Thermal ablation is an efficient method of medical treatment, such as cancer therapy, wound closure, laser cutting, freckle removal and other treatments. In order to guarantee the curative effect and the safety of the patients, the thermal response of the tissue which is subjected to the heat source need to be carefully monitored. However, it is too difficult to achieve real-time monitoring on the full-field temperature. In the present study, efforts were made to build up a theoretical model for the prediction of the thermal response in the human skin. The Dual-Phase-Lag (DPL) bio-heat transfer model and the Henrique’s burn assessment model were employed to describe the interaction of multi-pulse heat source and the skin. The repeated multi-pulse laser is a common heat source in the thermal treatment and the thermal responses of the skin would be complicated under the common effects of the non-Fourier effects and the multi-pulse source. The Green’s function approach was used to solve the governing equations analytically. The closed-form solution for the temperature distribution of the skin was obtained and the thermal damage was estimated based on the temperature results. The influences of the biological parameters (the phase lags of the heat flux and the temperature gradient) and the heat source parameters (the pulse number and the duty ratio) on the temperature distribution, the burn degree and the irreversible burn depth of the irradiated region were discussed. However, the high temperature could hurt the healthy cells by loosening the cell membranes and denaturing the proteins2. In order to guarantee the curative effects and the patients’ safety, the temperature distribution in human tissue should be strictly controlled. Given that real-time monitoring on the full-field temperature cannot be achieved at present, it is extremely significant to make precise prediction of the temperature distribution before thermal treatment. The mechanism of the thermal responses in human tissue during the hyperthermia need to be investigated in detail.Thermal ablation is an efficient method of medical treatment, such as cancer therapy, wound closure, laser cutting, freckle removal and other treatments. In thermal ablation, human tissues are exposed to different kinds of heat producers, including microwave, radio-frequency, infrared radiation, magnetically excitable thermo-seeds and other thermal sources3 was the first to study the heat conduction in the living biological tissues by proposing a bio-heat transfer model on the basis of Fourier’s law. Since then, the Pennes model has been widely applied by researchers. For example, Yue4 developed a one-dimensional steady-state bioheat transfer model of living tissues in cylindrical coordinates and derived the analytical solution with the usage of Bessel’s equation. Shih5 investigated the thermal response of a semi-infinite biological tissue due to a sinusoidal heat flux at the skin. Laplace transform method was employed to obtain the semi-analytical solution for the governing equations. Fu6 explored the thermal response inside the skin in the process of thermal treatment on the basis of Pennes model. The influences of the tumor shape, including the location, geometry and size, on the temperature distribution were taken into account. Singh7 investigated the thermal effects on the breast tumor during the radio-frequency ablation. The effects of temperature-dependent parameters of the multi-layer breast model, such as electrical and thermal conductivities, were considered. Bhowmik8 carried out a numerical study on the thermal response of skin with different fat thickness. The Pennes model and the numerical simulation method were employed to derive the temperature distribution of the skin tissue. Marchandise9 probed the feedback from human nociceptive system on the thermal stimuli with finite element simulation and Fourier’s law employed. The influences of the water-content rate in both hairy skin and glabrous skin were discussed emphatically.Among the existing literatures, Pennes12. To make the thermal conduction model more accurate, Cattaneo13 and Vernotte14 established a thermal wave model (C-V model) with a finite thermal propagation velocity by involving the heat flux relaxation time τq. Ulteriorly, Tzou15 promoted this heat transfer model by introducing another relaxation time τT for the phase lag of temperature gradient. The improved model is called dual-phase-lag (DPL) bio-heat conduction model which allows either the temperature gradient to precede heat flux vector or the heat flux vector to precede temperature gradient. Researchers17 investigated the heat transfer problems in spherical coordinate system with DPL model and numerical method employed. The influences of the non-Fourier effects induced by the phase lags of the heat flux and the temperature gradient on the thermal behavior were considered. Afrin18 explored the non-equilibrium heat transfer in living biological tissues by considering the arterial and venous bloods. They found that the phase lags of the heat flux and the temperature gradient depend on the properties of artery, vein and tissue, blood perfusion rate and convective heat transfer rate. Nobrega19 studied the heat conduction in the skin subjected to the pulse laser heating and fluid cooling. The analytical solutions for the one-dimensional skin model were presented with three different heat conduction models employed. Jaunich20 analyzed the temperature distributions in the skin which is subjected to a short pulse laser. The results from the hyperbolic model and the parabolic model were compared.All the above researches were based on the traditional Fourier’s law which assumed that thermal propagation velocity is infinite. However, this assumption was found not suitable for some issues in which the non-Fourier effects shows up, such as heat transfer problems in low-temperature environment and in the biological tissue21 proposed an expression for the protein denaturation process based on the first order approximation of the Arrhenius equation, which has been widely used in the burn degree prediction. Lu and co-authors23 developed a computational approach to examine the heat transfer process, heat-induced mechanical response, as well as the associated pain level with numerical simulation and the finite difference method employed. Liu24 analyzed the thermal response for estimating thermal damage in laser irradiated biological tissue with DPL model employed. Afrin25 investigated the thermal damage in the living biological tissue induced by laser irradiation with a generalized DPL model employed. The results derived from the generalized and classical DPL model were compared and the influences of the laser parameters were discussed. Wang26 investigated the thermal response of the skin which is subjected to the pulse boundary heat flux with Laplace transform technique employed. Venugopalan27 explored the thermodynamic response of biological tissue by experimental and analytical methods.It is of great significance to explore the mechanism of thermal damage to make sure of the reliability and safety of thermal treatments. For this purpose, Henriques28 investigated the free vibrations of functionally graded circular plates with elastic supports on the basis of the classical plate theory. Green’s function method was employed to solve the boundary value problem. Chao29 presented the thermoelastic behavior of two circular inclusions in an infinite plane which is subjected to a point heat source. The solution obtained in the study can be treated as the Green’s functions for the crack problem associated with two circular inclusions. Seremet31 inspected the steady-state thermoelastic problems for domains described in cylindrical coordinate system and Cartesian coordinate system. A new method was proposed to derive the Green’s function for incompressible Lamé equations and the Green’s function method was used to solve the governing equations. Zhao32 investigated the coupled thermoelastic vibration of the Euler–Bernoulli beam with cracks which was subjected to heat flux. The explicit expressions of the temperature and displacement response in the beam were obtained with the usage of Green’s function method. Ma33 has solved some heat transfer problems by using Green’s function method, such as the three-dimensional living biological tissue subjected to a scanning laser beam, the DPL heat conduction in a finite medium subjected to a moving feat source, and the bi-layered circular plate irradiated by laser pulse.The complication of the transient heat conduction equations with non-Fourier effects under consideration makes the theoretical results very difficult to derive. Green’s function method can be beneficial to solve the complex differential equations of mathematical physics. Zur32. At present, the thermal response of human tissue under continuous laser or single pulse laser has been investigated widely. However, there is few reports about the response induced by repetitive laser. In this case, the thermal conduction equations should be solved respectively in different time segmentations because of the discontinuity of the thermal source, which makes it difficult to predict the temperature distribution during the thermal ablation. To address this issue, the present study will focus on the heat transfer conduction procedure in a three-dimensional model in vivo skin which is subjected to a repetitive pulse laser. The governing equations will be established on the basis of DPL model. Exact solution of temperature distribution will be obtained by employing the Green’s function method, which is meaningful for the further exploration on the physical mechanism of the thermal treatment. The thermal damage will be evaluated and the influences of non-Fourier effects will be discussed.In clinical medicine, the continuous laser or pulsed laser is selected for different disease. In addition, it is reported that both single pulse laser and repetitive pulse laser can be used34:τq and τT represent the phase lags of the heat flux and the temperature gradient, q the heat flux, k the heat conductivity, T the temperature, t the spatial and temporal coordinates, respectively.The heat conduction model employed in the present study is the DPL model, which is modified from the Fourier’s law with two relaxation times under consideration. This heat conduction model can be expressed as3 considered the influences as an average distribution in the biological tissue and established a bio-heat conduction model (Pennes model) which has been widely used for decades of years. With the combination of the Pennes model and the DPL model15, proposed a modified bio-heat transfer model, which can be written as:ρ and C are the density and specific heat of