Modelling of the heat accumulation process during short and ultrashort pulsed laser irradiation of bone tissue.

An analytical model is presented that qualitatively describes the cooling of a biological tissue after irradiation with short and ultrashort laser pulses. The assumption that the distribution of temperature at the initial moment of surface cooling repeats the distribution of the absorbed laser energy allowed us to use the thermal conductivity approximation in both cases. The experimental results of irradiation of dry bone with nanosecond and femtosecond laser pulses are compared with the calculated data. The necessity of taking into account the change in the optical parameters of hard tissue in the field of laser irradiation during its treatment by nanosecond and femtosecond laser pulses and the key role of residual heating in its carbonization around the exposure region is shown. The application of the model to a particular biological tissue can significantly simplify the search for optimal parameters of lasers for surgical procedures.


Introduction
Laser technologies are successfully used in a wide variety of medical fields such as surgery [1][2][3][4][5], therapy [5,6], cosmetology [7][8][9][10] and for effective treatment of medical instruments [11]. The use of lasers in medicine is based on the management of biological, physical and chemical processes during laser irradiation of biological tissue [12]. In this case, the considerable difference in the processes of laser interaction with the biological medium under the conditions of varying laser parameters is observed [12,13]. In recent years, a wide range of phenomena occurring during laser irradiation of biological tissues have been studied. However, the rapid development of laser systems with ultrashort pulses generates a need for additional investigations [14][15][16][17][18]. It should be noted that the most medical applications of laser radiation are based on the effect of heating of biological tissues [13].
The thermal model, where four stages of laser interaction with material: absorption of light, heating, destruction, afteraction are separately considered, was developed in 1970s. The model successfully describes the effect of laser radiation on metals, semiconductors and dielectrics for continuous radiation, long and short pulses (pulse duration τ > 10 −9 s). With the advent of lasers generating ultrashort pulses (pulse duration τ < 10 −12 s), it became necessary to consider the ultrafast processes of energy exchange between electron gas and lattice, the characteristic times of which are comparable or longer than the duration of the laser pulses. The two-temperature model proposed for metals [19] required the development to describe such effects. According to the model, heating of free electrons and their interaction with atoms (ions) of the lattice, leading to heating of the metal, are considered. Due to high laser intensities, characteristic of ultrashort laser pulses, the probability of multiphoton light absorption in semiconductors and dielectrics increases [20]. In this case, a high concentration of electrons is generated, and the materials are metallized [21]. Based on this understanding of the physics of the process, several numerical models [21][22][23][24][25][26], that complement each other and allow to consider the contribution of different processes: recombination, saturation, external emission, plasmon excitation, etc., have been developed. Generally, the numerical models calculate the interaction of single ultrashort pulse with matter. However, in the laser treatment of different materials the employment of the regimes of multipulse irradiation is required. In this case, the models, that consider the process of heat accumulation or morphological changes of the surface between pulses, have a special place [27,28]. Such models can use a combination of quantitative and analytical methods for solving a system of equations [29].
The interaction of laser radiation with biological media is extremely complex [12,13]. The complex structure and multicomponent composition of biological tissues, high water content, light scattering by centers randomly distributed in volume contribute to the initiation of a variety of physical and chemical processes that occur in biological media at different temperatures [13]. The pulse-periodic laser irradiation of biological tissue can lead to the residual heating [18] and cause irreversible damage to surrounding tissues. In this connection, it is necessary to analyse the features of thermal impact of trains of short and ultrashort laser pulses on biological tissues.
In this paper an analytical model is proposed to describe the cooling of a biological tissue after irradiation by trains of short and ultrashort laser pulses. A dehydrated bone tissue was selected as a model object. The model object is considered as the quasi-homogeneous medium with averaged values of thermophysical and optical characteristics. The obtained spatial distributions of accumulated heat on the surface of model object as a function of the pulse repetition rate are compared with the data of experimental irradiation of model object conducted by the authors and given in [30], as well as the experimental data presented in [18]. As will be shown in this paper, the model is applicable for nanosecond and femtosecond laser pulses.

