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An analytical model of temperature regulation in human head
ARTICLE IN PRESS
Journal of Thermal Biology 29 (2004) 583–587 www.elsevier.com/locate/jtherbio
An analytical model of temperature regulation in human head A.L. Sukstanskii, D.A. Yablonskiy Department of Radiology, Washington University, St. Louis, Missouri 63110, USA
Abstract An analytical model of human brain temperature regulation is proposed. The model describes the distribution of brain temperature as a function of internal and external parameters, such as temperature of the incoming arterial blood, blood flow, oxygen consumption rate, ambient temperature, and heat exchange with the environment. It is shown that substantial changes in human brain temperature can be accomplished only through changes in the temperature of the incoming arterial blood or substantial suppression of blood flow. Other parameters can lead only to temperature changes near the brain surface. r 2004 Elsevier Ltd. All rights reserved. Keywords: Thermoregulation; Cerebral metabolism; Cerebral blood flow; Metabolic heat; Heat exchange; Brain temperature regulation; Selective brain cooling; Bioheat equation.
1. Introduction It is well accepted that temperature regulation in a human brain is accomplished through several major mechanisms. They are: blood flow, temperature of incoming arterial blood, oxygen consumption rate, and heat exchange with the environment. Several computersimulated models of temperature distribution in a human head have been developed (e.g., Nelson and Nunneley 1998; van Leeuwen et al., 2000). While these models provide important insights into the problem, their results substantially depend on input parameters that are not always known and may vary in broad ranges. The goal of our study is to develop an analytical model describing temperature distribution in a human head. The model provides analytical expressions that allow evaluation of changes in brain temperature under Corresponding author. Mallinckodt Institute of Radiology, 4525 Scott Avenue, St. Louis, MO 63110, USA. Tel.:+1+314362-1815; fax: +1-314-362-0526. E-mail address: [email protected] (D.A. Yablonskiy).
the influence of measurable input parameters. It can be used to predict a brain temperature response to such conditions as extreme heat or cold, external cooling with air or water flow, extensive exercise, cardiac by-pass surgery, etc.
2. Theoretical approach Our approach is based on a bio-heat equation originally proposed by Pennes (1948) for description of the temperature distribution in organs. We consider the head as a system consisting of cerebral tissue (brain) with overlaying layers of cerebrospinal fluid (CSF), skull and scalp. The temperature distributions in these four regions, Tj(r) (j ¼ 0; 1; 2; 3 corresponds to the brain, CSF, skull and scalp, respectively) can be found as a solution of a set of static bio-heat equations a0 r2 T 0 rb cb w0 ðT 0 T a0 Þ þ q0 ¼ 0; r2 T 1 ¼ 0;
r2 T 2 ¼ 0;
a3 r2 T 3 rb cb w3 ðT 3 T a3 Þ þ q3 ¼ 0;
0306-4565/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtherbio.2004.08.028
ð1Þ
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A.L. Sukstanskii, D.A. Yablonskiy / Journal of Thermal Biology 29 (2004) 583–587
where aj, wj, Taj, qj are thermal conductivity, blood flow, arterial blood temperature and metabolic heat production rate in the corresponding region, respectively; rb and cb are the density and specific heat of blood. In CSF, the blood flow and metabolic heat production are obviously absent; besides, in the skull, these quantities are very small (Olsen et al., 1985) and can be ignored. Also, we assume that the arterial blood temperature is constant across the brain. As shown by Yablonskiy et al. (2000), if surface effects are ignored, the temperature distribution in the brain is uniform with brain temperature equal to T b ¼ T a0 þ T m0 ;
T m0 ¼ qm0 =rb cb w0 ¼ 0:9z0 ½ C
(2)
where Tm0 is a temperature increase (shift) due to metabolic heat production and z0 is the so-called oxygen extraction fraction (OEF). This uniformity is a result of Eq. (2) and previous findings of positron-emission tomography (PET) (Lebrun-Grandie et al., 1983; Powers et al., 1984) that maps of the OEF in the normal human brain are largely flat despite considerable regional differences in the blood flow and oxygen consumption rate. Surface effects, obviously, violate uniformity of the brain temperature distribution, and their influences should be accounted for by boundary conditions to Eqs. (1) at the interfaces brain/CSF, CSF/skull, skull/ scalp and at the head surface (scalp/air). A problem of such a type can be analytically solved only in some simple geometries (e.g., plane, cylinder, sphere), whereas a real geometry of the head is rather complicated. However, the problem under consideration can be substantially simplified by the following consideration: as we show below, temperature inhomogeneities in the brain have a characteristic length D that depends on the blood flow w0: D ¼ ða0 =rb cb w0 Þ1=2 :
(3)
Typically, this characteristic length is about several millimeters which is much smaller than an adult human brain size of 15 cm or even neonatal brain size of
6 cm. It leads to the temperature in the brain being (a) practically homogeneous except of a narrow ( D) region in the vicinity of its surface and (b) practically independent of specific brain geometry. Indeed, numerical calculations (Nelson and Nunneley, 1998) and (van Leeuwen et al., 2000) of T(r) for adult and neonatal head clearly demonstrated that T(r) is non-uniform only near the surface. Because the typical curvature of the head surface is also much bigger than the characteristic length D, the temperature distribution near the brain surface can be treated as a one-dimensional problem, the brain being considered as a semi-infinite region (say, xo0), covered by three infinite layers of thickness d1, d2, d3, corresponding to CSF, skull and scalp, respectively.
