Biothermomechanical behavior of skin tissue

Transcription

1 Acta Mech Sin (2008) 24:1 23 DOI /s REVIEW Biothermomechanical behavior of skin tissue F. Xu T. J. Lu K. A. Seffen Received: 8 October 2007 / Accepted: 15 November 2007 / Published online: 10 January 2008 Springer-Verlag 2008 Abstract Advances in laser, microwave and similar technologies have led to recent developments of thermal treatments involving skin tissue. The effectiveness of these treatments is governed by the coupled thermal, mechanical, biological and neural responses of the affected tissue: a favorable interaction results in a procedure with relatively little pain and no lasting side effects. Currently, even though each behavioral facet is to a certain extent established and understood, none exists to date in the interdisciplinary area. A highly interdisciplinary approach is required for studying the biothermomechanical behavior of skin, involving bioheat transfer, biomechanics and physiology. A comprehensive literature review pertinent to the subject is presented in this paper, covering four subject areas: (a) skin structure, (b) skin bioheat transfer and thermal damage, (c) skin biomechanics, and (d) skin biothermomechanics. The major problems, issues, and topics for further studies are also outlined. This review finds that significant advances in each of these aspects have been achieved in recent years. Although focus is placed upon the biothermomechanical behavior of skin tissue, the fundamental concepts and methodologies reviewed in this paper may also be applicable for studying other soft tissues. The project supported by the Overseas Research Studentship (ORS) and Overseas Trust Scholarship of Cambridge University, the National Natural Science Foundation of China ( , ), the National 111 Project of China (B06024), and the National Basic Research Program of China (2006CB601202). F. Xu K. A. Seffen Engineering Department, Cambridge University, Cambridge CB2 1PZ, UK T. J. Lu (B) MOE Key Laboratory of Strength and Vibration, School of Aerospace, Xi an Jiaotong University, Xi an, China tjlu@mail.xjtu.edu.cn Keywords Review Skin tissue Bioheat transfer Biomechanics Biothermomechanics 1 Introduction Human body lives in a thermal environment with spatially heterogeneous temperatures. Even within a single organism, its temperature cannot be uniform internally due to spatial and temporal fluctuations. The non-uniform distribution of temperature then induces heat transfer both inside the organism and through the interface with external surroundings. This, in turn, causes non-uniform thermal stress distribution. Furthermore, in extreme hot or cold conditions, both nonuniform temperature and stresses may lead to pain sensation. Heat transfer is thus a primary component of biological activities. Being the largest single organ of the body, skin plays a variety of important roles including sensory, thermoregulation and host defense, etc. Amongst these, the most important one is thermoregulation: skin functions thermally as a heat generator, absorber, transmitter, radiator, conductor and vaporizer, thus acting as an important barrier for the human body to various outside conditions. However, in extreme environment, skin fails to protect the body when the temperature lies outside the normal physiological range. Furthermore, in medicine, with advances in laser, microwave and similar technologies, various thermal therapeutic methods have been used to cure skin disease/injury, such as the removal of port-wine stains [1 3], pigmented and cutaneous lesions [4 6] and tattoos [7]. The objective is to induce thermal injury precisely within tissue structures located up to several millimeters below the surface but without affecting the surrounding, healthy tissue. For effective treatment, it is essential to study the coupled thermal, mechanical, biological

2 2 F. Xu et al. and neural responses of the affected skin tissue: a favorable interaction results in a procedure with relatively little pain and no lasting side effects. Despite of the widespread use of heating therapies in dermatology, these typically do not draw upon the detailed understanding of skin bio-thermo-mechanics, for none exists to date, even though each behavioral facet has been well established and understood. A detailed understanding of the coupled biological mechanical response of skin tissue under thermal agitation will contribute to the design, characterization and optimization of therapeutic strategies for delivering better treatment. 1.1 What is skin biothermomechanics? Skin biothermomechanics is the study of the bio-thermal mechanical behaviors of skin tissue under thermomechanical loadings, which is highly interdisciplinary, involving bioheat transfer, burn damage, biomechanics and physiology Thermomechanical behavior of skin tissue The major constituent of dry skin is collagen. During heating, there appears thermally induced mechanical stress; when the temperature exceeds a certain threshold ( 43 C), thermal denaturation of collagen occurs, which in macro-scale is termed thermal shrinkage (correspondingly, pain is generated). It has been found that mechanical stressing is functionally as important as the temperature level during thermal shrinkage [8]. Furthermore, during denaturation, not only the structure, but also the hydration of collagen changes. Thermal denaturation of a collagenous tissue can therefore lead to remarkable changes in skin mechanical, thermal, electrical, and optical properties. Consequently, strain (displacement), stress and temperature are highly correlated: in other words, the problem is fully coupled. All of the above specialties necessitate the study on the underlying mechanisms of biothermomechanical behavior of skin. 1.2 Specialities of the problem Thermal behavior of skin tissue The transfer of heat in skin is mainly a heat conduction process coupled to complicated physiological processes, including blood circulation, sweating, metabolic heat generation and, sometimes, heat dissipation via hair or fur above skin surface. The thermal properties of skin vary across different layers (epidermis, dermis and subcutaneous tissue); even within the same