the skin. ρb, Cb, wb represent the density, specific heat and perfusion of blood, respectively. Ta is the temperature of arterial blood, Qm for the metabolic heat generation and Q for the external heat source.The thermal conduction properties of the skin are quite different from that in the classic homogeneous materials. The thermal response will be influenced by the convection between blood and the skin, blood perfusion in vascular beds and metabolic heat generation. Pennesl, b and h, respectively. The rectangular laser pulses irradiate the cuboid at the center of the top surface. The governing equation on the basis of DPL bio-heat conduction model can be expressed as:θ  = T–T0 is the temperature rise in the skin, T0 for the core temperature of human body. Ql represents the power density of the laser source, which can be written as:φ(z), Q1(x), Q2(y) and Q3(t) are expressions for the spatial and the temporal distributions of the laser energy, which are given byµa is the absorption coefficient of the skin, H(*) for the Heaviside function, Ra for the optical reflectivity, I0 for the power density of the input laser energy, R0 for the geometric dimension of the laser spot. toi)( and tei). Te is the ambient temperature and h1 is the convective heat transfer coefficient.The model is considered as a cuboid domain in human body. So temperatures of the side surfaces and the bottom surface are assumed equal to the core temperature The initial conditions can be expressed as:tperiod with the duration time being tpulse-width. It is clear that tpulse-width = toi)( − tei) yie yie\docuSo the temperature distribution of the skin model in the present study can be expressed with the usage of Green’s function as following:The Green’s function can be derived from Eqs. –27). Th. Th27). 36:s can be derived by the following equation:The mode functions satisfying the boundary conditions areBy operating on Eq.  with \dos of Eq. , one canThe inversion formula isWith Eq.  under coSubstitution of Eq.  into Eq.With the boundary conditions and the initial conditions under consideration, the solution for Eq.  can be eSubstitute Eqs. –8), , 38. The core temperature of human body is usually set as 37 °C, and the temperature of arterial blood and the initial temperature of the vivo skin can also be set as Ta = 37 °C and T0 = 37 °C. The ambient temperature is described as Te = 20 °C .The parameters of the heat source and the skin in the present study are shown in Tables 21 is used, which is based on the description of protein thermal denaturation rate. And the parameters of the model are shown in Table 319. It is accepted that: 39.The burn assessment model employed proposed by42. So the phase lags will be chosen based on this range.The exact values of the two phase lags for the skin are still under study. However, it is widely agreed that the values fall to the order of magnitude ~ sτq = τT = 0. The expression for the distribution of the temperature rise based on the Pennes model can be expressed as:The DPL heat transfer model can degenerate to the Pennes model by setting Substituting Eq.  into Eq.wb) is set to be zero and the analytical solution is reduced to the Pennes model. The Abaqus software is employed for the finite element analysis. The skin model is meshed into 160,000 elements, which are set to be 8-node linear heat transfer bricks. The grids is densified at the center point of the top surface, as is shown in Fig. A finite element simulation is carried out to validate the analytical solution derived in the present study . For ease of comparing the numerical and analytical solutions, the value of blood perfusion (npulse = 3) and the duty ratio is set to be 0.3 . The boundary condition on the top surface of the model is natural convection condition. And the temperature on the other surfaces is equal to the core temperature T0. The value of the convective heat transfer coefficient is set to be h1 = 10 W/(m2 K) with the assumption of natural convection between the skin and the air. It is shown in Fig. Figure In the following, the thermal response of the skin irradiated by repetitive laser pulses are investigated based on the DPL model.τq and temperature gradient phase lag τT on the temperature distribution induced by repetitive laser pulses. The pulse number is set to be npulse = 10 and the duty ratio is rduty = 0.5 . It is shown in Fig. 44. The non-heating stage is the latter ten seconds when the heat source is taken off and the temperature decreases due to the heat transfer and the convective heat dissipation. Figure x = l/2, y = b/2, z = 0). A great τq leads to a tardy heat conduction progress, which means the temperature variation shows up before the heat transfer starts. On the opposite, a greater τT leads to a faster heat balance progress, which means that more heat flux will be transferred from high-temperature region to the low-temperature region before the temperature variation appears. It is shown in Fig. τq takes a great value, the temperature rises rapidly during the first three pulse periods and then the growth tendency slows down. During the last three pulse periods, the tendency of temperature growth nearly disappears especially when τq = 10 s. In the non-heating stage, a greater τq leads to a faster cooling rate because of the fluctuation characteristic caused by τq. If τq is great enough, the temperature can be even lower than the initial temperature. Figure τT increasing from 0.1 s to 10 s. The larger the value of τT is, the more smoothly the temperature decreases during the non-heating stage. Figure t = 10 s) on τq and τT. It can be found that the peak temperature of the skin induced by the multi-pulse heat source increases with the increment of τq and the decrement of τT. The phase lag of heat flux strengthens the influence of external heat source and leads to intense temperature variation by hindering the heat transfer process. On the contrary, the phase lag of the temperature leads to gentle temperature change by promoting the thermal balance process.Figure npulse = 1 and it decreases obviously with the increment of npulse. Figure rduty = 1, the multi-pulse heat source regenerate to a constant heat source and the temperature changes smoothly. Figure npulse = 1 and then reduces rapidly. With the increasing of npulse, the reduction tendency keeps attenuating and the value of the peak temperature tends to be stable. Figure Figure 21, which is based on the protein denaturation rate estimation, is adopted to assess the burn degree. Figure x=l/2, y=b/2, z= 0). The parameters of the heat source are set to be I0 = 3 × 104 W/m3, rduty = 0.5, npulse = 10. The ordinate is set to be logarithmic to show the thermal damage more distinctly. Figure τq. The phase lag of heat flux postpones the thermal balance progress and aggravates the heat accumulation. Consequently, more intense temperature rise will be induced, which has been shown in Fig. τT is opposite to that of τq. A great τT will promote the heat equilibrium process and reduce the heat accumulation in the irradiated region. As a result, the burn degree decreases with the increasing of τT. It is shown in Fig. τq = 1 s, no irreversible burn occurs in the skin when the temperature gradient phase lag increases to τT = 10 s.During the laser therapy, the skin is subjected to the laser pulses so that the temperature rise and the thermal damage occur in the irradiated region and the tumor cells will be killed. In order to protect the healthy part and kill the tumor efficiently, it is significant to estimate the burn degree in the irradiated skin. In the present study, the Henrique’s modelnpulse. This tendency is tremendous especially when npulse ≤ 5. When npulse keeps increasing, the burn degree tends to that caused by a constant heat source. Figure t= 10 s). Figure npulse decreases from 5 to 1.The parameters of the heat source, including the pulse number and the duty ratio, show crucial effects on the thermal response of the skin. As a result, the thermal damage in the irradiated region is also under the influences of these parameters. Figure t= 7 s, extending downward for about eight seconds and then trends to be stable at the moment t= 15 s. The decrement of the pulse number leads to an increasing burn depth and takes a shorter time for the irreversible burn to appear, especially when npulse ≤ 5. The decreasing of the duty ratio also benefits the thermal damage. However, the appearing moment of the irreversible thermal damage changes little when the value of rduty increases from 0.1 to 1. Figure According to the Henrique’s model, the thermal damage is regarded to be irreversible when Ω reaches 1. In order to describe the irreversibly burnt region more clearly, the irreversible burn depth at the center point of the irradiated region has been calculated and the influences of the duty ratio and the pulse number are as shown in Fig. The temperature response and thermal damage of skin were derived when it is subjected to repetitive pulse laser based on the DPL biological heat transfer model and the Henrique’s burn assessment model.τq is great, the temperature rises rapidly in the initial period of the heating stage and the rise tendency of the temperature gradually disappears as time passes. In addition, the increment of τq also leads to the increasing of the maximum temperature and the thermal damage in the irradiated region. On the contrary, the increasing of τT results in the decrement of the maximum temperature and the burn degree.In the present study, the influences of the two phase lags and the heat source parameters on the temperature distribution and the thermal damage in the skin were researched. When npulse and rduty do not affect the quantity of the energy input but the energy temporal distribution. The peak temperature maximizes when npulse = 1 and decreases obviously with the increment of npulse. As a result, the burn degree and the depth of the irreversible thermal damage decreases with the increment of npulse. The influence of the pulse number is the most obvious when npulse = 1, keeping decreasing with the increment of npulse and gradually disappearing when npulse is great enough . The duty ratio has similar effects on the skin thermal response with the pulse number does. The increment of rduty causes the decrement of the peak temperature, the burn degree and the depth of irreversible burn. However, the influences of the pulse number is more obvious than the duty ratio.The duty ratio and the pulse number obviously affect the thermal response of the skin. The quantity of the total input energy is set to be constant. That is to say, the values of The validation of thermal damage parameter seems difficult because the quantitative description and measure is hard. However the temperature result validation is possible.


