Significance of Blood Vessels on Optimization of Absorbed Power and Temperature Distributions during Hyperthermia

Significance of blood vessels in optimization of absorbed power and temperature distributions during hyperthermia

Huang-Wen Huang

, Chihng-Tsung Liauh

, Tzu-Ching Shih

, Tzyy-Leng Horng

, Win-Li Lin

aDepartment of Innovative Information and Technology, Software Engineering Group, Langyang Campus, Tamkang University, I-lan, Taiwan

bDepartment of Mechanical Engineering, Kun-Shan University, Tainan, Taiwan

cDepartment of Biomedical Imaging and Radiological Science, China Medical University, Taichung, Taiwan

dDepartment of Applied Mathematics, Feng Chia University, Taichung, Taiwan

eInstitute of Biomedical Engineering, College of Medicine and College of Engineering, National Taiwan University, No. 1, Section 1, Jen-Ai Road, Taipei, Taiwan

a r t i c l e i n f o

Article history:

Available online 16 September 2010

Keywords:

Thermally significant blood vessels Hyperthermia

Temperature distribution Absorbed power deposition Optimization

a b s t r a c t

This study investigated the significance of blood vessels in the absorbed power and temperature distri- butions when optimization was employed during hyperthermia. The treated tumor region was simulated using a three-dimensional (3D) tissue model embedded with a countercurrent blood vessel network (Huang et al., 1996). 3D temperature distributions are obtained by solving the conduction equation in the tissues and the convective energy equation with specified Nusselt number in the vessels. 3D absorbed power depositions are obtained by using optimization to reach a uniform temperature of 43 °C for the desired treated region. Results show that the absorbed power deposition for optimization with fine spa- tial resolution produces a uniform temperature distribution maintained at 43 °C in the desired treated tumor region except for some cold spots and/or small cold strips caused by thermally significant large vessels. The amount of total absorbed power suggests that a region with thermally significant vasculature requires much more power deposited than one without vasculature. In addition, optimization with coarse spatial resolution results in a highly inhomogeneous temperature distribution in the treated region due to the strong cooling effect of blood vessels. Therefore, prior to hyperthermia treatments, ther- mally significant blood vessels should be identified and handled carefully to effectively reduce their strong cooling effect, particularly those vessels flowing into the treated region.

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1. Introduction

Hyperthermia cancer treatment requires precise thermal ab- sorbed power deposition to raise tumor tissue temperature up to the therapeutic range with a sufficient amount of time duration and to prevent overheating the normal tissues. Many researchers [1–4]have investigated noninvasive heating modality for explor- ing power deposition with fine spatial resolution and/or optimiza- tion within the tumor region. Hyperthermia applicator technology is currently one of the most important issues that can improve temperature homogeneity in the treated region as well as reach optimal applied power deposition.

The Pennes’ bio-heat transfer equation (BHTE)[5]has been a standard model for predicting temperature distributions in living tissues. The equation was established through conducting a se- quence of experimental temperature measurements of tissue and arterial blood temperatures in the resting human forearm. The

equation includes a special term that describes the heat exchange between blood perfusion and solid tissues. The blood temperature is assumed to be constant arterial blood temperature. The main limitation of this equation is its inability to describe the impact of thermally significant vessels. Some researchers[6–10]have also developed alternative equations having the same goal of attempt- ing to formulate a single, general field equation that can predict the overall characteristics of temperature distributions in tissues.

Those approximate field equations, which are intended to predict the average thermal behavior of the tissue, do not have the ability to accurately model the effects of isolated, large vessels. Thus, such blood vessels must be modeled using separate equations. The ef- fects of such vessels have been studied by Chato [11] and Huang et al.[12], and Huang[13]developed analytical models for single vessels. Other investigators [14–23] have done numerical and experimental hyperthermia studies for single vessels and/or coun- ter current vessel pairs embedded in either a purely conductive media (with either a normal thermal conductivity, or an enhanced, effective thermal conductivity) or in media modeled by the Pennes’

BHTE. One example, published by Rawnsley et al.[21], illustrated the experimental temperature data measured in the thighs of

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doi:10.1016/j.ijheatmasstransfer.2010.08.017

⇑ Corresponding author. Tel.: +886 2 2312 3456x81445; fax: +886 2 2394 0049.

