The Effects of Large Blood Vessels on Temperature Distributions During Simulated Hyperthermia Zong-Ping Chen Radiation Oncology Department.
Robert B. Roemer Aerospace & Mechanical Engineering Department. University of Arizona, Arizona Health Sciences Center, Tucson, AZ 85724
Several three-dimensional vascular models have been developed to study the effects of adding equations for large blood vessels to the traditional bioheat transfer equation of Pennes when simulating tissue temperature distributions. These vascular models include "transiting" vessels, "supplying" arteries, and "draining" veins, for all of which the mean temperature of the blood in the vessels is calculated along their lengths. For the supplying arteries this spatially variable temperature is then used as the arterial temperature in the bioheat transfer equation. The different vascular models produce significantly different locations for both the maximum tumor and the maximum normal tissue temperatures for a given power deposition pattern. However, all of the vascular models predict essentially the same cold regions in the same locations in tumors: one set at the tumors' corners and another around the inlets of the large blood vessels to the tumor. Several different power deposition patterns have been simulated in an attempt to eliminate these cold regions; uniform power in the tumor, annular power in the tumor, preheating of the blood in the vessels while they are traversing the normal tissue, and an "optimal" power pattern which combines the best features of the above approaches. Although the calculations indicate that optimal power deposition patterns (which improve the temperature distributions) exist for all of the vascular models, none of the heating patterns studied eliminated all of the cold regions. Vasodilation in the normal tissue is also simulated to see its effects on the temperature fields. This technique can raise the temperatures around the inlet of the large blood vessles to the tumor (due to the higher power deposition rates possible), but on the other hand, normal tissue vasodilation makes the temperatures at the tumor corners slightly colder.
Introduction It can be inferred from the clinical results of Dewhirst et al. (1984) that in order to have a successful hyperthermia cancer treatment, the temperatures in the entire hyperthermia target volume should be raised above a therapeutic value (e. g., 43 ° C), while at the same time the temperatures in the surrounding normal tissues should be kept at safe, low values. Due to the inhomogeneity of blood perfusion and other thermal properties inside and around tumors, as well as the limitations of current heating technologies, it is very difficult to reach that goal in clinical situations. In particular, when there are large blood vessels in a tumor the convective effect of the blood flow can carry away a significant amount of the deposited energy, thus possibly forming steep temperature gradients around the vessels with associated cold tissue regions near the vessels (Overgaard et al., 1980). If such cold regions are present, power deposition patterns must be devised to heat them without overheating normal tissues (Roemer, 1991). In order to find Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENOINEERING. Manuscript received by the Bioengineering Division March 5, 1991; revised manuscript received February 1, 1992. Associate Technical Editor: K. R. Diller
and effectively utilize such patterns, a quantitative mathematical model to predict the temperatures inside of and in the vicinity of large blood vessels is needed. Several vascular models currently exist which explicitly account for the convective effects of large blood vessels on the tissue temperature distributions. Some of these are concerned with hyperthermia. First, Lagendijk et al. (1984) derived a three-dimensional finite difference model, which included both the convective effect of a single large blood vessel and the effects of perfused blood, to predict hyperthermia temperature fields under uniform heating conditions. No counter-current vessel pairs were simulated. Second, Baish et al. (1986a) simulated counter-current vessel pairs in a one dimensional study, and calculated the artery, vein and average tissue temperatures for uniform heating. In related hyperthermia studies Charny and Levin (1989) also used a one-dimensional model to calculate the variation of temperature in the axial direction for a counter-current vessel pair plus its surrounding tissue. Although temperature differences existed among the three compartments of their lumped parameter model (vein, artery, tissue) which was based on Baish et al.s' previous work (Baish et al., 1986b), radial conduction was not included and thus the effect
Journal of Biomechanical Engineering
NOVEMBER 1992, Vol. 114 / 473
Copyright © 1992 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/29/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