The boundary conditions on the interfaces between the regions and at the external head surface are T 0 ð0Þ ¼ T 1 ð0Þ; a0 T 0 0 ð0Þ ¼ a1 T 0 1 ð0Þ ðbrain=CSFÞ; 0 0 T 1 ðd 1 Þ ¼ T 2 ðd 1 Þ; a1 T 1 ðd 1 Þ ¼ a2 T 2 ðd 1 Þ ðCSF=skullÞ; T 2 ðd 1 þ d 2 Þ ¼ T 3 ðd 1 þ d 2 Þ; a2 T 0 2 ðd 1 þ d 2 Þ ¼ a3 T 0 3 ðd 1 þ d 2 Þ 0
a3 T 3 ðdÞ ¼ h ðT 3 ðdÞ T e Þ þ qev
ðskull=scalpÞ; ðscalp=airÞ; (4)
where prime denotes a derivative with respect to x; d ¼ d 1 þ d 2 þ d 3 : These boundary conditions reflect the fact that no heat dissipation takes place on the interfaces between internal surfaces (brain/CSF/skull/scalp), hence temperature and heat flow should be continuous on each of these interfaces. The boundary condition between scalp and environment takes into account heat exchange with the environment. Here Te is an ambient temperature; h is a heat transfer coefficient, which effectively takes into account heat transfer due to air convection and radiation. It is easy to see that the evaporation term qev in the last boundary condition leads only to a renormalization of the ambient temperature: T e ! T~ e ¼ T e qev =h:
(5) ~ Note that the effective ambient temperature T e is always less than Te.
3. Results and discussion A solution of Eqs. (1) is sought in the form T 0 ðxÞ ¼ T a0 þ T m0 þ A0 expðk0 xÞ; T 1 ðxÞ ¼ A1 x þ B1 ; T 2 ðxÞ ¼ A2 x þ B2 ; T 3 ðxÞ ¼ T a3 þ T m3 þ A3 expðk3 xÞ þ B3 expðk3 xÞ; (6) 1=2 and T mj ¼ where for j = 0 and 3, kj ¼ rb cb wj =aj qj =rb cb wj are metabolic temperature shifts. Note that k0 ¼ 1=D with D defined by Eq. (3). The coefficients Aj, Bj can be found by substituting Eqs. (6) into the boundary conditions (4). After some straightforward but tedious algebra, the coefficient A0 can be written in the form r0 A0 ¼ 1 ðh þ r0 þ r1 þ r2 þ r~ 3 Þ Z sinh Z þ cosh Z 1 ;
T a0 þ T m0 T~ e þ DT hr3 DT ¼ T a0 þ T m0 T a3 T m3 ;
ð7Þ
where, Z ¼ d 3 k3 ; rj ¼ d j =aj ðj ¼ 1; 2; 3Þ are thermal resistances (per unit surface area) of the corresponding layers, r0 ¼ 1=a0 k0 is an effective thermal resistance of the brain surface, and r~ 3 is the effective thermal
ARTICLE IN PRESS A.L. Sukstanskii, D.A. Yablonskiy / Journal of Thermal Biology 29 (2004) 583–587
resistance of the scalp, sin h Z Z2 r0 þ r1 þ r2 r3 þ r~ 3 ¼ hr3 Z
þ ðcosh Z 1Þ h1 þ r0 þ r1 þ r2 :
ð8Þ
This effective resistance depends on the actual thermal resistance of the scalp, r3, the thermal resistances of all the other layers, and also on the blood flow in the scalp. If the blood flow in the scalp is small enough and 1=2 Z k3 w3 1; the effective thermal resistance r~ 3 of the scalp reduces to r~ 3 ’ r3 þ Z2
r3 1 1 1 r0 þ r1 þ r2 þ OðZ4 Þ: þ þ þ hr3 2 6 2h
(9) Naturally, r~ 3 ! r3 at w3 ! 0: Apparently, if the coefficient A0 in Eq. (7) were zero, the temperature of the brain would be constant everywhere. The deviation of A0 from zero creates an inhomogeneous distribution near the brain surface, with the temperature exponentially approaching a constant deep-brain value with a characteristic length D defined by Eq. (3). The first term in Eq. (7) is responsible for
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brain temperature inhomogeneities due to the difference between the deep brain temperature and an effective ambient temperature. The second term is responsible for brain temperature inhomogeneities due to the differences in metabolic temperature shifts and temperatures of the arterial blood flowing between brain and scalp. The deep brain temperature is defined by Eq. (2) which suggests that a major factor influencing brain temperature is the temperature of incoming arterial blood. This is in agreement with experimental findings of Nybo et al. (2002) that brain temperature during prolonged exercise with hyperthermia