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Measurement of blood flow using temperature decay: Effect of thermal conduction
Inr J Radialion Oncology Biul. Phys.. Vol. Printed in the U.S.A. All rights reserved.
t .I0
0360-3016/&X6 $3.00 Copyright 0 I986 Pergamon Press Ltd.
12, pp. 373-378
??Original Contribution
MEASUREMENT
OF BLOOD FLOW USING TEMPERATURE EFFECT OF THERMAL CONDUCTION TALJIT
Physics Division, Department
S. SANDHU,
DECAY:
PH.D.
of Therapeutic Radiology, Henry Ford Hospital, Detroit, M’F48202
Within the framework of the bioheat equation, we studied the effect of conduction on the thermal washout curve for a model tissue configuration subjected to a thermal perturbation such as hyperthermia treatment. In particular, we studied the implications of the assumption made by many investigators to neglect the effect of thermal conduction while analyzing the temperature decay curve for measuring blood perfusion. The present analysis suggests that during the localized hyperthermia treatments, this assumption can lead to inaccurate values for the blood perfusion parameter. This is particularly so under non-steady state conditions when the temperature distribution is changing. In addition to the value of blood flow, the shaue of the temperature decay curve depends on the temperature _ distribution at the start of temperature’decay. Hyperthermia,
Blood flow, Thermal conduction.
ues to specifically indicate the presence of conduction effects. However, as is demonstrated in this paper, the conduction effects are highly dependent on thermal gradients that exist at the onset of thermal decay and are, therefore, transducer dependent. The effective value, therefore, does not represent the characteristics of the tissue alone. The purpose of this paper is to quantitatively investigate the effects of thermai conduction on the temperature decay curve. We studied the temperature decay curve for a model tissue configuration using the bioheat equation. In limited cases, the temperature decay curve analysis may be used to measure blood flow. The shape of this curve is dependent on the temperature distribution at the time of power cut off in addition to the value of the blood flow. We stress that unless some tests are performed, the estimated values of blood flow parameter can be in error by as much as 100%. The shape of the temperature decay curve and, therefore, the blood flow extracted from it is also affected by the position of the point of measurement within the temperature distribution.
INTRODUCTION
There have been a few attempts’.3.5-8.10.1’ to measure blood flow in tissues under hyperthermic conditions. Blood flow is an important parameter to measure for two reasons. First, it can aid in understanding the physiological response of tissue to hyperthermia treatment and, second, it is crucial that we know the value of blood flow to construct reasonable theoretical models for predicting the temperature distribution within the tissue under treatment. Investigations in developing these theoretical models to predict temperature distribution are important now because there are no practical methods of measuring the temperature throughout the tissue volume. An important input parameter for the theoretical models is the value of blood flow to account for the convective transport of heat due to blood perfusion. Current methods for accurately measuring blood flow are not suitable in a clinical environment such as during hyperthermia treatments. Some investigators 6~10have attempted to extract the blood flow value from the temperature decay curve obtained by turning off the power input to the hyperthermia transducer. These investigators have assumed an exponential decay of the temperature and have thus neglected the effect of thermal conduction. Roemer et al.* have reported a formalism to evaluate blood perfusion from both steady state and transient temperature data. These investigators have termed the perfusion values as effective val-
METHODS
AND
MATERIALS
Theory
The theoretical analysis of the temperature decay curve is based on the assumption that the temperature is described by the bioheat equation:
Presented at the Annual Meeting of American Association of Medical Physicists, Chicago, Illinois, July 15-19, 1984. Acknowledgments-The author would like to acknowledge useful discussions with Dr. Robert Roemer, Dr. J. Strohbehn and
Dr. F. Waterman, and to thank Ms. Viktoria Leonavicius for her help in the preparation of this manuscript. Accepted for publication 30 October 1985. 373
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=
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:SE(I)uba aluMa uw aM .patDa@auSTunai uogelaua% IcayDqoqelauIaql‘aJouIlaqUnJ ~~o~n~laMOdaq~%h~~o~ -10~ ainwadwa~aqi%u!q~~sapQuoa~l?aMa~u!spaddo~p uaaq seq wai a31nos Ghaua aql ‘(I)uba %upl_r~UI Q,/Fiy/ru/r) anss!i~o Li!Ag3npuoD@way1 = y put!‘(3as/%y/,ur) M~U poolq 3yaturqo~ = m (3JWr) PoofqJo vaq q=ds = 9%~ (3,$3y/l)anss!lJolt?aqxpacis = J (,w/t?$ poo]qjo +uaa = qd (,w/%y) anssiijo &!suaa = d (3,) ainwadrual anssgauqasea = 0~ ‘(aas)? auq E 1 uogrsodle(&)am~wadura~ = (1‘1)~ =aqM
:OJpa3npaJSI(1)uba ‘h$aurtulCs Iesgaqdslapun
375
Temperature decay measurement of blood flow 0 T. S. SANDHU ~~12
perature, and To = 7’(0, 0) - To is the maximum temperature rise above the baseline at the center of this heated volume. This initial spatial temperature distribution (Fig. 1) at the onset of temperature decay is a reasonable approximation to the measured steady state distributions reported in the literature2,6 for localized hyperthermia treatments with microwave transducers (e.g., see Figs. 5, 7, and 9 in reference 2). The values of other physical parameters used in the present analysis are: k=
0.44 J/m/“C/s,
P’
1070 kg/m3
mlllOOglmin
r =o
Pb = 1060 kg/m3 0
120
60
160
240
C= 3500 J/kg/“C, and cb =
3900 J/kg/“C.
Results of the calculations are presented in Figs. 2-5. Figure 2 shows the percentage temperature rise ( T/F0 - 1) X 100, at the center, as a function of time for different size heated tumor volumes within the normal tissue. The different curves are for different ‘u’values (i.e., size of the heated tumor volume). The value of ‘a’(in cm) is indicated to the right of each curve. It should be noted that: (a) temperature decay is not exponential, and (b) the exact solution approaches exponential decay shown by a dotted line as the value of ‘a’ is increased. Figure 3 shows the temperature (eqn 5) rise at different points within the heated volume for a fixed value of ‘a’. The shape of the temperature decay curve changes with the position of the temperature measurement point and the curves are drastically different from the exponential solution shown by a dotted line. Figure 4 compares the value of the blood perfusion parameter ‘6’as calculated from the initial slope of the temperature (eqn 6) plotted on the semi-logrithmic graph for different tumor sizes, ‘a’, with the slope of the exponential solution obtained by neglecting the thermal conduction. The two values approach each other as the value of the parameter ‘u’is increased. To determine the circumstances under which the exponential solution is a reasonable approximation to the exact solution, eqn (6) was evaluated for different values 47 46
300 t
360
420
460
540
600
(set9
Fig. 2. Percentage temperature rise above the baseline temperature plotted as a function of time after the onset of temperature decay. The dotted line represents the values calculated when thermal conduction is neglected. The solid lines represent the values calculated using eqn (6). The number at the end of each curve indicates the value of the parameter ‘a’ in cm, which is also the radius of the heated spherical volume surrounded by tissue at normal temperature. All other parameters are identical for these curves.
of blood perfusion keeping the size parameter fixed. The results are given in Fig. 5. Here, as the value of blood perfusion increases, the thermal conduction effects become less and less important. DISCUSSION The temperature decay curves presented by solid lines in Figs. 2-3 represent the exact solutions of the bioheat WI 12 ml/ 1 OOglmin
20 3 B I-
lo
F I
1 0
60
I
I
120
160
I
240 t
Fig. 1. The initial temperature distribution f(r) at the onset of temperature decay. The dotted and solid lines are for the size parameter a = 3 cm and a = 10 cm, respectively.
