Thermal conductivity measurements and predictions for biological fluids and tissues

("~ 1{ ! O 1 1 1 0 1 , O ¢ I

"

Vol. 3, No 4, 196~J

TIIERMAL C O N D U C T I V I T Y M E A S U R E M E N T S A N D P R E D I C T I O N S FOR BIOLOGICAL FLUIDS AND TISSUES'" 1[. F. POPIq,;NI)ll,/K,

R. R.~,NI)AI,I,. J. A. BRE1,;DI,'N, ,1. E. C I I A M B E I I S ,

AN1) J. H. M U R P H Y Geo,~cwnce, l, ld. S,,bma Beech, C~,liJurnia The objectives of the l))'esent research have been 1) to measure the therm'tl conducti\'ities of the more imporlant, biologieal fluids and t.i.,sues f)'on~ hunmns and animals, 2) to in'edict, the eonduetivil,ies on tlw l)~
'l')t1.:ou)"rjcaL COXDUCq')ox :-k[OD.Ui,-+ It is desirable to be ahle to predict, the (hermal conductivit.ies of biological .-l)et:imens Mmwing o:dy the chemical compositions invo!ve(l. Analytical conductivity relations also make it. lmssibh' to corn:late more effectively bhe experimental data l)eing ol)tained for biologieM materials. Three different conduct.ion models have been eotLsidered. One model is based on the postulate thai. the consti~,uents are positione(1 in laminae parallel to tlm heat flow; in the ,,~e(:ond one (he laminae are positioned normal to the heat flow; the third one is based on the 1)ostulate thai; small particles are uniformly distributed in a second

(6 q

- -

t,)

(1)

=

where q is heat flow in the thermal circuit; (11 - G) is teml)er'~tm'e diffezence (i)otential) a('ros~ lira tbermal circuit; and R, u is tim e(lui\'alenl, thermal resistance o1' a lmr'tllel re.-i..,l,ance eirm~it. From elemm~tary eircuil, em~('el)ts, 1

" ~l

_

IG,t

(2)

z.~n R,,

where R.,, is tim ~th individual circuit, resistance. The q~mntit,ies 1~,, aud R,q cml b,, exl)ressed itl terms of lira x'ariables .-how1~ in Fig. 1, ~,,

=

.........

(',{)

I,.,,3y,, I R,,, -

(-I)

'{',,,~Y l

Ul)on combining ,g(ltmtions 2, 3, and -1 there results tl

= Z

j,.,,

where k y , , / A y is tile volume fraction of tile nth constituent. Ttm weight, fraction of lhe nbh constituen t is defined by

* The thermal conductivity research ozt bioIogieal fluids and tissues presented here was supported by the Medicine and l.)ei)tistry Branch of the Office of Naval Ilesem'eh. AJt early version of the thermal conductivity at)l)ar;tl.us used "in t.his

y . l & x -~,, ,.,,,,

=-

'&Y,, "r,, -

zXy l .Xx ~.

.~g ~.

((i)

where %, is density of nth constituent, and 3' is density of the composite ma0erial. Thus, I,;qua-

research was first, d e v e l o p e d in e o m t e e t i o n with

Grant I-I6699 from the National

eomponenL and that the volume of all tl~e lmrl,ities is smetl) compared lo the total. Constituents in. laminae parullel to Imal .flow. The (ieri\'ation of the relation t~mw,.en the overall thermal ,.(mductivity of ,t model ia whid~ lamizv,w are l)ositioned l)m'allel (o the heat flow (see Fig. I) follows.

tion 5 can also be e x p r e s s e d as

Institutes of

H e a l t h where the e q u i p m e n t was used to nteasure the then'hal e o n d u e t i v i t y of blood. R e c e i v e d July 1, 1966.

rt

(7)

= v E: o

318

"gn

TH El/M AL CO NDUCTIV1TY 5I EA SI.TP,I';M lgNTS

•

-

%{t" i

&x

....

.

\ ~

AX,

\ \ \

.~y,

Rn

T H E R M A L CIRCUIT Ay

-

I

LENGTH

UNIT HE~T T R A ~ S F E R . ~ R E ~

i"m. I. "I'lmrnml con(lucl.i\'ily h('nt llow.

model

l);,.~:~l ()n con.,-litu(.'nt }aznin:t:' t}mt It(. lmrallel

Conslitue~d.s i~l taminac ~wrmal lo heat ./low. A sitnilar d(.,)'ivati~)n can I)e mmle f~))" the c~L-e in whi(:l~ tim hear, t-t
3' E

,,

¢0,~

(S)

0 k,o 5'n.