layer, there exists large nonhomogeneity and anisotropy due to the presence of blood vessels. Both the physiological processes mentioned above and the thermal properties of skin are influenced by a variety of factors such as temperature, damage, pressure, and age, etc. To complicate matters further, skin is an active, selfregulating system: heat transfer through the skin dramatically affects the state of skin, which can lead to the redistribution of blood flow within the cutaneous vascular network and, counteractively, influence heat transfer in skin tissue Mechanical behavior of skin tissue The mechanical behavior of skin tissue is found to be heterogeneous, anisotropic, non-linear, and viscoelastic in vivo because of its non-homogenous structure and composition. It is affected by many factors such as age, gender, site, and hydration, etc. Furthermore, the classical electrometric constitutive models are not suitable to describe the complicated mechanical behavior of skin tissue. 1.3 Importance of skin thermomechanics and thermal pain Medical applications With advances in laser, microwave, radio-frequency, and similar technologies, a variety of thermal methods have been developed and applied to the treatment of disease/injury involving skin tissue. These thermal treatment methods normally involve either a raising or lowering of temperature in targeted skin area to kill or thermally denaturize necrotic cells, wherein precise monitoring of the spatial and temporal distribution of temperature and stress in skin tissue is required. Meanwhile, to maintain the surrounding healthy skin at a safe temperature level, selective cooling techniques on skin surface are adopted during these treatments. Notwithstanding these important and widely used medical applications, an understanding of the responsible thermal mechanical mechanisms remains limited. Further refinement of current medical treatments and innovation in thermal treatments, requires a thorough understanding of this interdisciplinary area. Therefore, comprehension of the phenomena of heat transfer and related biothermomechanics in soft tissues is of great importance and can contribute to a variety of medical applications: (1) design and characterization of strategies for delivering thermal therapies; (2) optimization of a thermal treatment by maximizing the therapeutic effect while minimizing unwanted side effects; (3) comparison of various treatment parameters by modeling instead of extensive parametric studies; (4) new treatment strategies proposing predictions and evaluation of their outcome by developing models and simulation tools.

3 Biothermomechanical behavior of skin tissue Contribution to pain study and pain relief A noxious thermal stimulus (heat or cold) applied to human skin is one of the three main causes of pain; solving how to relieve the thermally induced pain implies further study and development of the thermal methods outlined above. Thermally induced damage plays an important role in causing pain (thermal pain) and, therefore, a better understanding of the temperature distribution, heat transfer process and thermomechanics in skin will contribute to the study of pain causes and its relief Other applications Besides biomedical applications, advances related to space and military missions may benefit from the proposed study. Extreme environments encountered in space travel and military activities necessitate the provision of sophisticated garments to astronauts and military personnel for thermal protection. As the interface between external environment and human inside body, skin certainly plays a significant role. Furthermore, challenges are also posed by the need to understand possible thermal effects on military personnel exposed to irradiation, such as incidental radio frequency radiation. Fig. 1 Structure of human skin [234] 1.4 Outline of this review A detailed literature review is carried out. As the biothermomechanical behavior of skin is highly interdisciplinary, involving the subjects of heat transfer, mechanics and biology, the review is divided into four parts: skin structure, skin bioheat transfer and thermal damage, skin biomechanics, and skin biothermomechanics. The major problems, issues and topics for further studies are also outlined. 2 Skin structure Skin is the largest organ of the body, making up approximately 14 16% of human adult body weight, and plays a variety of important roles. It is thus of great necessities to appreciate the structure, function and properties of the skin. Skin generally consists of three layers: epidermis, dermis and subcutaneous tissue, as shown in Fig. 1. The thickness of these layers varies depending on the location of the skin. Since dermis is the main part of the skin, its components are introduced below. 2.1 Collagen fibers in dermis Collagen is the major dermal constituent and accounts for approximately 60 80% of the dry weight of fat-free skin Fig. 2 Molecular/fibrillar configuration of Type I collagen [235] [9,10] and 18 30% of dermis volume of [9]. Collagen in human dermis is mainly consisted of periodically banded, interstitial collagens (types I, III and IV), where about 80 90% is type I and 8 12% type III. Three polypeptide chains make up type I collagen molecules and are stabilized in a triple-helix arrangement by intramolecular cross-links, as shown in Fig. 2. These molecules are, in turn, aggregated into a parallel pattern to form collagen fibrils, which is maintained by intermolecular crosslinks and provides the tissue with its tensile properties. Type IV collagen co-distributes and assembles into fibrils with both types I and III collagen in which it assists in regulating the fibril diameter. The wavy