of their vessels in producing cold spots was not realistically modeled. That study was instead used to investigate certain assumptions in the Weinbaum-Jiji's modified bioheat transfer equation (Weinbaum and Jiji, 1985). Their related one-dimensional radial study, which did not explicitly include large blood vessels, but did look at hyperthermia conditions (Charny et al., 1990), was also used primarily to study bioheat transfer equation formulations. Several vascular models have also been developed for nonhyperthermia studies. In particular, by introducing several simplifications Chato (1980) could analytically solve for the temperature fields for a single vessel and a counter-current vessel pair imbedded in a nonperfused homogeneous media. That counter current pair analysis was recently extended by Zhu et al. (1990) in an approximate analytical solution, but again, this was a basic study, not concerned with hyperthermia situations. In an extension of those works to multiple vessels and more complicated situations, Williams (1990) recently developed a vascular model, which numerically modeled multiple single vessels and multiple counter-current blood vessel pairs imbedded in a homogeneous, nonperfused tissue. None of these last three models included hyperthermia power deposition, and none of them considered the thermal effects of perfused blood in smaller vessels. A few studies also exist which look at various effects caused by variable mass flow rate in counter-current models. These include the above work of Chato (1980), Charny and Levin (1989), Charny et al. (1990), Weinbaum and Jiji (1985), and Wissler (1987) among others. However, no two or three-dimensional study on the thermal effects of the variable mass flow rate in "supplying" arteries which supply the perfused blood to the smaller vessels of tissue, and "draining" veins which drain the perfused blood from the tissue has been reported for hyperthermia conditions. Such supplying and draining vessels exist in tumors, and their thermal effects should be studied specifically. Finally, no hyperthermia studies exist in which the arterial temperature used in the bioheat transfer equation (which is used to model the perfused tissue in which the large blood vessels are imbedded) is calculated as a function of position along a supplying artery. The few related studies of this factor are the whole body temperature regulatory studies such as those of Wissler (1985) and Gordon et al. (1976). In this study, several different vascular models, including supplying arteries and draining veins, have been developed. Both (1) the convective mechanism of the directional blood flow in the large blood vessels and (2) the bioheat transfer equation thermal sink mechanism for the perfused blood, which is used in an attempt to account for the collective heat transfer effects of the blood flow in the smaller vessels, are simulated. A spatially variable arterial temperature is used in the bioheat transfer equation for the supplying artery case. These models have been used to demonstrate the influence of the different vessel models on hyperthermia temperature fields for several different heating patterns. To eliminate cold regions in tumors and to obtain improved hyperthermia temperature fields, optimal, spatially variable power deposition patterns must be found. Since the present ' study involves three dimensional inhomogeneous models with different types of blood vessels, a complete optimization study would involve many parameters and be very complicated. Therefore, an exhaustive optimization study is beyond the scope of this work. Nevertheless, it does attempt to find improved power deposition patterns for several ideal, simple, finite sized power deposition fields, and for proposed vasodilation manipulations. These simple patterns will allow clearer interpretations of the predicted results, and will allow the effects of the various blood vessel models to be emphasized and seen clearly. The simple power deposition patterns will, however be more realistic than those of both (a) Ocheltree and Frizzell (1987, 1988), who analyzed the optimal power depo474 / Vol. 114, NOVEMBER 1992
sition problem based on the predicted temperature distributions by the bioheat transfer equation using a power deposition field with infinitely fine resolution, and (b) Roemer (1991) who also studied the optimization problem based on the one dimensional bioheat transfer equation, but with only a prespecified (not a calculated) amount of arterial blood preheating. The present modeling efforts represent an extension of the above studies. Improved but still idealized (rather than completely realistic) cases are still utilized in an attempt to see major trends arid to study limiting cases. These modeling efforts are steps toward the eventual solution of the complete optimization problem for realistic vessel and tumor perfusion models and power deposition fields. Models and Simulations Six different three-dimensional vascular models were developed and simulated in this study (Fig. 1). The dimensions of the overall tissue region for the bioheat transfer equation and single vessel models (Figs. 1(a), 1(b), and 1(c) are 10 x 