One species uniformly dislributcd in a second species. The third heat conduction model that has been studied i.~ clefilmd 1)3, ,-,nm}l paz'lieles or mm eomponeut uniformly disl,ributed i~ a second colnponent,; the volume of ail the l~,%L'tielesis small compared to the total. An equation which relates the thermal con.duclivity of the mixture to the eondu~:ti'vil,ies of the particles azu{ era'tier aml the volume fraction was derived t~y Euckert,'--' 1-- ( 1 - - a k i n ) b /,'~

. R2

S

"-

LAMINAE

I,

~

II

\\\

.

,I

!b

\\

- - ~ - - _ =--

--

RI

.

I tz

_

-~, \ k\\

'

k , ~\\

/~Yl

•

"°

I 7 -1 [

i?//~\\

IIt1~\\ I / / /

.

~

:J 1!t

I +

(a -

(9)

~)b

where k is conduetivit4," of tlie mixf.ure" ,% '"p is conductivity of lhc carrier; k~ is eonduel.ivit,y of small partit,.lc,.~;' b is ratio of parlicle volume to the total volume; trod a = 3/(2 + kJkv). .lpplications. Although the number of individ-

to

ual ('ousti/alent:~ thai eOinl)ris(; biolo.L,'ical l,mterials is l'u'ge, the three most iml~ortant ones, as hu" ,',,.~ tlw/'mal condnclix'ily is coneerne~l, are, water, i)rol.ein, and fat. Mso, these three matevials constitute over 97.% (by weight) of Ml of the tis.-ues and fluids l~rfisently ex'unined. Equtl,lions 7 and s have been used to predict the fl~erma! con~luct.ivitic,-: of e.-,scntiMly "tll the biologi?.al fluids and ti.~sues ,~tudied in lhc present progr'am. The results which are presented in a mzl~sequent, section of l.he lmper are l)~secl on the po~,lulate that the specimens are only composed of water, l~rotein, and f"tt. One would expect Jgqtmlion 7 (laminae jmrallel to heat flow) to be more realistic than Equation 8 (laminae normal to heat flow) in representing an aettmI system eomtmsed of a uniformly mixed heterogeneous grouping of two or three constituents. A checkerboard grouping of two constituents, for example, show.,." that parallel conduction paths exiat in a mtmber of directions. Equation 8, however, wotfld be expected to represent satbffaetorily a layered specimen such as skin. Equation 9, which defines heat conduction in only two components (one species uniformly distributed in ~ second one), can be used to (Iciermine the thermal eonduetivities of erythroeytes in blood from whole blood conductivity measure-

320

POPPENDIEK ET AL. f- ! N S U L A T I O N

. WATER COOLING COILS

/ /

/ //

/

t

/

Y

/

, /

.]. /

) /

/

/

t]

I

,

/

~

. COPPER //COOLING

PLATE

//

='_O O! O! O

INSULATION ~

~ / / # / / / / / / / / / / / / / / / / ~

"r.. {.-.-,','-- ".,:- -:=..':-.~ .%.'-%-z.~---~,'-,Z.-.-'_%2-~: 1 / N U L L ELECTRIC ~ , . ~ - - - - . - - - ' ~ ~ - ~ - ~ = £ - 5 #"J HEAT METER HEATERS

"~////I/////////~_ "-::. : -?.'-,':':'.: i:.:.':-::..."'.::.:..':i:..'.:'.".i':.-.'.:. :'.".-:-.:.'i:.::..~.• _. :'. '-" LL'-':~. "." :" "': ':.:'.'.-', ,':'.' "'. ,v :..'~. :'::, V- .;. ; -".: ".:,l

9"

OfO /

/

1 /

oou,.