1
1
I
300
360
420
I
460
I
540
I 600
(set)
Fig. 3. The temperature decay curves for different spatial points within the heated volume as calculated using eqn (5). The position is indicated at the end of each curve by the distance (in cm) from the center. The initial slope is position dependent indicating that it is influenced by the local temperature gradients via thermal conduction. The dotted line is the decay curve calculated by neglecting the thermal conduction and is position independent.
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a (cm) Fig. 4. The solid line represents the apparent value h’ of the parameter b = ucbpb/c as obtained by equating it to the initial slope of the exact solution (eqn 6) plotted on a log-linear graph.
The dotted line represents the exact value that is the same as the initial slope of the curve obtained when thermal conduction is absent.
equation, and are significantly different from the simple exponential form (dotted lines), obtained by neglecting the thermal conduction term. For a fixed value of blood perfusion, the greater the thermal gradients (small ‘a’ value), the greater the deviation of the exact solution from the exponential form. As the thermal gradients at the point of measurement are decreased by increasing the size parameter (Fig. I), the exact solution shifts toward the simple exponential form. The exact solution in general, however, can not be represented by an exponential function. This fact not only affects the blood perfusion measurements, but also the temperature measurement techniques used by some investigators. The measurement of temperature in a microwave field with thermocouples’ is based on exponential backward extrapolation after microwave power shut-off. This can lead to significant errors because, as shown here, the actual temperature decay is not necessarily an exponential function of time. The magnitude of these errors will, of course, depend on local thermal gradients and the value of blood perfusion. Milligan et a1.6in their attempts to extract the blood perfusion from the temperature decay curves found that the experimentally measured data was not describable by an exponential curve. Instead of including the effects of thermal conduction to overcome this difficulty, they introduced a rather arbitrary dimensionless temperature dependent parameter into the bioheat equation. It should be pointed out that the thermal gradients at the onset of temperature decay reported in
March
1986. Volume
12, Number
3
reference 6 are quite similar in form to the one used here. In light of the present results, it is not surprising that the experimental temperature decay curves were not exponential. To investigate quantitatively the errors introduced by assuming an exponential temperature decay, apparent values for the parameter b = wcbpb/cwere evaluated in the following manner. For a fixed value of blood perfusion, w, the exact solution (eqn 6) was evaluated for different values of the parameter ‘a’. The assumption was made that these values of T vs t follow an exponential form r = T&” and the parameter b’was evaluated from initial slope of r vs t plot on a semi-logrithmic graph. The results are shown in Fig. 4 for the calculations performed with w = 6 ml/l00 g/min that represents the blood perfusion for resting muscle. The dotted line indicates the actual value of the parameter b = wchph/c.These results indicate that the value of the blood perfusion parameter evaluated in this manner can be in error by as much as an order of magnitude if large thermal gradients are present. To investigate whether the assumption of neglecting thermal conduction is reasonable for tissues with large values of blood flow, the exact solution (eqn 6) was evaluated for varying values of blood flow, w, with a fixed value of ‘a’, that is, same thermal gradient conditions. The results are shown in Fig. 5. The percentage error shown in the ordinate is the percentage difference between the value of ‘b’ evaluated as described in the previous paragraph and the actual value b = Wcbpb/c. In the presence of large thermal gradients (small ‘a’values) the error is large (~40%) even in the presence of high perfusion rates (84 ml/ 100 g/min). The error reduces to less than 10% when the thermal conduction effects are reduced by decreasing thermal gradients (large ‘a’values). This suggests that for the determination of blood perfusion from temperature decay, thermal conduction plays an important role. The effects of thermal conduction can be neglected only if changes in thermal gradients at the point of measurement are negligible. For small values of blood perfusion (6 ml/ 100 g/min), even when the temperature distribution at the onset of temperature decay is relatively uniform (small thermal gradients), the effects of thermal conduction introduce an error of more than 10% in blood perfusion. The present analysis, therefore, suggests that a simple exponential form for the decay curve can be assumed for blood flow analysis if the measurement probe is located in the heated volume where changes in gradients are not significant. This will necessitate thermal mapping’ of the heated volume. CONCLUSION In summary, the analyses presented here demonstrate that the thermal conduction term in the bioheat equation cannot, in general, be neglected without grossly compromising the accuracy of its solution for temperature distribution. Furthermore, as shown in Figs. 4-5, the value
Temperature decay measurement of blood flow 0 T. S. SANDHU
12
24
36
48
60
72
84
wfml/ lOOg/minl
Fig. 5. The percentage error introduced in determining effects in the analysis of temperature decay curve.