grows in parallel with the arterial blood temperature. The temperature shift due to metabolism, Tm, is at most 0.9 1C but for a typical OEF of 0.3 (30%), Tm is about 0.3 1C. This assessment is in a good agreement with previous theoretical estimates (Nelson and Nunneley, 1998; Yablonskiy et al., 2000) and experimental data (Nybo et al., 2002). The structure of other coefficients in Eqs. (6) is similar but we are not providing them here as we are interested only in the brain temperature. Eqs. (7)–(8) (plus similar expressions for other coefficients appearing in Eqs. (6)) provide an analytical solution to the problem and can be used to calculate the temperature distribution in the
Fig. 1. Temperature distribution near the head surface (x ¼ 0 on horizontal axis corresponds to the brain surface) at different values of (a) heat transfer coefficient h, (b) effective ambient temperature T~ e ; (c) blood flow in the scalp w3, and (d) blood flow in the brain w0. The following parameters are fixed and identical for all graphs: d 1 ¼ 0:2 cm; d 2 ¼ 0:5 cm; d 3 ¼ 0:3 cm; Ta0=Ta3=37 1C, Tm0=0.3 1C, Tm3=0.05 1C, rb=1.05 g/cm3, cb=3.8 J/(g1C), a0-3=(5.03, 5.82, 11.6, 3.4) 103W/(cm 1C) (cited from Table 1 in (Nelson and Nunneley 1998)). These values are typical for a human brain. Lines 1 in all graphs correspond to h=4 104 W/(cm2 1C), T~ e ¼ 20 C; w3=8.4 104 s1, w0=9.45 103 s1. Lines 2–4 in (a) correspond to h ¼ 0; 8 104 and N [W/cm2 1C)]. Lines 2–5 in (b) correspond to T~ e ¼ 30; 40; 10 and 0[1C]. Lines 2–4 in (c) correspond to w3=16, 32 and 0[ 104 s1]. Lines 2–5 in (d) correspond to w0=19, 5, 2.5, and 1[ 104 s1].
ARTICLE IN PRESS A.L. Sukstanskii, D.A. Yablonskiy / Journal of Thermal Biology 29 (2004) 583–587
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brain. Several important temperature distributions in a human head are presented in Fig. 1. Fig. 1a demonstrates the temperature distribution at different values of the heat transfer coefficient h. Line 1 corresponds to the generally accepted value h=4 104W/(cm2 1C). This value can be easily experimentally changed by insulating the head from the external medium (h decreases) or by maintaining conditions for more effective heat exchange (h increases). Line 2 corresponds to h ¼ 0; that can be achieved simply by isolating the head from the external medium. In this limit, the general expressions can be substantially simplified, in particular, the temperature at the head surface, T s T 3 ðdÞ; and the temperature at the brain surface, T bs T 0 ð0Þ; takes the form T bs ¼ T a0 þ T m0 DT
Z r0 sinh Z ; r0 þ r1 þ r2 Z sinh Z þ r3 cosh Z
(10) T s ¼ T bs DT
r3 ðcosh Z 1Þ þ r1 þ r2 Z sinh Z : r0 þ r1 þ r2 Z sinh Z þ r3 cosh Z (11)
Obviously, if DT ¼ 0; the temperature distribution under condition h=0 becomes homogeneous across the whole head. Lines 3 and 4 in Fig. 1a correspond to h=8 104W/ (cm2 1C) and h ¼ 1: The latter limit can be achieved, for example, by using a powerful fan or by surrounding the head by running cold water. In this extreme, the temperature at the head surface obviously coincides with the temperature of surrounding media, T s ¼ T e ; whereas the temperature at the brain surface, Tbs, takes the form T bs ¼ T a0 þ T m0
Z r0 T a0 þ T m0 T~ e ðcosh Z 1Þ DT r0 þ r1 þ r2 Z cosh Z þ r3 sinh Z
(12) Fig. 1b illustrates the temperature distribution at different values of the effective ambient temperature T~ e ; Eq. (5). The variation in T~ e from 0 to 401C leads to substantial changes in the scalp and skull temperature, whereas the temperature at the brain surface varies only slightly (less than 1 1C). The influence of the blood flow in the scalp is shown in Fig. 1c: line 1 corresponds to w3=8.4 104 s1 (Vietla et al., 1993), line 2 and 3 correspond to doubled and quadrupled blood flow, and line 4 corresponds to w3 ¼ 0: As we see, a substantial change in w3 only slightly affects the temperature distribution in the brain (several tens of 1C). It should be noted that there is the main common feature of Figs. 1a–c: a substantial change in the heat transfer coefficient (from 0 to N), in the ambient temperature (from 0 to 40 1C), or in the scalp blood flow