4 4 F. Xu et al. Fig. 3 Schematic of skin mechanical behavior under a uniaxial and b biaxial tension and unaligned collagen fiber bundles form an irregular network almost parallel to the epidermis surface, which allows considerable deformation in all directions without requiring elongation of individual fibers and thus provide both tensile strength and elasticity [11,12]. 2.2 Elastin fibers in dermis Elastin fibers are a minor structural component of the dermis structure, accounting for 4% of dermis dry weight and 1% of dermis volume [9,13]. Elastin fibers are considerably thinner and more convoluted than collagen fibers. The base unit of elastin is a long protein chain that is cross-linked by lysine molecules, with four elastin chains joined at each cross-link by the covalent bonding of a lysine molecule from each elastin chain. Although direct connections between elastin and collagen fibers have not been established, collagen fibrils appear to wind around elastin cores. 2.3 Ground substance The amorphous ground substance can be considered as a highly viscous, thixotropic liquid whose fluid properties are determined by a low concentration (0.05% wet weight of human dermis) of mucopolysaccharides, proteoglycans and glycoproteins [14]. The most important mucopolysaccharides in ground substance is hyaluronic acid, which is analogous to polymer molecules found within rubbers. The hyaluronic acid chain has proteoglycans side-chain, which in turn is cross-linked by glycoproteins to additional structures within dermis, such as collagen or elastin fibrils. It is this cross-linking function that is responsible for forming fibers from collagen fibrils and providing the dermis with its rubber-like constitutive behavior. Together they form a gel which does not leak out of the dermis, even under high pressure. 3 Skin biomechanics 3.1 Experimental study Due to great importance in clinical and cosmetic applications, the mechanical properties of skin have been experimentally studied in vivo and in vitro for a long time since the study of Langer in In vitro tests are often used in lieu of in vivo measurements, as the latter is affected by both skin tissue itself and other structures it is attached to, which means it is very difficult to obtain a uniform strain field in the sample and control boundary conditions when performing Skin behavior under stretch The tensile behavior of skin is the most studied and there exist several comprehensive reviews [15 18]. Typical stress versus stain relationships under uniaxial and biaxial tension are shown in Fig. 3a, b, respectively. Three obviously different regions can be observed: (1) low modulus portion, which is caused by the gradual straightening of an increasing fraction of the wavy collagen fibers and the stretching of elastic fibers; (2) linear region, which is attributed to the stretching and slippage of collagen molecules within crosslinked collagen fibers and collagen fibril slippage; (3) final yield region, which is due to the loss of fibrillar structure resulting from the defibrillation of collagen fibrils. Compared with uniaxial stretch, the stress strain curve is shifted left in biaxial stretch due to the two-directional stretch of collagen fibers, as shown in Fig. 3b Skin behavior under compression In vivo, skin is under compressive load from different media such as chairs, shoes and sockets of prosthetic limbs. Although there are abundant published experimental data for skin

5 Biothermomechanical behavior of skin tissue Skin failure Understanding the factors that control the extent of tissue damage as a result of material failure in soft tissues may provide means to improve diagnosis and treatment of soft tissue injuries. The most common failure tests of skin are tensile tests [25,29 41], piercing tests [25,42 53] and tearing tests [54 56]. In spite of these studies, the failure characteristics of skin are still not clearly established yet. This is attributed to the difficulty in obtaining reliable measurements on soft biological tissues. Furthermore, there is almost no reported study of skin failure characteristics under dynamic loadings. 3.2 Skin mechanical models Fig. 4 Representative stress strain responses from a compressive test and comparison with published tensile behavior in literature in tension, the mechanical behavior of skin in compression has been rarely studied. The pressure in skin is composed of solid tissue pressure and interstitial fluid pressure [19], where the first one is carried mainly by the solid elements in the tissue and is the important factor and the main cause of the tends to occlude tile blood vessels and the second one determines the motion of tissue fluid and the diffusion coefficient of blood into the capillaries. This point has also been confirmed by Oomens et al. [20], who found a good agreement between experimental results of skin response under compression and finite element calculations with mixture elements (solid/fluid mixtures). The compressive response of skin is highly viscoelastic and also non-linear [21 23]. Both the viscous and non-linear aspects were found to diminish greatly by preload [24] and depend on age, sex, site, hydration and obesity [21]. Further, almost all previously published mechanical models of skin tissue are based on experimental data under tension and few are based on data under compression. A representative compressive stress versus strain response of pig back skin from our previous test is presented in Fig. 4 (dot). The stress strain relationship exhibits a three-stage strain hardening: toe region with low stiffness at low strain levels, transition region from low to high stiffness, and high stiffness region at large strain levels. Compared with the mechanical characteristics of skin in tension [25], the compressive stress strain curves are similar in trend; however, the transition from low to high stiffness in tension occurs mostly at a strain level of 10 20% [26 28], which is much smaller than that in compression (about 20 30% strain), as shown in Fig. 4. Similar results have also been reported for pig back skin [23]. Besides experimental studies, many theoretical models have been proposed to characterize the mechanical behavior of skin, which can be broadly classified as three main types [57]: continuum models, phenomenological models, and structural models Continuum models The continuum models use general material theories to demonstrate the multiaxial behavior of skin [28,58 61]. The main drawback of these approaches is that the various parameters have no physical or biological meaning, except for the stiffness coefficient [62] Phenomenological models The phenomenological models