving In the mesenteric blood vessels flow is much reduced and it is often arrested m the flippers (Irving et al., 1941). The kid- neys are effectively isolated from blood circulation during prolonged submersion (Murdough et al, 1961) or reduced (Stone et al, 1973) I

In this model, it is assumed that during the diving the eyes, ears, cranial nerves and the base of the brain receive blood flow at the basal level The organs in the trunk core such as liver, etc., receive blood flow at half the basal blood flow. This sums to about 37 5~ of the total blood flow at steady state

DESCRIBING EQUATIONS

Energy balances for the systems previously described lead to parUal differential equations having

Mathematical blothermal model of Cahforma sea hon 41

the same general form for each of the four concentric subeyhnders of each of the four sections

rk,- r l +

BF, + -E-~(T=- 7"3 (15)

where p, = density of ah layer (kg/cm 3) C, = specific heat of tth layer (kcal/kg-°C) T, = T, (t,r) temperature of the lth layer at hme t

and radms r (°C) t = time (hr)

k, = thermal conductivity of the ith layer (Kcal • cm/cm 2 hr. °C)

r = radms (cm) Q, = metabolic rate of the ah layer (Kcal/hr) V, = volume of the tth layer (cm 3)

Ev, = evaporative water loss of tth layer (Kcals/hr) BFi = blood flow to lth layer (cm3/hr)

2 = dimensionless function accounting for the effect of counter current heat exchange between veins and arteries (assumed to be unity)

~, = product of density and specific heat of blood (Kcal/cm 3 °C)

TB~ = temperature of the central blood compartment (°C)

In this equation, the term on the left represents energy accumulation throughout the ith section. The first term on the right represents energy diffusion through the tissue by conduction The term Q,/V, is the metabolic heat generation term while Ev,/V~ ts the heat loss from the section by evaporauon Note that evaporatwe loss is non-zero only at wetted surface sectmns and in those head and trunk core sec- t i n s that simulate the respiratory system The last term in equation (15) represents convective heat exchange with the central blood pool In fact, this term was the prmclpal means for heat distribution within the model Note that a factor was mcluded to allow for possible countercurrent heat exchange between venous and arterial blood However, since the control system was capable of producmg a zero blood flow to the surface, this factor was not needed for heat conservatmn and hence was assigned the constant value of unity.

In order to solve these sixteen partial differential equations plus that descnbmg the blood pool, con- straming condmons must be specified Some of these take the form of mltial condmons which specify all of the temperatures at the instant the transient begins:

T,(O,r) = To,(r) (16)

TBL(O) = TBt, 0 (17)

Also needed are the boundary condmons which relate the ammal to its environment. In general, these are based on the fact that the local rate of conduction of heat to the surface through the s k i is equal to the rate of heat transfer from the surface to the environment

Finally since each central element possesses axial symmetry

OrJ,=o = 0 (19)

The partial differential equations are solved using a fimte-difference technique since this method allows variation of physical properties with position Basi- cally, the procedure consists of subdividing each of the circular elements into a number of annular shells and assigning a single characteristic temperature to the material in each of the shells A marching procedure is employed in which the initial temperatures are used to compute the temperatures after a short Interval of time At These new temperatures are then used to compute the temperatures at tu'ne 2At and so on, for as long as necessary

The Crank-Nlcholson formulation scheme, which is an lmphclt-recurrenee formula with the weighing factor of 1/2 is used here This method is theoretically stable for all net sizes However, because of difficulties at interfaces between sections, the second order par- tmi derivatives were first reduced to first order by Integration once with respect to r by assuming that the temperature m each cell to be homogeneous (Luecke et al, 1970; Wlssler, 1964) The resulting first order equations were then solved without abnormal difficulty usmg standard techmques

Evaluation of heat lost to the environment could be made using the outsMe heat transfer coefficient and the ski-to-environment temperature gradient, or by computation of the rate of change of total internal energy in all of the layers m the body It ~as found that in many cases these two numbers did not agree probably because of the accumulation of round-off error m the numerical solution of the outer differential layer of the s k i The data reported was computed on the basls of total mternal energy change.

Effectively, there were four sets of unknown parameters which needed to be fitted usmg expermaental data These were the feedback gams for vasoconstriction, vasoddatlon and sweatmg and the effective set pomts The latter were chosen by assummg thermal neutrality at 20°C in air The uncontrolled temperatures reached m the various sections under these condmons were taken as set pomts The method of com- puting an overall set point used by Wissler (1964) for the human model was adopted, feedback was apportioned on the basis of weighted average of head core and skin surface temperatures. The former, of course, predommated m the welghtmg

Feedback gams were determined by assuming linear relations and that maximal coolmg occurs (sweating and vasoddatlon) m 28°C air and maxmaal heat conservation (vasoconstriction) m 0"C air The maxunal peripheral blood flow was determmed by matching experimental steady state core temperatures in 25°C air. Feedback gains, control equations and set points are summarized m Table 5.