10x20 cm in the X, Fand Z directions, respectively, and the tumor dimensions are 4 x 4 x 4 cm. In the counter-current pair and the parallel flow vessel models (Figs. 1(d), 1(e), and 1(f), the Y and Z overall tissue and tumor region dimensions are the same as in the previous three models. However, in order to have geometrically symmetrical models when pairs of vessels are present, the X direction dimensions for both the overall tissue and the tumor are increased by 2 mm. This is necessary since a 2 mm grid spacing is used for all calculations. This grid spacing gives a 2 mm centerline to centerline distance for all vessel pairs. The boundary conditions are a constant temperature of 37°C on all surfaces of the parallelepiped. The blood flow velocities for most cases studied are set at 10 cm/ s, and the vessel's diameters are set at 1 mm, as based on Chato (1980). The tumor blood perfusion values for those cases are set at 1.23 kg/m 3 /s. This value is calculated from the model for the single artery supplying the tumor (SAST, Fig. 1(c), in which all of the blood perfusion in the 64 cm3 tumor is supplied completely by the artery (1 mm diameter and 10 cm/s). In order to facilitate comparisons among all the vascular models, this tumor perfusion value is then used for all studies. A blood flow velocity of 1 cm/s and the corresponding tumor perfusion value of 0.123 kg/m 3 /s are also used for some cases. Normal tissue blood perfusions vary for different calculations, as specified later. The tissue thermal conductivity is set at 0.5 W / m / °C, the specific heat of the blood is 4185 J/kg/°C, and the Nussselt number is set at 4 (Chato, 1980). An important feature of these models concerns the supplying arteries and draining veins. A supplying artery supplies all of its blood to the tumor, and a draining vein collects all of its blood from the tumor. The mass flow rates in the arteries decreases linearly in the flow direction from the full value (10 cm/s for most cases) at the inlet to the tumor, to zero at the far side of the tumor. Similarly, the flow rates in the draining veins increase linearly along their flow direction from zero at the origin of the vein to the full value (10 cm/s for most cases) at their outlet from the tumor. Additionally, in these models the arterial blood temperature, which varies along the flow direction, is then used in the steady state bioheat transfer equation outside the vessel in the tumor region, i.e., in the tumor Tar= Tar(z), so that the bioheat transfer equation is kV2T+Qap-Wcb(T-Tar(z))=Q
(1)
where T is the tissue temperature, k is the tissue thermal conductivity, Qap is power deposition, W is the blood perfusion rate, cb is the specific heat of blood, Tar is the arterial temperature, and z is the coordinate in the arterial flow direction. The use of Tar(z) for the arterial temperature in the bioheat transfer equation represents an improvement over the use of a constant (37°C) value since it accounts for some, but not Transactions of the ASME
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Robert B. Roemer Aerospace & Mechanical Engineering Department. University of Arizona, Arizona Health Sciences Center, Tucson, AZ 85724
Several three-dimensional vascular models have been developed to study the effects of adding equations for large blood vessels to the traditional bioheat transfer equation of Pennes when simulating tissue temperature distributions. These vascular models include "transiting" vessels, "supplying" arteries, and "draining" veins, for all of which the mean temperature of the blood in the vessels is calculated along their lengths. For the supplying arteries this spatially variable temperature is then used as the arterial temperature in the bioheat transfer equation. The different vascular models produce significantly different locations for both the maximum tumor and the maximum normal tissue temperatures for a given power deposition pattern. However, all of the vascular models predict essentially the same cold regions in the same locations in tumors: one set at the tumors' corners and another around the inlets of the large blood vessels to the tumor. Several different power deposition patterns have been simulated in an attempt to eliminate these cold regions; uniform power in the tumor, annular power in the tumor, preheating of the blood in the vessels while they are traversing the normal tissue, and an "optimal" power pattern which combines the best features of the above approaches. Although the calculations indicate that optimal power deposition patterns (which improve the temperature distributions) exist for all of the vascular models, none of the heating patterns studied eliminated all of the cold regions. Vasodilation in the normal tissue is also simulated to see its effects on the temperature fields. This technique can raise the temperatures around the inlet of the large blood vessles to the tumor (due to the higher power deposition rates possible), but on the other hand, normal tissue vasodilation makes the temperatures at the tumor corners slightly colder.
