Bulletin q/M~thematical Biology Vol, 53, No. 4, pp. 525 536, 1991. Printed in Great Britain.
0092 8240/9153.00+0.00 Pergamon Press plc © 1991 Society for Mathematical Biology
EFFECT OF DERMAL TUMOURS ON TEMPERATURE DISTRIBUTION IN SKIN WITH VARIABLE BLOOD FLOW V. P. SAXENA and KAMAL RAJ P A R D A S A N I School of Mathematics and Allied Sciences, Jiwaji University, Gwalior 474 011, India Several investigations have been made for the heat flow problems in skin and subdermal tissues under normal physiological and atmospheric conditions. This paper considers the existence of a malignant tumour in the underlying tissues of epidermis of a human body. The surrounding tissues are assumed to have normal physiological functions, namely self-controlled metabolic activity, variable blood flow and perspiration. For the malignant portion the metabolic activity is taken to be continuous and uncontrolled. The effect of this factor is studied on the temperature profiles of the skin.
1. Introduction. Under normal physiological conditions, the human body maintains a thermal balance with the environment. To maintain this balance, the skin and subcutaneous tissues (SST) undergo various changes in their physiological activities. The SST region consists of dead tissues in the form of stratum corneum on one side and subcutaneous tissues with continuous metabolic activity and dense blood vessels on the other. In between are stratum germinativum and dermis. The stratum germinativum consists of some living cells but almost no blood vessels. The dermis consists of connective tissues, blood vessels, fat cells, sweat glands, oil glands and nerve fibres, etc. The population density of blood vessels increases as we go down the dermis and becomes almost uniform in the subcutaneous part. Any abnormality in the SST region or its physiological processes could disturb the heat balance of the body with the environment. This abnormality may be in the form of disorder of growth, inhibition or hyperfunction of enzymes (Boyd, 1970, p. 214) which is found in the growing tumours. The tumours are of two types: benign and malignant. Malignant tumours are characterized by uncontrolled growth and hyperfunction of enzymes. The establishment and progressive growth of malignant tumours is possible when the supply of essential nutrients is adequately maintained (Song et al., 1984) through vascular networks. During the initial stage of tumour growth, the tumour cells are supported by the nutrients supplied from the host vasculature. The number of host arteries or arterioles seldom grows while the number and length of tumour capillaries fed by the same host arteries increase as the tumour grows. Such an increase in the 525
526
V . P . SAXENA AND KAMAL RAJ PARDASANI
capillary network and an increase in demand for blood in excess of the capacity of the host arterioles would result in a decline in arteriolar pressure. At the same time, the extra vascular pressure in the t u m o u r increases as a proliferation of t u m o u r cells within a limited space. An increase in the extra vascular pressure exceeding the arteriolar pressure results in regional vascular stasis and necrosis. The limitation of diffusion length of oxygen and possibly other nutrients is believed to be another cause of necrosis in the tumours. The t u m o u r blood flow varies significantly depending on the type, age and size of turnouts. Under normal environmental conditions the blood flow in tumours may or may not be greater than that in normal tissues. Earlier, experimental investigations were made by Patterson (1976, 1978) to obtain temperature profiles in the SST region. Saxena (1983), Saxena and Arya (1979, 1981) and Saxena and Bindra (1984) also obtained temperature distributions in the SST region using analytical and numerical techniques. These investigations were performed under normal physiological and atmospheric conditions. Barker (1976) used finite differences to obtain the temperature distribution in breast having a uniformly perfused t u m o u r at various positions in the tissues. He took all the parameters like blood mass flow rate, rate of metabolic heat generation and thermal conductivity as constants. Also, Jain (1980) has presented simple mathematical models to predict temperature distributions in the normal and