E-mail address:winli@ntu.edu.tw(W.-L. Lin).

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anesthetized greyhound dogs under hyperthermic conditions heated by scanned focused ultrasound. When small numbers(3) and (4) of blood vessel pairs were included in Pennes’ bio-heat transfer thermal model, the model showed significant improve- ment in its ability to predict experimental temperatures. Leeuwen et al.[22]also stressed that efforts to obtain information on the positions of the large vessels in an individual hyperthermia patient could be rewarded with a more accurate prediction of the temper- ature distribution. Finally, a few studies[22,24–26]have modeled the effect of collections of a large number of parallel vessels or of networks of vessels on the resulting temperature distributions.

However, those studies did not consider optimization in hyper- thermia treatments.

This study investigated the impact of thermally significant blood vessels on hyperthermia treatment using a thermal conduc- tion matrix model embedded with a network of blood flow vessels (fully conjugated blood vessel network model (FCBVNM), Huang et al.[24]). Optimization was used to determine the power deposi- tion to achieve an ideal 3D therapeutic temperature distribution with a uniform temperature of 43 °C for the desired treated region.

Optimized absorbed power deposition with fine/coarse spatial res- olution was considered to study the effects of thermally significant blood vessels on the resulting temperature distributions and the limitation of heating ability for power systems.

2. Methods

2.1. Vessel network geometry and fully conjugated blood vessel network model (FCBVNM)

FCBVNM is a model formulation which describes the solid tis- sue matrix having thermally significant vessel generations (seven levels). The effects of all vessels smaller than the terminal (level se- ven) vessels are not explicitly modeled in FCBVNM. Thus, those smaller vessels (connected to the terminal arteries and the termi- nal veins in the network) are implicitly assumed to be thermally insignificant in the FCBVNM.

The tissue geometry used in this study consists of a regular, branching vessel network as partially shown (only the arterial ves- sels are shown) inFig. 1that is embedded in a control volume, which is an (approximate) cube of dimensions L = 8.2 cm and W = H = 8 cm in the x, y, and z directions, respectively. All vessels

are straight-line segments parallel to one of the three Cartesian axes. There are up to seven levels of arteries, beginning with the main artery (level one) which lies along the central, lengthwise (x) axis of the cube. Table 1 of Huang et al.[24]lists the basic vessel network properties used in this study. The diameters of the arteries decrease by a constant ratiocbetween successive levels of branch- ing vessels (the ratio of diameters of successive vessel genera- tions), i.e.

c

Di ð1Þ

where Diand Di+1 are the diameters of two successive levels of branching arteries. When two successive levels of numbered vessels do not branch but only change direction (i.e., levels six and seven in this model), the vessel diameter does not change. In this study, we usedc= 0.9 in the presented results.

The desired treated tumor region is a cube described inFig. 2(a) with 20 mm in x, y and z dimensions. The locations and paths of arterial vessels inside the treated region are described inFig. 2(b) and (c). The geometric arrangement of the countercurrent veins is essentially identical to that of the arteries, with all of the veins offset from the arteries by one finite difference node in x, y, and z dimensions as appropriate to avoid intersections of vessels.

2.2. Mathematical equations for the thermal model

The governing equation in tissues is described below,

r ðkrTðx; y; zÞÞ  _wbcbðTðx; y; zÞ  TaÞ þ qs¼ 0 ð2Þ where k, cb, wb, and qsare the thermal conductivity of soft tissue, specific heat of blood, blood perfusion rate and absorbed thermal power density, respectively. Ta is the arterial blood temperature and may be replaced by T_b(x, y, z) only if blood temperature is from terminating arterial vessels. The metabolism effect is neglected in Eq.(2)due to its limited effect on temperature distribution during hyperthermia.