apply mathematical formulas, such as using a mathematical formation [63 66] or a strain energy density function [59,67], to imitate the response of skin to various types of mechanical loading based on the data from mechanical experiments, without detailed internal structural consideration. A problem with the phenomenological approach, however, is the wide variability in material parameters for: (a) different experimental protocols for a given specimen, and (b) slight cycle-to-cycle variations within a single protocol [68]. Consequently, it is difficult to make reliable interpretations of the behavior of a tissue [69] since this method provides little insight into how the constitutive properties of skin affect the mechanics. Moreover, orthotropic strain energy density functions are not readily available in commercial finite element codes Structural models The continuum and phenomenological models are useful in describing the macro-mechanical response of skin, but unlike the structural models, they lack physically significant

6 6 F. Xu et al. parameters [70]. The structural models describe the gross mechanical behavior of skin by combining and analyzing the behavior of its individual components on the basis of microstructural geometry and properties, with the alignment and straightening of the fiber network taken into consideration [70 75]. Although structural models involve physically significant parameters, providing an insight into how tissue structure affects the constitutive behavior of skin [70], they are mathematically complex requiring precise quantification of the tissue architecture including constituent interactions. Accordingly, these microstructural relations have not described the data better than phenomenological relations [68]. 4 Skin bioheat transfer and thermal damage 4.1 Skin bioheat transfer Skin bioheat transfer has been studied extensively, due to its huge importance in thermoregulation. Most of the earlier studies emphasized on the study of skin temperature around heat sources for tumor diagnose [76 78], which is based on observations of Lawson [79] that the skin temperature over a malignant tumor is higher than that of surrounding skin due to increased blood flow in tumor. Later, more attention was paid to the transfer of heat across skin in contact with hot material or cold material [80 85] to obtain quantitative relationship between the threshold temperature for pain sensation and the thermal properties of contact material. Nowadays, with applications of laser, microwave and similar technologies in the thermal treatments for disease and injury involving skin, emphasis has been shifted to electromagnetic heating, such as microwave [86,87], radiofrequency [88,89] and laser [90 92]. However, most of these works are based on theoretical analysis and numerical simulation due to the difficulty of performing heat transfer experiments in skin tissue in vivo. Since the appearance of Pennes bioheat equation [93], a variety of models on heat transfer in different tissues of human body have been proposed, with the body tissue represented, from the standpoint of heat transfer, as a homogeneous continuum material where a hierarchical vascular network is embedded [94]. According to the different ways for considering the influence of blood flow in the vascular network on heat transfer, the proposed models can be classified into four categories: continuum models, vascular models, hybrid models, and models based on porous media theory [95 99]. Since the effect of blood vessels on heat transfer is strongly related to their sizes [ ], a thermal equilibration length of blood vessels, L eq, is defined, which represents the length at which the difference between blood and tissue temperature decreases to 1/e of the initial value. Similar equations for L eq have been proposed by different researchers [101,102, ], with a typical one given as: ( ) [ ρb c b 1 L eq = VDv 2 8k b 2 + k b ln k eff ( Dc D v )], (1) where ρ b, c b and k b denote the density, specific heat and thermal conductivity of blood, respectively; D c and D v denote the mean tissue cylinder and vessel diameter; and V is the mean vessel flow velocity. The ratio of L eq to the actual vessel length demonstrates the distinction of thermal significance, ε: ε = L eq /L. (2) If ε 1, blood will exit the vessel at essentially the tissue temperature, whereas for ε 1, the blood temperature will not decay and will leave the tissue at the same inflow temperature [101,104]. Since ε for blood vessels in the skin tissue has a value of about to [108], blood will exit the vessel at essentially the tissue temperature and thus the Pennes equation suffices for describing skin heat transfer, given as [93]: ρc T = k 2 T + ϖ b ρ b c b (T a T ) + q met + q ext, (3) where ρ, c, k are the density, specific heat and thermal conductivity of the skin tissue, respectively; ρ b, c b are the density and specific heat of blood, ϖ b is the blood perfusion rate; T a and T are the temperatures of blood and skin tissue, respectively; q met is metabolic heat generation in the tissue and q ext is heat generation due to external heating sources. It should be noted here that the Pennes equation is built upon the classical Fourier s law. According to its anatomical structure of Fig. 1, skin tissue is generally considered as a layered structure, as shown in Fig. 5. In most of early studies, single-layer skin model was used due to its simplicity, where the skin is treated as a single homogeneous layer with constant thermal properties. This is obviously far from reality since the very important thermal effect of dermal blood flow is significantly underestimated by treating skin as a single region [109], although the one-dimensional model is a good approximation when heat mainly propagates in the direction perpendicular to skin surface (e.g., laser heating). Therefore, various types of multiregional model were proposed. Since introducing more than one region into a thermal model of the skin tissue makes the analytical solutions of these problems very difficult, many numerical methods have been used, with Finite Difference Method (FDM) and Finite Element Method (FEM) the most often used.