neoplastic tissues of various mammals during normothermia and hypothermia. Here we consider a growing turnout situated in the dermis. This turnout consists of a necrotic core and peripheral region. The turnout periphery is assumed to have increased rates of blood mass flow and metabolic heat generation. The tumour core may or may not have any metabolic activity and blood mass flow due to necrosis in that region. The outer skin surface is assumed to be exposed to the atmosphere and heat loss takes place due to convection-radiation and evaporation. This study has been performed for a one-dimensional steady state case. 2. Theory. The well-known partial differential equation for heat flow in the SST region, which was initially given by Perl (1962), is pg ~
= Div(K grad T) +
mbCb(T A --
T) + S
(1)
where p, g and K are density, specific heat and thermal conductivity of the tissues respectively, m b and c b are mass flow rate and specific heat of blood respectively. T A is the arterial blood temperature. S = S + IV, where S and W denote self-controlled and uncontrolled metabolic heat generation. The first and second terms on the r.h.s, of (1) represent the heat transfer due to conduction and perfusion respectivley. This perfusion term in (1) is an approximation of the following:
E F F E C T O F D E R M A L T U M O U R S O N T E M P E R A T U R E IN S K I N
527
4'o o)
(2)
where Tv and Pb are respectively the venous blood temperature and density of the blood. ~bA and ~o represent the tissue perfusion due to arterial and venous blood respectively. Under normal conditions for a peripheral region there is hardly any difference between the values of q5A and q5o. Also, Tv is always dominated by the tissue temperature. Thus for the SST region, To can be logically taken equal to T, due to the low rate of blood flow and size of the vessels. The mass blood flow rate m b is defined as the product of blood density Pb and tissue perfusion qS. This perfusion term holds at almost every point in the peripheral region. The heat carried away by the part of blood used for nutritional purposes is not incorporated in this model. Considering the SST region as a continuum takes care of this aspect to some extent. In spite of all these drawbacks, it is still the widely accepted model. We consider the following boundary conditions:
K oT = h(T- T~)+ LE
(3)
at the skin surface and T=
(3)
Tb
at the inner boundary. Here n, h, Ta, L and E are respectively normal to outer surface, heat transfer coefficient, atmospheric temperature, latent heat of evaporation and rate of sweat evaporation. Equation (1) for a one-dimensional steady state case, along with boundary conditions (3) and (4), is compared with the Euler-Lagrange equation and transformed into the following equivalent variational form (Myers, 1971; Heubner, 1975; Hildebrand, 1972):
]
I=½,,ao C \ d x } +mbcb(TA--T)2--2(S+W)T dx + ½[h(r- Ta) 2 + 2LET].
(5)
We divide the SST region into nine layers (see Fig. 1), of which the epidermis makes up two layers, the dermis five layers and the subdermal part two layers. The malignant t u m o u r under study is assumed to occupy three of the five layers of the dermis. Such a division provides us with flexibility in assigning suitable and independent values to quantities like thermal conductivity, rate of blood mass flow and rate of metabolic heat generation. We assign values T~ (i = 0, 1,
528
V . P . SAXENA A N D K A M A L RAJ P A R D A S A N I
I
ATMOSPHERE
[ S T R A , UM
To
: :
CORNEUM ,, o o
o
STRATUM GERMINATIVUM
TI
,I *(a - a ). . . . . . . . . . . . . . . . . . / 1o , o * o 2ow,
l
*
.
-,.,,f,,
,,
.
o
~
.
.
(a.,a~)".'.',':",',:. . : ~ ' - ' "
,~,v-/+~ ,:.
:,,,:,,:,,~
• .,.
.....
.... .:
o
o
,
.
.
.
.
.,_._'.-,,-..-,..-...:.
, - " - : - ' ~-: ' . . - . -
. . . . . . .
T
. o
2
,",:
_ .....
,
2;,///d_///j,j,/////////////////////~
T3
t a[, - a 3 ) "//." T urn o ur P er i p h e r y / / / / / / / / / / / / / / /
T5T4
DERMIS
~
"
/
Q
y y/////////////
o
~ / / / / / / / / / / / / / / / / / / / T •
-q':
----
( a 7 - a 6)_" ".-J,--.-L'.-
S U B CU TA NE OUS
T,SSUES.
:-
.
.
.
.
.
.-
-'.-..:,'-_":-'_-_" :":'-_"
. . . .
.-. ....
..,,:-.-_
"."-:' L---.
-"
6
:'.-:,.1
_.-: L'-:'.,'"
::-
I ~ / / / / ~ ' ~ x ~ / / / / Q ' x , ~
T7
L ~ 9 -~ B ' / / 2 , . \ \ ' ~ / / / ~ . \ \ ' , : / / / / ~ , \ ' . . : ~ / ~ K .