The convective energy equation is solved for the FCBVNM model,

mb;icbrTbðx; y; zÞ ¼ Nu  kb

p

p

Nomenclature

c specific heat capacity, J kg^1C^1 C cost function

D diameter of blood vessel, mm hcoef updated power coefficient

H length of control volume in z direction, cm k thermal conductivity, W m^1C^1

L length of control volume in x direction, cm m mass flow rate, kg s^1

Nu Nusselt number of blood vessels P absorbed power deposition, W m^3 q absorbed thermal power, W m^3 R radius of blood vessel, mm T temperature, °C

w perfusion rate, kg m^3s^1

W length of control volume in y direction, cm x coordinate in the x direction

y coordinate in the y direction z coordinate in the z direction

DCn cost function increment at nth iteration

DT temperature difference, °C

Greek symbols

c ratio of diameters of successive levels of branching arteries

p mathematical constant

q density, kg m^3

Subscripts

adj adjacent tissue temperature

b blood

bv blood vessel

i index of node number; index of level number of blood vessels

n iteration

s absorbed source power

vr venous returning terminal vessels w vessel wall

A total of 682 vessels in the model need to be calculated using Eq.

(3). That is, i is from 1 to 682 for Eq.(3). Constant blood bleeding

or collection along the vessels is considered for 128 arterial and 128 venous terminal ends of level 7 vessels.

1

y

x z

2

3 4 5 6

7

Fig. 1. Schematic diagram to show portion of the arterial vessel network used in this study. All seven vessel levels (levels 1–7) for the arterial network are shown, and the venous network, which is not shown, is parallel to the arterial network, with a grid size in the x, y, and z dimensions away from the arterial network.

X=62 mm plane

(x, y, z)=(0,40,40) mm

X=0 plane

X=42 mm plane

(0,0,0) (42,40,40)

(62,40,40) (62,40,60) (62,60,60)

(62,20,40)

(42,60,40)

(42,60,60) (42,40,60) )

b ( )

a (

4

1 2 3 5

6 7 (42,60,60)

(52,60,60)

5 6 7

4

1 (62,60,60)

(52,60,60) (c)

5 6 7

4

1 (62,60,60)

x z

y

Fig. 2. (a) A transparent view of parallelepiped showing the desired heated tumor region, which is a cube of 20 mm in each dimension. The level 1 blood vessel (the largest) runs through the desired heated cube’s edge from (42, 40, 40) to (62, 40, 40). The inlet temperature of the level 1 vessel starting at (0, 40, 40) is set at 37 °C. (b) shows the coordinates of 8 corners of the desired treated cube (unit: mm). (c) is a dissected transparent view showing all associated arterial blood vessel paths (or segments) in the cubic volume, and venous vessels do not appear in the figure. There are two branches of levels 5–6–7 blood vessels and one of levels 5–6 on the back boundary as the dissected view indicates.

To simulate the relationship between blood flow and perfusion in tissues, the following equation is used:

mb;icbrTbðx; y; zÞ ¼ Nu  kb

p

p

dmb;i

dx  cbTbðx; y; zÞ ð4Þ

In Eq.(4), dmb,i/dx is the constant blood bleeding or collecting mass rate at any cross-section along terminal ends, as the vessels linearly perfuse blood into local tissue environments (i.e. subvolumes) or collect blood from local tissue environments. That is, for the small- est terminal arterial vessels (level 7), a decreasing blood flow rate is present.

For venous return thermal model, assuming all of the blood which has perfused a terminal subvolume is collected at any cross-section by the terminal vein associated with that subvolume, the following equation is used to calculate the blood temperature in terminal veins

T_vr¼1 4

X⁴

i¼1

Ti;adj ð5Þ

where Ti,adjis the tissue temperatures adjacent to venous vessel. As a terminal vessel runs in any x, y or z straight direction, there are four neighboring tissue nodes considered in terms of computational scheme (i.e. finite difference method). For graphically illustrating the models and assumptions above,Fig. 3shows models depicting Eqs.(3)–(5).

The conservation of energy equation is applied for those loca- tions where branching vessels occur. Calculation of the blood tem- perature in vessels with a steady uniform velocity profile for blood

flow has been used widely and accepted in many studies. Recently, Horng et al.[27]reconfirmed the usage of steady uniform velocity profile approach. Conduction occurs in all three directions in the tissue matrix, and the outer control volume surface is held at a con- stant reference temperature (i.e. identical to the inlet artery temperature).