Introduction It can be inferred from the clinical results of Dewhirst et al. (1984) that in order to have a successful hyperthermia cancer treatment, the temperatures in the entire hyperthermia target volume should be raised above a therapeutic value (e. g., 43 ° C), while at the same time the temperatures in the surrounding normal tissues should be kept at safe, low values. Due to the inhomogeneity of blood perfusion and other thermal properties inside and around tumors, as well as the limitations of current heating technologies, it is very difficult to reach that goal in clinical situations. In particular, when there are large blood vessels in a tumor the convective effect of the blood flow can carry away a significant amount of the deposited energy, thus possibly forming steep temperature gradients around the vessels with associated cold tissue regions near the vessels (Overgaard et al., 1980). If such cold regions are present, power deposition patterns must be devised to heat them without overheating normal tissues (Roemer, 1991). In order to find Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENOINEERING. Manuscript received by the Bioengineering Division March 5, 1991; revised manuscript received February 1, 1992. Associate Technical Editor: K. R. Diller
and effectively utilize such patterns, a quantitative mathematical model to predict the temperatures inside of and in the vicinity of large blood vessels is needed. Several vascular models currently exist which explicitly account for the convective effects of large blood vessels on the tissue temperature distributions. Some of these are concerned with hyperthermia. First, Lagendijk et al. (1984) derived a three-dimensional finite difference model, which included both the convective effect of a single large blood vessel and the effects of perfused blood, to predict hyperthermia temperature fields under uniform heating conditions. No counter-current vessel pairs were simulated. Second, Baish et al. (1986a) simulated counter-current vessel pairs in a one dimensional study, and calculated the artery, vein and average tissue temperatures for uniform heating. In related hyperthermia studies Charny and Levin (1989) also used a one-dimensional model to calculate the variation of temperature in the axial direction for a counter-current vessel pair plus its surrounding tissue. Although temperature differences existed among the three compartments of their lumped parameter model (vein, artery, tissue) which was based on Baish et al.s' previous work (Baish et al., 1986b), radial conduction was not included and thus the effect
Journal of Biomechanical Engineering
NOVEMBER 1992, Vol. 114 / 473
Copyright © 1992 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/29/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
of their vessels in producing cold spots was not realistically modeled. That study was instead used to investigate certain assumptions in the Weinbaum-Jiji's modified bioheat transfer equation (Weinbaum and Jiji, 1985). Their related one-dimensional radial study, which did not explicitly include large blood vessels, but did look at hyperthermia conditions (Charny et al., 1990), was also used primarily to study bioheat transfer equation formulations. Several vascular models have also been developed for nonhyperthermia studies. In particular, by introducing several simplifications Chato (1980) could analytically solve for the temperature fields for a single vessel and a counter-current vessel pair imbedded in a nonperfused homogeneous media. That counter current pair analysis was recently extended by Zhu et al. (1990) in an approximate analytical solution, but again, this was a basic study, not concerned with hyperthermia situations. In an extension of those works to multiple vessels and more complicated situations, Williams (1990) recently developed a vascular model, which numerically modeled multiple single vessels and multiple counter-current blood vessel pairs imbedded in a homogeneous, nonperfused tissue. None of these last three models included hyperthermia power deposition, and none of them considered the thermal effects of perfused blood in smaller vessels. A few studies also exist which look at various effects caused by variable mass flow rate in counter-current models. These include the above work of Chato (1980), Charny and Levin (1989), Charny et al. (1990), Weinbaum and Jiji (1985), and Wissler (1987) among others. However, no two or three-dimensional study on the thermal effects of the variable mass flow rate in "supplying" arteries which supply the perfused blood to the smaller vessels of tissue, and "draining" veins which drain the perfused blood from the tissue has been reported for hyperthermia conditions. Such supplying and draining vessels exist in tumors, and their thermal effects should be studied specifically. Finally, no hyperthermia studies exist in which the arterial temperature used in the bioheat transfer equation (which is used to model the perfused tissue in which the large blood vessels are imbedded) is calculated as a function of position along a supplying artery. The few related studies of this factor are the whole body temperature regulatory studies such as those of Wissler (1985) and Gordon et al. (1976). In this study, several different vascular models, including supplying arteries and draining veins, have been developed. Both (1) the convective mechanism of the directional blood flow in the large blood vessels and (2) the bioheat transfer equation thermal sink mechanism for the perfused blood, which is used in