COMPUTED RESULTS

(a) Air

Figure 3 shows the average temperature of the trunk core computed using the mathematical model

42 RICHARD H LUECKE, V NATARAJAN AND FRANK E SOUTH

Table 5 Control system equatmns

(1) Total evaporative coohng

(2) Blood flow (a) Muscle

(b) Blubber

(c) Skin

Ev = EV~(Tnc - Tnc)(T, - T,)

BF 3 = BF3.NoR. + Exer

BF2 = BF2,NORM + Cs(Tnc - 7"nc)

warm BFt = BFI,NOR. + Vo(Tnc --Tin)Tin > 7"nc

cool BFI = BFI,NORM - - Vc(Tnc - 7"nc) Tnc < "Fuc

Tnc =average head core temp, 7~.c = set point head core = 37 12YC, T~ = weighted average skm temp, T~ = set pomt skm = 31 00°C, Exer = exercise rate, BF, = total blood flow rate, BF,.soR. = total blood flow with no control, Eve = evaporatwe gain = 1000 kcal/hr/°C 2, Cs = blubber feedback gam = 5000 cm'/hr/°C, Vo = vasodllatlon gain = 5000 cm2/hr/°C, V c = vasoconstriction gain = 7000 cm2/hr/°C

A restriction on all blood flow ]s BF, > 0

as a function of time when the animal zs hauled out dry and at rest at 0 °, 5 °, 10 °, 15 °, 20 °, 25 ° and 28°C mr temperatures The computed values closely reproduce measured deep body temperatures when the ammal at rest was exposed to the air temperatures of 10 °, 15 °, 20 ° and 25°C in a metabolic chamber (South et al., 1973). The body temperature of the ammal is relatwely constant at 5 °, 10 °, 15 ° and 20 ° mr temperatures. But at 0°C a~r temperature, the body temperature drops by 1.5°C over a period of 4 hr. Under these condmons the anunal would resort to increased pen. pheral vasoconstriction and shivering and non-shwer- ing thermogenesls, as well as behavtoral and postural adjustments to maintain tts body temperature.

Along with the rise m core temperature, the experimental ammals exhtbited &scomfort m atr temperatures above 25°C. Since the effectiveness of c~rculatmg blood to the skin ~s reduced by the small temperature gradient between core and ambtent, the restricted abv hty to lose heat by evaporatwe cooling gives these ammals luntted tolerance to high temperatures. Un- der such circumstances the animal might be expected to spend most of the t]me in water

The measured (South et a l , 1975) and the computed heat loss by radlatmn, convection, resptrat,on and skin evaporation as a functmn of atr tempera-

3 9

o

• 3 8

,"-:-.-~-- - - - - - =5--_--_ . . . . . . . "~ . . . . . . . . .

~ -

. . . . EXPERIMENTAL DATA, et al COMPUTED TEMPERATURES

3.~) , A i i i i i , I 2 3 4

TIME- HOU RS

Fig 3 Average temperature of the trunk core of the an,- ma lm mr as a functmn of time

tures are shown m Fig. 4. It can be seen that agam the measured values are qmte well reproduced by the computatmns These values indicate that the radmnt loss was the most important route (as high as 50 or 55%) of the total heat loss to the environment until very high temperatures are reached. Hart & Irving (1959) reported the skm temperature of harbor seals (mean of two points posterior to axdla and differing less than 0 5°C) at an air temperature of 20 ° was 30°C. The model stmulated value was 28°C

(b) Swimming

The computed average trunk core temperature of the antmal swmammg m water at temperatures of 0 °, 5 °, 10 °, 15 ° and 20 ° at 160 and 190m/mm are shown m Figs 5 and 6, respectwely. The same feedback control equations were used for the anunal on land and swtmming It can be seen from the Figs 5 and 6, that the animal ts able to mamtam Its body temperature m 0°C water by swunmmg at the higher veiocay

3 2

F.XPERIMENTAL 0ATA S(XffH p, ~ al

2~ ~ 0 / ~ COMPUTED OATA

• ',, ) t

0 5 IO 15 20 25 28 ~R T£MPIrRATURE - *C

Ftg 4 Heat loss as a funct,on of mr temperature

Mathematical blothermal model of Cahforma sea hon 43

38C

37."

37C F

1-

35.1

WATER ZO'C

' a4 ' ~ 8 ' ,'2 ' ,'e ' i o T l l d E - H O ~ S

Fig 5 Computed average trunk core temperature of the ammal swimming at 160 m/mm as a functmn of time

400 WATER 2 0 " C ~

3S5

F ~ 3SC

! 3 8 . e

38.( WAI"ER 5" 8, IO'C

WATER O'C

37.'

014 ' 0 '8 ' 1'2 ' 1'6 ' 2 '0 TIME- HOURS

Fig 6 Computed average trunk core temperature when the animal is swimming at 190 m/mm as a function of time

These computatmns indicate that durmg swimming the heat loss due to convection is the smgle most ]mportant route for heat loss (as high as 80%) Surface skin temperatures sbowmg the small temperature gradient for the swimming animals are given m Table 6.