T9 = T b
I ~4 0 w h e n a b n o r m a l processes are taking place. Accordingly, we assume e~= 0 and 2i = 0 in t u m o u r regions. e i = 1, 2z = 1,/?i = 0 and W~= 0 in n o r m a l and benign tissues.
530
V . P . SAXENA A N D KAMAL RAJ PARDASANI 37
1]=3 1]=S 36
ut
"/// sit~" 35
34
~':i
I
o'.2
i
o'.~
I
XCrn
o~-6
I
o'.8
I
1'.o
Figure 3. Graph between temperature T and position X for T, = 23°C and E= 0. Broken line is for benign tissues. As discussed earlier, there m a y or m a y not be any blood flow and metabolic activity taking place in the t u m o u r core due to necrosis and stasis in that region. However, we assume that complete necrosis and stasis has not taken place in the core of the growing t u m o u r and a small amount of blood flow and metabolic activity is taking place in the turnout core. Accordingly, we take fli = 1 and IVii= ½s in t u m o u r core. Also, we take fli = t / a n d W~= t/s in the tumour periphery, where ~/is any non-negative real number. Here q takes a value between 0 and 1 if the blood flow and metabolic activity in the tumour periphery is less than or equal to that of the surrounding normal tissues, while q > 1 when the b l o o d flow rate and metabolic activity in the turnout periphery are greater than that in normal tissues. Also, it is difficult to assign any value to arterial blood
EFFECT OF D E R M A L T U M O U R S O N T E M P E R A T U R E IN SKIN
531
36 rl:
35 [/"
"~= 1
////
o 34 *1--
!I
I
I
/ 7" I
/
t, !
32
!
I
Figure
0.1
012
013
014 015 01.6 X Crn. ----=,=-
0!7
[
0-6
0.9
4. Graph between temperature T and position X for T,=23°C and E=0.24 x 10 -a g/cm2--min. Broken line is for benign tissues.
temperature. The arterial blood temperature decreases with the increase in distance from the body core as the blood loses heat to the tissues while travelling through the vessels in the region. Therefore, we assign the following values to the arterial blood temperature in each element, depending on its position from the skin surface and body core: T ( ~ ) = ~Ti T b --~ ( 1 - - o ' i ) r
/
i=3(1)9
(7)
where a i is a non-negative ratio and 0~
0092 8240/9153.00+0.00 Pergamon Press plc © 1991 Society for Mathematical Biology
EFFECT OF DERMAL TUMOURS ON TEMPERATURE DISTRIBUTION IN SKIN WITH VARIABLE BLOOD FLOW V. P. SAXENA and KAMAL RAJ P A R D A S A N I School of Mathematics and Allied Sciences, Jiwaji University, Gwalior 474 011, India Several investigations have been made for the heat flow problems in skin and subdermal tissues under normal physiological and atmospheric conditions. This paper considers the existence of a malignant tumour in the underlying tissues of epidermis of a human body. The surrounding tissues are assumed to have normal physiological functions, namely self-controlled metabolic activity, variable blood flow and perspiration. For the malignant portion the metabolic activity is taken to be continuous and uncontrolled. The effect of this factor is studied on the temperature profiles of the skin.