Experimental setup
The experiments were performed using a system (MiniMarker-2, Laser Center Ltd., Russia) based on an ytterbium pulsed fibre laser with a pulse duration of 50 ns, wavelength of 1070 nm and a variable pulse repetition rate from 1 kHz to 100 kHz. The radiation power and energy were measured with a microprocessor-based meter (Gentec-EO SOLO2, Gentec Electro-Optics, Inc., Canada). We also used a microscope (Axio Imager.Aim, Carl Zeiss, Germany) with a CCD camera (AxioCam ICc3, Carl Zeiss, Germany). The sample was a deer bone, preliminary dried at room temperature during six months after extraction. The thickness of the sample was 4 ± 0.2 mm. The bone sample was irradiated in the focal plane of the longfocus objective with a beam spot radius in the waist region 30 ± 3 µm during 10 seconds with varied pulse repetition rate. The fluence was 3.5 Jcm −2 when the laser was in focus.

Theoretical modelling: the model of heat accumulation
The regimes of laser irradiation of biological tissues are selected depending on a desired biomedical result. The regimes of multipulse irradiation are often used for laser treatment of soft and hard biological tissues. In this case, laser irradiation can lead to the residual heating around the exposure region, causing irreversible damage to surrounding tissues.
In this section we will consider the heating of biological tissue outside the area of laser exposure and the heat accumulation during multipulse irradiation. In this case, the determining factor is the amount of absorbed laser energy and its distribution over the volume of the biological tissue that depends on its optical properties. The fraction of the absorbed energy is determined by the absorptivity of biological tissue, and the depth of light penetration is inversely proportional to the effective absorption coefficient.
It is well known that the optical properties of a biological tissue can change during an ultrashort laser pulse at power densities of more than 10 13 Wcm −2 [31]. It was shown in [32] that such changes occur mainly due to the generation of free electrons in water, which is the main component of biological tissues. The absorption coefficient of the formed electron plasma (according to estimates [31]) can reach α = 900 cm −1 by the middle of the pulse. All these parameters correspond to the modes of destruction of biological tissues containing a large percentage of water with the fluence is approximately 1 Jcm −2 [33]. In the absence of water, the absorption of light by biological tissues can occur due to the excitation of organic molecules, that is, the transfer of bound electrons from level to level. The multiphoton absorption is possible, but it does not lead to an increase in absorption, since no free electrons are generated. In this case, the absorption of light is determined only by the energy structure of the molecules in the same way as for nanosecond and longer laser pulses.
There are two mechanisms for the conversion of absorbed laser energy into heat during irradiation of biological tissues: 1) the interaction of excited π-electrons (but not free electrons [34]) with tissue molecules, 2) the relaxation of excited tissue molecules (as for long laser pulses) [35]. In both cases, the time of conversion of laser energy into heat (time of heating) t r is typically from units to tens of picoseconds [34][35][36].
The absorbed energy of laser radiation in dehydrated biological tissues can be estimated under the condition that the absorptivity and absorption coefficient remain constant during the pulse. On the one hand, it simplifies the mathematical formulation of the problem, and on the other hand, it allows us to investigate in detail the processes developing in the protein complex. In order to simplify the analysis of thermal processes and eliminate the influence of water, we will consider the dehydrated bone as an object of research.
The nature and intensity of the effect of laser radiation on biological tissue significantly depends on wavelength. In the case of laser interaction with bone tissue in the near infrared region of the spectrum, apart from the absorption there occurs strong scattering of radiation [37,38]. The problem becomes significantly complicated due to the necessity of additional investigations of the effects that occur not only at the tissue surface, but also in its volume, which is beyond the scope of the present paper. In this connection, we will use the effective absorption coefficient α, which allows for the absorption proper and the scattering of light in the inhomogeneous medium [12], and we will consider the thermal processes at the bone surface. In the calculations we will use the optical characteristics of the bone tissue at wavelength of about 1 μm and experimental results, obtained using laser systems with similar wavelengths: λ = 1.07 μm for a nanosecond laser, λ = 1.03 μm for a femtosecond laser.
