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A Generic Bioheat Transfer Thermal Model for a Perfused Tissue

Devashish Shrivastava and J. Thomas Vaughan

Center for Magnetic Resonance Research, University of Minnesota, Minneapolis, MN 55455 Phone:

626 2001, FAX: (612) 626 2004, Email: dev@cmrr.umn.edu

Abstract

A thermal model was needed to predict temperatures in a perfused tissue, which satisfied the

following three criteria. One, the model satisfied conservation of energy. Two, the heat transfer rate

from blood vessels to tissue was modeled without following a vessel path. Three, the model applied

to any unheated and heated tissue. To meet these criteria, a generic bioheat transfer model (BHTM)

was derived here by conserving thermal energy in a heated, vascularized, finite tissue and by making

a few simplifying assumptions. Two linear, coupled differential equations were obtained with the

following two variables: tissue volume averaged temperature and blood volume averaged

temperature. The generic model was compared to the widely employed, empirical Pennes’ BHTM.

The comparison showed that the Pennes’ perfusion term wCp(1−ε) should be interpreted as a local

vasculature dependent heat transfer coefficient term. Suggestions are presented for further

adaptations of the general BHTM for specific tissues using imaging techniques and numerical

simulations.

Introduction

Several diagnostic and therapeutic protocols require the determination of in vivo temperatures

to plan and optimize the protocols [1–5]. To manage computational costs, temperatures in a

tissue embedded with ‘small’ (< 1 mm in diameter) more frequent blood vessels are determined

using approximate thermal models known as the bioheat transfer models (BHTMs) [6–8].

Temperatures in the ‘large’ (vessel diameter ≥ 1 mm), less frequent blood vessels are

determined using the exact thermal model the convective energy equation (CEE) [9].

Several BHTMs have been presented before. Many of them have been proposed based on

intuition with undefined and empirical parameters and variables [10–14]. Lack of a formal

derivation made it difficult to relate the parameters and variables of these BHTMs to in vivo

temperatures and parameters (e.g., blood flow, blood vasculature geometry, thermal properties

of tissue and blood vessels). This made implementation and interpretation of the BHTMs and

their results controversial and less reliable [15].

Other BHTMs have been derived by conserving thermal energy in a heated, vascularized, finite

tissue. However, to model the heat transfer rate from blood vessels to tissue, several of these

derived BHTMs assumed local thermal equilibrium (i.e., the tissue and blood volume averaged

temperatures were equal to each other) [15–22]. The assumption of the local thermal

equilibrium is, in general, not valid in a heated tissue [23–27]. Rest of the derived BHTMs

used parameters that required following paths of thermally important vessels [18,20,28,29].

Blood vessels smaller than ≤ 500 μm in diameter are difficult to image in vivo [7,30]. Thus,

following paths of thermally important vessels ≤ 500 μm is difficult. Excellent reviews for

several of the above BHTMs were presented in [31,32].

Correspondence to: Devashish Shrivastava.

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J Biomech Eng. Author manuscript; available in PMC 2010 January 1.

Published in final edited form as:

J Biomech Eng. 2009 July ; 131(7): 074506. doi:10.1115/1.3127260.

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Several BHTMs have been derived by conserving thermal energy in a heated, vascularized,

finite tissue and using volume averaging theorem of the theory of porous media.[33–36] Use

of the volume averaging theorem introduced unknown, spatially varying continuous vector

functions to relate the local, point-wise variation in the averaged variables and parameters to

volume averaged variables and parameters. The determination of the unknown vector

formulations requires the complete, continuous vascular tree.

This paper presents a basic and general BHTM formulation that satisfied the following three

criteria. One, the model satisfied conservation of energy. Two, the heat transfer rate from blood

vessels to tissue was modeled without following vessel paths. Three, the model applied to any

unheated and heated tissue.

Mathematical Derivation of a Generic Bioheat Thermal Model

General Derivation

A finite, vascularized, heated tissue was considered. It was assumed that blood stayed in

vasculature and everything surrounding the blood was tissue. Conserving energy for a point in

the tissue and blood resulted in the following convective energy equation (CEE) [9]. Note that

the velocity of tissue uT was zero in equation (1).

(1)

Equation (1) was integrated over the tissue and blood sub-volumes, separately. Both equations

are presented below.

(2)

The following equation (3) was obtained by applying Green’s divergence theorem to the first

terms on the left hand side (LHS) and right hand side (RHS) of equation (2).

(3)

where, i = j = T, Bl and i≠ j.

Assuming a) constant density and specific heat and b) incompressibility for the blood in the

averaging blood volume, the second term on the LHS reduced to zero due to conservation of

mass. (Note that this term was zero for a tissue since uT = 0). Thus, equation (3) was simplified

as follows.

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(4)

where, i = j = T, Bl and i≠ j.

In equation (4), the first term on the RHS represented the energy gained by the ith control

volume from adjacent control volumes. In other words, this term represented the energy gained

by a tissue (or blood) sub-volume from adjacent tissue (or blood) sub-volumes. The second

term on the RHS represented the energy gained from the interaction between the tissue and

blood sub-volumes in the vascularized tissue volume.

Next, normalizing equation (4) by volume Vi the following general form of the new BHTM

(equation (5)) was obtained. This form satisfied the energy equation and was valid for any

unheated and heated tissue. Note that equation (5) represented two equations; one for tissue

and another for blood.

(5)

where, i = j = T, Bl and i≠ j and

(6)

Simplifications

The following simplifications were made to obtain a simplified BHTM.