of blood perfusion obtained from the temperature decay is affected significantly if thermal conduction is ignored. The exact solution which includes the effects of both the thermal conduction and convection due to blood perfusion, however, approaches the simple exponential form (obtained by neglecting thermal conduction) under the conditions of negligible thermal gradients. During hyperthermia treatments the point of measurements with low temperature gradients can be chosen if the heated volume is mapped for temperature distribution as discussed by Gibbs2 Furthermore, it should be stressed that in addition to the blood perfusion, the shape of the temperature decay curve depends highly on the initial temperature distri-
the blood perfusion by neglecting thermal conduction
bution at the onset of decay. Therefore, the blood flow values obtained from the temperature decay curves during the period of heating before the steady state is reached are meaningless since the temperature distribution at the onset of each decay curve will be different. Further, if the hyperthermia transducer produces a highly non-uniform temperature distribution, the blood flow values obtained by neglecting the thermal conduction effects will be transducer (size, energy deposition pattern) dependent and, therefore, not very valuable. In conclusion, the method of determining blood flow from temperature decay curves is of limited value and can be used only under special conditions.
REFERENCES Cater, D.B., Petrie, A., Watkinson, D.A.: Effect of 5-Hydroxytryptamine and Cyproheptadine on tumor blood flow: Estimation by rate of cooling after microwave diathermy. Acta Radiol. Ther. Phys. Biol. 3: 109-128,
Gibbs, F.A.: “Thermal
1965.
mapping” in experimental
cancer
treatment with hyperthermia description and using a semiautomatic system. Int. J. Radiat. Oncol. Biol. Phys. 9: 10571063, 1983. 3. Hand, J.W., Hopewell, J.N., Foster, J.L., Field, S.B.: Microwave heating of musculature in the pig by two 9 15 MHz
378
4.
5. 6.
7.
1. J. Radiation Oncology 0 Biology0 Physics direct-contact applicators. In Hyperthermia in Radiation Oncology, G. Arcangeli and F. Mauro (Eds.). Milano, Italy, Masson. 1980, pp. 37-44. Ingersoll, L.R., Zobel, O.J., Ingersoll, A.C.: Heat Conduction. With Engineering and Geological Applications. New York, McGraw-Hill. 1948, pp. 140- 14 1. Milligan, A.J.: Intestinal blood flow in Chinese hamster during hyperthermia (Abstr.). Rad. Res. 91: 318, 1982. Mill&m, A.J., Conran, P.B., Ropar, M.A., McCullough, H.A., Ahuja, R.K., Dobelbower, R.R.: Predictions of blood flow from thermal clearance during regional hyperthermia treatment. Int. J. Radial. Oncol. Biol. Phys. 9: 1335- 1343, 1983. Milligan, A.J., Panjehpour, M.: Mathematical predictions of tumor and normal tissue blood flow during hyperthermia (Abstr.). Rad. Res. 94: 534-535. 1983.