(from 0 to 4 times that of the normal value 8.4 104 s1), result in significant changes of the temperature on the head surface Ts and of the temperature distribution in the CSF, skull and scalp, whereas the temperature in the brain, T0(x), is affected only within a narrow interval near the interface with the CSF layer. In the brain interior, i.e. at, xo D the function T0(x) is practically flat and is equal to T b T a0 þ T m0 ; as expected. The region where the temperature in the brain is nonuniform plays the same role as a skin-layer in metals in electro-dynamic problems; its size is equal to D ¼ k1 0 and is given in Eq. (3). Just as the electromagnetic ‘‘skindepth’’ D is inversely proportional to square root of frequency, here the skin-depth D increases with decreasing blood flow w0 and vice versa. Thus, a substantial change in temperature within the brain can be achieved by varying the blood flow in the brain, as shown in Fig. 1d. Of course, when the brain blood flow becomes small and D increases substantially, the real geometry of the brain and the curvature of its surface should be incorporated into the calculations for precise temperature distribution information. 4. Conclusion A model of the temperature distribution in the human head is developed. The analysis predicts changes in the brain temperature as a function of major internal and external parameters: the temperature of incoming arterial blood, blood flow, oxygen extraction fraction, ambient temperature, and heat exchange with the environment. In particular, the model can be used for predicting a head temperature response to extreme conditions such as heavy exercise or exposure to heat or cold and for estimating the extent of possible changes in brain temperature during selective head cooling during bypass surgery. Besides, our results illustrate the fallacy of attempting to induce isolated brain hypothermia by head surface cooling: even extreme changes in ambient temperature leads to temperature changes only in the vicinity of the head surface whereas in the brain the temperature remains practically constant. Acknowledgements The authors are grateful to Professors Joseph J. H. Ackerman, Marcus E. Raichle and Mark S. Conradi for discussion and helpful comments. This work was supported by NIH Grant R01 NS41519. References Lebrun-Grandie, P., Baron, J.-C., Soussaline, F., Loch’h, C., Sastre, J., Bousser, M.-G., 1983. Coupling between regional
ARTICLE IN PRESS A.L. Sukstanskii, D.A. Yablonskiy / Journal of Thermal Biology 29 (2004) 583–587 blood flow and oxygen utilization in the normal human brain. A study with positron tomography and oxygen. Arch. Neurol 40, 230–236. Nelson, D.A., Nunneley, S.A., 1998. Brain temperature and limits on transcranial cooling in humans: quantitative modeling results. Eur. J. Appl. Physiol. 78, 353–359. Nybo, L., Secher, N.H., Nielsen, B., 2002. Inadequate heat release from the human brain during prolonged exercise with hyperthermia. J. Physiol. 545 (Pt 2), 697–704. Olsen, R.W., Hayes, L.J., Wissler, E.H., Nikaidoh, H., Eberhart, R.C., 1985. Influence of hypothermia and circulatory arrest on cerebral temperature distribution. J. Biomech. Eng. 107, 354–360. Pennes, H.H., 1948. Analysis of tissue and arterial blood temperature in the resting human forearm. J. Appl. Physiol. 1, 93–122.