(7)

(8)

where, i = j = T, Bl and i≠ j, and

(9)

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(10)

where,

The first simplification (equation (7)) was similar to the simplification proposed by other

investigators [18, 19, 29]. This simplification needs to be verified numerically and

experimentally in vascularized tissues. A new, to be determined parameter Ci1 was introduced

in equation (7) to keep the simplification general.

The second simplification (equation (8)) defined the thermal interaction between a tissue and

the embedded vasculature using the tissue and blood volume averaged temperatures and a heat

transfer coefficient. Note that the heat transfer coefficient was different from conventional heat

transfer coefficients since it was defined based on the volume averaged temperatures.

Conventional heat transfer coefficients were defined using a tissue boundary temperature and

a mixed mean blood temperature. To compute these new heat transfer coefficients, Shrivastava

et al. derived Poisson conduction shape factors (PCSFs) for heated tissues embedded with one

and two vessels using the area averaged tissue temperature and vessel boundary temperatures

[37–40]. Poisson conduction shape factors for tissues embedded with more than two vessels

are needed. Additionally, a quantification of Nusselt number is desired, which is defined based

on the volume averaged blood temperature and the vessel wall temperature.

Regarding an appropriate number of thermally important vessels to be considered in an

averaging volume to obtain US values, Xu et al. demonstrated that 1–3 vessels of diameter ≥

50μm were present in every 1 mm2 of a perfused kidney tissue in a pig. [41]. Similar studies

are needed for other organs including human organs.

The third simplification (equation (9)) was justifiable since the convective term was at least

~10 times larger than the conductive term. This could easily be verified using the following

values for the blood: k = 0.5 W/(mK), ρ = 1000 kg/m3, Cp = 4180 J/(kg K), u = 100 D m/s,

vessel diameter D ≥ 10 μm, tissue length scale ~ O(1 mm) [42, 43].

The fourth simplification (equation (10)) was similar to the simplification proposed by equation

(7). This simplification needs to be verified numerically and experimentally in vascularized

tissues. A new, to be determined parameter Ci2 was introduced in equation (10) to keep the

simplification general.

Further, note that P was defined as a blood volume averaged vector mass flux in an averaging

perfused tissue volume. This definition was appropriate for an averaging tissue volume with

≤1.0 thermally significant vessel. Xu et al. reported vessel number density of ≤1.6 per mm2

for vessels with diameter ≥100 μm in a pig kidney cortex. In most of the cortex (~76%), the

vessel number density was ≤1.0. Less than 30% of the vessels were found paired [41]. For

averaging perfused tissue volumes with two or more, arbitrarily flowing thermally important

vessels; a new, more appropriate definition of P may need to be identified. Next, the constant

CBl2 should be evaluated using numerical simulations and experiments such that equation (10)

was always satisfied. The new P should be defined such that it could be spatially and temporally

quantified using imaging methods.

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With the above simplifications implemented the following simplified BHTM was obtained.

Equations (11) and (12) were coupled equations. Equation (11) was valid for the tissue sub-

volume. Equation (12) was valid for the blood sub-volume.

(11)

(12)

Note that in equation (12) the local perfusion P and the blood volume VBl were unknown. These

values should be determined for different tissue types using imaging methods; e.g., magnetic

resonance imaging, Positron emission tomography, ultrasound, etc [44–47]. Alternatively, the

local perfusion P can be estimated by developing geometric vascular network maps and

assuming mean blood flow of ~100(diameter of vessel) m/s [42]. The tissue volume VT can be

estimated as the difference between the imaging voxel volume and blood volume.

Equation (11) and (12) together with equations (7–10) presented a generic BHTM framework.

To develop the BHTM completely for a given organ and application, all the parameters in

equations (11) and (12) should be estimated based on a real vasculature of the organ,

physiological conditions, and expected variation in those physiological conditions for that

application in the organ. This is so since vessel-tissue heat transfer rates and thermally

important vessels are strong functions of the relative distribution of the vessels of different

diameters and blood flow rates in an organ. [27, 37–40] Subject to subject variation may also

need to be quantified. However, this variation may not be significant for a given organ and

application. Shrivastava et al. measured unique, normalized temperature contours in porcine

brains under RF heating conditions relevant for ultra-high field magnetic resonance imaging

applications. [3] Sensitivity study needs to be performed on the relative importance of different

parameters in predicting temperatures. Further, confidence intervals and error estimates in

predicting temperatures need to be determined. Once all the associated parameters are

determined, the use of the general BHTM adapted for a given organ and application will provide

reasonably accurate volume averaged temperature field without solving for the point-wise true

temperature field in a vascularized tissue. The tissue volume VT should be determined on a

case by case basis to establish the relationship between the values of all parameters and the

local vascular geometry and blood perfusion rate.

Next, the new BHTM requires the diameter of the vessels, locations, flow rate in the vessels

for each averaging voxel volume. These pieces of information are needed to compute US and

P values for each averaging volume and can be obtained using current imaging methods.

However, the model does not require the path of the vessel from one averaging volume to

others. The effect of the vessel eccentricity on the vessel-tissue heat transfer rate and Poisson

Conduction Shape Factors was found relatively less compared to the effect of the vessel radius.

[37–40] Following a certain vessel and building an imaging based vascular tree is challenging.

The model presents a way to obtain temperatures by circumventing the need for knowing a

continuous vascular tree.

A continuous vascular tree is required to develop confidence intervals and error estimates

everywhere in an organ. Since building a continuous vascular tree is challenging, discrete

temperature measurements need to be performed to estimate confidence intervals and error

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