Background:

Ultrasound imaging became a diagnostic tool in the 1970s with the introduction of the gray-scale ultasonography. In this scale, the reflected echoes from tissues are nonlinearly compressed so that specularly reflected and diffusely scattered echoes can be captured to construct the image. Prior to that, only specularly reflected echoes were used as the primary building component of the ultrasound image. Recent research in the fundamental knowledge of ultrasound tissue interaction has lead to a better understanding of its mechanism and side effects. Studies were directed to the understanding of ultrasound scattering off tissues. Experimental methods for measuring scattering of ultrasound waves in tissue have been developed to generate models that characterize tissue-ultrasound interaction. These studies on tissue-ultrasound interaction revealed the nature of the undesired side effects of ultrasound therapy and imaging.

Problem Statement:

Ultrasound waves consist of cycles of compression and expansion that exert a positive and negative pressure. These cycles of pressure work on the molecules by pulling them together and pushing them away. During the negative cycle, given sufficient ultrasound intensity, cavities are introduced into the subjected body. While pure liquids have very high tensile bond strength that prohibit ultrasound waves from creating cavities, body tissue contains trapped gas in small solid particles that greatly reduce the tensile bond strength and enable ultrasound waves to create cavities.

The small gas pockets are constantly being worked out by the ultrasound waves during positive and negative pressure cycles until they reach a critical size. The growth of the cavity mainly depends on the intensity of the ultrasound waves. A very high intensity ultrasound can cause the cavity to implode during a single pressure cycle while a low ultrasound intensity require several pressure cycles to reach the critical size.

Figure 1: Bubble growth and implosion process

This figure illustrates the growth process of a cavity under a low intensity ultrasound where the size grows and shrinks until it implodes. While the cavity goes through cycles of expansion and compression, the net effect is growth in size due to the fact that gas diffusion in the expansion cycle is greater than that of the compression cycle.

This process continues until the cavity reaches a critical size where the cavity starts to grow rapidly due to high absorption capacity. The critical size mainly depends on the intensity of the ultrasound. Shortly after reaching the critical size, the cavity implodes.

Figure 2: Imploding Cavity

The implosion of the cavities sets up the environment for unusual chemical reactions. This is mainly due to the extreme temperature created by the implosion that can reach up to 5,500 degrees. During the collapse, pressure inside the cavities can reach levels equivalent to thousands of atmospheric pressure. During the implosion stage, the cavity bursts and expels liquid traveling at roughly 400 km/h. This implosion is so extreme that it can send small particles with such a high speed and cause them to melt at the point of impact and reformat.

Figure 3: Zinc powder after a cavity implosion

These extreme conditions are responsible to several biological effects. One well known effect is the cell suspension changes due to the implosion of cavities. Cavitations occur in the fluid suspending cells. Studies clearly shown the direct relationship between cavitations caused by ultrasound and distortion of cell’s membrane suspended by the fluid in which the implosion occurred. It is unclear, however, if the internal structure or function of the cell that survived a nearby cavity implosion is affected.

Project Objective:

Through this project, my work will be directed towards studying the effect of ultrasound waves on healthy cells and developing model for the effect of the ultrasound on biological tissue. Mainly, a heat model will be developed to describe the process of ultrasound heating of the biological tissue during imaging and treatment.

Model:

Energy transported via ultrasound waves can be characterized by several terms depending on purpose. Generally, energy of ultrasound is described by pressure, temperature, density and particle displacement. Developing a model for ultrasound waves requires a strong understanding of the wave’s nature. Specifically, for a bio-heat model of ultrasound waves, a special attention should be directed to the understanding of the attenuation, scattering and absorption of ultrasound waves traveling through biological tissue.

Starting from the basics of physics, a wave propagating through a medium loses its energy as a function of distance via reflection, absorption and scattering. This lost energy is converted to heat that causes the unwanted effects of ultrasound. Consider a sound wave traveling in the x direction, the pressure of the wave is expressed as:

Where Pxo is the pressure at time zero and B is the attenuation coefficient of the medium. The attenuation coefficient varies greatly depending on the medium. For an example, attenuation coefficient for bone =1.3 neper/cm and 0.11 for kidney. This general pressure equation can be used to describe the scattering of the ultrasound upon striking tissue.