In order to estimate the heat accumulation during the multipulse laser treatment of hard biological tissue, it is necessary to consider the afteraction stage. The afteraction mechanisms are activated immediately after the end of the laser pulse and develop in different scenarios depending on the pulse duration. This is well illustrated by the diagram presented in Fig. 1. The light absorption, heating, ablation, and related chemical and mechanical processes develop during the nanosecond laser pulse. The maximum temperature of heating is reached at the end of the pulse. After that the cooling of the surface begins, which is determined by the thermal conductivity. During the femtosecond laser pulse, only photoexcitation processes develop. After the end of the laser pulse an afteraction stage begins, which includes heating the biological tissue to the maximum temperature and its cooling. The cooling of the surface is the longest process, the time of which is typically exceeds nanoseconds [39].Thus, the time of the thermal afteraction of femtosecond laser pulse does not depend on the pulse duration, is determined by the thermophysical properties of the biological tissue, and is comparable with the time of the effect of the nanosecond pulse [39]. Therefore, one can use the thermal conductivity approximation during consideration of the cooling process of the biological tissue surface after irradiation with a femtosecond laser pulse.
Without considering the complex processes of photoexcitation and relaxation, we assume that all absorbed laser energy is spent on heating, and the cooling begins from the moment the maximum temperature of heating is reached. At low thermal diffusivity a (2.3 × 10 −3 -3.8 × 10 −2 cm 2 s −1 ) typical of biological tissues [18,[40][41][42][43] the thickness of the region heated due to thermal conductivity for nanosecond and femtosecond laser pulses is usually much smaller than the irradiation zone size 0 r at r << . In addition, the most biological tissues, in particular bone tissue, have low absorption coefficients 1 / r at α >> in the near infrared region of the spectrum. Therefore, one can use the special case of solving the one-dimensional semibounded solid heating model. In this case, the distribution of temperature in the laser impact zone at the initial moment of surface cooling repeats the distribution of the absorbed laser energy. We assume that the cooling is mainly due to heat dissipation along the radius and the temperature has a Gaussian distribution over the spot, which is conserved at any depth and exponentially decreases with z (the z axis is directed deep into the biological tissue): where T max is the maximum temperature, to which biological tissue surface is heated before the beginning of cooling, α is the effective absorption coefficient of bone tissue, r 0 is the radius of the laser beam spot and r is the radial distance from the centre of the beam.
where E is the laser fluence, A and C are the absorptivity and volumetric heat capacity of the bone tissue, respectively. The surface temperature distribution (z = 0) before the beginning of cooling Considering Eq. (3) as the initial temperature distribution, we solve the thermal conductivity equation describing the cooling of the heated region where a is the thermal diffusivity of bone tissue. The radial surface temperature distribution over time before the next pulse is obtained The surface of biological tissue may not have time to cool to initial temperature between pulses during laser treatment at pulse repetition rate f. In this case, the residual temperature of the surface after the impact of N pulses at the time of arrival of the N + 1th pulse can be evaluated, using It should be noted that Eq. (6) describes the change in surface temperature relative to the initial (room) temperature. The dynamics of the surface temperature depending on the number of laser pulses at the cooling stage is schematically shown in Fig. 2. It is seen that the residual temperature ΔT is the initial one for the next pulse and depends on the pulse repetition rate. As seen in Eq. (6), the residual temperature does not depend on the pulse duration, which means that this expression can be used to estimate the accumulated heat on the surface of biological tissue for short and ultrashort laser pulses.

Results
The validity of the proposed model of heat accumulation was checked by comparing the calculated radial distributions of the residual temperature of dry bone surface according to Eq. (6) for various laser pulse durations with the experimental data.
The radial distributions of the residual temperature of the dry bone surface irradiated with the nanosecond laser at pulse repetition rate of 1 and 10 kHz, calculated for the constant effective absorption coefficient α = 10 cm -1 and absorptivity A = 0.22 according to Eq. (6), are presented in Fig. 3. As can be seen in Fig. 3 the residual temperature increases with increasing pulse repetition rate. It is the initial temperature by the arrival of the next pulse in the region of the laser spot, and it is decisive in changing the surface morphology at the periphery, where laser radiation does not reach. In the experiment at pulse repetition rates below 15 kHz the morphology of bone surface is not changed [30], which is typical for small values of the residual temperature obtained in the calculation.