2.3. Optimization with fine spatial power deposition: based on local temperature response in the treated region

Fig. 4is the flow chart to describe continuously adjusting ab- sorbed power deposition in the desired treated tumor region in or- der to reach ideal temperature (uniform temperature throughout the treated tumor region with a temperature of 43 °C). The evalu- ation criterion of absorbed power deposition is shown in Eq.(6), which states that root mean square of difference of ideal tempera- ture (43 °C) and calculated temperature of all heated target nodes divided by (43  37) °C reaches less than the criterion value (set to be 10% of the temperature difference of (43  37) °C). If the crite- rion is achieved, we obtain the optimization of absorbed power deposition such that the heating temperature distribution is close to the ideal temperature distribution. Otherwise, the absorbed power deposition will be adjusted according to the local tempera- ture. The readjusted power deposition (Pn+1) is described in Eq.(7)

Evaluation criterion :

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

all target nodesðDTðx;y;zÞÞ² Total number of target nodes

r

ð43  37Þ^C 60:1 ð6Þ

P_nþ1ðx; y; zÞ ¼ Pnðx; y; zÞ þDPðx; y; zÞ ð7Þ withDPðx; y; zÞ ¼ Coef DTðx; y; zÞ, Coef is 10,000, n is the iteration number and D T(x, y, z) is the difference of ideal temperature (43 °C) and calculated temperature.

2.4. Optimization with lumped power deposition: uniform absorbed power deposition in the treated tumor region

To investigate the spatial resolution of absorbed power deposi- tion on the temperature distribution, a uniform power deposition in the entire desired treated tumor region is applied. Two impor- tant parameters need to be introduced for this optimization. One is cost function and the other is the power coefficient. Cost func- tion is set to be Cn¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

all target nodesðTðx; y; zÞ  43^CÞ²

q at nth iter-

ation, and the absorbed power is Pn+1= Pn+ hcoefDCn, with DCn+1= Cn+1 Cn and the coefficient hcoef. hcoef is the updated power coefficient, with a constant value of 3050. It is chosen based on a smoothly converging (i.e., no oscillating) search and with less computational time required during optimization. The optimiza- tion process will be terminated whenDCn is smaller than 10^4. With optimization of uniform power deposition, it allows the attention to be focused on the effect of the vasculature on the tem- perature distribution.

2.5. Numerical methods

The numerical scheme used to calculate the temperatures was a black and red finite difference SOR method[28]with upwind dif- ferencing used for the vessels. The numerical details are described by Chen [29] and Hunag[13]. Special algorithms are used to ac- count for the vessel corners where arteries and veins change direc- tion, and where two or more arteries divide, or two or more veins join. The thermal resistances around the circular vessels were cal- culated using the logarithmic resistance approach as described by Chen and Roemer[30]. The property values used in treated tumor- ous and non-treated normal tissues were k = 0.5 W m^1°C^1, (a)

(b)

(c)

Ti,adj (i=1~4)

Vein vessel Tissue

Tvr

Artery vessel segment

Mb,i Mb,i

Tissue

Control volume Vessel wall

Linearly perfused into tissue dMb,i

dx

x

0 Artery vessel

segment Mb,i

Tissue

Fig. 3. (a) The model used for non-terminal artery blood vessels in which the convective energy equation(3)is solved for the FCBVNM model. Mb,iis the blood flow rate entering vessel i and leaving with identical mass flow rate Mb,i. (b) The model used for the artery terminal vessels (level 7) as explained in Eq.(4). Mb,iis the blood flow rate entering vessel i with linearly perfused mass flow rate into tissue of dMb,i/dx. (c) The model of calculating returning vein blood temperature, as explained in Eq.(5).

c = cb= 4000 J kg^1°C^1 and q= 1000 kg m^3. The vessel heat transfer coefficient was calculated using a constant Nusselt num- ber of four for all vessel levels. In all cases, a finite difference nodal spacing of 2 mm was used. Test results with a nodal spacing of 1 mm for test cases using either the arterial vessel network (when no veins were present) or the countercurrent vessel network showed no significant differences with the results of the compara- ble 2-mm nodal spacing models. This 2-mm spacing gives an inter- vessel centerline to centerline diagonal spacing of 2.8 mm for the countercurrent vessels due to the 2-mm offsets in x, y, and z. The boundary temperature was set to 37 °C at the outer control volume surfaces. Inlet temperatures of (level 1) vessels to the control vol- ume were also set to 37 °C.