When mammalian spermatozoa are released in the female reproductive tract, they are incapable of fertilizing the oocyte. They need a prolonged exposure to the alkaline medium of the female genital tract before their flagellum gets hyperactivated and the acrosome reaction can take place, allowing the sperm to interact with the oocyte. Ionic fluxes across the sperm membrane are involved in two essential aspects of capacitation: the increase in intracellular pH and the membrane hyperpolarization. In particular, it has been shown that the SLO3 potassium channel and the sNHE sodium-proton exchanger, two sperm-specific transmembrane proteins, are necessary for the capacitation process to occur. As the SLO3 channel is activated by an increase in intracellular pH and sNHE is activated by hyperpolarization, they act together as a positive feedback system. Mathematical modeling provides a unique tool to capture the essence of a molecular mechanism and can be used to derive insight from the existing data. We have therefore developed a theoretical model formalizing the positive feedback loop between SLO3 and sHNE in mouse epididymal sperm to see if this non-linear interaction can provide the core mechanism explaining the existence of uncapacited and capacitated states. We show that the proposed model can fully explain the switch between the uncapacitated and capacited states and also predicts the existence of a bistable behaviour. Furthermore, our model indicates that SLO3 inhibition, above a certain threshold, can be effective to completely abolish capacitation. Despite continuous research in reproductive biology over the last two decades, the prevalence of couple infertility (over 12 months) remains around 15%, among which 30% is due to a male infertility factor . The spein vitro studies of mammalian sperm showed that the presence of albumin and bicarbonate in the physiological incubation medium is essential for capacitation to occur   , membrani)  and a cai) , and sini) .i and hyperpolarization observed during capacitation are dependent on the activity of CATSPER, sNHE and SLO3  and the transmembrane electrical potential (Vm) of a mouse epididymal spermatozoon, which includes the feedback between the increase in pHi and hyperpolarization resulting from the activations of sNHE and SLO3.Our model is a minimal two variables model describing the evolution in time of the intracellular pH  is a function of the concentration of HCO3−, The first term on the right side of e sensor  and is ae sensor   and [Ca2+]i , j, for which data is not yet available for mouse sperm, have been set to the values obtained for sodium-protons’ exchangers in fibroblasts .The next contribution to the variations in pHes CAII,  catalysei is the acid loading resulting from the metabolism, taken as a constant. This acid loading, which allows the sperm to reach an equilibrium state by compensating the pHi increase due to the protons’ extruders, includes the contribution of the leak of protons from the acrosome which is an acidic organelle that has been shown to alkalize during capacitation  from an increase in [HCO3−]i, which is followed by the activation of the protein kinase A (PKA) and finally leads to the auto-phosphorylation of the cSrc kinase  is fixed at the value of 7.4 and the temperature is set at 37°C, as usually the case in sperm incubating media. Original source code is available on GitHubThe simulations, using the evolution 3−]e = 15 mM).In order to find if the model reproduces the transition from a depolarized acidic state (not promoting capacitation) to a hyperpolarized alkaline state  of the sperm, we performed a simulation of the incubation of a sperm cell as it is usually done in the laboratory, consisting of the incubation into a solution containing 15 mM bicarbonate  and Vm(t) on i,Vm) ≃. After the 15 mM pulse of bicarbonate at time t = 2 min, the state of the sperm shifts to a pHi of 7.1 and a Vm of −75 mV. This transition of the sperm’s state is here fast, of the timescale of 5 min; however, adding the phosphorylation delay of SLO3 results in a transition’s timescale of 15 min (data not shown), in accordance with data of the literature for capacitation of mouse sperm cells  ≃. This latter state is hyperpolarized and alkaline, and so corresponds to a sperm’s capacitating state, as the one found in 3−]e = 15 mM.The top graph of 3−]e = 10 mM we find three intersections of the nullclines, which are shown in the middle graph of m ≃−70 mV and pHi ≃7.1. In order to gain insight of what is happening here we computed the steady states for all values of [HCO3−]e and plotted these steady states in two diagrams shown on 3−e. Below 8 mM HCO3−e spermatozoa are in uncapacitating states and above 12 mM HCO3−e are found the capacitating states of the sperm, and between these two appears a region (shaded in the figure) where three steady states of the sperm coexist. The analysis of the dynamics of the system shows that the steady states in the middle  ≃, but if the sperm is prepared on the right of the diagram, decreasing the [HCO3−]e to the gray region will keep the sperm at the capacitating state near .Between these two bicarbonate concentrations of 5 and 15 mM however, at e for some time is able to bring the sperm back to the uncapacitating state.That is what we show here with a simulation of an experiment where a hysteresis effect resulting from this bistability arises in the state of the sperm. The simulation, shown on We would like to insist here that a capacitating state does not mean a capacitated state, we are rather assessing the intracellular ionic state consistent with sperm incubated in capacitating conditions.3−e when SLO3 is inhibited at a level of 70% is presented on The proposed minimal model can also be used as a tool in order to predict how the capacitation process can be blocked efficiently. We simulated the time evolution of the sperm’s states when either SLO3 or sNHE is inhibited and have examined the transition between the uncapacitating and capacitating states. The result of the simulation of the incubation of sperm in 15 mM HCOi will increase no more than 0.2 unit, from 6.6 to less than 6.8, and the transmembrane potential will not hyperpolarize below −45 mV. The existence of this inhibition threshold above which the sperm does not capacitate anymore is due to the fact that the system is brought into the bistable region and will therefore remain blocked to the uncapacitated state during incubation.By carrying on the same simulations with different inhibition percentages of SLO3 we find a threshold at 60% above which this ions’ fluxes aspect of capacitation is prevented ; the pHi3−e but with 70% inhibition of sNHE, the effect on capacitation is similar to the SLO3 inhibition  found in the literature for capacitated spermatozoa (Our minimal model gives account of the experimentally observed hyperpolarization and alkalization of the mouse sperm’s cell during capacitation. Indeed the hyperpolarization of the sperm’s cell to −75 mV lies in the range (Vrmatozoa . We emphrmatozoa .i during capacitation, our model shows an increase in pHi from 6.6 before incubation to 7.1 after incubation. The uncapacitated value given by the model was in fact adjusted to the value of 6.6 obtained by i values of 6.6 before capacitation and 7.1 after capacitation are not fixed and do depend on the exact value of the acid load from the metabolism which is a free parameters as mentioned here above.Concerning the evolution of the sperm’s pHi and Vm. This capacitation switch can be reversed back as a sufficiently strong decrease in [HCO3−]e will bring the sperm back to an uncapacitating state. The existence of this hysteresis phenomenon resulting in a capacitation threshold which is different from the decapacitation threshold is a prediction of the model that can be tested experimentally. On a functional level, the predicted hysteresis, inducing a shift in bicarbonate sensitivity, allows the capacitation process to be more robust in presence of local variations of bicarbonate concentration. This ensures that, once capacitated, sperm cells are less sensitive to these fluctuations, providing a possible explanation for how sperm persist capacitating even when they travel through various parts the female genital tract.Our analysis of the sperm’s steady states as a function of the bicarbonate concentration in the incubating medium showed the existence of a bistability region where the sperm can be found in two different states and we showed that once the sperm has reached the hyperpolarized and alkaline state of capacitated sperm, it will remain capacitated regardless of small fluctuations in the bicarbonate concentration. In addition, we show that this hysteresis phenomenon in the cellular response allows the encoding of transient signals, like a pulse in the external bicarbonate concentration, by long-lasting changes both in pHThe reversibility of capacitation is still a matter of debate. Even though the reversibility of capacitation could be abolished by other factors not included in the model, such as the degradation of Catsper , severalConcerningthe inhibition of capacitation, when the sperm is incubated in a 15 mM bicarbonate solution we find a switch at 60% inhibition of SLO3 above which capacitation is prevented. Even though the inhibition of sNHE has similar effect on capacitation in an incubation medium of 15 mM bicarbonate, we find that inhibiting SLO3 is more effective than inhibiting sNHE when the bicarbonate concentration is increased. Indeed, a 95% inhibition of SLO3 prevents capacitation for all possible bicarbonate concentrations that could be found in the genital tract of the mammals , which ii-sensitive Catsper channel activated during capacitation, and moreover, the reversibility of the switch implies that this capacitation’s aspect of calcium influx can be reversed back. In addition, as in a first approximation we have modeled the soluble adenylyl cyclase activity only through the increase in intracellular bicarbonate concentration and not by explicit increases in intracellular calcium concentration which are known to activate this adenylyl cyclase (2+ signal propagation following the activation of Catsper, as well as the subsequent sustained calcium increase in the sperm’s head. To address this, they take into account the Ca2+ release from the redundant nuclear envelope which was modeled to be indirectly gated by variations in intracellular Ca2+ concentration through the production of inositol 1,4,5-trisphosphate from the phospholipase C. Recently, the inclusion of Catsper into a more complex mathematical model of mouse sperm capacitation has been carried out by in silico SLO3’s loss of function, it can not be recovered by overactivation of sNHE. On the contrary, they show that the overactivation of SLO3 can recover the capacitation response of sNHE’s loss of function. This result is now corroborated by our modeling study, highlighting that SLO3 is essential for the capacitation process. Yet we emphasize here again that our model describes the states of the sperm reached after long durations of incubations.The proposed minimal model has allowed us to identify the core molecular mechanism which is likely to control murine sperm capacitation. It may be extended to include calcium dynamics and further refined to the human case. Concerning the increase in intracellular calcium during capacitation, the feedback loop between sNHE and SLO3, which robustly switches the state of the sperm to an elevated intracellular pH, is likely to keep the pH cyclase , an exteTo conclude, using a modeling approach, we have identified a possible core molecular mechanism underlying the transition between the uncapacitated and capacitated state in mouse sperm and studied its dynamic as well as identified pharmacological targets regulating this process. These results may be relevant in the possible development of treatments of male infertility and non-hormonal male contraception, as the inhibition or stimulation of these actors of sperm’s capacitation can modulate the ability of sperm to fertilize the egg. For this, the proposed model should be extended to the human case.Such a minimal model gives direction for further experimental studies aimed at understanding various aspects of sperm dynamics, and moreover, additional phosphorylation and transport mechanisms of interest could be studied by adding explicitly their contribution to this model.