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Powers, W.J., Grubb Jr, R.L., Raichle, M.E., 1984. Physiological responses to focal cerebral ischemia in humans. Ann. Neurol. 16, 546–552. van Leeuwen, G.M.J., Hand, J.W., Lagenduk, J.J.W., Azzopardi, D.V., Edwards, A.D., 2000. Numerical modeling of temperature distribution within the natal head. Pediatr. Res. 48, 351–356. Vietla, S., Eberhart, R.C., Meyer, D.M., 1993. The influence of hypothermia and tissue perfusion on temperature distributions in simulated cranial surgery. In: Roemer, R.B. (Ed.), Advances in Bioheat and Mass Transfer. American Society of Mechanical Engineers, New York, pp. 9–17. Yablonskiy, D.A., Ackerman, J.J.H., Raichle, M.E., 2000. Coupling between changes in human brain temperature and oxidative metabolism during prolonged visual stimulation. PNAS 97 (13), 7603–7608.
Journal of Thermal Biology 29 (2004) 583–587 www.elsevier.com/locate/jtherbio
An analytical model of temperature regulation in human head A.L. Sukstanskii, D.A. Yablonskiy Department of Radiology, Washington University, St. Louis, Missouri 63110, USA
Abstract An analytical model of human brain temperature regulation is proposed. The model describes the distribution of brain temperature as a function of internal and external parameters, such as temperature of the incoming arterial blood, blood flow, oxygen consumption rate, ambient temperature, and heat exchange with the environment. It is shown that substantial changes in human brain temperature can be accomplished only through changes in the temperature of the incoming arterial blood or substantial suppression of blood flow. Other parameters can lead only to temperature changes near the brain surface. r 2004 Elsevier Ltd. All rights reserved. Keywords: Thermoregulation; Cerebral metabolism; Cerebral blood flow; Metabolic heat; Heat exchange; Brain temperature regulation; Selective brain cooling; Bioheat equation.
1. Introduction It is well accepted that temperature regulation in a human brain is accomplished through several major mechanisms. They are: blood flow, temperature of incoming arterial blood, oxygen consumption rate, and heat exchange with the environment. Several computersimulated models of temperature distribution in a human head have been developed (e.g., Nelson and Nunneley 1998; van Leeuwen et al., 2000). While these models provide important insights into the problem, their results substantially depend on input parameters that are not always known and may vary in broad ranges. The goal of our study is to develop an analytical model describing temperature distribution in a human head. The model provides analytical expressions that allow evaluation of changes in brain temperature under Corresponding author. Mallinckodt Institute of Radiology, 4525 Scott Avenue, St. Louis, MO 63110, USA. Tel.:+1+314362-1815; fax: +1-314-362-0526. E-mail address: [email protected] (D.A. Yablonskiy).
the influence of measurable input parameters. It can be used to predict a brain temperature response to such conditions as extreme heat or cold, external cooling with air or water flow, extensive exercise, cardiac by-pass surgery, etc.
2. Theoretical approach Our approach is based on a bio-heat equation originally proposed by Pennes (1948) for description of the temperature distribution in organs. We consider the head as a system consisting of cerebral tissue (brain) with overlaying layers of cerebrospinal fluid (CSF), skull and scalp. The temperature distributions in these four regions, Tj(r) (j ¼ 0; 1; 2; 3 corresponds to the brain, CSF, skull and scalp, respectively) can be found as a solution of a set of static bio-heat equations a0 r2 T 0 rb cb w0 ðT 0 T a0 Þ þ q0 ¼ 0; r2 T 1 ¼ 0;
r2 T 2 ¼ 0;
a3 r2 T 3 rb cb w3 ðT 3 T a3 Þ þ q3 ¼ 0;
0306-4565/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtherbio.2004.08.028
ð1Þ
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A.L. Sukstanskii, D.A. Yablonskiy / Journal of Thermal Biology 29 (2004) 583–587