/"

/ /

-

O--',iO / (~, .// ,' " ,

//

-

/

9-Y~ ,'

_/\\' / \~

o0 /

' '

I

CELL END

g

;0, '

~

' "~-COPPER ' ' COOLING ' / PLATE

" -

THERMOCOUPLES F](;. 2. Cross-sectional view of tl}ermM conduct.ivil,y cell.

rnent~. The results for the present program are given in a subsequent, section of the paper. DESCRII"TION OF APP-~ltATUS

The apparatus for meu.suring the thermal conductivities of biological tissues and fluids consists of a number of components. The heat source is composed of tu'o flat, electrical heating elements with a mill heat flow meter inserted between them (see Fig. 2). The eieetric powers for the two heaters were so adjusted that the heat flow through the null heat flow meter was zero; under these circmvustances, all the heat, generated in the bottom heater was forced to flow through the cell containing the biological specimen. A piece of insulating material was p l ~ e d between the top heater and the cooling plate to keep the current voltage requirement of the top heater at a mininmm. Spacers were added m the plastic insulatkm around the cell to ensure cori'eet positioning of ~tm lower heating element. Because of the small difference in temperature between the cell and the container environment, the heag losses from the ceil ends were small , %

(<0.25%). The thermal conductivity cells, one for liquids and the other for tissues and viscous fluids, consist of two highly conductive plates. 1Removable thermocouples were inserted in holes drilled through portions of the plates parallel to

the cell surface." The top plate of the tissue cell is removable so tha.t t,he eell can be filled with the biological specimen; the conductivity eel! used to study liquids has permanently positioned plates with holes in the top plate for filling. Cell plates m'e spaeed a short distance apart to maintain a large width lo thicMmss ratio and thus ensure one-dimensional heat flea' through the biological specimen. The heat flux, tempera,tare difference across the sample, and the sample thickness are measured in this apparatus at stettdy state. The thermal conductivity which is defined as the heat, flux divided by the temperature derivt~tive with respect to distance along the heat flow p'd,h is then determfimd. Figure 3 shows a photograph of the emnplete thermal conductivity apparatus; its main componeuts are (1) the insulating housing which contains the conductivity cell, (2) the constant head water cooling system which removes heat from the cooling plate, (3) the electrical power and metering unit, and (4) a lmtentiometer used to measm'e temperature differences and heat meter voltages. EX/?ERIMEN'/kakla PROCEDURES

The thennM conductivity cells used to study biological liquids contained transparent plastic end windows which made it possible to detect the presence of small air bubbles that may have

oq~ u! saoa~ot[ ot{~ o~ -~ndu.z [-ea!,z~,oa[o.oW, ou.zm.zo~ap tIa]qat sao~tu3toa pu~ ,saalomtu~ l~a!aq~aOlOat[& "saldmvs punoa~un puv puno.z:~ u~oxt~c[ 'aaaaamq ;ptmoj aaaa~ sa.t~!suap pu~ so!~,!a.t~onpuoa lvtu./oq), qaoq uz ,,,'oauoao~!p ilmUS , XlaaP,~taa 9tuo S "~aldm~g snoauo~oumq aao~tt popla.t£ 9,znpaaoad au~,~,~I oq& '.topu!a$ ~ o t u uomuloa ~ q~!at puno.t~ o,taa~ soldm~.s aoq.]o

pu~ alias [~an,~gu a!aql u.~ SlIaa aq'I olu.~ poae,[d o.~aa~ guommods ong~!~ aq'l jo omo£ ")uo~aad a.za.~ spt.ng ssaaxa ao soa~ds a~ ou ~vti~ o~ o.ma ~o.z$ ql!'a Patlg OSlV, aaaa~ sans~l jo s~uauzaan,~;atu ,~!a!~anpuoa [~tu,rot[:l o~"ettt oa pasn SlIaa aqj, 'aln,~aa plnoa~ .~}uomoans~atu £1!at.~anpuoa snoouo.uo on ~m{l os ,Q~ssoaou s,,~ a,mpaaoad >iaotla s!q& "IlOa ottl. u! padd~a.vl omoaoq

'm~lu,tudd~ X~.~,ttlanpuoa [~tuaaqq o'lo[dtuoa oq} jo t[de,a,~oaot[d "g "olX

},%1

t

i

l

g ,i

7 "1

"

'

! , 2, ,

%

"

}"

;?,,

4,

,%

•

I ,

i _, /%,

,,,,

,,

,'

,,;1;

.,

,,

,

' 5'i

q,

"

, t

,[

,

o

g,'7'

f

,

~rt

• ~ 4

"'L

i'; 2:.