The article analyses drivers as well as coordination mechanisms and instruments for the energy transition in Italy from a multilevel governance perspective. It addresses the structural constraints that influenced the decision-making processes and organisation of the Italian energy sector and the socio-technical challenges opened up by enhancing renewables. The current energy system is making the move from a centralised, path-dependent institutional and organisational structure to a more fragmented and pluralistic one. Renewables and decentralised patterns of production and consumption are key elements of this paradigmatic shift, which is paralleled by a multiplication of decision-making arenas and actors. These actors follow different interests, problem understandings and green growth narratives, increasing the complexity of governing the energy transition. Against this background, community-based renewable energy policy is assuming a very important role and Italy is putting efforts to establish an enabling framework in line with the requirements of the European Union. The goal of this strategy is to foster a positive link between acceptance of the energy transition and decentralised local activities. In the conclusion we address problems and barriers to new modes of governance, and discuss possible approaches to improved cooperation. The Italian energy transition is widely affected by multi-level dynamics. The impact of these dynamics, however, is ambiguous. As in France, direct state intervention in industrial policy  and centralised models had been considered for long time key to achieve economic development and international competitiveness. Over the years and in the era of the liberalisation of the energy sector, centralised frameworks showed their flaws and are nowadays no longer considered to be able to meet the challenges of the energy transition. The decentralisation of energy policy-(making) started in the late 1990s coupled with generous support schemes have been instrumental in fostering the growth of renewables. Decentralisation has favoured the emergence of bottom-up experimentations as well the activation of mutually reinforcing dynamics that have increased the scope and reach of the Italian energy transition. At the same time, the rearrangement of competences over energy policy-making—especially after the 2001 constitutional reform that has included ‘energy’ as matter of competitive legislation between the state and the Italian regions—has somehow caught up Italian renewable energy policy. The country can be described as a ‘quasi-federal’ state in this policy field  dimension of multilevel governance. We ask whether and to what extent decentralisation has been  beneficial to the sector as the lack of effective coordination and scarce public involvement combined with incoherent views on energy transition are dampening the Italian renewable energy policy. Moreover, we ask whether and how innovative bottom-up approaches involving local actors—i.e. energy communities—can help enhancing energy transition pathways in Italy and foster a positive link between acceptance of (renewable) energy projects and decentralised local activities. Indeed, these bottom-up approaches are regarded as a strong means to implement a more democratic and inclusive energy transition  governments in several policy areas granting them legislative and administrative functions. As a result, Italy has been depicted as a state with a ‘quasi-federal’ institutional structure  privatisation of the Italian energy sector continued. These processes were paralleled by the gradual increase of the role of renewables in the Italian energy mix, especially in the electricity sector .The Environment Ministry coordinates the activities concerning greenhouse gas emissions, the transposition of the EC-directives for emission trading, and is also the co-signer of policy measures promoting renewable energy and energy efficiency within the responsibility of the MSE. The responsibility for biomass lies within the Ministry of Agriculture that promotes the utilisation of RES in agriculture sector and provides financial backing for the cultivation of energy crops. Beside these institutions, some responsibilities also rest within the Ministries of Transport, Finance, Culture and National Heritage. Additionally, CIPE  which is chaired by the President of the Council of Ministers is responsible for the co-ordination and horizontal integration of national policies. Its competencies, among many others, include climate change. The committee approves national GHG emissions reduction programs. In addition, there is also the Inter-Ministerial Technical Committee for Emissions of GHGs (CTE) established in 2012 to support the CIPE’s climate-related work.At the sub-national level, key institutional actors are 19 Regions and the two autonomous Provinces (the Provinces of Trento and Bolzano) that are assimilated to regional governments. The role of these actors in Italian energy policy has increased since the 1990s (see below). Coordination between the central level and the subnational level is managed by a ‘System of conferences’ that are institutional bodies designated by the Italian constitution for inter-institutional vertical coordination.A new influential actor was established following the liberalisation of the energy sector, the regulatory authority for electricity and natural gas (at the time named AEEG). Later its mission  was enlarged to include also water and the regulator was renamed in AEEGSI. In the field of research, there are a number of public organisations like the National Research Council (CNR) and the National Agency for New Technologies, Energy and Environment (ENEA) as well as RSE that carries out research in the electricity field with a focus on national strategic projects funded through the Fund for Research into Electrical Systems in energy-efficiency and renewable.Although ENI and ENEL are still key actors in this sector, new players have entered into the scene as decentralisation has enlarged the spectrum and weight of the relevant actors. New entrants with different stakes in energy decentralisation also increased the complexity of the governance of the electricity sector. Differently from France, where the majority of the French political class remains anchored to the “historical” model based on the role of the State . GSE is a publicly owned company that provides incentives for and develops renewables. GSE is also the parent company of Acquirente Unico (AU) that acts as a national wholesaler to protect energy consumers. Finally, Gestore dei Mercati Energetici (GME) is in charge of the electronic platform to exchange power, natural gas and energy tradable certificates . On the market, the number of distributors and retailers has grown and counts on 128 DSO, eight of which have been unbundled . Law No. 10/1991 also provided for energy planning at provincial and municipal levels. Later on, in 2001, with the so-called ‘Turin Protocol’, the Italian regions fully recognised the environmental and climate dimensions of energy policy. Since then, these dimensions have been integrated in Regional Energy and Environmental Plans . In 2001, Italy also enacted an important reform of the Constitution . This reform modified the Title V of the Italian Constitution granting important functions to Regions over energy matters. In particular, the national level stipulates the general guidelines and principles of the Italian energy policy, and Regions and Provinces legislate accordingly. Within this new institutional setting, the process of administrative decentralisation over the Italian energy policy making has been strengthened and consolidated. Especially in the area of renewables, regional and local governments have obtained a very important role in the authorisation regime for the localisation of RES‑E installations as well as in the environmental impact assessment (EIA) procedures.The authorisation regime for the localisation of RES‑E installations is regulated by Legislative Decree No. 387/2003 and No. 28/2011. These laws grant Regions and municipalities the competences over all types of RES‑E installations a part from those offshore, which fall under national competencies. Regions are typically responsible for larger installations whereas municipalities for smaller one, although Regions can also delegate functions to the provincial level. The environmental impact assessment over RES‑E installations regulated mainly by Legislative Decree No. 152/2006 and the subdivision of responsibilities follow the same pattern. This fact further contributed to complicate the governance of renewable development. The siting and deployment of renewables plants has become a very long and complex process. Regional and local governments have a veto power over this process, whereas mechanisms for inter-institutional coordination and public involvement have been poorly developed.Rather than addressing these issues, the national government has sought to recentralise energy policy making. In 2014, the Renzi centre-left government proposed a new Constitutional reform, which among other novelties provided for a recentralisation of decision-making over energy by curtailing the role of the Regions. As the attempts for establishing an upper chamber for regional representatives were spoiled by the result of a failed referendum, the constitutional and institutional setting of the Italian energy policy maintained its quasi-federal structure. This structure is no longer completely centralised because of the redistribution of powers between the State and local authorities, but many features of the old model are still present. The resulting institutional setting has contrasting effects on the Italian energy transition as it triggers both negative and positive policy feedbacks.Although the evolution of renewable energy in Italy has been characterised by incoherent policies inhibiting the development of a national RES industry, the last decades witnessed a sustained growth of RES. Today Italy has the second-largest solar photovoltaic (PV) installed capacity in Europe, but the achievement of some of the 2020 European targets can be considered to be due to the effects of the economic crisis, rather than to the implementation of coherent policies for the promotion of renewable energy and energy efficiency Deloitte . Indeed,In particular, the development of the Italian support schemes has been affected by multilevel dynamics in two ways. In a first period (2001–2010) the problems related to the authorisation procedures have led the government to grant substantial financial supports for renewables. This strategy was intended to ‘compensate’ developers for the administrative costs and risks they faced in investing in a new sector owing an uncertain regulatory framework. In a second period (2011–2019), however, the very high costs reached by the Italian support schemes prompted the government to introduce stringent measures that resulted in a slowdown of renewables development Prontera . These tThe early 2000s witnessed a low level of expansion of renewables despite the introduction of the Green Certificates support scheme which proved unfavourable particularly for PV and smaller installations. In 2005, the Berlusconi government introduced a new feed in premium scheme for