where aj, wj, Taj, qj are thermal conductivity, blood flow, arterial blood temperature and metabolic heat production rate in the corresponding region, respectively; rb and cb are the density and specific heat of blood. In CSF, the blood flow and metabolic heat production are obviously absent; besides, in the skull, these quantities are very small (Olsen et al., 1985) and can be ignored. Also, we assume that the arterial blood temperature is constant across the brain. As shown by Yablonskiy et al. (2000), if surface effects are ignored, the temperature distribution in the brain is uniform with brain temperature equal to T b ¼ T a0 þ T m0 ;
T m0 ¼ qm0 =rb cb w0 ¼ 0:9z0 ½ C
(2)
where Tm0 is a temperature increase (shift) due to metabolic heat production and z0 is the so-called oxygen extraction fraction (OEF). This uniformity is a result of Eq. (2) and previous findings of positron-emission tomography (PET) (Lebrun-Grandie et al., 1983; Powers et al., 1984) that maps of the OEF in the normal human brain are largely flat despite considerable regional differences in the blood flow and oxygen consumption rate. Surface effects, obviously, violate uniformity of the brain temperature distribution, and their influences should be accounted for by boundary conditions to Eqs. (1) at the interfaces brain/CSF, CSF/skull, skull/ scalp and at the head surface (scalp/air). A problem of such a type can be analytically solved only in some simple geometries (e.g., plane, cylinder, sphere), whereas a real geometry of the head is rather complicated. However, the problem under consideration can be substantially simplified by the following consideration: as we show below, temperature inhomogeneities in the brain have a characteristic length D that depends on the blood flow w0: D ¼ ða0 =rb cb w0 Þ1=2 :
(3)
Typically, this characteristic length is about several millimeters which is much smaller than an adult human brain size of 15 cm or even neonatal brain size of
6 cm. It leads to the temperature in the brain being (a) practically homogeneous except of a narrow ( D) region in the vicinity of its surface and (b) practically independent of specific brain geometry. Indeed, numerical calculations (Nelson and Nunneley, 1998) and (van Leeuwen et al., 2000) of T(r) for adult and neonatal head clearly demonstrated that T(r) is non-uniform only near the surface. Because the typical curvature of the head surface is also much bigger than the characteristic length D, the temperature distribution near the brain surface can be treated as a one-dimensional problem, the brain being considered as a semi-infinite region (say, xo0), covered by three infinite layers of thickness d1, d2, d3, corresponding to CSF, skull and scalp, respectively.
The boundary conditions on the interfaces between the regions and at the external head surface are T 0 ð0Þ ¼ T 1 ð0Þ; a0 T 0 0 ð0Þ ¼ a1 T 0 1 ð0Þ ðbrain=CSFÞ; 0 0 T 1 ðd 1 Þ ¼ T 2 ðd 1 Þ; a1 T 1 ðd 1 Þ ¼ a2 T 2 ðd 1 Þ ðCSF=skullÞ; T 2 ðd 1 þ d 2 Þ ¼ T 3 ðd 1 þ d 2 Þ; a2 T 0 2 ðd 1 þ d 2 Þ ¼ a3 T 0 3 ðd 1 þ d 2 Þ 0
a3 T 3 ðdÞ ¼ h ðT 3 ðdÞ T e Þ þ qev
ðskull=scalpÞ; ðscalp=airÞ; (4)
where prime denotes a derivative with respect to x; d ¼ d 1 þ d 2 þ d 3 : These boundary conditions reflect the fact that no heat dissipation takes place on the interfaces between internal surfaces (brain/CSF/skull/scalp), hence temperature and heat flow should be continuous on each of these interfaces. The boundary condition between scalp and environment takes into account heat exchange with the environment. Here Te is an ambient temperature; h is a heat transfer coefficient, which effectively takes into account heat transfer due to air convection and radiation. It is easy to see that the evaporation term qev in the last boundary condition leads only to a renormalization of the ambient temperature: T e ! T~ e ¼ T e qev =h:
(5) ~ Note that the effective ambient temperature T e is always less than Te.