1. Introduction. Under normal physiological conditions, the human body maintains a thermal balance with the environment. To maintain this balance, the skin and subcutaneous tissues (SST) undergo various changes in their physiological activities. The SST region consists of dead tissues in the form of stratum corneum on one side and subcutaneous tissues with continuous metabolic activity and dense blood vessels on the other. In between are stratum germinativum and dermis. The stratum germinativum consists of some living cells but almost no blood vessels. The dermis consists of connective tissues, blood vessels, fat cells, sweat glands, oil glands and nerve fibres, etc. The population density of blood vessels increases as we go down the dermis and becomes almost uniform in the subcutaneous part. Any abnormality in the SST region or its physiological processes could disturb the heat balance of the body with the environment. This abnormality may be in the form of disorder of growth, inhibition or hyperfunction of enzymes (Boyd, 1970, p. 214) which is found in the growing tumours. The tumours are of two types: benign and malignant. Malignant tumours are characterized by uncontrolled growth and hyperfunction of enzymes. The establishment and progressive growth of malignant tumours is possible when the supply of essential nutrients is adequately maintained (Song et al., 1984) through vascular networks. During the initial stage of tumour growth, the tumour cells are supported by the nutrients supplied from the host vasculature. The number of host arteries or arterioles seldom grows while the number and length of tumour capillaries fed by the same host arteries increase as the tumour grows. Such an increase in the 525
526
V . P . SAXENA AND KAMAL RAJ PARDASANI
capillary network and an increase in demand for blood in excess of the capacity of the host arterioles would result in a decline in arteriolar pressure. At the same time, the extra vascular pressure in the t u m o u r increases as a proliferation of t u m o u r cells within a limited space. An increase in the extra vascular pressure exceeding the arteriolar pressure results in regional vascular stasis and necrosis. The limitation of diffusion length of oxygen and possibly other nutrients is believed to be another cause of necrosis in the tumours. The t u m o u r blood flow varies significantly depending on the type, age and size of turnouts. Under normal environmental conditions the blood flow in tumours may or may not be greater than that in normal tissues. Earlier, experimental investigations were made by Patterson (1976, 1978) to obtain temperature profiles in the SST region. Saxena (1983), Saxena and Arya (1979, 1981) and Saxena and Bindra (1984) also obtained temperature distributions in the SST region using analytical and numerical techniques. These investigations were performed under normal physiological and atmospheric conditions. Barker (1976) used finite differences to obtain the temperature distribution in breast having a uniformly perfused t u m o u r at various positions in the tissues. He took all the parameters like blood mass flow rate, rate of metabolic heat generation and thermal conductivity as constants. Also, Jain (1980) has presented simple mathematical models to predict temperature distributions in the normal and neoplastic tissues of various mammals during normothermia and hypothermia. Here we consider a growing turnout situated in the dermis. This turnout consists of a necrotic core and peripheral region. The turnout periphery is assumed to have increased rates of blood mass flow and metabolic heat generation. The tumour core may or may not have any metabolic activity and blood mass flow due to necrosis in that region. The outer skin surface is assumed to be exposed to the atmosphere and heat loss takes place due to convection-radiation and evaporation. This study has been performed for a one-dimensional steady state case. 2. Theory. The well-known partial differential equation for heat flow in the SST region, which was initially given by Perl (1962), is pg ~
= Div(K grad T) +
mbCb(T A --
T) + S
(1)
where p, g and K are density, specific heat and thermal conductivity of the tissues respectively, m b and c b are mass flow rate and specific heat of blood respectively. T A is the arterial blood temperature. S = S + IV, where S and W denote self-controlled and uncontrolled metabolic heat generation. The first and second terms on the r.h.s, of (1) represent the heat transfer due to conduction and perfusion respectivley. This perfusion term in (1) is an approximation of the following:
E F F E C T O F D E R M A L T U M O U R S O N T E M P E R A T U R E IN S K I N
527
4'o o)
(2)
where Tv and Pb are respectively the venous blood temperature and density of the blood. ~bA and ~o represent the tissue perfusion due to arterial and venous blood respectively. Under normal conditions for a peripheral region there is hardly any difference between the values of q5A and q5o. Also, Tv is always dominated by the tissue temperature. Thus for the SST region, To can be logically taken equal to T, due to the low rate of blood flow and size of the vessels. The mass blood flow rate m b is defined as the product of blood density Pb and tissue perfusion qS. This perfusion term holds at almost every point in the peripheral region. The heat carried away by the part of blood used for nutritional purposes is not incorporated in this model. Considering the SST region as a continuum takes care of this aspect to some extent. In spite of all these drawbacks, it is still the widely accepted model. We consider the following boundary conditions:
K oT = h(T- T~)+ LE
(3)
at the skin surface and T=
(3)
Tb
at the inner boundary. Here n, h, Ta, L and E are respectively normal to outer surface, heat transfer coefficient, atmospheric temperature, latent heat of evaporation and rate of sweat evaporation. Equation (1) for a one-dimensional steady state case, along with boundary conditions (3) and (4), is compared with the Euler-Lagrange equation and transformed into the following equivalent variational form (Myers, 1971; Heubner, 1975; Hildebrand, 1972):
]
I=½,,ao C \ d x } +mbcb(TA--T)2--2(S+W)T dx + ½[h(r- Ta) 2 + 2LET].