PV: the so-called Energy Account (‘Conto Energia’). The subsequent new Prodi government further enhanced the Italian renewable energy policy. Within the context of the implementation of Directive 2009/28/EC, the Italian government set its targets for renewables development in its National Action Plan (NAP): 17% share of renewables in gross final energy consumption by 2020 and 26% for renewables consumption in the electricity sector. To uphold these targets, the government updated the Energy Account, maintaining its generous incentives, and introduced a new feed-in tariff—the so-called All-Inclusive Tariff (AIT) (‘Tariffa Onnicomprensiva’)—for non-PV small generating plants. Also, in 2008, the government introduced some adjustments in order to improve the Green Certificates system and increase the attractiveness of investments in renewable technologies. These innovations proved to be effective in promoting renewables in the country: already in 2010–11 Italy has been able to outdo its EU obligations —retroactively reduced the existing feed-in incentives for PV. Overall, the changes made since the early 2010s have been effective in putting under control the costs of RES‑E support schemes, but affected the development of renewables , with a + 80% increase with respect to 2017. The highest contribution was expected from PV and wind power. In 2018, more than 60% of RES energy was supported by public spending GSE . The redIn 2019, the Italian National Energy and Climate Plan (NECP) confirmed the 55% RES‑E target for 2030. In the same year, with Ministerial Decree of 4 July 2019, the government further updated the current support schemes : policy advice and administrative support for local governments and business actors; licensing and Environmental Impact Assessment (EIA). Local governments are involved in planning, authorisation regime as well as EIA. These latter areas are those that have been more problematic for the Italian energy transition.With regard to planning, since the ‘Turin Protocol’ of 2001, the Italian regions have included European and national climate and decarbonisation targets and objectives in their planning process and the implementation of related policy measures. This has favoured mutually reinforcing dynamics among EU, national and regional level governance. From 2000 to 2009 all the Italian Regions have issued their PEAR and set targets for renewables development. Regional energy planning has been institutionalised later on as regions started a regular process of revision or updated their plans following the EU and national developments.This trend has been reinforced with Law No. 13/2009 that has provided for the redistribution of the EU targets on renewables among the Italian regions. The Ministerial Decree of MISE of 15 March 2012  specified the contribution expected by each Region for achieving the national targets. Against this background, all Italian Regions have enacted regional laws and programmes to further support the development of renewables. Moreover, regional governments increasingly involved in their energy planning local stakeholder and business actors. This has facilitated the embedding of energy transition in local development strategies and aided public acceptance for decisions over energy policy.Regarding the support for renewables, starting from 2007, the EU structural funds begun to play an increasingly important role for implementing regional energy policy. This role has been further enhanced with the 2014–2020 EU cycle of structural funds. The Italian regions have actively used these funds to promote local development strategies based on green economy and renewables. In this case as well, mutually reinforcing dynamics proved to be important in accelerating renewables development. In 2017, almost all the Italian Regions (and the autonomous Provinces of Trento and Bolzano) surpassed the targets assigned them by the national government, the only exceptions being Liguria and Latium.Regional and Provincial governments have also played an important role providing policy advice and administrative support for local governments and business actors. For example, they have acted as territorial coordinators for the implementation of the EU Covenant of the Mayors and the European Local Energy Assistance (ELENA) programmes in Italy , the national government sought to re-centralise some legislative and administrative competences in the energy field, including the area of renewables.These innovations, however, were hardly effective in accelerating authorisation and siting procedures that remain a hurdle for the Italian energy transition. Often, local actors and citizens have a negative attitude towards larger renewables installations. Wind parks or large PV installations—can be at odds with local development strategies as well as with frames of energy transition based on bottom-up practices and decentralised system of production and consumption. Local politics further exploit these tensions which are nurtured by a long legacy of scarce public involvement in energy policy making. These problems are aggravated by an inefficient system of vertical coordination. In particular, the coordination between national and regional levels is assured by the works of the ‘Conferenza Stato-Regioni’, whereas all the administrative actors involved in Italian energy policy  are involved within the ‘Conferenza-Unificata’. As matter of fact, the general mandate of these bodies proved to be poorly suitable for the challenges posed by the energy transition. They neither provide incentives to reconcile different frames and interests between the national and sub-national actors nor they offer appropriate expertise to support complex technical decisions.In 2017, the National Energy Strategy provided for the establishment of a special body—gathering different Ministries as well as regional and local governments—with the aim of steering the implementation of the energy transition. This proposal was relaunched with the 2019 NECP, which bolstered a new governance system—with a political body and a technical body—for increasing coordination among national ministries and regional and local governments. These provisions, however, have not been implemented yet and new mechanisms for public consultation are still missing.As we have described above, the current energy system is in the process of passing from a centralised, path-dependent institutional and organisational structure to a more pluralistic one. The increased share of renewables and decentralised patterns of production and consumption are key elements of the paradigmatic shift, which is paralleled by a multiplication of decision-making arenas and actors. Against this background, community-based renewable energy unfolding different levels of citizens’ participation and ownership of renewable projects has left the niche status and is assuming a more important role. Renewable energy projects led and financed by citizen can be also considered an innovative medium to bridge the investment gap  and off-grid forms of energy production and self-consumption, it was convenient for the Italian government to implement the transposition of the Directive (EU) 2018/2001 (RED II) as soon as possible WEC‑I . A speciFor Legambiente, the largest Italian environmentalist association, the implementation of RECs needs to be accompanied by policies to enhance larger renewable energy plants of varying sizes. These not only contribute to improve the Italian energy balance and enable to achieve the goal of zero net emissions, but also, associated with storage systems, will ensure grid flexibility and security  up to 110% over five years in the field of energy efficiency and new RES installations. However, this has proved to be detrimental rather than incentivising activities, because of additional bureaucratic procedures and the measures aimed at decarbonisation do not seem to be integral part of a broader, long-term strategyWhilst policy targets for energy communities are established at national level, regional administrations provide rules and guidelines to implement RES and energy communities in their territories. In this context, the Piedmont Region played a pioneer role and issued the Regional Law n. 12 of 3 August 2018, promoting the institution of energy communities. Following the establishment of the national regulation in 2020, other Regions issued their regulatory framework for energy communities covering some specific local issues while remaining within the context established by national legislation. Regional legislative initiatives have been pursued in Piedmont, Apulia, Liguria, Calabria and Campania.There are currently various models of community energy ranging from collective self-consumption groups, such as a block of buildings that share a PV system to renewable energy communities as a free association of consumers and prosumers at the level of the same electrical substation. However, there are still only a few examples of energy communities; Legambiente  surveyedAlthough the minimum requirements for an enabling framework (i.e. points a) to e) in RED II Article 22(4)) have been met, there are still issues to be improved. They concern in particular regulatory and capacity-building support provided to public authorities in enabling and setting up renewable energy communities, and in helping authorities to get engaged directly as well as rules to secure the non-discriminatory treatment of consumers that participate in RE ; infrastructural (large projects); institutional  and instrumental (subsidies) (Di Nucci and Russolillo Recently, the Italian government has concentrated on creating an infrastructure for supporting sustainable clean generation, with the aim of meeting carbon neutrality by 2050. However, the Italian energy system has been and still is partially Against the background of persisting path dependences, the emergence and development of community energy can be seen as a decisive step to enhance the energy transition towards renewables. At national governmental level there are some challenges especially concerning the enhancement of a truly “enabling” framework and the lacking availability of the necessary infrastructure, which puts a stop to many activities or innovative schemes that energy communities are working to implement. But this is not a task for the central government alone. Local governments are also called to enhance measures to promote energy communities in achieving energy efficiency and meeting at the same time also mitigation of energy poverty objectives. Local governments and system operators need to get more acquainted with the specific needs and capabilities of the energy communities. Despite these problems, the process of decentralisation has triggered an activism of regional and local governments. This engagement has been instrumental in promoting positive practices in terms of local energy strategies for enhancing renewables.In our excursus we have seen that the new Italian multilevel institutional system has been conductive to both negative policy feedbacks that hindered the energy transition, and mutually reinforcing dynamics that fostered the renewables development. The real conundrum concerns the coordination among different scales as coordinated actions are a prerequisite for low-carbon paths (see also Sarrica et al.