7 Biothermomechanical behavior of skin tissue 7 Fig. 5 Idealized skin model 4.2 Non-Fourier heat conduction phenomena in skin tissue In many situations, heat conduction has been treated according to the classic Fourier s law, which assumes that any thermal disturbance on a body is instantaneously felt throughout the body or, equivalently, the propagation speed of thermal disturbance is infinite. Although this assumption is reasonable in the majority of practical applications, it fails in particular thermal conditions or heat conduction media, where the heat conduction behaviour shows a non-fourier feature such as thermal wave phenomenon, or hyperbolic heat conduction as defined mathematically. The non-homogeneous inner structure of biological tissue suggests the existence of non-fourier heat conduction behavior, as temperature oscillation (an unusual oscillation of tissue temperature with heating) and wave-like behavior are commonly observed. Temperature oscillation in living tissue was first observed by Richardson et al. [110] and later by Roemer et al. [111], who subjected canine thigh muscle to an abrupt application of microwave heating at different power levels. Subsequently, Mitra et al. [112] carried out four different experiments with processed meat for different boundary conditions and also observed the wave-like behavior. On the other hand, Davydov et al. [113] experimentally observed that heat transfer in a muscle tissue under local strong heating exhibits substantial anisotropy, which cannot be explained by the standard Fourier-theory based heat diffusion model. Banerjee et al. [114] measured the thermal response of meat under laser irradiation, and found that the non-fourier hyperbolic heat conduction equation is a better approximation than the classical parabolic Fourier heat conduction formulation. However, there exist different viewpoints on the non- Fourier behavior of biological materials. For example, Tilahun et al. [115] and Herwig and Beckert [116] questioned the experimental results of Mitra et al. [112]. The former tried to reproduce the experiment of Mitra et al. [112] with processed meat but did not observe the expected non-fourier behavior. In turn, they cited several issues associated with the experiments of Mitra et al. [112] that might have caused the observed temperature jumps. Herwig and Beckert [116] also found no evidence of non-negligible non-fourier heat conduction effects. They pointed that the thermal lag effect can be explained by the Fourier heat conduction rather than by the wave behavior. Unfortunately, it was not possible to reconcile the conflicting measurements of Mitra et al. [112] with those of Tilahun et al. [115] and Herwig and Beckert [116], for the experiments were performed differently and there was no information in either study about the internal details of the processed meats [117]. Although a wave-like heat transfer behavior in living tissue is intriguing, no ultimate conclusion can be drawn at present due to the complexity of biological systems [118]. Theoretically, treating the non-homogeneous biological material as a porous medium under the same boundary conditions as in one of Mitra s experiments, Xu and Liu [119] found that the wave-like thermal behaviour in the meat may be caused by the convection of water inside the tissue. This aspect was thought to be induced in the experiments of Mitra et al. [112] by pressing the meat samples together at the start of each experiment and by the subsequent development of temperature gradients across the samples. The temperature jumps were attributed to the arrival of warm water at the measurement locations in the colder samples before the effect of pure conduction became noticeable at these locations. However, Xu and Liu [119] did not directly compare their predictions with measurements. Davydov et al. [113] also attributed their observation of anomalous heat transfer behaviour in muscle tissue to the flow of interstitial liquid as a result of non-uniform heating. Despite the resemblance between the findings of Mitra et al. [112] and Xu and Liu [119], to interpret the experiments of Mitra in the context of convection rather than pure conduction requires additional study. Alternatively, the temperature oscillation phenomenon has been attributed to blood perfusion oscillation due to heating [111, ]. However, this explanation was subsequently questioned by others [118,]. Using an artificially simulating construction similar to a bioheat transfer system, Liu et al. [124,125] carried out a series of theoretical experiments and found that the temperature oscillations can be well fitted with the thermal wave analysis.

8 8 F. Xu et al. In our previous study [126], different non-fourier heat conduction models are explored to investigate the relationship between thermal relaxation times and the thermomechanical response in skin tissue, as outlined below. (1) Pennes bio-heat transfer equation (PBHTE) As is well known, the conduction term in the traditional Pennes bioheat transfer equation [93] is based on the classical Fourier s law: q(r, t) = k T (r, t), (4) where q is the heat flux vector representing heat flow per unit time, per unit area of the isothermal surface in the direction of the deceasing temperature; k is the thermal conductivity which is a positive, scalar quantity; T is the temperature gradient; r stands for the position vector. The general bioheat transfer equation is given as: ρc T = q + ϖ b ρ b c b (T a T ) + q met + q ext. (5) Combining the above two equations, one gets the Pennes bio-heat transfer equation: ρc T = k 2 T + ϖ b ρ b c b (T a T ) + q met + q ext. (6) (2) Thermal wave model of bioheat transfer (TWMBT) Ever since the experimental observation of a finite thermal wave speed in liquid helium [127], the wave behavior in heat conduction has been argued from various physical points of view [ ]. The necessity of a finite heat propagation speed has also been demonstrated from a microscopic point of view [131,132]. Using the concept of a finite heat propagation velocity, Cattaneo [129] and Vernott [130] independently formulated a modified unsteady heat conduction equation, which is a linear extension of the unsteady Fourier equation, where additional τ q is added to Eq. (4) in order to account for the thermal wave behaviour not captured