3. Results

3.1. Optimization with fine spatial power deposition: based on local temperature response

To investigate the significance of blood vessels in the tempera- ture distribution for optimal hyperthermia treatment, an optimiza- tion scheme as shown in the method section was employed to determine the absorbed power deposition, which included the heat- ing of blood vessels to achieve an optimal treatment.Fig. 5(a)–(e) is the optimal temperature distributions on the planes 4 mm away from the front boundary, the front boundary, the middle, the back boundary, and 4 mm away from the back boundary of the treated re- gion, respectively, andFig. 5(f)–(h) is the absorbed power deposi- tions on the planes of the front boundary, the middle, and the back boundary of the treated region, respectively, for a blood perfusion of 0.5 kg m^3s^1and a blood flow velocity of 320 mm s^1in the level 1 vessel.Fig. 5(a) shows that the temperature is approximately 40.0 °C near the treated region and displays a cold spot at the center which is due to level 1 artery blood vessel running perpendicular in- wards to the plane. At its southeastern diagonal direction about 2.8 mm away from the level 1 artery, a level 1 vein blood vessel is running in an opposite direction outwards to the plane. The vein ap-

pears to be collecting some thermal energy by convection through treated region.Fig. 5(b) shows that the temperature on the boundary of the treated region is close to ideal temperature (43 °C), and there are steep thermal gradients near the level 2 artery running upwards from the center point. As seen inFig. 5(f), large amounts of thermal power were deposited on levels 2, 3 and 5 arteries. The maximum thermal power deposition is approximately 3.7  10⁶W m^3. Fig. 5(c) shows that the temperature in the treated region is close to ideal temperature, while high temperature appears outside of the treated region due to a level 6 artery carrying convective thermal energy leftwards. It illustrates that thermally significant blood ves- sels within the treated region have effectively been heated and car- ried the convective thermal energy out of the region.Fig. 5(c) also shows maximum temperature located on (or near) left-side artery branch node of levels 5 and 6 arteries in the treated region.

Fig. 5(g) shows large thermal power is deposited on the corner of treated region, which is an area with dense blood vessels (levels 3, 4 and 5 arteries and veins).Fig. 5(d), the temperature distribution at the back boundary of the treated region, shows that a cold spot is found near the northwestern corner of heated region. The spot is 1.7 °C below the ideal temperature, and it is caused by the level 4 vein flowing into the heated region. As expected,Fig. 5(h) shows large amount of thermal power deposited on the same corner as shown inFig. 5(g) to compensate the heat loss caused by vessels.

Dense blood vessels act as energy sinks, and large amount of thermal power deposition is required in that area in order to maintain the lo- cal temperature at the desired level.Fig. 5(e), temperature on the plane 4 mm away from the back boundary of the heated region, shows some hot spots, and these spots are approximately 42 °C.

One spot, located in the northeastern direction more than 4 mm away from the heated region, has a temperature of about 38.3 °C.

Those hot spots are caused by arteries carrying hot blood flow.

3.2. Effect of blood perfusion and blood flow rates on the optimization

As blood perfusion increases, the flow rates in vessels get higher due to the conservation of blood mass, and the higher flow rate will produce a stronger thermal impact on the treated region. Fig. 6 Governing equations

(FCBVNM) to calculate temperature distribution

Difference between calculated and ideal temperature

<=10% of (43-37) °C

Adjust power deposition in treated region No

Yes Initial and boundary

conditions

Ideal temperature distribution

Initial input:

uniform power deposition in treated region

Pn+1(x,y,z) = Pn(x,y,z) +Δ P(x,y,z) Δ P(x,y,z) =Coef*Δ T(x,y,z)

Optimized temperature and absorbed power deposition

Fig. 4. The flow chart of optimization used in this study. The absorbed power deposition in the desired treated cube (20 mm in each dimension) is adjusted locally in order to achieve an ideal therapeutic temperature of 43 °C uniformly for the entire cube.