(c) Dwmg The computed average core temperature while div-

ing at 240 m/min is given m Fig. 7. Surprisingly, temperatures increase with time even in 0°C water and, m fact, the core temperatures are nearly independent of environmental water temperature This unexpected and prev]ously unreported behav]or is the result of several factors"

(1) Since the ammal is under water, respiration obviously cannot be a source of heat loss

39-"

39. •

39

. 3871

3 8 3

~, 3 i ~ 9

3 7 7

TIME - MINUTES

Fig 7 Computed average temperature of the trunk core of the ammal when it is diving at 240 m/rain m water of temperatures 0 °, 5 °, 10 °, 15 ° and 20°C as a functmn of

time

(,) Due to bradycardta and vasomotor reflexes, blood flow is confined almost entirely to the core Little convectwe heat is transported across the blubber to the skin The blubber layer becomes an effioent insulator.

(m) The swunmmg effort to maintain 240 m/mm liberates about 100 kcal/hr in addition to the 70 kcal! hr of basal metabolic heat.

Thus, the dwmg anunal generates about 2 4 times the basal heat while havmg only one inefficient mechamsm for heat removal.

This type of temperature behavior has never been observed or reported, but was lmphed in the expectations of Bartholomew & Wflke (1956) that during an actwe dive, skin vasodllatlon mlght occur should the requirements for heat loss become more imperatwe than control of CO2 and 0 , m the blood It would appear that this phenomena could occur m 0°C water was well as m 20°C water they suggested Of course, 15-20min represent long dive durations for sea hons. More frequent surfacing would be fol- lowed by resumptmn of respiration, increased per]- pheral vasodflation, and termination of bradycardm These responses would rapidly transfer large quanti- ties of heat to the enwronment. One objecuve of experimental studies will be to investigate thermal be- hawor during this critical period

A temperature profile of the trunk of the ammal at steady-state diving is shown m Fig 8 Notice that maximal temperatures occur, not m the core, but m the muscle layer where the heat of exercise work is liberated but where blood perfuslon is very low In this reglon heat flow reverses direction so that the

Table 6. Computed temperature of the last (outer) differential layer of the skin in 2 0 hr and at various water temperatures when the ammal is swimming at 160m/mm

Water Head skin Trunk skin Fore-flippers Hind-flippers temp temp temp skin temp skin temp (°c) (oc) (*c) (oc) (oc)

0 4 76 1 87 2 88 3 45 5 9 26 6"68 760 8 10

10 13 77 11 75 1244 12 80 15 18 29 1690 17 66 17 75 20 22 83 2204 22 86 22 70

44 RICHARD H LLECKE, V NATARAJAN AND FRANK E SOUTH

_ _ J

o Qt ttl

2(

i i t i

CORE .L I ~ L I [ I ,~t~BLUBBER IN 1

DIF'F'ERENTIAI. LAYERS

Fig 8 Computed temperatures at 16 2 mln as a function of differential layers in the trunk of the animal, when it

is diving at 240 m/ram

core, instead of transferrmg heat out, actually recetves a flow of heat inward from the muscles

South et al (1975) observed apparent heat seques- termg durmg htgh temperature calorimetric deter- mlnat tons Sudden mcreases in mternal temperature were noted as the animals were removed from the calorimetric chambers ; temperatures which exceeded continuously measured deep core temperatures These model diving calculations mdtcate one mode whereby heat can be stored at temperatures higher than the core Elucidatton and predtctton of such unexpected phenomena is one of the valuable results of modeling studies

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PIERCE R W (1970) Design and Operation of Metabolic Chamber for Marine Mammals Ph D Dlss, University of Cahfornla, Berkeley

RIDGEWAY S H tEd ) (1972) Mammals of the Sea, Biology and Med,one Thomas, Springfield, Illinois

ROBINSON S, MEYER F R, NEWTON J L . TS'AO C H tg: HOLGERSEN L O (1965) Relations between sweating, cutaneous blood flow, and body temperature m work J appl Physlol 20. 275

SCHOLANDER P E (1940) Experimental investigations of the respiratory function m dlwng mammals and birds Hvalradskr 22. 1-131

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SEAGRAVE R C (1971) Biomedical Apphcatlons of Heat and Mass Transfer The Iowa State Umverslty Press, Ames, Iowa

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SOLTH F E. LUECKE R H. SHANKLIN M D & ZATZMAN M L (1975) Calorimetry air temperature and possible heat sequestration by the California sea lion Comp Blo- chem& Physlol In press

SPOTILA J R, LOMMEN P W. BAKKEN G S & GATES D M (1973) A mathematical model for body temperatures of large reptiles, lmphcatlon for dinosaur ecology Am Naturalist 107. 301-404

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Mathemaucal blothermal model of Cahfornla sea lion 45

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a mathematical biothermal model of the california sea lion

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