Statistically, road accidents involving pedestrians occur in the autumn and winter months, when outdoor temperatures reach −30 °C. The research presented in this paper investigates the impact of a pedestrian’s head on laminated windscreen, taking into account the effects of external temperature, heating of the windscreen from the inside, and fatigue of the glass. The automotive laminated windscreen under study is made from two layers of glass and a Polyvinyl Butyral (PVB) resin bonding them together. PVB significantly changes its properties with temperature. The Finite Element Method (FEM) simulations of a pedestrian’s head hitting the windscreen of an Opel Astra II at <−30 °C, +20 °C> were performed. The obtained Head Injury Criterion (HIC) results revealed an almost twofold decrease in safety between +20 °C and −20 °C. The same test was then performed taking into account the heating of the windscreen from the inside and the fatigue of the glass layers. Surprisingly, the highest HIC value of all the cases studied was obtained at −30 °C and heating the windscreen. The nature of safety changes with temperature variation is different for the cases of heating, non-heating, and fatigue of glass layers. Glass fatigue increases pedestrian safety throughout the temperature range analysed. The safety of so-called Vulnerable Road Users (VRU), i.e., pedestrians, cyclists, motorcyclists, and all those who are potential victims during a collision with a motor vehicle, is currently one of the leading topics in the field of vehicle safety. In 2007, pedestrian safety tests were made compulsory worldwide and in the European Union  for the One of tests of this type is when a pedestrian’s head strikes the windscreen of a car . The accA car windscreen is a laminated panel usually made up of two layers of glass with Polyvinyl Butyral (PVB) resin bonding them together. The issues of verification of phenomena for laminated sandwich systems with the use of two-dimensional and solid elements have been the subject of scientific studies ,6,7,8,9.In 2002 D’Haene  describeOther studies  were foBased on the author study, the mechanical properties of PVB were found to be strongly temperature dependent. The PVB tensile tests for temperatures from −30 °C up to +20 °C are presented in the author’s publication . It descThe validity of research related to pedestrian safety at temperatures much lower than +20 °C is reflected in the statistics of accidents involving pedestrians in Poland in 2019, prepared by the Police Headquarters .The highest number of accidents involving pedestrians was recorded in the months from October to January. The highest number of such accidents was recorded in December, i.e., 13.8%. In November it was 10.9%. The number of injured and killed is also highest during the autumn and winter months. The highest number of killed was recorded in December, i.e., 16%. In January it was 12%. The highest number of injured was recorded in December, i.e., 13.7%, and in November 11.1%. The pedestrian fatal injuries are mainly related to head injuries during the collision. It is necessary to carry out a verification of the problem of pedestrian safety in various thermal conditions, with particular attention paid to the temperatures prevailing in the autumn and winter months.(1)collision with VRU at a laminated windscreen temperature of −30 °C, all vehicle heating systems were switched off;(2)collision with VRU at a laminated windscreen temperature close to −30 °C, the vehicle heating systems were on during vehicle start-up but the windscreen was not fully heated;(3)collision with VRU at a laminated windscreen temperature close to −30 °C, the vehicle heating systems were on during vehicle start-up but the windscreen was not fully heated, the laminated windscreen subjected to additional fatigue loads from under-window frame operation related to the vehicle body torsional rigidity.The issues of pedestrian head impact on laminated windscreen during an accident involving VRU for operating conditions covered by the temperature range <−30 °C, +20 °C> have been analysed. The following cases have been characterised:When considering cases 2 and 3, attention should be paid to the complex state of operational loads to which the laminated vehicle windscreen is subjected when operating at temperatures characteristic for glass transition of the PVB polymer bonding layer. The additional factors influencing temperature reduction and non-uniform heating of the windscreen include: the derivative of vehicle speed and resistance to motion, the effect of wind, precipitation. The windscreen shown in To investigate the phenomena described in The windscreen model consists of two layers of glass 2.1 mm thick and a PVB layer 0.76 mm thick. Based on the results of works ,19, the The glass material model represented the Johnson–Holmquist material model. That model has a failure criterion based upon the Hugoniot condition. It describes the behaviour of a solid during the sudden action of an external forcing. The Johnson–Holmquist model is described by the following equations.Stress equation:State equation:The Hugoniot limit stresses are expressed by the equation:A, C, n—material constants, t—time, The stresses before failure are described by the following equation:B, C, m—material constants.The uniaxial stresses of complete fracture are represented by equation:D—damage accumulation factor.The current stresses are calculated according to the following equation:The evolution of failure is described by the equation:Characteristics of experimental laminated glass works described in articles ,20,21. TThe characteristics of the polyvinyl butyral resin are described in ,25,26,27t = −30 °C: actual and simulation.The tensile tests have been performed in a virtual environment for an analogous polymer operation range. A Marlow material model has been used to describe the operation of polyvinyl butyral. The engineering stress–strain curve obtained from the simulation was compared with the curve obtained from material tests. The result of the analyses performed using the Finite Element Method (FEM) is presented in It has been proven that the PVB behaviour changes with the temperature change. The stress–strain relationship for the temperature of +20 °C was described by a polynomial of the second order, for the temperature range of +10 to −30 °C by a polynomial of the third order, while the influence of the first extremum increased with decreasing temperature.A very good level of convergence of the solution has been obtained. The Marlow model has been proven to be adequate to describe the material behaviour with such unique course as the revealed one. Used Marlow model allowed to achieve a convergence despite the non-linearity of the curve over the entire range of analysed temperatures.The simulations of head impact on the laminated windscreen have been performed for operation temperatures of +20 to −30 °C without taking the influence of the additional energy stream coming from the heating system into account. The numerical description of the modelling process was based on the information contained in the publications ,30,31,32a—head impactor acceleration, The criterion for assessing the pedestrian safety is the Head Injury Criterion (HIC) described by the relation (9):t2 − t1 below 15 ms are omitted. The lower the maximum HIC value, the smaller the head injury and better VRU safety.It is desirable that the levels of HIC values are as small as possible. The measurable consequences of the analysed solution are: reduction of head injuries and increase of VRU safety. During calculation of the maximum HIC value the time intervals t = 8 ms. The results are shown in It has been found that the maximum displacement of the laminated windscreen at the impactor impact point in the Z-axis direction was x = 34.0 mm for A summary of the results of the maximum windscreen displacement in the Z-axis direction resulting from the head impactor impact, for the temperatures analysed, is shown in After analysis of verified cases it has been proven that the susceptibility of laminated windscreen with a PVB membrane decreases with decreasing temperature which is in line with . The winThe maximum HIC values for the temperature ranges considered are shown in The maximum value for the head injury criterion (HIC) has been obtained for temperature of −20 °C. For all analysed cases of HIC change over time, excluding the temperature of −30 °C, a single extremum was found. For the temperature of −30 °C two extrema have been determined. The phenomenon is related to the nature of changes in the polymer structure, resulting from the transition of PVB to the vitreous—brittle state, which was confirmed by the results of the differential scanning calorimetry (DSC) tests performed by the authors. The representation of the phenomena in the test samples was very high stress value for the first maximum, as shown in t < 20 °C, a very large increase in the HIC values was observed for +10 °C. Moreover, as temperatures drop throughout the year, there is an increase in conditions that adversely affect the course of accidents, i.e., reduced visibility, rain, snow, and fog. There are also pedestrian visibility restrictions. The above-mentioned factors increase the number of accidents involving VRU.Comparing the HIC values at +20 °C—the reference level for which vehicle tests related to pedestrian safety are performed—with the results obtained for tests in range of −30 °C < The analyses of phenomena occurring during the impact of a pedestrian’s head on laminated windscreen in the temperature range of <−30 °C, 0 °C> have been carried out, taking into account the blow heating of the windscreen. The phenomena occurring in the structure of laminated windscreen at negative temperatures, subjected to the influence of a stream of warm air, have been analysed. Based on the heat wave propagation diagram shown in S—cross-sectional area through which heat flows, t—flow time, d—thickness of the body through which heat flows.The amount of heat applied from the heating system to the windscreen (zone A marked in A gradient value of 20 °C was assumed for the purpose of the calculations. The PVB polymer properties for each zone were defined upon the basis of the tests discussed in The character of the HIC curve as a function of temperature was found to be different for the cases of unheated and heated windscreen. In the first case, there is a non-linear relationship between HIC and temperature. The consequence of heating the windscreen was a change to a near linear relationship. The HIC values for the heated windscreen were lower compared to the corresponding temperature levels for the case of unheated windscreen. The value of The effect of fatigue in laminated windscreen has been determined. The assessment of the effect of fatigue of laminated glass panel has been analysed on the example of a car windscreen, considering loads from torsion of the car body during its operation  and consThe analyses of the head impactor hitting the central point of the windscreen were performed for the same sets of temperatures as described in Comparing the HIC values for the cases of windscreen heating and windscreen heating with fatigue, it was found that over the entire temperature range analysed, the safety of VRU would be higher for the glass subjected to operational fatigue. The curve describing the temperature dependence of HIC with fatigue is of a different course and nature to that determined for laminated windscreen with no fatigue loads history. It is a second order curve. The maximum HIC value for the case of heating and fatigue has been obtained for external temperature of −20 °C, i.e., The studies of VRU safety in the case of a pedestrian head collision with a laminated windscreen of a passenger car, taking into account the effects of thermo-mechanical loads, allowed for the formulation of conclusions and observations which have not been known and published so far. Starting with the case of an impact in the absence of windscreen heating and comparing the HIC values for all temperatures analysed, it was proven that pedestrian safety significantly decreases with decreasing temperature. Starting at +10 °C, an increase of 43.9% in the HIC maximum was recorded, and at −20 °C the HIC value was 183.5%, compared to the value obtained at the reference temperature of +20 °C.Analysing the case of VRU head impact on the laminated windscreen at temperatures below the PVB glass transition temperature, taking into account the heating of the windscreen from the inside by the vehicle heating system, a different character of HIC changes was found than the one observed in the analyses without windscreen heating. HIC for windscreen heating increased as the temperature decreased. The nature of changes in HIC parameter is linear. The effect of the decrease in stiffness of the glass layers following the fatigue loads of the ceramic material on the stiffness of the laminated windscreen and the implications of this on the safety of VRU were determined.