[g,g

8&NXI4~XIIfISVZI'V X&IAI&DflGNOD 'IV1~NXH&

')')') o.-.,

j P O P I ' E N D I I ' i K .~"P AL,

0,40

conduetivil,y al)lmratus were ealil)r,'tted ( < 1% (l('vi,'ztions from true vtd~les). In addilion, l h e al)l)ar'~tus wt~s used io measure lhe th(,rmal conduetivitics of water at, regul'u' intervals (luring the entire ]'(,se.arch p r o g r a m lo gu'u'tmi~pe the aecurncv of the s~:~tezn. Wetter cond,wtivitv measurenwnt,s differed from literature values ~ b y less t,lmn m 3 % . It.

EXPER~,~i~¢N'~)~b ]:~J'SUL'r8

Q

Condu('Kvities of unstressed specimens. "rhc thermal conductivity "q)pactttus wt~,~ use(l t,o s t u d y 20 differeul, biological fluids attd tissues over the lelnperature r'an,L,'e 75--100°F. T h e results are presenled in 'l','fl)le 1. T h e conductivity m e a s u r e m e n t s for h u m a n blood require special commenl,. Some 30 l.herm~l met~suremmds were m a d e using five sel)m'ate lots of blood from healthy donors. Also, d a t a were obtained for p l a s m a and high l}enmtoerit, samples whi('.h were prepared with the aid of lal)oraiory c('ntrifuges; FiRure 4 simws ,~ graph of llmrmal con(lu(',tivitv oi' t)loo(l over ~t r'mge

~L U_

-.,..

c0 I,-,,,

0 20

-

---

~

1

~'e xlm, ime ntal I~TU/h~ fl °Y

,',d/on sec ~'C X lOa

Wa, I er. . . . . . . . . . . . . . . . . . . . . . Bovine vitreous humor , . . . . .

0.346 0. 343

B o v i n e .,tque(ms h u m o r . . . . . .

0.33.1 0.330 0.324 0.315

1,3,10 !. 3O3

0.306 O. 305 O, 303 0. 287 0.282

1,265 1,26l 1. 253 1..187 1.16G

:Egg y o l k . . . . . . . . . . . . . . . . . . . .

Chicken skin . . . . . . . . . . . . . . . . B o v i n e lung . . . . . . . . . . . . . . Bovine fat . . . . . . . . . . . . . . . . . . . Bovine bone marrow . . . . . . . . . * 75°F < , i < 100°F.

-

C~

-I

010

I

I

f

rI'IIEIi:MA I, CONI)UCTI VITI ES O]," UNSTRE.q'qND ]~IOL(IGICAI.~ .]~LUIDS AND rPISSUI~S*

l I u m a n plasma . . . . . . . . . . . . I h l m a n urine . . . . . . . . . . . . . . c~lr muscle . . . . . . . . . . . . . . . . . t l u m a n blood (hem.tlocrit,, 43%) . . . . . . . . . . . . . . . . . . . . . . . Bovine musele . . . . . . . . . . . . . . Bovine kidney . . . . . . . . . . . . . . Bovine brain . . . . . . . . . . . . . . Bovine liver. . . . . . . . . . . . . . . P o l y v i n y l p y r r d i d o n e additive (20 ml of aet.inomyosin D ; 146.8 g of polyvin3qpyrrolidoneT; 0.6/% NaC1 to aaa ml) . . . . . . . . . . . . . . . . . . . . . . . . I-Iuman gtls~.a'ie juice . . . . . . . .

----

o lJ

TAB L E 1

Nalllple

. . . . . .

1.'t;~1

1..118 l.aSl I. 365

0

-=

0

=

---=

-

20

40 60 HEMATOCRIT, PERCENT

80

IO0

Fro. 4. Therm,d conductivity moasurenwnts of blood a.s a function of hematoc, rit, (80°F < t < 92°F). of hem al oerit values.* N o t e t1~af, the em~ lu etiv i ty (teerettses about, 14% a.~ t~l~e hema, toeri(, value f r o m 0 lo ~.a¢v ~,,, ;'c.. Beemlse the blood cells •lrlele,lse~ ,.... ,u'e expected to h a v e b e t t e r thermal resi,%m~ec thm~ plasma, the function shown in Figure 4 seems reasonable. D u r h l g tim course of the experim e n > , attempt,~ were m a d e to find tt~e influence of cell setlJing on t h e r m a l eonductivit,y; none could be detected, as had previously been predicted from theory.