Fig. 5. Temperatures and absorbed power depositions for a blood perfusion rate of 0.5 kg m^3s^1after optimization with fine spatial power deposition. The ideal temperature is set to be 43 °C, and the blood flow velocity is about 320 mm s^1in level 1 vessel. (a)–(e) are the temperature distributions at x = 38 mm (4 mm away from the front boundary), x = 42 mm (the front boundary), x = 52 mm (the middle of the treated region), x = 62 mm (the back boundary) and x = 66 mm (4 mm away from the back boundary) planes, respectively. (f)–(h) are the absorbed power depositions at x = 42 mm, 52 mm and 62 mm planes, respectively, after optimization (units in figure, temperature: °C and power density: W m^3).

shows temperature and power depositions for a blood perfusion of 2.0 kg m^3s^1and a blood flow velocity about 1280 mm s^1in le- vel 1 blood vessel.Fig. 7shows temperature and power depositions

for a blood perfusion of 0.123 kg m^3s^1and a blood flow velocity of about 80 mm s^1in level 1 blood vessel, which in size and blood flow velocity of vessel is identical to dog data from Chen and Fig. 6. Temperatures and absorbed power depositions for a blood perfusion rate of 2.0 kg m^3s^1after optimization with fine spatial power deposition. The ideal temperature is set to be 43 °C, and the blood flow velocity is about 1280 mm s^1in level 1 vessel. (a)–(e) are the temperature distributions at x = 38 mm (4 mm away from the front boundary), x = 42 mm (the front boundary), x = 52 mm (middle of the treated region), x = 62 mm (the back boundary) and x = 66 mm (4 mm away from the back boundary) planes respectively after optimization. (f)–(h) are the absorbed power depositions at x = 42 mm, 52 mm and 62 mm planes, respectively, after optimization (units in figure, temperature: °C and power density: W m^3).

Holmes[6]. The optimized power deposition pattern is similar to but with a higher or lower value than that shown inFig. 5, and the temperature distribution displays that uniform temperature

close to the ideal value can be obtained in the treated region except for some cold spots which are produced by the arteries same as the case of 0.5 kg m^3s^1.

Fig. 7. Temperatures and absorbed power depositions for a blood perfusion rate of 0.123 kg m^3s^1after optimization with fine spatial power deposition. The ideal temperature is set to be 43 °C, and the blood flow velocity is about 80 mm s^1in level 1 vessel. (a)–(e) are the temperature distributions at x = 38 mm (4 mm away from the front boundary), x = 42 mm (the front boundary), x = 52 mm (middle of the treated region), x = 62 mm (the back boundary) and x = 66 mm (4 mm away from the back boundary) planes, respectively, after optimization. (f)–(h) are the absorbed power depositions at x = 42 mm, 52 mm and 62 mm planes, respectively, after optimization (units in figure, temperature: °C and power density: W m^3).

3.3. Optimization without thermally significant blood vessels in the tissues

Pennes’ bio-heat transfer equation is used to investigate tem- perature and absorbed power deposition in the treated region for the condition without thermally significant blood vessels. A uni-

form blood perfusion rate of 0.5 kg m^3s^1 in the entire tissue was studied using the optimization.Fig. 8(f)–(h) shows the opti- mized power deposition on the front boundary, the middle and the back boundary planes, respectively. Most of the power is deposited on the corners and edges of the treated region to com- pensate for thermal energy loss through conduction due to the

Fig. 8. Temperatures and absorbed power depositions for a blood perfusion rate of 0.5 kg m^3s^1with no vasculature present and after optimization with fine spatial power deposition. The ideal temperature is set to be 43 °C. (a)–(e) are the temperature distributions at x = 38 mm (4 mm away from the front boundary), x = 42 mm (the front boundary), x = 52 mm (middle of the treated region), x = 62 mm (the back boundary) and x = 66 mm (4 mm away from the back boundary) planes respectively after optimization. (f)–(h) are the absorbed power depositions at x = 42 mm, 52 mm and 62 mm planes, respectively, after optimization (units in figure, temperature: °C and power density: W m^3).

strong conductive effects near corners and edges. The deposited power pattern of Fig. 8(h) is identical to that shown inFig. 8(f) and (g) (the middle plane of the treated region) displays that there

is less power deposited on corners and center area as compared to Fig. 8(f) (the front boundary plane). It indicates that the thermal diffusion rate is much smaller in the middle region.Fig. 8(b)–(d)