March 1986, Volume 12, Number 3 8. Roemer, R.B., Fletcher, A.M., Cates, T.C.: Obtaining local SAR and blood perfusion data from temperature measurements: Steady state and transient techniques compared. Int. J. Radiat. Oncol. Biol. Phys. 11: 1539-1550, 1985. 9. Scott, R.S., Johnson, R.J., Kowal, H.S., Krishnamasetty, R.M., Story, J., Clay, L.: Hyperthermia in combination with radiotherapy: A review of five years experience in the treatment of superficial tumors. Int. J. Radiat. Oncol. Biol. Phys. 9: 1327-1333, 1983. 10. Waterman, F.J., Fazekas, J., Nerlinger, R.E., Leeper, D.B.: Blood flow rates in human tumors during hyperthermia treatments as indicated by thermal washout (Abstr.). Rad. Res. 91: 426, 1982. 11. Waterman, F.J., Nerlinger, R.E., Leeper, D.B., Moylan, D. J. : Blood flow in human tumors during local hyperthermia (Abstr.). Rad. Rex 94: 598, 1983.
t .I0
0360-3016/&X6 $3.00 Copyright 0 I986 Pergamon Press Ltd.
12, pp. 373-378
??Original Contribution
MEASUREMENT
OF BLOOD FLOW USING TEMPERATURE EFFECT OF THERMAL CONDUCTION TALJIT
Physics Division, Department
S. SANDHU,
DECAY:
PH.D.
of Therapeutic Radiology, Henry Ford Hospital, Detroit, M’F48202
Within the framework of the bioheat equation, we studied the effect of conduction on the thermal washout curve for a model tissue configuration subjected to a thermal perturbation such as hyperthermia treatment. In particular, we studied the implications of the assumption made by many investigators to neglect the effect of thermal conduction while analyzing the temperature decay curve for measuring blood perfusion. The present analysis suggests that during the localized hyperthermia treatments, this assumption can lead to inaccurate values for the blood perfusion parameter. This is particularly so under non-steady state conditions when the temperature distribution is changing. In addition to the value of blood flow, the shaue of the temperature decay curve depends on the temperature _ distribution at the start of temperature’decay. Hyperthermia,
Blood flow, Thermal conduction.
ues to specifically indicate the presence of conduction effects. However, as is demonstrated in this paper, the conduction effects are highly dependent on thermal gradients that exist at the onset of thermal decay and are, therefore, transducer dependent. The effective value, therefore, does not represent the characteristics of the tissue alone. The purpose of this paper is to quantitatively investigate the effects of thermai conduction on the temperature decay curve. We studied the temperature decay curve for a model tissue configuration using the bioheat equation. In limited cases, the temperature decay curve analysis may be used to measure blood flow. The shape of this curve is dependent on the temperature distribution at the time of power cut off in addition to the value of the blood flow. We stress that unless some tests are performed, the estimated values of blood flow parameter can be in error by as much as 100%. The shape of the temperature decay curve and, therefore, the blood flow extracted from it is also affected by the position of the point of measurement within the temperature distribution.