ConducKviKes of specime~,s exposed lo freezethaw conditions. T h e therma,l eonductivities of 0. 274 O. 257 0. 243 0. 206 0.163 0. 133 O. 127

]. 133 1. 053 1.017 0.852 0,674

0.550 0.52,5

bovine brain, kidney, and liver were me~sm'ed Mter the specimens were first, slowly frozen to 32°F and then slowly thawed by exposure to • The thermal conduc~ivities of blood from different donors (for the same hematocrit, value and temperature level) were round to differ by as much ~s 6%. The experimental points shown in Figure 4 represent mean values for the several lots of blood used.

TI-IIgRMAL CONDUC I; IVi iF5 MEASUI:H",MENTS room temper'dure; both ground -rod whole specimens wero studied. 8igaaifieant, inere:~ses (of the order of 10 to 29%) in eonduetivil,y resulted for liver, kidney, and },wain tissues that ha(l undergone slow freezing /,o a little below 32°1'` ml(l subsequent storage at, 32°F for more than 4 or ,5 days. Freeze-thaw experiments with bovine muscle also indicated increases in thermal conductivity of tJm order of 10 to 20%. For a storage period of less than 3 days, the conductivity of liver was found to exhibit a smaller change than that for 5 days. Conductivity mea~surenwnts were also made oa samples of bovine liver which were frozen ral~idly by immersion in liquid nitrogen (no pr~teet.ive additive) and then thawed quickly by agitation in warm wt~ter. Thermal conduelivity increases as a result of liquid nitrogen freezing and relatively rapid warming we.re not as great (.--~5%) ~s slow fl'eeze-l,haw eN)osu)'e, ~tthough more. visible structural ~lamage to the l.issue was ot~served. Ital)id freeze-thaw eonduetivil,y measuremmlls for lmvine kidney, brain, and ~lltlsele are now in progress. The eon
Conductivilies of unstressed biological specimens. At the time of the literaau'e search on lhermal conduetivities of biological fluids and tissues, two appm'ently eont!'adietory values for muscle were tabulated; one value 9 w,~,s 0.31 B T U / m r hr ft OF and the other value ~ was 1.33 B T U per hr f~ °F. The latter measurement apl)em'ed to be anomalously high for.a uonrne I:allie substance. Compm'ison,~ with other materials and theoretical eonsideratkms ma(le the lower value apl~ear t,o be more reasonable. In Ihe 1)resent study, therm'd conductivity mea.~urements were made on beef muscle in the directions 1)erpendieular and parallel to the grain. A mean value of 0m eonduel, ivity

323

TABLE 2 r~tlERMAIJ CONI)UCTIVITIES OF STRESSED BIOLOGICAL FLUIDS AND TISSUES* Sample

Slow freeze-thaw, 32°F Bovine liver, ground, frozen <3 d a y s . . . . . . . . . . . . . Bovine liver, ground, frozen > 5 days . . . . . . . . . . . Bovine k i d n e y , grmmd, frozen 5 days . . . . . . . . . .

Bovine bruin, gromad, frozen 4 days . . . . . . . . . . . . . . Rapid freeze-lhaw, -320°F Boviae liver. . . . . . . . . . . . . '.

BTU/hr

ytoF

r a t / o n see °C X 10 ~

0.287

1.187

0.324

1.340

0.333

1.377

0.34[i

1.431

0.305

1.261

* 75°F < t < 100°F.

w:~s found to be 0.305 BTU per hr ft °F which was in good agreement with the conductivity reported by Spells:); the v@ms for heat flow paralM aim perlmndicular to the muscle grain differed 195"only a few per cent,. Two thermal conductivity measurements for hunmn blood were found reported in the liI,eratm.e.~,, 9 Spells 9 used a relative meszurement method rather t,han an absolute method to make the determination; a thermal eonductivi[y of 0.293 BTU per hr ft °F was reported. Ponder G used an "~bsolu t e n te&suremell t method; a thermal conductivity" of 0.:.t9 BTU per hr ft °P was reported. From the present study, a vM~m of 0.306 BTU per hr ft °F resulted at a hemat~erit value