3. Results and discussion A solution of Eqs. (1) is sought in the form T 0 ðxÞ ¼ T a0 þ T m0 þ A0 expðk0 xÞ; T 1 ðxÞ ¼ A1 x þ B1 ; T 2 ðxÞ ¼ A2 x þ B2 ; T 3 ðxÞ ¼ T a3 þ T m3 þ A3 expðk3 xÞ þ B3 expðk3 xÞ; (6) 1=2 and T mj ¼ where for j = 0 and 3, kj ¼ rb cb wj =aj qj =rb cb wj are metabolic temperature shifts. Note that k0 ¼ 1=D with D defined by Eq. (3). The coefficients Aj, Bj can be found by substituting Eqs. (6) into the boundary conditions (4). After some straightforward but tedious algebra, the coefficient A0 can be written in the form r0 A0 ¼ 1 ðh þ r0 þ r1 þ r2 þ r~ 3 Þ Z sinh Z þ cosh Z 1 ;
T a0 þ T m0 T~ e þ DT hr3 DT ¼ T a0 þ T m0 T a3 T m3 ;
ð7Þ
where, Z ¼ d 3 k3 ; rj ¼ d j =aj ðj ¼ 1; 2; 3Þ are thermal resistances (per unit surface area) of the corresponding layers, r0 ¼ 1=a0 k0 is an effective thermal resistance of the brain surface, and r~ 3 is the effective thermal
ARTICLE IN PRESS A.L. Sukstanskii, D.A. Yablonskiy / Journal of Thermal Biology 29 (2004) 583–587
resistance of the scalp, sin h Z Z2 r0 þ r1 þ r2 r3 þ r~ 3 ¼ hr3 Z
þ ðcosh Z 1Þ h1 þ r0 þ r1 þ r2 :
ð8Þ
This effective resistance depends on the actual thermal resistance of the scalp, r3, the thermal resistances of all the other layers, and also on the blood flow in the scalp. If the blood flow in the scalp is small enough and 1=2 Z k3 w3 1; the effective thermal resistance r~ 3 of the scalp reduces to r~ 3 ’ r3 þ Z2
r3 1 1 1 r0 þ r1 þ r2 þ OðZ4 Þ: þ þ þ hr3 2 6 2h
(9) Naturally, r~ 3 ! r3 at w3 ! 0: Apparently, if the coefficient A0 in Eq. (7) were zero, the temperature of the brain would be constant everywhere. The deviation of A0 from zero creates an inhomogeneous distribution near the brain surface, with the temperature exponentially approaching a constant deep-brain value with a characteristic length D defined by Eq. (3). The first term in Eq. (7) is responsible for
585
brain temperature inhomogeneities due to the difference between the deep brain temperature and an effective ambient temperature. The second term is responsible for brain temperature inhomogeneities due to the differences in metabolic temperature shifts and temperatures of the arterial blood flowing between brain and scalp. The deep brain temperature is defined by Eq. (2) which suggests that a major factor influencing brain temperature is the temperature of incoming arterial blood. This is in agreement with experimental findings of Nybo et al. (2002) that brain temperature during prolonged exercise with hyperthermia grows in parallel with the arterial blood temperature. The temperature shift due to metabolism, Tm, is at most 0.9 1C but for a typical OEF of 0.3 (30%), Tm is about 0.3 1C. This assessment is in a good agreement with previous theoretical estimates (Nelson and Nunneley, 1998; Yablonskiy et al., 2000) and experimental data (Nybo et al., 2002). The structure of other coefficients in Eqs. (6) is similar but we are not providing them here as we are interested only in the brain temperature. Eqs. (7)–(8) (plus similar expressions for other coefficients appearing in Eqs. (6)) provide an analytical solution to the problem and can be used to calculate the temperature distribution in the
Fig. 1. Temperature distribution near the head surface (x ¼ 0 on horizontal axis corresponds to the brain surface) at different values of (a) heat transfer coefficient h, (b) effective ambient temperature T~ e ; (c) blood flow in the scalp w3, and (d) blood flow in the brain w0. The following parameters are fixed and identical for all graphs: d 1 ¼ 0:2 cm; d 2 ¼ 0:5 cm; d 3 ¼ 0:3 cm; Ta0=Ta3=37 1C, Tm0=0.3 1C, Tm3=0.05 1C, rb=1.05 g/cm3, cb=3.8 J/(g1C), a0-3=(5.03, 5.82, 11.6, 3.4) 103W/(cm 1C) (cited from Table 1 in (Nelson and Nunneley 1998)). These values are typical for a human brain. Lines 1 in all graphs correspond to h=4 104 W/(cm2 1C), T~ e ¼ 20 C; w3=8.4 104 s1, w0=9.45 103 s1. Lines 2–4 in (a) correspond to h ¼ 0; 8 104 and N [W/cm2 1C)]. Lines 2–5 in (b) correspond to T~ e ¼ 30; 40; 10 and 0[1C]. Lines 2–4 in (c) correspond to w3=16, 32 and 0[ 104 s1]. Lines 2–5 in (d) correspond to w0=19, 5, 2.5, and 1[ 104 s1].
