Chapter 2 Literature Review
Previous work in the flelds of hyperthermia, bioheat transfer, thermal damage modeling and quantiflcation, optimal control and hyperthermia treatment planning is summarized in this section. The section gives an overview of relevant research done in the major points of interest of this thesis.
Hyperthermia and Thermal Ablation Therapies
Hyperthermia is generally regarded as elevating temperatures of a cancerous region of tissue to cytotoxic temperature in the range of 41 to 44 oC for a duration of 30 to 45 minutes. It is often used in conjunction with other treatment modalities such as radiotherapy and chemotherapy. The treatments aim to maintain surrounding healthy tissue at physiologi-cally normal temperatures and target only regions where malignancies are found.
Thanks to recent technological gains in flelds ranging from medical imaging to energy delivery mechanisms, thermal ablation strategies have become an attractive alternative to standard surgical therapies in the treatment of malignant cancerous tumors (Ahmed and Goldberg (2002)). These treatment modalities have the added beneflt of being only minimally invasive or completely non-invasive. There are a wide variety of energy sources currently used, but some of the most common are high-intensity focused ultrasound (HIFU), radio frequency(RF), laser and microwave heating. The heating simulations presented in this work are based on HIFU treatments because they are non-invasive and can be modeled as volumetric or boundary heating in the simulation.
HIFU is a transcutaneous, minimally invasive technique that focuses ultrasound energy with a high peak intensity (Ahmed and Goldberg (2002)). This technology is capable of generating a power intensity of 5,500 kW=m2 and temperatures as high as 80oC in rat liver with the use of a 4 MHz transducer. Ultrasound heating is limited by interference with regional bone and air and the fact that only small volumes of tissue that can be thermally ablated with this technology during one treatment (Ahmed and Goldberg (2002)). This information is given to provide a brief background of ultrasound heating. The focus of the paper is on the optimization of the protocol not on the particular modality itself.
The Bioheat Transfer and Thermal Damage Models
The foundation of this research are models which predict both the transient temperature distribution within a section of tissue or biological material, and the corresponding extent of thermal damage. Since the particular modality is not the focus of this work, models which describe how a particular modality such as HIFU heats tissue are not included. Instead, energy enters the control volume as volumetric or boundary heating. The Bioheat transfer and thermal damage models are now presented.
The Bioheat equation
The one component common to nearly any hyperthermia simulation is the Pennes Bioheat Transfer Equation, (BHTE), which is shown below in generalized form (Pennes (1948)). This equation was developed as a means of quantifying the efiect of blood within the mi-crovasculature of the tissue on the local temperatures. The presence of blood in the tissue is considered a convective term with the blood temperature as the heat sink temperature. Pennes asserts that accurate modelling of the temperatures within the body, particularly the human forearm, relies on accurate knowledge of the blood °ow or perfusion, !(kgbl=m3tis s) in that area. Nearly 60 years later, it remains a di–cult parameter to quantify. The BHTE is shown in detail in chapter 3.
Alternate approaches do exists for modeling heat transfer in the body. For example if a large scale number of vessels are included as a velocity vector, ~u, the equation can be written as shown in Eqn. 2.2. A solution to this model would require detailed knowledge of the vasculature in the area being modeled and would be much more computationally intensive than the BHTE.
The focus of this work is on the optimization itself, not on which variation of the BHTE is used. Loulou and Scott (2002) and Alifanov (1994) have shown that the conjugate gradient method with the adjoint problem can be applied to a wide variety of difierent governing equations. For this reason the BHTE shown in Eqn. 2.1 will be used in this thesis to model the transient temperature fleld.
Thermal damage models
Two difierent thermal damage models are used in this study which both quantify the ir-reversible rate process of thermal injury. The models require a priori knowledge of the temperature distribution in the area of interest. Since the full transient temperature of the area of interest is calculated with the BHTE, they are a natural flt. The reasoning behind using this type of model is that the complexities of thermal denaturation can be somewhat overlooked if the temperature history is taken as a summary of the more complicated story going on at the molecular level (Diller (1992)).
The Henriques damage model
The flrst model of thermal damage is commonly known of as the Henriques model (Henriques and Moritz (1947)). Henriques and Mortiz formulated this model by investigating the extent of thermal injury to porcine skin. The damage was caused by applying a constant temperature boundary condition to a portion of porcine skin for a specifled portion of time. The skin was investigated to quantify the extent of thermal damage. A summation of instantaneous damages gives the complete damage equation below.
The equation requires material-speciflc values of the activation energy, E (J/mol) which is an energy barrier molecules must overcome to denature, and the molecular fre-quency factor, A (1/s) (Pearce and Thomsen (1992)). Here, < is the universal gas constant.
The activation energy and frequency factor are usually found via constant temperature ex-periments. First a threshold of permanent damage, › = 1 is selected. Then samples of a material are placed in a constant temperature bath of a non-reactive °uid such as water which is held at a constant temperature. Then precise measurements are taken for the time taken to achieve permanent thermal damage. These ordered pairs of time and temperature are then used in a least-squares flt to estimate E and A (Yang et al. (1991)). The method works well except at the extremes of the constant temperature bath range. Over a relatively small temperature range, denaturation times can vary by many orders of magnitude. For instance it may take hours or even days to burn skin at 39oC but only seconds at 80oC.