Fig. 9. Temperatures and absorbed power depositions for a blood perfusion rate of 0.5 kg m^3s^1after optimization with lumped power deposition. The ideal temperature is set to be 43 °C, and the blood flow velocity is about 320 mm s^1in level 1 vessel. (a)–(e) are the temperature distributions at x = 38 mm (4 mm away from the front boundary), x = 42 mm (the front boundary), x = 52 mm (the middle of the treated region), x = 62 mm (the back boundary) and x = 66 mm (4 mm away from the back boundary) planes, respectively. (f)–(h) are the absorbed power depositions at x = 42 mm, 52 mm and 62 mm planes, respectively after optimization (units in figure, temperature: °C and power density: W m^3).

shows that a very uniform therapeutic temperature distribution in the treated region can be achieved as there are no thermally signif- icant blood vessels present.

3.4. Optimization with lumped power deposition in the treated region

To investigate the effect of lumped power deposition on the resulting temperature distribution, a uniform power deposition for the entire treated region was used. This represents the limita- tion of the heating system to tune its power spatially fine enough to meet the treatment requirement.Fig. 9(a)–(e) is the optimized temperature distributions on the planes of 4 mm away from the front boundary, the front boundary, the middle, the back boundary, and 4 mm away from the back boundary of the treated region, respectively, andFig. 9(f)–(h) are absorbed power depositions on the planes of the front boundary, the middle and the back bound- ary, respectively, for a blood perfusion of 0.5 kg m^3s^1 and a blood flow velocity of 320 mm s^1in the level 1 vessel. The initial guess of the uniform power deposition in the treated region was 10⁵W m^3, and the optimized absorbed power deposition was ob- tained as the difference (DCn) between two successive cost func- tion values was smaller than 10^4. With this optimization of lumped power deposition, the temperatures shown inFig. 9(b)–

(d) indicate that the temperatures in the treated region are highly inhomogeneous, with a temperature about 3 °C below the desired therapeutic temperature in the places near the boundary planes.

These temperature distributions display that blood flow of vessels results in a significantly lower temperature strips along the vessels in the treated region, particularly a large vessel located at the boundary of the treated region.

3.5. Comparison of total absorbed power in the treated region

Fig. 10shows the total absorbed power in the treated region for different conditions of optimization. It illustrates that the power deposited for optimization with blood vessel (BV) present is much higher than that without blood vessels. The total power for optimi- zation based on local temperature response to tune its power deposition to compensate the convective effect of blood vessels was greater than that for optimization with lumped power deposi- tion. For a blood perfusion rate of 2.0 kg m^3s^1in the tissue, the total absorbed power was about 3.4 times of that for the case of 0.5 kg m^3s^1.

4. Discussion

With the purpose of investigating the optimized temperature distribution for hyperthermia treatment with thermally significant blood vessels in the treated region, we employed a vascular model with blood vessels embedded in the tissues and using optimization to tune power deposition to meet the treatment needs. In clinical practice, the location and geometry of vessels need to be obtained first either by magnetic resonance imaging (MRI) or ultrasound, and then optimization can be applied. Obtaining vessel informa- tion is a complicated and time-consuming task. Although the pres- ent study employed a simple vessel network model, it helped us develop the basic approach and illustrate the effect of blood ves- sels, which PBHTE could not describe.

Two optimization schemes were used in this study to represent the optimization for two extreme heating systems used in hyper- thermia treatment: one with fine spatial power deposition, able to tune its local power according to each local temperature re- sponse, and the other one with lumped uniform power deposited in the entire treated region and able to be tuned its power level according to the cost function based on the entire temperature re- sponse. The former scheme was able to deposit enough power to heat the blood vessels and their adjacent areas to effectively limit the blood vessel cooling effect, while the latter one was not able to deliver enough power to cover the effect of blood vessels and resulting large cold strips along the vessels. This shows interesting heating situations in hyperthermia treatment: (1) blood vessels can carry convective energy and circulate to produce cold spots/

strips in the desired treated region and hot spots/strips in the nor- mal tissues and (2) a heating system with high resolution power deposition ability is required to effectively reduce the cooling ef- fects of blood vessels during hyperthermia.