It was found that at the repetition rate of 15 kHz the ablation crater is formed, surrounded by the larger areas of melted and carbonized tissue, at the sixth second of irradiation [30] ( Fig. 4(a)). A discrepancy between the observed surface changes and the calculated values of residual temperature was obtained (curve (1) in Fig. 4(b)). The solid frame of bone tissue is heterogeneous, which leads to its non-uniform heating and destruction at different temperatures. The carbonization (charring) leads to an increase in effective absorption of hard biological tissue [12]. Therefore, the obtained discrepancy can be eliminated, if we allow for a change in optical characteristics of bone tissue (α and A) at the repetition rate of 15 kHz due to its carbonization in Eq. (6). It should be noted that in different papers the carbonization temperatures have significantly different values from more than 100 to 300 °C [12,18,[44][45][46][47].This is due to the effect of the exposure time on the result of laser treatment and the specificity of particular biological tissue. During short-term exposure to laser radiation on tissue, localized areas of carbonization may be observed near the surface. Further increase in the exposure time results in a shift of the carbonization front deep into the biological tissue. The highest temperatures of carbonization were reported for the biological tissues, initially containing water [48]. In the case of dehydrated bone tissue irradiation, the energy is not used for water evaporation, which reduces the energy required to achieve the carbonization temperature. Therefore, it is quite probable that the carbonization of dehydrated bone can begin at a temperature approximately 150 °C. The carbonized frame burns away at temperatures higher than 500 °C [12]. At temperatures above 800 °C the crystalline formations, generated by the increase in temperature, melt into bigger crystals. The melting point of bone is approximately at 1630 °C [49]. We replaced the initial values of α and A in Eq. (6) with 15 cm -1 and 1, respectively, at the time from the sixth to the tenth second of irradiation ( Fig. 4(b)). The temperature of carbonization in Fig. 4(b) is presented without considering the room temperature (20 °C), i.e. 130 °C. According to the calculation, the residual temperature (ΔТ) of the dry bone surface exceeds the carbonization temperature at the distances from the centre of the laser beam spot, approaching 0.5 mm. This corresponds to the outer radius of the carbonized area, observed in the experiment at the room temperature ( Fig. 4(a)).
It should be noted that all experiments were performed on thin deer bone dried at room temperature for six months after extraction. There is always water in a porous bone in an amount equal to the equilibrium moisture content, which is determined by the temperature and humidity of the surrounding air. However, this amount is much less than the amount of water in fresh bone.

Dry bone: irradiation with femtosecond laser pulses
In the modelling we used the optical and thermophysical characteristics of bone tissue, presented in Section 3.1. The results of calculation obtained within the framework of the model of heat accumulation are compared with the experimental data presented in [18]. The radiation from the femtosecond laser (Tangerine, Amplitude Systemes, France) with a pulse duration of 320 fs, center wavelength of 1030 nm, and a variable pulse repetition rate from 1 Hz to 2 MHz, was focused into a spot with the diameter 12.6 µm. The fluence was 40 Jcm −2 when the laser was in focus. The sample was a dry bovine bone a few millimetres thick. The bone sample was irradiated during 10 seconds with different pulse repetition rates. By means of a thermal camera (FLIR SC7650e, FLIR Systems Inc., USA) the average temperature of the bone surface was measured [18].