Pearce and Thomsen (1992) expanded on the damage model in their study of tissue fusion processes. They state that the damage integral is also equal to the natural log of the ratio of the concentration of undamaged material at the beginning of a treatment to the concentration of undamaged material at the end of a treatment.
If a damage coe–cient of › = 1 is used and the assumption that the material is 100% undamaged at the beginning of a treatment, complete thermal damage occurs when 63.2% of the local material has been permanently damaged.
The thermal dose model
Nearly 40 years after the work of Moritz and Henriques, Sapreto and Dewey (1984) pub-lished a model which was an efiort to draw comparisons between difierent hyperthermia treatments. The comparison works by using a reference temperature of 43oC and relating any time-temperature history to the \iso-efiect » at that reference temperature. A thermal dose of one represents one hour at 43oC. The equation is shown below and explained fur-ther in chapter 3. The thermal dose has been used to asses the extent of treatments and to optimize treatments particularly in the fleld of HIFU (Dorr (1992), Loulou and Scott (2002)).
In efiect this model difiers from the Henriques model in one major way. In their paper the authors comment that a one degree increase in temperature over 43oC required a two-fold decrease in exposure time. Since R is actually a function of the activation energy, E the implication of this statement is that the energy barrier described in the previous model varies with temperature (Dewey (1994)).
To account for this variation, the term R is allowed to change with temperature, with R = 0.5 for temperatures greater than the 43 oC reference temperature and R = 0.25 for temperatures below 43o. While research has shown that the activation energy does depend on temperature (Wright (2003)) the reported values for R which are widely used may not apply to all cell lines and biological materials. Loulou and Scott (2002) propose a linear variation of R between its high and low bounds, however successful operation of the algorithm does not depend on knowledge of the exact values of R, or even how R changes with temperature.
Both models have strong points. Given the uncertainty that already will be present in any simulation due to imprecise thermal property and perfusion values used, the gains by adding a temperature dependent activation energy may be outweighed by the computational simplicity of the Henriques model. The Arrhenius parameters included in both models (E, A and R) are available in the literature for a wide variety of materials cells and proteins (Dewey (1994), Diller (1992)). Additionally this model uses parameters which are calculated based on an observable change, making it much more practical to use in a laboratory setting. On the other hand, the unit of the thermal dose model is \exposure time at 43oC, which can be valuable knowledge in a clinical setting.
It is important to note that these models are simpliflcations of a much more complex phenomenon. Wright and Humphrey (2002) assert that despite the current practice of using damage and dose models used in research a detailed understanding of the biomechanics of tissue denaturation is lacking. The review gives some key background of protein denatu-ration at the molecular level. One point which Wright makes, which is reiterated later in this research is that a very speciflc level of denaturation can achieved via a wide range of time-temperature curves, and mechanical stresses; the latter being a factor which is not treated in this research.
The results of any optimized heating protocol should be validated experimentally. This validation requires a tissue phantom. Madden and Scot (2003) presented an agar based tissue phantom that could be used to simulate physiologically accurate blood perfusion. While this model is experimentally applicable, tissue will undergo denaturation whereas agar will not. Other researchers have however used albumen (egg white) as a tissue phantom because like tissue it is composed of proteins that denature permanently when heated. This heating follows a flrst order rate process reaction and can be modelled via the Henriques model (Pfefer et al. (2000)), (Pearce and Thomsen (1992)). The Arrhenius properties of albumen can be found in the literature (Yang et al. (1991)). Albumen is composed of roughly 80% water and 14% protein known as ovalbumin (Yamamoto et al. (1997)). Ablated albumen is opaque to the naked eye, facilitating measurements of the extent of permeant thermal damage.
Chapter 1 Introduction
1.1 Goals and Objectives
Chapter 2 Literature Review
2.1 Hyperthermia and Thermal Ablation Therapies
2.2 The Bioheat Transfer and Thermal Damage Models
2.3 Experimental Phantom
2.4 The Minimization Algorithm
2.5 Hyperthermia Treatment Planning
2.6 Closing Remarks
Chapter 3 Theoretical Considerations
3.1 The Pennes Bioheat Transfer Model
3.2 Finite Di®erence Formulation
3.3 Finite Di®erence Validation
3.4 Thermal Damage Models
Chapter 4 Control Problem Formulation
4.1 The Direct Problem Formulation
4.2 Objective Function Formulation
4.3 Conjugate Gradient Method
4.4 The Solution Algorithm
Chapter 5 Experimental Methods
5.1 Experiment Goal
5.2 Test Stand Introduction
5.3 Experiment Equipment
5.4 Modeling Considerations
5.5 Test Stand Selection
5.6 Setting the Baseline
Chapter 6 Results and Discussion
6.1 Validation Experiment
6.2 Simulation Results
Chapter 7 Conclusions
7.1 Objective 1: Solving the Optimal Control Problem
7.2 Objective 2: Perform an Experiment to Validate the Algorithm Output
Chapter 8 Recommendations
GET THE COMPLETE PROJECT
Optimal Control of Thermal Damage to Biological Materials