or 43%. In 1960, Meryman (personal communication) measured the thermM conductivity of rabbit liver; his value of 0.218 BTU per hr ft °P compares reasonably with the value of 0.282 determined in tim presen~ study for bovine liver. Spells '~ has proposed thaf, a correlation exists between water eontent ~nd thermal conductivity, which al~pears to be a good first approximation. For the materials which he inchMed in-his st.udy, the correlation seemed to hold for sampIes with a watei' content of 50% or higher. When several additional biological materials were examined in the present study, however, and their t,lmrmal conduetivit.ies were plotted vs. water content, the linear correlation was no longer found to bold. The most. notable deviations occur for chicken skin and bovine hmg samples (which fall ,¢

--

-

Composition by Weight Protein

\\'nler

I

I

]

Fat

!

-----

-.---..

---.-

Tllertnal Conductivity

iI Rcfer-

I

Water. . . . . . . . . . . . . . . . . . . . . Protein. . . . . . . . . . . . . . . . . . Fat . . . . . . . . . . . . . . . . . . . . . . . Bovine vitreous I ~ ~ i m o.r... Bovine aqueous humor. . . . Hurnrin plasrria.. . . . . . . . . . . Calf nniscle. . . . . . . . . . . . . . . II-furnn,nblood (hemttocrit, 43y3 . . . . . . * . . . . . . . . . . . . . Reef muscle.. . . . . . . . . . . . . Bovine kidney. . . . . . . . . . . . Boviuc brain., . . . . . . . . . . Bovine liver. . . . . . . . . . . . . Egg yolk.. . . . . . . . . . . . . . . Chicken skin.. . . . . . . . . . . . Bovine lung. . . . . . . . . . . . . . . Bovine fat. . . . . . . . . . . . . . . . Bovine bone r ~ ~ n r r o w . .. . .

* 1 BTU per hr f t OF

=

4.135 X

cnl per cnl sec "C.

in the region of water con ten& greater than ' i O y;,). The water colhtsllt correlathn docs not, appear to be complete in a general sense. Consider the case of lung tissue. l u this specimen &m exists a large volumc of cziptured gas. Sincc, the thermal conductivitics or" gases are comiderttbly srnrtller than those for solids and liquids, the gases must contribute to the low value of thermal conductivity obtained for'lung tissue. It thus supports the idea, that not only water content h u t thc content of all inajor species is impor tan t in col-relating thermal conductivity values of various sa~nples. 111this paper, it is proposed as t,hc w s t upprosimation that all biological materials axe composed of w ~ t e r protein, , and fat as far ns thermsl conductivity is concerned.* Table 3 gives the composition of d l biological smz1des studied in term? of these three comI~onenb.1Vhe.e reference d u e s for three constituents did not tots1 100%, adjustments were made, rhtaining- the same relative percentages, so that tlhe total would be 100f~;. Experimental densities. of ),iological materials in sorrie e k e s ttTere not available or as sccurately known $I .&sired. Thus densities were computcd for each &&&rial and then used in tho ;heoretical thermal conductivity equations presented pre-

* Lung tissue is an exception discussed below.

whi,~rcy is density of the ~vlwlcsample; y l , y 2, and ya arc densities of water, protein, and fatr, respectively ; , w2 , rtrlci w3 arc wight per cell t of m t m , protein, and fat;, rcq~ectively; and P I , v 2 , and v 3 are VOIUIUC per cent of wa.ter, protein, and fat, respectively. The theoretical espression for the clensi ty of a I)iological ~natcrinl in terms of the ~ e n s i t i cand ~ w i g h t fsxction of its three prim(? constituents is then

Et was necessary to consider a fourth conponcnt,. air, in computing the clcnsity mid thermal r:ontluctivity of lung tissue, 13y r:nlculating theoretical volunzes for each of tall(: threc other con~ponents (rvatcr., protein, m d ' fnl) and a theoretical v o l u m for the entire. sample based on the esperirnental density, it was found thatl 57.474, hy volume of the tissue was air. rllt~hough the weight per cent of air was only 0.08 %, its