INTRODUCTION
There have been a few attempts’.3.5-8.10.1’ to measure blood flow in tissues under hyperthermic conditions. Blood flow is an important parameter to measure for two reasons. First, it can aid in understanding the physiological response of tissue to hyperthermia treatment and, second, it is crucial that we know the value of blood flow to construct reasonable theoretical models for predicting the temperature distribution within the tissue under treatment. Investigations in developing these theoretical models to predict temperature distribution are important now because there are no practical methods of measuring the temperature throughout the tissue volume. An important input parameter for the theoretical models is the value of blood flow to account for the convective transport of heat due to blood perfusion. Current methods for accurately measuring blood flow are not suitable in a clinical environment such as during hyperthermia treatments. Some investigators 6~10have attempted to extract the blood flow value from the temperature decay curve obtained by turning off the power input to the hyperthermia transducer. These investigators have assumed an exponential decay of the temperature and have thus neglected the effect of thermal conduction. Roemer et al.* have reported a formalism to evaluate blood perfusion from both steady state and transient temperature data. These investigators have termed the perfusion values as effective val-
METHODS
AND
MATERIALS
Theory
The theoretical analysis of the temperature decay curve is based on the assumption that the temperature is described by the bioheat equation:
Presented at the Annual Meeting of American Association of Medical Physicists, Chicago, Illinois, July 15-19, 1984. Acknowledgments-The author would like to acknowledge useful discussions with Dr. Robert Roemer, Dr. J. Strohbehn and
Dr. F. Waterman, and to thank Ms. Viktoria Leonavicius for her help in the preparation of this manuscript. Accepted for publication 30 October 1985. 373
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=
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Temperature decay measurement of blood flow 0 T. S. SANDHU ~~12
perature, and To = 7’(0, 0) - To is the maximum temperature rise above the baseline at the center of this heated volume. This initial spatial temperature distribution (Fig. 1) at the onset of temperature decay is a reasonable approximation to the measured steady state distributions reported in the literature2,6 for localized hyperthermia treatments with microwave transducers (e.g., see Figs. 5, 7, and 9 in reference 2). The values of other physical parameters used in the present analysis are: k=
0.44 J/m/“C/s,
P’
1070 kg/m3
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r =o
Pb = 1060 kg/m3 0
120
60
160
240
C= 3500 J/kg/“C, and cb =
3900 J/kg/“C.
Results of the calculations are presented in Figs. 2-5. Figure 2 shows the percentage temperature rise ( T/F0 - 1) X 100, at the center, as a function of time for different size heated tumor volumes within the normal tissue. The different curves are for different ‘u’values (i.e., size of the heated tumor volume). The value of ‘a’(in cm) is indicated to the right of each curve. It should be noted that: (a) temperature decay is not exponential, and (b) the exact solution approaches exponential decay shown by a dotted line as the value of ‘a’ is increased. Figure 3 shows the temperature (eqn 5) rise at different points within the heated volume for a fixed value of ‘a’. The shape of the temperature decay curve changes with the position of the temperature measurement point and the curves are drastically different from the exponential solution shown by a dotted line. Figure 4 compares the value of the blood perfusion parameter ‘6’as calculated from the initial slope of the temperature (eqn 6) plotted on the semi-logrithmic graph for different tumor sizes, ‘a’, with the slope of the exponential solution obtained by neglecting the thermal conduction. The two values approach each other as the value of the parameter ‘u’is increased. To determine the circumstances under which the exponential solution is a reasonable approximation to the exact solution, eqn (6) was evaluated for different values 47 46
300 t
360
420
460
540
600
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Fig. 2. Percentage temperature rise above the baseline temperature plotted as a function of time after the onset of temperature decay. The dotted line represents the values calculated when thermal conduction is neglected. The solid lines represent the values calculated using eqn (6). The number at the end of each curve indicates the value of the parameter ‘a’ in cm, which is also the radius of the heated spherical volume surrounded by tissue at normal temperature. All other parameters are identical for these curves.
of blood perfusion keeping the size parameter fixed. The results are given in Fig. 5. Here, as the value of blood perfusion increases, the thermal conduction effects become less and less important. DISCUSSION The temperature decay curves presented by solid lines in Figs. 2-3 represent the exact solutions of the bioheat WI 12 ml/ 1 OOglmin
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360
420
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460
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540
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Fig. 3. The temperature decay curves for different spatial points within the heated volume as calculated using eqn (5). The position is indicated at the end of each curve by the distance (in cm) from the center. The initial slope is position dependent indicating that it is influenced by the local temperature gradients via thermal conduction. The dotted line is the decay curve calculated by neglecting the thermal conduction and is position independent.
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a (cm) Fig. 4. The solid line represents the apparent value h’ of the parameter b = ucbpb/c as obtained by equating it to the initial slope of the exact solution (eqn 6) plotted on a log-linear graph.