This figure illustrates the basic scattering of the ultrasound wave. With an initial direction of j and initial pressure of Pj, a scattered wave Ps is generated. For the scattered wave in the Z direction, the following expression can be developed.

Where A is the scattering amplitude that depends on the object and O is the direction of the observation point.

Bio-Heat Model7:

The wave scattering equations enable us to model heating of tissue during ultrasound imaging or treatment. A thermal model for ultrasound treatment and imaging is important is it enables us to picture the temperature distribution. The model simply provides a mathematical equation to calculate the tissue temperature during treatment or imaging. To develop such a model, the following properties should be closely studied:

- Thermo-physical properties: thermal conductivity and heat capacity of tissues.
- Geometry of tissue.
- Heat production during absorption of ultrasound energy.

Also, a decision must be made prior to develop such a model regarding whether to include the boundary or not. If boundary is included, the heat transfer at the tissue surface must be considered. If boundary is not included, the heat flow between tissue particles is only considered. For the purpose of this paper, the boundary conditions will be included. Another consideration is the mechanisms of heat flow to be modeled. Mainly, almost all heat is transferred through conduction and convection. The heat transferred by conduction inside the tissue is governed by Fourier laws that links the amount of heat transferred to cross section and differential temperature.

Heat conduction inside tissue:

This is the basic law of heat transfer as described by Fourier equation. In it, amount of heat transferred is directly proportional to differential heat over time and inversely proportional to medium length. The heat flow can be easily derived from this equation by simply dividing by unit area * unit time.

Where: f: heat flux

k: coefficient of heat conductivity

T: magnitude of maximum change in temperature

Applying this for tissue, small length, the equation is reduced to:

Conducted heat in tissue is a major component in building the bio-heat equation.

Where P is the density of tissue, C is the specific heat, qs absorbed ultrasound energy, qp rate of heat flow by blood perfusion and qm is the metabolic activity.

This form represents the general form of the bio-heat equation that describes the heating of tissue in ultrasound treatment. A complete description of the effect requires solving for absorbed ultrasound energy, blood perfusion and metabolic activity.

Blood Perfusion: Represents the net thermal energy transported by blood flow to tissue being studied. Blood transports thermal energy in and out of the tissue as blood flows. For the bio-heat equation, I am only interested in the net transported energy.

Several factors affect thermal energy transported by blood. Mainly, transporter and tissue characteristics which include the following:

- Density of blood
- Specific heat of blood
- Temperature of blood
- Blood perfusion factor (blood volume per tissue mass unit per time unit)

So, we can conclude that the net energy transported by blood is:

Where:

Substituting blood perfusion equation into the bio heat equation gives:

Metabolic Activity: For the purpose of this paper, the metabolic activity will be neglected. The bio heat equation reduces to:

Absorbed Ultrasound Energy: Due to the enormous difficulty of modeling the absorbed ultrasound energy, the term is left unexpanded in the bio heat equation. The absorption level of ultrasound energy by a body varies greatly depending on the homogenousity, density and many other factors.

The final version of the bio heat equation describing the heating of the tissue is:

Reference:

- Kenneth S. Suslick, the Chemical Effects of Ultasound, Scientific AmeriCAn, 1989.
- K. Hynynen, Nonlinear Absoprtion during Scanned Focused Ultrasound Hyperthermia, Ultrasound Symposium, 1985.
- Rafael V. Davalos, Boris Rubinsky, Lluis M. Mir, Theoretical Analysis of the Thermal Effects during in Vivo Tissue Electroportion
- http://en.wikipedia.org/wiki/Medical_ethics
- http://www.sdms.org/
- http://208.42.83.77/ethics/long.htm
- Stefan Andersson, How to model heating of tissue during laser treatment irradiation.
- Frank P. Incropera, David P. Dewitt, Introduction to Heat Transfer, Interactive Heat Transfer 2.0, John Wiley & Sons, Inc, 2002.