Measurement of local tissue perfusion through a minimally invasive heating bead

Yong-Gang Lv; Jing Liu

doi:10.1007/s00231-007-0233-z

Heat and Mass Transfer

December 2007, 44:201 | Cite as

Measurement of local tissue perfusion through a minimally invasive heating bead

Authors
Authors and affiliations

Yong-Gang LvEmail author
Jing Liu

Original

First Online: 16 February 2007

Received: 15 July 2006

Accepted: 19 January 2007

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3 Citations

Abstract

A minimally invasive approach was proposed to measure local blood perfusion rate in living tissues, based on the well-known Pennes bioheat equation. The measuring probe consists of a heater covered with conductive epoxy and temperature sensor deposited on the probe–tissue interface. By monitoring the probe–tissue interface’s temperature response before and after employing the constant heat flux, the tissue blood perfusion rate can be obtained. A theoretical model was developed to describe the measurement system. In vivo experiments were performed on the rabbit’s thighs to validate this method. At last, uncertainties implied in the temperature measurement and voltage across the heater was evaluated. The results point out the way to improve the accuracy of the present method and its appropriate application occasion.

Keywords

Local blood perfusion Penne’s equation Constant heat flux Minimally invasive In vivo measurement Bioheat transfer

List of symbols

C: Specific heat of tissue (J/kg K)
C_b: Specific heat of blood (J/kg K)
K: Thermal conductivity of tissue (W/m K)
Q_m: Metabolic rate of tissue (W/m³)
q₀: Heat flux passing from the bead to the tissue (W/m²)
R: Resistance of the heating wires embedded in the bead (Ω)
R₀: Bead radius (m)
T: Tissue temperature (°C)
T₀: Initial tissue temperature (°C)
T_a: Artery temperature (°C)
t: Time (ms)
U: Voltage across the heating wires (V)
ΔU: Uncertainty of the voltage (V)
W_b: Blood perfusion rate (kg/m³ s)
ΔW_b: Uncertainty of the predicted blood perfusion (kg/m³ s)
r: Coordinate (m)

Greek letters

α: Thermal diffusivity of tissue (m²/s)
θ: Temperature elevation due to external heating (°C)
${\theta_{{R_{0}}}}$: Temperature elevation at the bead-tissue interface due to external heating (°C)
${\Delta \theta_{{R_{0}}}}$: Uncertainty of the temperature (°C)
ρ: Density of tissue (kg/m³)
ρ_b: Density of blood (kg/m³)

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Notes

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China.

Appendix: Temperature response of living tissue subjected to constant surface flux heating

To analyze the transient temperature response of living tissues subjected to the constant surface flux heating, the theoretical model was established as follows,

$$ \frac{1}{r}\frac{{\partial ^{2} (rT)}}{{\partial r^{2}}} + W_{b} C_{b} (T_{a} - T)/K + \frac{{Q_{m} (t)}}{K} = \frac{1}{\alpha}\frac{{\partial T}}{{\partial t}} $$

(A1)

At the distant site far from the bead, the constant heating almost has no influence on the temperature there. In order to solve the question, at r = R′ = 0.02 m is assumed as the distant site far from the bead. The equilibrium temperature under a constant voltage is the initial temperature at the following heating process. Then boundary and initial conditions to Eq. (A1) can then be expressed as:

$$ - K \cdot \frac{{\partial T}}{{\partial r}}\bigg| {_{{r = R_{0}}}} = q_{0}, \quad r = R_{0} $$

(A2a)

$$ \frac{{\partial T}}{{\partial r}} = 0, \quad r = {R}^{\prime} $$

(A2b)

$$ T(r,0) = T_{0}, \quad t = 0 $$

(A2c)

Assuming

$$ T{\left({r,t} \right)} = T_{0} + \psi {\left({r,t} \right)}\exp {\left({- \frac{{W_{b} C_{b}}}{{\rho C}}t} \right)} $$

(A3)

Then Eqs. (A1–A2a, b, c) are transformed into

$$ \frac{{\partial ^{2} \psi}}{{\partial r^{2}}} + \frac{2}{r}\frac{{\partial \psi}}{{\partial r}} = \frac{1}{\alpha}\frac{{\partial \psi}}{{\partial t}} $$

(A4)

$$ - K \cdot \frac{{\partial \psi}}{{\partial r}}\bigg| {_{{r = R_{0}}}} = q_{0} \exp {\left({\frac{{W_{b} C_{b}}}{{\rho C}}t} \right)}, \quad r = R_{0} $$

(A5a)

$$ \frac{{\partial \psi}}{{\partial r}} = 0,\quad r = {R}^{\prime} $$

(A5b)

$$\psi (r,0) = 0,\quad t = 0 $$

(A5c)