(5)
We divide the SST region into nine layers (see Fig. 1), of which the epidermis makes up two layers, the dermis five layers and the subdermal part two layers. The malignant t u m o u r under study is assumed to occupy three of the five layers of the dermis. Such a division provides us with flexibility in assigning suitable and independent values to quantities like thermal conductivity, rate of blood mass flow and rate of metabolic heat generation. We assign values T~ (i = 0, 1,
528
V . P . SAXENA A N D K A M A L RAJ P A R D A S A N I
I
ATMOSPHERE
[ S T R A , UM
To
: :
CORNEUM ,, o o
o
STRATUM GERMINATIVUM
TI
,I *(a - a ). . . . . . . . . . . . . . . . . . / 1o , o * o 2ow,
l
*
.
-,.,,f,,
,,
.
o
~
.
.
(a.,a~)".'.',':",',:. . : ~ ' - ' "
,~,v-/+~ ,:.
:,,,:,,:,,~
• .,.
.....
.... .:
o
o
,
.
.
.
.
.,_._'.-,,-..-,..-...:.
, - " - : - ' ~-: ' . . - . -
. . . . . . .
T
. o
2
,",:
_ .....
,
2;,///d_///j,j,/////////////////////~
T3
t a[, - a 3 ) "//." T urn o ur P er i p h e r y / / / / / / / / / / / / / / /
T5T4
DERMIS
~
"
/
Q
y y/////////////
o
~ / / / / / / / / / / / / / / / / / / / T •
-q':
----
( a 7 - a 6)_" ".-J,--.-L'.-
S U B CU TA NE OUS
T,SSUES.
:-
.
.
.
.
.
.-
-'.-..:,'-_":-'_-_" :":'-_"
. . . .
.-. ....
..,,:-.-_
"."-:' L---.
-"
6
:'.-:,.1
_.-: L'-:'.,'"
::-
I ~ / / / / ~ ' ~ x ~ / / / / Q ' x , ~
T7
L ~ 9 -~ B ' / / 2 , . \ \ ' ~ / / / ~ . \ \ ' , : / / / / ~ , \ ' . . : ~ / ~ K .
T9 = T b
I ~4 0 w h e n a b n o r m a l processes are taking place. Accordingly, we assume e~= 0 and 2i = 0 in t u m o u r regions. e i = 1, 2z = 1,/?i = 0 and W~= 0 in n o r m a l and benign tissues.
530
V . P . SAXENA A N D KAMAL RAJ PARDASANI 37
1]=3 1]=S 36
ut
"/// sit~" 35
34
~':i
I
o'.2
i
o'.~
I
XCrn
o~-6
I
o'.8
I
1'.o
Figure 3. Graph between temperature T and position X for T, = 23°C and E= 0. Broken line is for benign tissues. As discussed earlier, there m a y or m a y not be any blood flow and metabolic activity taking place in the t u m o u r core due to necrosis and stasis in that region. However, we assume that complete necrosis and stasis has not taken place in the core of the growing t u m o u r and a small amount of blood flow and metabolic activity is taking place in the turnout core. Accordingly, we take fli = 1 and IVii= ½s in t u m o u r core. Also, we take fli = t / a n d W~= t/s in the tumour periphery, where ~/is any non-negative real number. Here q takes a value between 0 and 1 if the blood flow and metabolic activity in the tumour periphery is less than or equal to that of the surrounding normal tissues, while q > 1 when the b l o o d flow rate and metabolic activity in the turnout periphery are greater than that in normal tissues. Also, it is difficult to assign any value to arterial blood
EFFECT OF D E R M A L T U M O U R S O N T E M P E R A T U R E IN SKIN
531
36 rl:
35 [/"
"~= 1
////
o 34 *1--
!I
I
I
/ 7" I
/
t, !
32
!
I
Figure
0.1
012
013
014 015 01.6 X Crn. ----=,=-
0!7
[
0-6
0.9
4. Graph between temperature T and position X for T,=23°C and E=0.24 x 10 -a g/cm2--min. Broken line is for benign tissues.
temperature. The arterial blood temperature decreases with the increase in distance from the body core as the blood loses heat to the tissues while travelling through the vessels in the region. Therefore, we assign the following values to the arterial blood temperature in each element, depending on its position from the skin surface and body core: T ( ~ ) = ~Ti T b --~ ( 1 - - o ' i ) r
/
i=3(1)9
(7)
where a i is a non-negative ratio and 0~