The radial distribution of accumulated temperature of the dry bone tissue surface during femtosecond laser irradiation at the repetition rate of 1 kHz and exposure time of 10 seconds calculated for the constant optical characteristics α = 10 cm -1 and A = 0.22 according to Eq. (6) is represented by the solid curve in Fig. 5. The residual temperature distribution obtained in [18] is shown by the dashed curve in the same figure. The satisfactory agreement between the theoretical and experimental curves confirms the efficiency of the model. In the experiment [18] the full width at half maximum of the temperature distribution for all pulse repetition rates was approximately 0.08 cm and it did not change even when carbonization occurred. In the calculations it was obtained that the full width at half maximum decreases from 0.12 cm to 0.06 cm with an increase in the pulse repetition rate from 1 kHz to 22 kHz. It should be noted that the obtained values fall into the confidence intervals, presented in [18].The observed difference can be attributed to the fact that the modelling does not take into account the change in the thermophysical parameters of bone tissue during laser irradiation. In the experiment presented in [18], the average accumulated temperature of the sample surface (above the room temperature) during femtosecond laser irradiation with different pulse repetition rates was also measured (Fig. 6). The dashed line in Fig. 6 shows a linear increase in temperature of bone surface from 2 °C to 40 °C with an increase in the pulse repetition rate from 1 to 20 kHz [18]. These values demonstrate a good fit with the straight line calculated for 10 seconds according to Eq. (6) (the solid line in Fig. 6).
At the repetition rate of 22 kHz a sharp rise in temperature was observed [18]. We replaced the initial values of α = 10 cm -1 and A = 0.22 in Eq. (6) with 18 cm -1 and 1, respectively. If the residual temperature is calculated using Eq. (6) with a change in optical characteristics of bone tissue due to its carbonization taken into account, the obtained value (400 о С) fall into the confidence interval, presented in [18] (cf. Figure 6).
A coincidence observed at 22 kHz allows us to conclude that during irradiation with both femtosecond and nanosecond laser pulses, the determining role belongs to the change in the optical properties of the bone caused by carbonization. Optical properties should change in the field of laser irradiation, where the laser beam is focused. This conclusion requires justification. According to the data presented in [18], the femtosecond ablation of dry bone occurred at repetition rates varying from 1 kHz to 20 kHz without traces of carbonization. The ring of carbonized tissue appeared only at repetition rate of 22 kHz. An analysis of the optical images of the irradiated bone, given in [18], showed the absence of traces of carbonization directly in the area of laser irradiation, only surface melting and ablation were observed. The melted surface turns white and it is difficult to assume that the absorption for the subsequent pulse will increase. At the same time, it is known that for realization of carbonization process, it is required a certain combination of temperature and time during which this temperature is maintained. The thermal processes inside the laser spot develop in few picoseconds at the stage of heating during irradiation with femtosecond laser pulse. Apparently, this time is not enough for the implementation of the carbonization process. In case of multipulse treatment, the next pulse will interact with the modified surface if the optical properties change with the previous pulse in the region of focused laser radiation. In contrast to the traditional solidphase carbonization mechanism, we can assume the vapor-phase carbonization mechanism, which starts with an active ablation of bone. The vapor-phase carbonization reaction occurs in ablated bone particles in the air, and its products in the form of soot are deposited back into the laser spot area and onto the walls of the laser crater, changing the optical properties of the surface for the next pulse.

Conclusion
The assumption that the distribution of temperature in the laser impact zone at the initial moment of biological tissue surface cooling repeats the distribution of the absorbed laser energy, allowed us to use the simple thermal conductivity approximation for the analysis of the residual heating of biological tissue surface by trains of nanosecond and femtosecond laser pulses.
An analytical evaluation of the heat accumulation during nanosecond and femtosecond laser irradiation of hard biological tissue allowed substantial confirmation of the key role of the residual heat in its carbonization around the area of laser impact and proved the necessity of external water or air cooling during the multipulse laser treatment.
The satisfactory coincidence of calculated values of accumulated heat with the experimental data presented in [18], devoted to the multipulse treatment of dry bone with femtosecond laser pulses, has shown the validity of using the proposed analytical model to estimate the average surface overheating at repetition rates below the carbonization threshold.
The necessity of taking into account the change in the optical parameters of hard biological tissue in the field of laser irradiation during its treatment by short and ultrashort laser pulses is shown.
The application of the model to a particular biological tissue can significantly simplify the search for optimal parameters of lasers for safe and effective surgical procedures. Further, it is of interest to investigate the effect of water on the process of heat accumulation in hard biological tissue.

Funding
Government of the Russian Federation (Grant 074-U01).