by Fourier s theory: q(r, t + τ q ) = k T (r, t). (7) The first order Taylor expansion of the above equation gives: q(r, t) + τ q q(r, t)/ = k T (r, t), (8) where τ q = α/c 2 t is defined as the thermal relaxation time, α is the thermal diffusivity, and C t is the speed of thermal wave in the medium [112,133]. The reciprocal of the relaxation time, f = 1/τ, is the critical frequency dictating the activation of thermal wave behavior [131]. Since both τ q and α are intrinsic thermal properties of the medium, the resulting thermal wave speed C t is also an intrinsic property [134]. Due to its similarity with acoustic wave, the proposed wavelike propagation of thermal signals is termed the second sound wave [127,131]. Table 1 Thermal relaxation times of cutaneous structures [218] Structure Size (µm) Thermal relaxation time (approximate) Melanosome µs Cell µs Blood vessel 50 1 ms ms ms Most biological materials that contain cells, superstructures, liquids, and solid tissue are non-homogeneous, so that their thermal relaxation times are much larger compared to engineering materials. Vedavarz et al. [135] found τ q of biological materials and tissue has a value in the range of 10 1,000 s at cryogenic temperature and s at room temperature. For meat products, τ q = s [136,137]. Mitra et al. [112] found that the value of τ q in processed meat was about 15.5 s while Roetzel et al. [138] found it to be 1.77 s. As for skin tissue, no data about the thermal relaxation time has been reported, but those of some important cutaneous structures have been presented by Stratigos and Dover [139], as given in Table 1. Applying the thermal wave theory to bioheat conduction equation, one can get the Thermal Wave Model of Bioheat Transfer, formulated as: τ q ρc 2 T 2 = k 2 T ϖ b ρ b c b T (τ q ϖ b ρ b c b + ρc) T ( q m + ϖ b ρ b c b T b +q m +q ext +τ q +τ q ) ext. (9) The above equation is known as a hyperbolic bioheat equation because there appears a two double-derivative term (called the wave term) that modifies the parabolic Fourier heat equation [140] into a hyperbolic partial differential equation. (3) Dual-phase-lag (DPL) model In order to account for deviations from the classical approach involving Fourier conduction and to consider the effect of microstructural interactions in the fast transient process of heat transport, an effect absent in the thermal wave model, a phase lag for temperature gradient, τ T, is introduced [133, 141,142]. Together with τ q, the corresponding equation is called the dual-phase-lag (DPL) equation, and is stated as: q(r, t + τ q ) = k T (r, t + τ T ), (10) where τ q and τ T can be interpreted as periods arising from thermal inertia and microstructural interaction, respectively, [143]: specifically, τ q is the phase-lag in establishing

9 Biothermomechanical behavior of skin tissue 9 the heat flux and associated conduction through a medium, while τ T accounts for the diffusion of heat ahead of sharp wave fronts that would be induced by τ q, and is the phase-lag in establishing the temperature gradient across the medium during which conduction occurs through its small-scale structures. Thus, Eq. (10) states that the gradient of temperature at a point in the material at time t + τ T corresponds to the heat flux vector at the same point at time t +τ q [144]. The Eq. (10) reduces to the thermal wave model by setting τ T = 0 and reduces to Fourier s heat equation by also setting τ q = 0. Through the first and second order Taylor expansions, the DPL model can be developed into several pertinent models, which are now summarized. Type 1 DPL model of bioheat transfer (DPL1MBT) The simplest example of DPL model is its first order expansions for both q and T, given as: q(r, t) + τ q q(r, t) = k [ ] T (r, t) T (r, t) + τ T. (11) Applying the above equation to bioheat conduction equation one can get type 1 DPL model of bioheat transfer, as: τ q ρc T 2 2 = k 2 T + τ T k 2 T ϖ b ρ b c b T (τ q ϖ b ρ b c b + ρc) T ( + + q met + q ext + τ q q met ϖ b ρ b c b T a + τ q q ext ). (12) The DPL1MBT model reduces to thermal wave model by setting τ T = 0 and reduces to Fourier s heat equation by setting τ q = τ T = 0. Type 2 DPL model of bioheat transfer (DPL2MBT) With first-order approximation for q and second-order approximation for T, one has: ( q T q + τ q = k T + τ T + τ ) T 2 2 T 2 2, (13) which leads to type 2 DPL model of bioheat transfer: τ q ρc T 2 2 = k 2 T + τ T k 2 T + k τ T T ϖ b ρ b c b T (τ q ϖ b ρ b c b + ρc) T ( + ϖ b ρ b c b T a + q met + q ext + τ q q met + τ q q ext ). (14) Type 3 DPL model of bioheat transfer (DPL3MBT) With second-order approximation for both q and T, one gets: q q +τ q + τ q 2 ( 2 q 2 2 = k T T +τ T + τ ) T 2 2 T 2 2, (15) from which type 3 DPL model of bioheat transfer is derived: τq 2 2 ρc 3 T 3 = k 2 T + kτ T 2 T + k τ T T + ( ϖ b ρ b c b )T + ( τ q ϖ b ρ b c b ρc) T ( + τ q 2 ) 2 ϖ 2 T bρ b c b τ q ρc 2 ( q m + ϖ b ρ b c b T b + q m + q ext + τ q + τ q q ext 4.3 Skin thermal damage + τ 2 q 2 2 q m 2 + τ 2 q 2 2 q ext 2 ). (16) When the skin temperature rises above a critical value ( 43 C), thermal damage will be induced. The Critical Thermal Load [145] approach was used earlier to quantify thermal damage in tissue; this assumes that the total damage is a function only of the total cumulative dosage, so that equal doses produce equal injury. A measure of the internal damage is defined as the cumulated dosage, q, equal to q = t 0 qdt, (17) where q is the heat flux in units of W/m 2. However, this method has been criticized by several researchers, for example, Stoll [145] has demonstrated that a large amount of energy delivered over an extended period of time may produce no damage whatever, where the same dose delivered nearly instantaneously may destroy the skin. Presently, the Arrhenius burn integration, proposed by Henriques and Moritz [146,147], is widely used. They assert that skin damage can be represented as a chemical rate process, which is calculated by using a first-order Arrhenius rate equation, whereby damage is related to the rate of protein denaturation, k, and exposure time, t, at a given absolute temperature, T. The dimensionless measure of thermal damage,, is introduced and its rate, k, is postulated to satisfy: k(t ) = d dt = A exp( E a /RT), (18)