The present paper investigates the update of an empirical probability distribution with the results of a new set of observations. The update reproduces the new observations and interpolates using prior information. The optimal update is obtained by minimizing either the Hellinger distance or the quadratic Bregman divergence. The results obtained by the two methods differ. Updates with information about conditional probabilities are considered as well. The present work is inspired by the current practices in Information Geometry ,2,3 wherAssume now that some observation or experiment yields new statistical data. The approach is then to look for a probability distribution that reproduces the newly observed probabilities and that interpolates the data with missing information coming from a prior.No further model assumptions are imposed. Hence, the statistical model under consideration consists of all probability distributions that are consistent with the newly obtained empirical data. Internal consistency of the empirical data ensures that the model is not empty. The update is the model point that minimizes the chosen divergence function from the prior to the manifold of the model.In the context of Maximum Likelihood Estimation (MLE) one usually adopts a parameterized model. The dimension of the model can be kept low and properties of the model can be used to ease the calculations. One assumes that the new data can lead to a more accurate estimation of the limited number of model parameters. It can then happen that the model is misspecified  and thatHere, the model is dictated by the newly acquired empirical data and the update is forced to reproduce the measured data. Finding the probability distribution is then an underdetermined problem. Minimization of the divergence from the prior probability distribution solves the underdetermination.B equalsA and its complement Jeffrey conditioning [In Bayesian statistics, the update B equalsq(B)=pempitioning . It implbabilism . This exThe present approach is a special case of minimizing a divergence function in the presence of linear constraints. See the introduction of  for an oA broad class of generalized Bregman divergences satisfies the Pythagorean equality ,9. PythaOne incentive for starting the present work is a paper of Banerjee, Guo, and Wang ,12. TheyThe next Section fixes notations. A of f given an event A with non-vanishing probability Consider a probability space A is denoted The probability space The outcome of repeated experiments is the empirical probability distribution of the events, denoted A defines a partition A in A is replaced by a partition The event g defined by g yields the empirical probabilitiesB.Introduce the random variable In this section two divergences are reviewed, the squared Hellinger distance and the quadratic Bregman divergence.For simplicity the present section is restricted to the case that the sample space Given two probability measures It satisfiesx belongs to i, with i in Let fined by . ConsideProposition 1.If with equality if and only if First prove the following two lemmas.Lemma 1.Assume that the probability measure andProof. One calculatesNow take Lemma 2.(Pythagorean relation) For any i isProof. The proof follows by taking Proof. (Proposition 1)i then it follows that for all iFrom the previous lemma it follows that A of A. This number is denoted f w.r.t. the probability measure In the present section the squared Hellinger distance, which is an f-divergence, is replaced by a divergence of the Bregman type. In addition let Given a partition of Similarly, conditional probability measures with probability mass function quadratic Bregman divergence.Fix a strictly convex function The following result, obtained with the quadratic Bregman divergence, is more elegant than the result of Lemma 2.Proposition 2.Consider the quadratic Bregman divergence Proof. One calculatesThe following result proves that the standard Kolmogorovian definition of the conditional probability minimizes the Hellinger distance between the prior probability measure Theorem 1.Ω satisfyingLet be given a partition given by . Let pi=and each of the Note that the probability measure Proof. With the notations of the previous section isi. □Proposition 1 shows that it is minimal if and only if Next, consider the use of the quadratic Bregman divergence in the context of a finite probability space.Theorem 2.Let be given a partition given by . Let pi=Then the following hold.a)(A probability distribution ν is defined by b)(Let σ be any probability measure on Ω satisfying c)(The quadratic Bregman divergence Proof. (a)The assumption (6) guarantees that the (b)One calculatesThe term(c)From (b) it follows that i and hence, that i. The latter implies Conversely, when i or that i. In the latter case the prior distribution The optimal update The entropy of the update when event If x in x belongs is larger than or equal to the prior probability Note that Theorem 2 cannot always be applied because it contains restrictions on the empirical probabilities. In particular, if the prior probability The two previous theorems assume that no empirical information is available about conditional probabilities. If such information is present then an optimal choice should make use of it. In one case the solution of the problem is straightforward. If the probabilities B, given the outcome of the measurement of the random variable g as introduced in Let us tackle the problem by considering the case that the only information that is available besides the probabilities The following result is independent of the choice of divergence function.Proposition 3.Fix an event B in Ω. Assume that the conditional probabilities Proof. An obvious choice is to take The optimal updates can be derived easily from Theorem 1. Double the partition by introduction of the following setsThe empirical probability of the set Next consider the optimization with the quadratic Bregman divergence. Probability distributions n is known with certainty. The probability mass function is given byAssume that the prior probability distribution is binomial with parameters k. A fit of the data with a binomial distribution is likely to fail because two accurate data points are given to determine a single parameter The experiment under consideration yields accurate values for the probability The geometric approach followed in the present paper yields an update from the binomial distribution to another distribution, one which is reproducing the data. The update is conducted in an unbiased manner. Quite often one is tempted to replace the model, in the case of the binomial model, by a model with one extra free parameter.Let us see what are the results of minimizing divergence functions. The probability space The optimal update according to Theorem 1, minimizing the Hellinger distance, is given by the probabilitiesThe optimal update according to Theorem 2, minimizing the quadratic Bregman divergence, is given by (7). The auxiliary measures A numerical comparison with It is well known that the use of unmodified prior conditional probabilities is the optimal way for updating a probability distribution after new data become available. The update procedure minimizes the Hellinger distance between prior and posterior probability distributions. For the sake of completeness a proof is given in Theorem 1.Alternatively, one can minimize the quadratic Bregman divergence instead of the Hellinger distance. The result is given in Theorem 2. The conservation of probability is handled in a different way in the two cases, either by multiplying prior probabilities with a suitable factor or by adding an appropriate term.The example of In the present research, the work of Banerjee, Guo, and Wang  was cons


The fascia thoracolumbalis (FTL) is an important component for stabilization and motion control of the lumbar spine. It coordinates the traction forces of the autochthonous muscles of the back (AM) and connects them to the muscles of the abdominal wall, shoulder, and buttocks.The aim of our study was to describe the assessment of the normal FTL and epimysium of the AM in MRI and to identify patterns associated with pathological changes in the lumbar spine.A total of 33 patients were retrospectively evaluated: 15 patients had no pathology at the lumbar spine; six patients had previous hemilaminectomy, three had spondylodesis, two had ventrolisthesis, and seven had scoliosis. The thickness of the FTL and EM was measured, and the adhesion of both structures was assessed.The fascial thickness at the levels of the lumbar vertebral bodies LVB 3 was 1.8, of LVB 4 it was 2.0, of LVB 5 it was 2.1, and at the sacral vertebra SVB 1 it was 1.8 mm. Fascial adhesions together with thickening of the EM occurred at the level of LVB 4 in 36% of the cases independently of the underlying disorder. Only thickening of the EM was seen in 48% of cases at the level of SVB 1. By contrast, adhesion of the FTL without epimysial changes occurred in 36% of cases at the level of LVB 3.Thickening and adhesions at the EM and FTL occurred both postoperatively and in the case of scoliosis. Furthermore, lipomatous and muscular herniation could be detected in the FTL postoperatively. Epimysial and fascial alterations may be imaging manifestations of chronic myofascial back pain and should be included in radiological assessments. The fascia thoracolumbalis (FTL) comprises an anterior and posterior leaflet thus creating a tubular system that surrounds the autochthonous musculature, functionally distributing mechanical forces to the lumbar spine and abdominal musculature. Since the FTL is a central component in segmental stability of the lumbar spine, we investigated the visualization of the FTL and the adjacent epimysium (EM) on magnetic resonance imaging (MRI) and correlated alterations of these anatomical structures with underlying disorders of the lumbar spine.Overall, 33 individuals, among them 18 patients with various pathologies of the lumbar spine (group A), and 15 with low back pain but without any pathologies detected on MRI (group B), were recruited from July to September 2020 and were retrospectively evaluated. All patients were referred for clarification of their clinical complaints and gave written informed consent for the MRI examination. Patients were randomly selected from the examination list (S.O.S) and anonymized for retrospective analysis. For anonymization, a number was assigned to each patient. The radiologist had no information about the clinical complaints. Six patients had previously been treated with hemilaminectomy, three patients had spondylodesis, two patients had ventrolisthesis, and seven patients were primarily notable for marked scoliosis. The median age was 72 years ; 16 patients were female, 17 male. Exclusion criteria were higher-grade stenosis of the spinal canal and neuroforamina.All MRI examinations were performed using a 3.0‑T MRI scanner  at the Radiological Center for Diagnostics and Therapy Munich (rdtm). Sagittal and oblique axial T2-weigthed (T2w) spin-echo (SE) parallel to the respective endplates, a coronary STIR, and a T1-weighted sagittal SE sequence were acquired.The thickness at the FTL and EM was measured at the level of LVB 3, 4, 5 and SVB 1 in the axial and oblique axial plane, respectively, in the Horos image viewing system. The diameter of the FTL and EM was determined in a plane perpendicular to the dorsal surface. An adhesion of the lamina posterior to