Existing thermally significant blood vessels in the treated region can generate cold strips or spots during optimization of hyperther- mia treatment. It is of vital importance to identify all thermally sig- nificant vessels because of drastic change of temperature along the blood vessels as shown inFig. 9(b)–(d) when a lumped power opti- mization was used, or some cold spots with smaller cold strips (Fig. 5(d)) when using optimization based on local temperature to tune its power at each location. The complexity of existing ther- mally significant blood vessels plays a very important role in suc- cessful hyperthermia treatments. This complexity is related to mass flow rates of vessels, inlet temperatures of vessels, and direc- tions of vessels. In addition, the results also indicate that a heating

2.323

0.954 1.215

7.622

0 1 2 3 4 5 6 7 8 9

BV and optimization(w=0.5)

No BV and optimization(w=0.5)

BV with lumped power optimization(w=0.5)

BV and optimization(w=2.0)

Power(watts)

Total absorbed power in the treated region

Fig. 10. Total absorbed power at the treated region for optimization with four different conditions. BV means vasculature presents in the treated region; w means the blood perfusion rate; optimization means optimization with fine spatial power deposition; and lumped optimization means optimization with lumped power deposition in the desired treated region on minimum cost function (units in figure, w: kg m^3s^1).

system with fine spatial power deposition ability is able to tune its power to effectively reduce the convective effect of thermally sig- nificant vessels to some local cold spots and/or small cold strips along the blood vessels, while a heating system with coarse spatial power deposition ability is not able to effectively reduce the con- vective effect and hence leave much larger cold strips along the blood vessels.

Comparing the temperatures and power depositions among blood perfusion rates of 2.0, 0.5 and 0.123 kg m^3s^1with a fine spatial power deposition optimization, their patterns were all sim- ilar, except the power level was higher and cold spots/strips were larger for a greater perfusion rate. And they all could result in homogeneous therapeutic temperature distributions in the treated regions. This indicates that a powerful heating system with high resolution of power deposition is important to overcome or effec- tively limit the cooling effect of blood vessels on the temperature distribution when a temperature imaging system is employed to map the temperature distribution and an optimization algorithm is used to determine the local power required. To further reduce the effect of large blood vessels on the temperature distribution in the desired treated region, preheating the upstream region for thermally significant vessels is a feasible way to raising the blood temperature before it runs into the desired treated region. A pow- erful heating system with fine spatial resolution and a good image/

temperature mapping system are necessary in order to preheat the blood vessels without overheating the surrounding normal tissues.

The amount of total absorbed power in the treated region shown inFig. 10indicates that the total power for the case with blood vessels was much higher than that without blood vessels present. The power depositions shown in Figs. 5(f)–(h) and 8(f)–(h) display that the powers deposited in the blood vessels and surrounding areas were the most significantly different. The total power for the blood vessels was even higher than that for per- fusion and boundary conduction.

5. Conclusion

To produce a uniform therapeutic temperature distribution in the desired treated region while minimizing the overheating of the surrounding normal tissue is desirable for hyperthermia treat- ment. To reach this goal requires a powerful heating system that is able to deposit power in the treated region to raise the tempera- ture of the entire treated region up to the desired value and over- come the energy taken away by blood perfusion, boundary conduction, and blood flow from the vasculature. This study em- ployed a 3D tissue model embedded with a countercurrent blood vessel network to simulate the treated tumor region and used opti- mization to determine the absorbed power deposition for different conditions. The temperature results after optimization show the cold spots and/or cold strips along the blood vessels. These temper- atures display the tremendous effects of blood vessels on the resulting heating temperature and the limitation of heating sys- tems. A powerful heating system with fine spatial resolution for power deposition has a better ability to deliver suitable power to locally overcome the convective effect caused by the thermally sig- nificant vessels. Although a heating system with fine power depo- sition is a very important factor during treatment, the complexity of existing thermally significant blood vessels plays a crucial role in successful hyperthermia treatments. This complexity is related to mass flow rates, inlet temperatures, and directions of vessels.

Therefore, prior to hyperthermia treatments, thermally significant blood vessels should be identified and handled carefully in order to

reduce their cooling effects on the desired treated region, particu- larly to those vessels flowing into the treated region.

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