an attempt to account for the collective heat transfer effects of the blood flow in the smaller vessels, are simulated. A spatially variable arterial temperature is used in the bioheat transfer equation for the supplying artery case. These models have been used to demonstrate the influence of the different vessel models on hyperthermia temperature fields for several different heating patterns. To eliminate cold regions in tumors and to obtain improved hyperthermia temperature fields, optimal, spatially variable power deposition patterns must be found. Since the present ' study involves three dimensional inhomogeneous models with different types of blood vessels, a complete optimization study would involve many parameters and be very complicated. Therefore, an exhaustive optimization study is beyond the scope of this work. Nevertheless, it does attempt to find improved power deposition patterns for several ideal, simple, finite sized power deposition fields, and for proposed vasodilation manipulations. These simple patterns will allow clearer interpretations of the predicted results, and will allow the effects of the various blood vessel models to be emphasized and seen clearly. The simple power deposition patterns will, however be more realistic than those of both (a) Ocheltree and Frizzell (1987, 1988), who analyzed the optimal power depo474 / Vol. 114, NOVEMBER 1992
sition problem based on the predicted temperature distributions by the bioheat transfer equation using a power deposition field with infinitely fine resolution, and (b) Roemer (1991) who also studied the optimization problem based on the one dimensional bioheat transfer equation, but with only a prespecified (not a calculated) amount of arterial blood preheating. The present modeling efforts represent an extension of the above studies. Improved but still idealized (rather than completely realistic) cases are still utilized in an attempt to see major trends arid to study limiting cases. These modeling efforts are steps toward the eventual solution of the complete optimization problem for realistic vessel and tumor perfusion models and power deposition fields. Models and Simulations Six different three-dimensional vascular models were developed and simulated in this study (Fig. 1). The dimensions of the overall tissue region for the bioheat transfer equation and single vessel models (Figs. 1(a), 1(b), and 1(c) are 10 x 10x20 cm in the X, Fand Z directions, respectively, and the tumor dimensions are 4 x 4 x 4 cm. In the counter-current pair and the parallel flow vessel models (Figs. 1(d), 1(e), and 1(f), the Y and Z overall tissue and tumor region dimensions are the same as in the previous three models. However, in order to have geometrically symmetrical models when pairs of vessels are present, the X direction dimensions for both the overall tissue and the tumor are increased by 2 mm. This is necessary since a 2 mm grid spacing is used for all calculations. This grid spacing gives a 2 mm centerline to centerline distance for all vessel pairs. The boundary conditions are a constant temperature of 37°C on all surfaces of the parallelepiped. The blood flow velocities for most cases studied are set at 10 cm/ s, and the vessel's diameters are set at 1 mm, as based on Chato (1980). The tumor blood perfusion values for those cases are set at 1.23 kg/m 3 /s. This value is calculated from the model for the single artery supplying the tumor (SAST, Fig. 1(c), in which all of the blood perfusion in the 64 cm3 tumor is supplied completely by the artery (1 mm diameter and 10 cm/s). In order to facilitate comparisons among all the vascular models, this tumor perfusion value is then used for all studies. A blood flow velocity of 1 cm/s and the corresponding tumor perfusion value of 0.123 kg/m 3 /s are also used for some cases. Normal tissue blood perfusions vary for different calculations, as specified later. The tissue thermal conductivity is set at 0.5 W / m / °C, the specific heat of the blood is 4185 J/kg/°C, and the Nussselt number is set at 4 (Chato, 1980). An important feature of these models concerns the supplying arteries and draining veins. A supplying artery supplies all of its blood to the tumor, and a draining vein collects all of its blood from the tumor. The mass flow rates in the arteries decreases linearly in the flow direction from the full value (10 cm/s for most cases) at the inlet to the tumor, to zero at the far side of the tumor. Similarly, the flow rates in the draining veins increase linearly along their flow direction from zero at the origin of the vein to the full value (10 cm/s for most cases) at their outlet from the tumor. Additionally, in these models the arterial blood temperature, which varies along the flow direction, is then used in the steady state bioheat transfer equation outside the vessel in the tumor region, i.e., in the tumor Tar= Tar(z), so that the bioheat transfer equation is kV2T+Qap-Wcb(T-Tar(z))=Q
(1)
where T is the tissue temperature, k is the tissue thermal conductivity, Qap is power deposition, W is the blood perfusion rate, cb is the specific heat of blood, Tar is the arterial temperature, and z is the coordinate in the arterial flow direction. The use of Tar(z) for the arterial temperature in the bioheat transfer equation represents an improvement over the use of a constant (37°C) value since it accounts for some, but not Transactions of the ASME
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/29/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
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