If the Green’s function for the above Eq. (A4) is obtained, the transient tissue temperature can easily be constructed [20]. Through introducing an auxiliary problem corresponding to Eq. (A4), the Green’s function can finally be obtained as (detailed derivation is omitted here)

$$ \begin{aligned} G{\left({\left. {r,t} \right|{r}^{\prime},\tau} \right)} &= 3/{{\left({b^{3} - a^{3}} \right)}} + \frac{1}{{r \cdot {r}^{\prime}}}{\sum\limits_{m = 1}^{\infty} {e^{{- \alpha \beta ^{2}_{m} {\left({t - \tau} \right)}}} {\big\{{\beta_{m} \cos {\left[ {\beta_{m} {\left({r - R_{0}} \right)}} \right]} + {\sin {\left[ {\beta_{m} {\left({r - R_{0}} \right)}} \right]}} /{R_{0}}} \big\}}}} \\ & \quad \times \frac{2}{{{\left({\beta ^{2}_{m} + 1/{R^{2}_{0}}} \right)}{\left[ {{\left({{R}^{\prime} - R_{0}} \right)} - 1/ {{\left[ {{R}^{\prime}{\left({\beta ^{2}_{m} + 1 / {{R}^{\prime 2}}} \right)}} \right]}}} \right]} + 1/ {R_{0}}}} \\ & \quad \times {\big\{{\beta_{m} \cos {\left[ {\beta_{m} {\left({{r}^{\prime} - R_{0}} \right)}} \right]} + {\sin {\left[ {\beta_{m} {\left({{r}^{\prime} - R_{0}} \right)}} \right]}} /{R_{0}}} \big\}} \\ \end{aligned} $$

(A6)

where β_m’s are the positive roots of

$$ \tan \beta_{m} {\left({{R}^{\prime} - R_{0}} \right)} = \frac{{\beta_{m} {\left({1 / {R_{0}} + 1 /{{R}^{\prime}}} \right)}}}{{\beta ^{2}_{m} - 1 /{{R}^{\prime}R_{0}}}} $$

(A7)

Then the tissue temperature field could be constructed with Eq. (A3)

$$ \begin{aligned} T{\left({r,t} \right)} &= T_{0} + \frac{{3\alpha R^{2}_{0} q_{0} {\left({1 - e^{{- {\left({{W_{b} C_{b}} /{\rho C}} \right)}t}}} \right)}}}{{K{\left({{R}^{\prime 3}- R^{3}_{0}} \right)}{\left({{W_{b} C_{b}}/ {\rho C}} \right)}}} + \frac{{\alpha R_{0}}}{{Kr}}{\sum\limits_{m = 1}^{\infty} {\frac{{\beta_{m}}}{{{W_{b} C_{b}} / {\rho C + \alpha \beta ^{2}_{m}}}}}}{\left({1 - e^{{- {\left({{W_{b} C_{b}} / {\rho C + \alpha \beta ^{2}_{m}}} \right)}t}}} \right)} \\ & \quad \times {\big\{{\beta_{m} \cos {\left[ {\beta_{m} {\left({r - R_{0}} \right)}} \right]} + {\sin {\left[ {\beta_{m} {\left({r - R_{0}} \right)}} \right]}} /{R_{0}}} \big\}} \\ &\quad \times \frac{2} {\left(\beta^2_m+1/R^2_0\right)\left[(R^{\prime}-R_0) -1/\left[R^{\prime}\left(\beta^2_m+1/R^{\prime 2}\right)\right]\right]+1/R_0} \end{aligned} $$

(A8)

The final solution for bead-tissue interface temperature T(R ₀, t ) is therefore in the form of

$$ \begin{aligned} T{\left({R_{0}, t} \right)} &= T_{0} + \frac{{3\alpha R^{2}_{0} q_{0} {\left({1 - e^{{- {W_{b} C_{b} t} /{\rho C}}}} \right)}}}{{K{\left({{R}^{\prime 3} - R^{3}_{0}} \right)}{\left({{W_{b} C_{b}} /{\rho C}} \right)}}} + \frac{\alpha}{K}{\sum\limits_{m = 1}^{\infty} {\frac{{2\beta ^{2}_{m}}}{{{W_{b} C_{b}}/ {\rho C + \alpha \beta ^{2}_{m}}}}}}{\left({1 - e^{{- {\left({{W_{b} C_{b}}/ {\rho C + \alpha \beta ^{2}_{m}}} \right)}t}}} \right)} \\ & \quad \times \frac{{\hbox{1}}}{{{\left({\beta ^{2}_{m} + 1 /{R^{2}_{0}}} \right)}{\left[ {{\left({{R}^{\prime} - R_{0}} \right)} - 1/ {{\left[ {{R}^{\prime}{\left({\beta ^{2}_{m} + 1/ {{R}^{\prime 2}}} \right)}} \right]}}} \right]} + 1/{R_{0}}}} \\ \end{aligned} $$

(A9)

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Authors and Affiliations

Yong-Gang Lv
- 1
Email author
Jing Liu
- 2

1.Department of Biomedical EngineeringThe City College of the City University of New YorkNew YorkUSA
2.Cryogenic Laboratory, Technical Institute of Physics and ChemistryChinese Academy of SciencesBeijingPeople’s Republic of China

Measurement of local tissue perfusion through a minimally invasive heating bead

Abstract

Keywords

List of symbols

Greek letters

Notes

Acknowledgments

References

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