t

t H E R M A L CONDUCTIVITY MEASUREMENTS

water and the mixtures were studied. Although some difficulty was experienced in remoying all the air from tile powdered protein used in making up the solution, it w ~ found that the extrapolated conductivity of the protein was within about 15% of the value for cellulose nitrate. A graph of predicted vs. experimental thermal conductivitics is given in Figure 5; all predictions with the exception of one value correspond to the model with laminae, parallel to the heat flow (calculated values are given in Table 3). :For 11 of the 14 samples studied, the deviations are less than 3%. Compared to the previous correlation of conductivity vs. water content suggested by Spells-" (in which the scatter of the correlation. was greater than 25%), the function represented

effect on density (lemanded its inclusion in the theoretical model. The experimental value for the thermal conductivity of fat given in Ta})le 3 was obtained in the present study. An experimentM density value for t'a~ was also measured. It should be noted that the eomlmsition values given for chicken skin are actually those for dog skin, since no values could be found for the former. The thermal conductivity given for protein in Table 3 is that for cellulose nitrate, the only substance similm" to pr()teins .for which a conductivity value is known. Further, the density of protein used in Table 3 is that of glutamic acid, an amino acid building block of l)roteins found in high concentration in animal tissues. Experimental me&stlremellI,s were made on several water and protein mixtures. It w~s hoped to extrapolate a conductivity value for pure p,'otein since flint sul~stm}ee could not be isolated in a usable form. Powdered protein w,'~s added to o,36o-.

I ......

325

4~

I

'1 ......

I ....

1

'

I-

ii1 F i g u r e 5 is

~;n,n; . . . . a_ :..~,.,.-rlC~l~

ilnprovemen~.

in the case of chicken skin, the conductivity prediction for the series flow model was closer to the experimental value than was the prediction for the parallel flow too(tel which satisfactorily

1 .... I

J

I

l-

i

l .... !

I

I

J ,;' A" _

ikfA --

i¢' 0.300

//

A/

//

//

I.L 0

0.200

h

/

Q: -r

//

2D t-El

/

a2

/

S"

i(

/A

IJJ Y_

/ 0.100

/

/

/

A

I

I

/ /

/

/

.ik PAtlALLEL YLOW MODEL It SERIES FLOW MODEL

/

/

/

/

/

.,

o

/

!

I . 1..... O.IOO

!.

!

1_

I

1

O.200

I

1

I

I

!

I

O.300

K THEORETICAL, B T U / H R FT ° F

Fro. 5. Predicted vs. experimental thermal eonductivities to biological fluids and tissues.

O ~')1'

o.,t)

])OPPENDIEK ET AI,.

predicted the eonduet,ivities for all the other t)Mogical fluids and tissues. It is felt that this result is probably exl)]ained ozt the b,~sis of the layered geometry (perpendicular to the heat flow). The. experimeni, al t~one tam'row dat'~ were lower than values predicted by both the series and paralM flow models (the one point wlfich significantly deviates h'om the/t5 ° line in fSg. 5). ]t is believed that, when the bone marrow w,'~-." removed fl'om the bone and positioned into the conductivity cell using a. nfierospatula, some air was trapped in the' thin layer, thus yMdhlg a lower conductivity value. Two sl)eeimens which have been st~tdied were ~)mitte~l from Table 3, namely, urine and gastric juice. An accurate composition in terms of the three eomponent, s (water, protein, and fat,) could not be determined for these samples. However, t.he high water content, (97 %) of urine ~is in good agreement with its 'high eonduel, ivity (0.324 BTU per hr fi) °F). A further refinement would most likely result in theoretied values quite dose to the experimental one. The high water eonten~, of ga.strie juice (99.0.o), however, is not in agreement with the low experimental eonduetivily (0.257 BTU per hr ft °F). It was fel~ at the time t!mt c.onduetivity measuremenl~s were made on the singh', ga~strie juice sample that it,s true composition was questionable in that a substantial amouut of foreign matter was noted to be pres:ent. A furl,her test of the model which is b ~ e d on !'tminae pm'allel to heat flow wa.s made by using it to calculate the eonductivit;y of erythroeytes m~d comparing the resulL.s wit.h values computed from the Eucken equation ~ for suspended particles. Using volume per cent of plasma and erythroeytes obtained from the hematocrit value, the parallel flow model gives a thermal conductivity value for erythrocytes of 0.274 BTU per hr ft °F, whereas the Eucken equation yields ~ conductivity of 0.260 BTU per hr f~ °F. The 5% difference between these ~wo values is considered a.s further support, of this pm'alleI flow model. Conductivilies of specimens exposed to freezelhaw conditions. The freeze-t.haw exposure conductivity measurements ob rained to date indicate that the ratios of exposed I~) unexpbge(l" eonductivities for bovine liver, kidney, brain, and muscle fall in the range 1.12 to 1.27 for slow freeze-thaw rate experiments (freezing to a lit, tie below 32°F and slow thawing to room tempera¢ure). Only bovine liver h~s been studied after