The dotted line represents the exact value that is the same as the initial slope of the curve obtained when thermal conduction is absent.
equation, and are significantly different from the simple exponential form (dotted lines), obtained by neglecting the thermal conduction term. For a fixed value of blood perfusion, the greater the thermal gradients (small ‘a’ value), the greater the deviation of the exact solution from the exponential form. As the thermal gradients at the point of measurement are decreased by increasing the size parameter (Fig. I), the exact solution shifts toward the simple exponential form. The exact solution in general, however, can not be represented by an exponential function. This fact not only affects the blood perfusion measurements, but also the temperature measurement techniques used by some investigators. The measurement of temperature in a microwave field with thermocouples’ is based on exponential backward extrapolation after microwave power shut-off. This can lead to significant errors because, as shown here, the actual temperature decay is not necessarily an exponential function of time. The magnitude of these errors will, of course, depend on local thermal gradients and the value of blood perfusion. Milligan et a1.6in their attempts to extract the blood perfusion from the temperature decay curves found that the experimentally measured data was not describable by an exponential curve. Instead of including the effects of thermal conduction to overcome this difficulty, they introduced a rather arbitrary dimensionless temperature dependent parameter into the bioheat equation. It should be pointed out that the thermal gradients at the onset of temperature decay reported in
March
1986. Volume
12, Number
3
reference 6 are quite similar in form to the one used here. In light of the present results, it is not surprising that the experimental temperature decay curves were not exponential. To investigate quantitatively the errors introduced by assuming an exponential temperature decay, apparent values for the parameter b = wcbpb/cwere evaluated in the following manner. For a fixed value of blood perfusion, w, the exact solution (eqn 6) was evaluated for different values of the parameter ‘a’. The assumption was made that these values of T vs t follow an exponential form r = T&” and the parameter b’was evaluated from initial slope of r vs t plot on a semi-logrithmic graph. The results are shown in Fig. 4 for the calculations performed with w = 6 ml/l00 g/min that represents the blood perfusion for resting muscle. The dotted line indicates the actual value of the parameter b = wchph/c.These results indicate that the value of the blood perfusion parameter evaluated in this manner can be in error by as much as an order of magnitude if large thermal gradients are present. To investigate whether the assumption of neglecting thermal conduction is reasonable for tissues with large values of blood flow, the exact solution (eqn 6) was evaluated for varying values of blood flow, w, with a fixed value of ‘a’, that is, same thermal gradient conditions. The results are shown in Fig. 5. The percentage error shown in the ordinate is the percentage difference between the value of ‘b’ evaluated as described in the previous paragraph and the actual value b = Wcbpb/c. In the presence of large thermal gradients (small ‘a’values) the error is large (~40%) even in the presence of high perfusion rates (84 ml/ 100 g/min). The error reduces to less than 10% when the thermal conduction effects are reduced by decreasing thermal gradients (large ‘a’values). This suggests that for the determination of blood perfusion from temperature decay, thermal conduction plays an important role. The effects of thermal conduction can be neglected only if changes in thermal gradients at the point of measurement are negligible. For small values of blood perfusion (6 ml/ 100 g/min), even when the temperature distribution at the onset of temperature decay is relatively uniform (small thermal gradients), the effects of thermal conduction introduce an error of more than 10% in blood perfusion. The present analysis, therefore, suggests that a simple exponential form for the decay curve can be assumed for blood flow analysis if the measurement probe is located in the heated volume where changes in gradients are not significant. This will necessitate thermal mapping’ of the heated volume. CONCLUSION In summary, the analyses presented here demonstrate that the thermal conduction term in the bioheat equation cannot, in general, be neglected without grossly compromising the accuracy of its solution for temperature distribution. Furthermore, as shown in Figs. 4-5, the value
Temperature decay measurement of blood flow 0 T. S. SANDHU
12
24
36
48
60
72
84
wfml/ lOOg/minl
Fig. 5. The percentage error introduced in determining effects in the analysis of temperature decay curve.