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brain. Several important temperature distributions in a human head are presented in Fig. 1. Fig. 1a demonstrates the temperature distribution at different values of the heat transfer coefficient h. Line 1 corresponds to the generally accepted value h=4 104W/(cm2 1C). This value can be easily experimentally changed by insulating the head from the external medium (h decreases) or by maintaining conditions for more effective heat exchange (h increases). Line 2 corresponds to h ¼ 0; that can be achieved simply by isolating the head from the external medium. In this limit, the general expressions can be substantially simplified, in particular, the temperature at the head surface, T s T 3 ðdÞ; and the temperature at the brain surface, T bs T 0 ð0Þ; takes the form T bs ¼ T a0 þ T m0 DT
Z r0 sinh Z ; r0 þ r1 þ r2 Z sinh Z þ r3 cosh Z
(10) T s ¼ T bs DT
r3 ðcosh Z 1Þ þ r1 þ r2 Z sinh Z : r0 þ r1 þ r2 Z sinh Z þ r3 cosh Z (11)
Obviously, if DT ¼ 0; the temperature distribution under condition h=0 becomes homogeneous across the whole head. Lines 3 and 4 in Fig. 1a correspond to h=8 104W/ (cm2 1C) and h ¼ 1: The latter limit can be achieved, for example, by using a powerful fan or by surrounding the head by running cold water. In this extreme, the temperature at the head surface obviously coincides with the temperature of surrounding media, T s ¼ T e ; whereas the temperature at the brain surface, Tbs, takes the form T bs ¼ T a0 þ T m0
Z r0 T a0 þ T m0 T~ e ðcosh Z 1Þ DT r0 þ r1 þ r2 Z cosh Z þ r3 sinh Z
(12) Fig. 1b illustrates the temperature distribution at different values of the effective ambient temperature T~ e ; Eq. (5). The variation in T~ e from 0 to 401C leads to substantial changes in the scalp and skull temperature, whereas the temperature at the brain surface varies only slightly (less than 1 1C). The influence of the blood flow in the scalp is shown in Fig. 1c: line 1 corresponds to w3=8.4 104 s1 (Vietla et al., 1993), line 2 and 3 correspond to doubled and quadrupled blood flow, and line 4 corresponds to w3 ¼ 0: As we see, a substantial change in w3 only slightly affects the temperature distribution in the brain (several tens of 1C). It should be noted that there is the main common feature of Figs. 1a–c: a substantial change in the heat transfer coefficient (from 0 to N), in the ambient temperature (from 0 to 40 1C), or in the scalp blood flow
(from 0 to 4 times that of the normal value 8.4 104 s1), result in significant changes of the temperature on the head surface Ts and of the temperature distribution in the CSF, skull and scalp, whereas the temperature in the brain, T0(x), is affected only within a narrow interval near the interface with the CSF layer. In the brain interior, i.e. at, xo D the function T0(x) is practically flat and is equal to T b T a0 þ T m0 ; as expected. The region where the temperature in the brain is nonuniform plays the same role as a skin-layer in metals in electro-dynamic problems; its size is equal to D ¼ k1 0 and is given in Eq. (3). Just as the electromagnetic ‘‘skindepth’’ D is inversely proportional to square root of frequency, here the skin-depth D increases with decreasing blood flow w0 and vice versa. Thus, a substantial change in temperature within the brain can be achieved by varying the blood flow in the brain, as shown in Fig. 1d. Of course, when the brain blood flow becomes small and D increases substantially, the real geometry of the brain and the curvature of its surface should be incorporated into the calculations for precise temperature distribution information. 4. Conclusion A model of the temperature distribution in the human head is developed. The analysis predicts changes in the brain temperature as a function of major internal and external parameters: the temperature of incoming arterial blood, blood flow, oxygen extraction fraction, ambient temperature, and heat exchange with the environment. In particular, the model can be used for predicting a head temperature response to extreme conditions such as heavy exercise or exposure to heat or cold and for estimating the extent of possible changes in brain temperature during selective head cooling during bypass surgery. Besides, our results illustrate the fallacy of attempting to induce isolated brain hypothermia by head surface cooling: even extreme changes in ambient temperature leads to temperature changes only in the vicinity of the head surface whereas in the brain the temperature remains practically constant. Acknowledgements The authors are grateful to Professors Joseph J. H. Ackerman, Marcus E. Raichle and Mark S. Conradi for discussion and helpful comments. This work was supported by NIH Grant R01 NS41519. References Lebrun-Grandie, P., Baron, J.-C., Soussaline, F., Loch’h, C., Sastre, J., Bousser, M.-G., 1983. Coupling between regional
ARTICLE IN PRESS A.L. Sukstanskii, D.A. Yablonskiy / Journal of Thermal Biology 29 (2004) 583–587 blood flow and oxygen utilization in the normal human brain. A study with positron tomography and oxygen. Arch. Neurol 40, 230–236. Nelson, D.A., Nunneley, S.A., 1998. Brain temperature and limits on transcranial cooling in humans: quantitative modeling results. Eur. J. Appl. Physiol. 78, 353–359. Nybo, L., Secher, N.H., Nielsen, B., 2002. Inadequate heat release from the human brain during prolonged exercise with hyperthermia. J. Physiol. 545 (Pt 2), 697–704. Olsen, R.W., Hayes, L.J., Wissler, E.H., Nikaidoh, H., Eberhart, R.C., 1985. Influence of hypothermia and circulatory arrest on cerebral temperature distribution. J. Biomech. Eng. 107, 354–360. Pennes, H.H., 1948. Analysis of tissue and arterial blood temperature in the resting human forearm. J. Appl. Physiol. 1, 93–122.
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