10 10 F. Xu et al. or, equivalently: = t A exp( E a /RT)dt, (19) 0 where A is a material parameter equivalent to a frequency factor, E a is the activation energy, and R = J/mol K is the universal gas constant. Equation (18) indicates that a reaction proceeds faster with larger values of T or A for the same E a, or with smaller values of E a for the same A. The constants A and E a are obtained experimentally. Many researchers have proposed other models, but most of them have similar format, where the only differences are in the coefficients used in the burn damage integral, arising from the different experimental databases used to define the models and the different emphasis when analysing the burn process. The available Arrhenius parameters (A, E a ) used to calculate thermal damage for skin tissue from the literature has been reviewed, as given in Table 2, and is fitted with the method used by Wright [148], as shown in Fig. 6. The results of Fig. 6 clearly suggest a linear relationship after a least-square fit between the Arrhenius parameters for skin tissue, given by: E a = ln(a). (20) With the above relation, the special reaction rate for the thermal damage process can be written as: k(t ) = A exp( E a /RT) = exp[(e a )/ ] exp( E a /RT). (21) Fig. 6 Cross-plot of Arrhenius parameters (A, E a ) with regression fit (scattered points values are taken from references [146,219, ,236] which are indicated; line best fit line by least squares) If the temperature is constant and E a is specified, then the thermal damage can be calculated as: = t 0 k(t )dt = k(t )t. (22) Let = 1.0 denote the beginning of irreversible damage, the time for the appearance of irreversible damage at temperature T can be calculated as: Table 2 Review of activation energy and frequency factor for skin tissue Temperature Sample and Site Activation energy, E a /R (K) Frequency References range ( C) assay E a (J/mol) factor (per s) 44 T 70 Necrosis Epidermis , [146] T 55 Necrosis Unknown , [219] T > 55 Necrosis Unknown , [219] 44 T 50 Necrosis Epidermis , [220] T > 50 Necrosis Epidermis , [220] Necrosis Bulk skin [221] Whole range Necrosis Epidermis , [222] Whole range Necrosis Dermis , [222] 44 T < 50 Necrosis Dermis , [223] 50 T 60 Necrosis Dermis , T 50 Necrosis Unknown , [224] T > 50 Necrosis Unknown to (T-53) 48 T 57 Purpura formation Porcine epidermis , [225] 40 T 60 Birefringence loss Rat skin collagen , [226]

11 Biothermomechanical behavior of skin tissue 11 Fig. 7 Effect of activation energy E a on a thermal damage process rate and b the time for the irreversible damage t =1 = 1/k(T ) = 1/(exp[(E a )/ ] exp( E a /RT)). (23) The cross-plot between reaction rate and E a for constant temperature is presented in Fig. 7a, whilst the time corresponding to the appearance of irreversible damage is plotted as a function of E a is shown in Fig. 7b. Based on the analysis of enzyme-catalyzed reactions in living tissue, Xu and Qian [149] presented a thermal damage model where the thermal stability of the substrate enzyme complex was taken into account. Based on the experimental results of Moritz and Henriques [146], a new damage function was given, which better fitted the data: = t 0 Ae αz dt, (24) 1 + Be βz where z = (1 T 0 /T ) is the dimensionless temperature with T 0 representing the reference physiological temperature of tissue; α is an empirical dimensionless constant related to enzyme denaturation and β is an empirical dimensionless constant related to complex formation and discomposition; A and B are numerical constants; the constants were given as: α = 100, β = 195, A = /s, B = The Arrhenius burn damage model has been shown to be a useful tool by which the time temperature history and damage accumulation are connected. However, it has several shortcomings [98,150]: (1) the models are mainly based on the burns created by surface heating; (2) these models do not account for the history of thermal insult; (3) these models do not consider many other factors besides temperature, such as ph variations, preheating, and mechanical loading et al.; (4) these models are based on the Arrhenius chemical reaction rate equation, but often applied in a context not matched; (5) the experiments performed to derive A and E a are only for rather long times of hyperthermic exposure covering 10 1,000 of seconds. 5 Skin biothermomechanics Skin biothermomechanics here is defined as the response of skin under thermomechanical loading, which leads to damage the thermal denaturation of collagen. As noted, skin also contains a small mount of elastin but which is very thermally stable [151], for example elastin can survive boiling for several hours with no apparent change, and does not need attention here. 5.1 Mechanism of thermal denaturation of collagen The collagen in human dermis is mainly type I collagen. Type I collagen has a domain within the triple helix that is completely devoid of hydroxyproline. Since hydroxyproline readily forms hydrogen bonds that stabilize the molecule, its absence makes this domain particularly susceptible to thermal damage [152]. There are two levels of organization where breakdown is thermodynamically significant [153]: one is the collagen molecule itself, in which three peptide chains are twisted around each other to form a helical, rod-shaped molecule; the other is the semi-crystalline fibril in which collagen molecules are assembled side-by-side in a staggered manner with the long axis of each molecule aligned with axial orientation of the fibril. When collagen is heated, the heat-labile intramolecular crosslinks are broken, as shown in Fig. 8, and the collagen undergoes a transition from a highly organized crystalline structure to a random, gel-like state, which is the denaturation process [154]. Collagen shrinkage occurs through the cumulative effect of the unwinding of the triple helix, due to the destruction of the heat-labile intramolecular cross-links, and the residual tension of the heat-stable intermolecular cross-links [ ]. The effects of heating on collagen can be reversible or irreversible and the precise heat-induced behaviour of collagenous tissue and shrinkage depend on several factors, including the collagen content [157], the maximum temperature reached and exposure time [156], the mechanical stress