of blood perfusion obtained from the temperature decay is affected significantly if thermal conduction is ignored. The exact solution which includes the effects of both the thermal conduction and convection due to blood perfusion, however, approaches the simple exponential form (obtained by neglecting thermal conduction) under the conditions of negligible thermal gradients. During hyperthermia treatments the point of measurements with low temperature gradients can be chosen if the heated volume is mapped for temperature distribution as discussed by Gibbs2 Furthermore, it should be stressed that in addition to the blood perfusion, the shape of the temperature decay curve depends highly on the initial temperature distri-
the blood perfusion by neglecting thermal conduction
bution at the onset of decay. Therefore, the blood flow values obtained from the temperature decay curves during the period of heating before the steady state is reached are meaningless since the temperature distribution at the onset of each decay curve will be different. Further, if the hyperthermia transducer produces a highly non-uniform temperature distribution, the blood flow values obtained by neglecting the thermal conduction effects will be transducer (size, energy deposition pattern) dependent and, therefore, not very valuable. In conclusion, the method of determining blood flow from temperature decay curves is of limited value and can be used only under special conditions.
REFERENCES Cater, D.B., Petrie, A., Watkinson, D.A.: Effect of 5-Hydroxytryptamine and Cyproheptadine on tumor blood flow: Estimation by rate of cooling after microwave diathermy. Acta Radiol. Ther. Phys. Biol. 3: 109-128,
Gibbs, F.A.: “Thermal
1965.
mapping” in experimental
cancer
treatment with hyperthermia description and using a semiautomatic system. Int. J. Radiat. Oncol. Biol. Phys. 9: 10571063, 1983. 3. Hand, J.W., Hopewell, J.N., Foster, J.L., Field, S.B.: Microwave heating of musculature in the pig by two 9 15 MHz
378
4.
5. 6.
7.
1. J. Radiation Oncology 0 Biology0 Physics direct-contact applicators. In Hyperthermia in Radiation Oncology, G. Arcangeli and F. Mauro (Eds.). Milano, Italy, Masson. 1980, pp. 37-44. Ingersoll, L.R., Zobel, O.J., Ingersoll, A.C.: Heat Conduction. With Engineering and Geological Applications. New York, McGraw-Hill. 1948, pp. 140- 14 1. Milligan, A.J.: Intestinal blood flow in Chinese hamster during hyperthermia (Abstr.). Rad. Res. 91: 318, 1982. Mill&m, A.J., Conran, P.B., Ropar, M.A., McCullough, H.A., Ahuja, R.K., Dobelbower, R.R.: Predictions of blood flow from thermal clearance during regional hyperthermia treatment. Int. J. Radial. Oncol. Biol. Phys. 9: 1335- 1343, 1983. Milligan, A.J., Panjehpour, M.: Mathematical predictions of tumor and normal tissue blood flow during hyperthermia (Abstr.). Rad. Res. 94: 534-535. 1983.
March 1986, Volume 12, Number 3 8. Roemer, R.B., Fletcher, A.M., Cates, T.C.: Obtaining local SAR and blood perfusion data from temperature measurements: Steady state and transient techniques compared. Int. J. Radiat. Oncol. Biol. Phys. 11: 1539-1550, 1985. 9. Scott, R.S., Johnson, R.J., Kowal, H.S., Krishnamasetty, R.M., Story, J., Clay, L.: Hyperthermia in combination with radiotherapy: A review of five years experience in the treatment of superficial tumors. Int. J. Radiat. Oncol. Biol. Phys. 9: 1327-1333, 1983. 10. Waterman, F.J., Fazekas, J., Nerlinger, R.E., Leeper, D.B.: Blood flow rates in human tumors during hyperthermia treatments as indicated by thermal washout (Abstr.). Rad. Res. 91: 426, 1982. 11. Waterman, F.J., Nerlinger, R.E., Leeper, D.B., Moylan, D. J. : Blood flow in human tumors during local hyperthermia (Abstr.). Rad. Rex 94: 598, 1983.