rapid fl'eeze-thaw exposure (freezing to -320°.F and rapid thawing in warm water); under these conditions the ratios of exposed to unexposed eonduetivities for this tissue were found to fall near 1..05. The fact tlmt slow freezing causes a greater thermal conductivity change thaa~ liquid nitrogen freezing in bovine liver may result, tee'rose of greater damage of the protein molecules by the dehydration and growth of ext.racellular ice tha~ accompany slow freezing. Because fl~e fl'eezing time is l(mger, ttiere is more time for eonecntrated salt solutions in the t,issue to ret)ct with the other components. The observed increase in conductivity ~ts storage time at 32°F inerea.ses mighl, be explained by the growth of extraeellulm" ice crystals. On thawing, this extraeellular ice may nol, redistribute i t.~elf between intraeeliular and extraedlula.r environments in the stone proportion as prior to freezing. If an excess of wa,ter remains outside the cells, a new heat flow path having Tess thermal resistance than t,he natural environmenl, may be l~rovide(l; as a result, the conduetivil, y would incre~use. SUMMARY The thermal conductivii,ies of normal samples of biological fluids and tissues determined experimentally tmve been presented. A special, unidirectional lleat flow al)l)ar~t~zs which yMds absolute conductivity results w~s described. Conductivity measurements were reporled for the following specimens: human blood and phmma; bovine nmsch:, vitreous humor, aqueous humor, hmg, liver, kidney, bone ma.rrow, fat, and brain; chicken skhl and egg yolk. The thermal conductivities of several tissues me,muted after these specimens were slowly frozen to "~ little below 32°F and then slowly ,hawed were presented; the eonduetivities incre,~ed 10 to 20% above values obt.ained for correslmnding unstressed st)ecimens. The corresponding conductivity changes for samples that were cryogenically h'ozen and more quiddy warmed were smaller. The flmrmal eonductivities of all the biologieal fluids and tissues studied also, were predicted using mat,hematieal heat conduction models b~sed on the premise that all biological sl)eeimens are con> posed of three primary materials, namely, water, fat, and protein. With the exception of one stratified biological species and one contaminated q3ecinmn, the predicted conduetivities for one of

THEIt 5I AL CONDUCTIVITY M E . \ S U R E M E N T S

the lnathematieal models differed from measured values by + 3 ~ , or les,~.

1.

ILEFE i:~E N CES J)o('tmmnta Geigy, 5t.h ed. Geigy Phat'nmceuticals, l)ivision of (;eigy Chemical Uorpo_"atlon, New York, 1950. J';uckcn, A. l"()rschuvxg auf dcm Gel)let des lngenicurwesens, vol. 11, p. 6. VDI Vcrlag, 1940. J. Food Sci., 28: 507z 1963. Langley, L. I,. and Cheraskin, E. The physiology of man, 2nd ed. Reinhold Publishing Corporation, New York, 1958. McAdums, W. H. Ifcnt transmission, 2n(l ed. McGraw-J]ill Book Oompany, New York, 1942. q

2. 3. 4. 5.

•

"

327

6. Ponder, E. Tile (.oefIicieni, of thermal condu(:tivity of blood and of various tissues. ,f. Gen. Physi[d., ~5: 915-551, 1962. 7. l(iohards, V., Br:~ve:'nmn, M., Floridi~, R., Persidsky, M., m~d Lowensteivi, J. Ini(.ial clinieai experiences with liquid nitrogen preserved I)lood, employing PVP as a protective additive. Am. J. Surg., I08: 313322, 1964. 8. Spector, W. S., ed. ]landl)ook of biological data. W. ]3. Saundcrs Company, PhiladelI)bia, I956. 9. Spells, K. E. The (hermal conductivitics of some biological tluids..1. Phys. Med. Biol., 149-153, 1960. 10. Stuhlman, O., Jr. An introduction to biophysics. John Wiley and Sons, New York, 1960.