12 12 F. Xu et al. Fig. 8 Schematic of thermal denaturation of collagen [155] applied to the tissue during heating [158], and aging [157, 159]. Different metrics have been used to characterize the thermal denaturation and heating-induced damage of collagen and collagenous tissues including biological metrics such as enzyme deactivation [160] and extravasation of fluorescenttagged plasma proteins [161], thermal metrics such as changes in enthalpy [162,163], mechanical metrics such as thermal shrinkage [8,158,164,165], and optical metrics such as thermally induced loss of birefringence [ ]. Although the shrinkage of collagen due to thermal denaturation have been widely used and suggested to be used as a convenient continuum metric of thermal damage [150,170], Wells et al. [171] point out that equilibrium shrinkage may not be a universal metric to measure thermal damage. Rather, there is a need to identify an independent metric by which one can determine the extent of thermal damage [172]. Diller and Pearce [150] point out that the dimensionless indicator of damage,, is, in fact, the logarithm of the relative concentration of reactants, or un-denatured collagen, in the collagen denaturation process, where can be alternatively considered as: [ ] C(0) (t) = ln, (25) C(t) C(0) and C(t) are the initial concentration and the concentration remaining at time t of un-denatured collagen. The degree of thermal denaturation, defined to be the fraction of denatured collagen, and denoted by Deg(t), can be calculated as: Deg(t) = C(0) C(t) C(0) = 1 exp[ (t)]. (26) also the hydration of collagen changes, which may involve an initial liberation of, but subsequent absorption of, water via water bridges. Not surprisingly, then, thermal denaturation of a collagenous tissue can result in marked changes in the thermal [173], mechanical [8,158, ] and optical properties [ ]. For example, the increased extensibility of soft tissues due to thermal treatment has been observed in both uniaxial [174,184,185] and biaxial studies [171,186]. However, there are comparatively few studies of the effects of thermal denaturation on the mechanical response of skin tissue [159, ], despite skin dermis being mainly composed of collagen. In view of this, we have carried out a series of experimental study on the temperature/thermal damage dependent mechanical behavior of skin tissue [ ]; the main findings are summarized below Tensile tests Uniaxial tensile tests of pig ear skin samples under different temperatures have been performed and the stress strain relationships are given in Fig. 9. There are two of behaviour regimes for all the curves. First, when ε<50%, the curves almost overlap although the data set is swamped by larger values. The thermal stability of skin is because the elastin dominates the mechanical behaviour of skin tissue in this low modulus region which is very thermal stable (surviving in boiling water for hours), although there is gradual straightening of an increasing fraction of the wavy collagen fibers and stretching of elastic fibers. When ε>50%, the stress increases almost linearly with strain; and the slopes under different temperatures are different, and reduce with increasing temperature. This effect is due to the stretching and slippage of collagen molecules within crosslinked collagen fibers and 5.2 Property variations due to thermal denaturation of collagen With the increase of temperature, the denaturation of collagen occurs. During denaturation, not only the structure, but Fig. 9 Stress strain relation of uniaxial hydrothermal tensile tests under different temperatures

13 Biothermomechanical behavior of skin tissue 13 Fig. 10 Stress-thermal damage degree relation of hydrothermal tensile tests under different temperatures to collagen fibril slippage: with an increase in temperature, the highly organized crystalline structure of collagen changes to a random, gel-like state, which results in the corresponding decrease in stiffness. Furthermore, the hydration change with temperature may also make a contribution. Water plays a significant role in governing the gross properties of skin tissue [195,196], which consist primarily of water. However, due to heating, the level of hydration may vary, e.g., Luescher et al. [197] suggested that primary hydration water is set free during the process of thermal denaturation. It should be noted that when T 60 C, there is relatively little change in the modulus with temperature, which can be explained by the thermal damage process: when the strain rate and temperature are given, the relationship between the strain and thermal denaturation degree (Deg) calculated according to the heating history can be obtained, as shown in Fig. 10. Theburn degree stands for the concentration of denatured collagen. They show that when T 60 C, the collagen is fully denaturized, while at T = 45 and 50 C, denaturation is much slower Compressive tests The hydrothermal compressive tests of back skin under different temperatures have also been performed and results are given in Fig. 11. It is interesting that, contrary to the tensile tests of Sect , the compressive stiffness increases with increasing temperature as shown in Fig. 12 although the thermal damage degree also increase. One key difference is that the compressive tests are performed in a direction normal to the principal orientation of collagen and elastin fibres. Even though there may exist dehydration effects and denaturation of collagen, which appear in the tensile tests, the compressive behaviour is governed by the mechanical properties of Fig. 11 Stress strain relation of hydrothermal compressive tests under different temperatures Fig. 12 Stress-thermal damage degree relation of hydrothermal compressive tests under different temperatures the gel-like ground substance, inside which the fibres are located [198,199]. Very little is known about the mechanical properties of this substance; it can only be speculated that its stiffness increases with increasing temperatures Viscoelastic behavior Uniaxial tensile tests of pig ear skin under four hyperthermal temperatures(t = 50, 60, 70, 80 C) have been performed. Together with the result obtained at body temperature (T = 37 C), the relaxation function R(t) = σ(t)/σ max is plotted in Fig. 13, with σ(t) denoting stress at time t and σ max representing the maximum stress corresponding to the peak of loading. Within physiological ranges, it has been