the FTL was assessed when the fascia could not be distinguished from the EM. Thickening of the interspinous ligament (IS) was considered to be present when it was wider than the corresponding spinous process. When the EM appeared compressed and wavy near the spinous process, this was documented as denticulation. All measurements were performed by a radiologist with more than 10 years’ experience in musculoskeletal MRI (B. A.). The evaluation was carried out without knowledge of the clinical findings. Descriptive statistical analyses were performed using Excel.The MRI examination was clinically indicated for all patients and informed consent was given. The ethical principles of the fifth revision of the Helsinki Declaration on ethical principles for research involving human subjects were observed.The normal anatomy of the FTL and EM was analyzed in group B. The lamina posterior of the FTL paralleled the EM of the erector spinae muscle in a dorsally convex arc. A thin fat band separated the FTL from the EM. The FLT and EM were inseparable near the spinous processes and at the level of SVB 1. The EM and FTL had an average thickness of 1.8 mm , in 19 at the level of LVB 4 (58%), in 12 at the level of LVB (36%), and in two at the level of SVB (6%). As shown in Table When the descriptions at FTL and EM are correlated with the particular pathologies, the following can be noted. In three patients with spondylodesis a long-distance adhesion of the EM with the FTL at the level of the spondylodesis was found. The EM was thickened with a diameter of more than 2 mm (Fig. In both patients with ventrolisthesis at the level of LVB 4–5 and LVB 5‑SVB 1, the EM and FTL were thickened symmetrically and bilaterally Fig.  and no fIn four of six patients with hemilaminectomy, a defect of the FTL was found in the area of the surgical access route. Herniation of fatty and muscular tissue, respectively, through this defect into the subcutaneous tissue was detected Fig. . In all In the seven patients with scoliosis, thickening of the interspinous ligament on the one hand and unilateral thickening of the EM with a lack of differentiation from the FTL on the other hand were noted Fig. .To date, little attention has been paid to imaging of the fascia of the lumbar spine. Langewin et al. employed ultrasound elastography on 71 patients with chronic back pain and found that the FTL plays a critical role in the normal movement of the spine . RestricFor the interpretation of MRI findings, it is important to consider the underlying anatomy , 16, 19.Various observations of this study have not been reported previously, such as the increase in thickness of the EM associated with adhesion or scarring of the FTL. Segmental movement restrictions, as found in the patients with spondylodesis or spondylolisthesis, result in bilateral thickening of the EM and adhesions between the EM and the FTL. In patients with scoliosis, this occurred unilaterally. Another observation not previously described is denticulation of the EM near the spinous process Fig. . Since tThis observational study has various limitations. A small and heterogeneous collective of patients was enrolled. We aimed at generating hypotheses concerning the visualization of the fasciae and the EM and their potential role in myofascial back pain. Prospective studies of larger and homogeneous patient collectives are mandatory in order to assess the diagnostic utility and reproducibility of our findings. Such studies may also elucidate the natural history of the evolution of the alterations of the fascia and EM of the lumbar spine and their impact on myofascial back pain. It will also be indispensable to address changes in the discovertebral unit and facet joints in further studies and to correlate the imaging findings with the clinical assessment of the patients.For the first time, changes at the EM and FTL in the lumbar spine as visualized on MRI were described. Thickening of the EM and FTL occurred postoperatively after spondylodesis and hemilaminectomy as well as in scoliosis, and both structures were sometimes inseparable. Furthermore, postoperative lipomatous and muscular herniation could be detected in the FTL. The EM and fascial changes may be the cause of chronic myofascial back pain and should be given more attention in radiological evaluations of the lumbar spine. Our findings may contribute to better diagnosis of patients suffering from unclear and chronic back pain.


Being able to estimate and predict future microplastic distributions in the environment is one of the major challenges of the rapidly developing field of microplastic research. However, this task can only be achieved if our understanding of the decay of individual microplastic particles is significantly enhanced. Here, we show by using a rate equation model that currently available data of size distributions measured at single times cannot provide useful insights into this process. To analyze what data contains more information we generated more complex artificial data mimicking subsequent measurements using a stochastic simulation algorithm. Applying our model to this data revealed the following minimal requirements for future experimental data: (1) data should be collected as time series at identical spots and (2) size measurements should be combined with mass measurements. In contrast to currently available data, flux rates and decay parameters of individual particles can be extracted from such data. It can be divided into two groups, depending on its origin: primary microplastic is already produced in microscopic size, whereas secondary microplastic originates from the fragmentation of larger plastic items through chemical, physical or biological processes4. It is unclear which type of microplastic is more common in the ocean, but the secondary microplastic is assumed to dominate in the open sea6. It is undisputed, that all types of plastic eventually decay into smaller fragments with time7. However, so far little is known about the decay processes of microplastic particles in the environment. Open questions include the time scales of microplastic decay and its dependence on particle size, material, environmental conditions etc. In a recently developed modeling approach, the decay of buoyant macroplastic has been studied8 providing estimates for the decay rates of microplastic. Furthermore, the transport of microplastic has been included in hydrological models10. However, modeling had so far only a small impact on the field of microplastic research. Nevertheless, we are convinced and as we demonstrate in this study, modeling can provide important approaches to extract more information from field data. Furthermore, ways for standardized monitoring could be paved, that could significantly enhance the field of microplastic modeling. The goal of our approach is to analyze if useful information about the decay process of microplastic particles can be extracted from currently available data. With this in mind, this study works towards three broad-gauged tasks: First, we want to find possibilities to extract decay rates from field data. Second, we aim at creating models based on decay rates that have the potential to project microplastic particle numbers in the future. Third, we seek to identify guidelines for quantifying field data to provide sufficient information for the extraction of decay rates and modeling the fate of microplastic in the environment.Microplastic is commonly defined as plastic particles below five mm in sizeIn our study we first extract decay parameters from single-time field data using a very simplified model that consists of a set of coupled differential equations. Those parameters are fed as decay rates into a simulation based on the Gillespie-algorithm. The Gillespie-algorithm is commonly used for the stochastic simulation of chemical reactions, especially in biochemistry. Here, we apply it to microplastic decay, where the decays are viewed as chemical reactions of first order. This simulation then aims at reconstructing information from field data, based on the identified decay parameters, starting from an initial particle distribution. Finally, we develop a more sophisticated generalized model that we use to extract decay rates from time-series field data and artificially generated time-series data. We use the information of these studies to develop guidelines, how empirical data can meet requirements for modeling purposes.11, observed particle sizes were represented using three bins for size ranges 0.63-5 mm (bin 1), 0.2-0.63 mm (bin 2) and 0.063-0.2 mm (bin 3). However, since time series are seldom available, field data does not yet reflect how such a size distribution changes with time.In most current studies, size distributions of microplastic particles are provided by using a discrete scheme for a single time and location. For example, in a study by Klein et al.Neglect all particle characteristics apart from size.There is no external influx or outflux of particles.As the decay of particles in bin 3 is not further monitored we assume that they do not decay.Upon decay, particles from bin 1 and 2 split up into two particles, both directly leaving their bin in order to go into the next bin.The corresponding set of coupled differential equations is then given byi and To analyze which information about decay rates of microplastic can be extracted from single-time field data, we first set up a simple model consisting of three coupled differential equations and two decay rates. They describe the time evolution of the number of plastic particles in three size categories (bins). To simplify matters first, we made the following assumptions: While the model given by Eq.  does not11 and field data from Cabernard et al.12 which both consist of particle size distributions in three size categories to calculate values for the ratio of decay rates Because our approach is independent of size category boundaries, it is applicable to any size categorization and arbitrary size distributions. To illustrate this, we exemplary used the aforementioned field data from Klein et al. The results for the Klein et al. data (where particle numbers at ten different sites were reported) are shown in Table 11 one sees that the sampling sites having a value of As can be seen from Table The average ratio is In contrast, for the Cabernard et al. dataset, we obtain a value of g to Eq.  a low va14. The observation that the Cabernard et al. microplastic distribution formed at a later point than the Klein et al. size distribution fits well to transport pathways of microplastic, where particles are typically transported from rivers to the open sea3.Thus, the decay rates for the Cabernard et al. data show a much stronger size dependence than the Klein et al. data. In the case of Cabernard et al. large particles tend to decay faster than smaller particles. This is plausible as larger particles have a larger surface and are thus exposed to stronger physical forces. In the case of Klein et al., decay might rather be influenced by factors other than size, that so far have not been taken into account. A possible explanation for this difference is that the Klein et al. data was collected in sediment while the Cabernard et al. data was taken at the water surface, where particle decay is generally stronger and more likely to be influenced by size, due to sun exposure and other size related physical forcesn size bins determine Generally, however, the above analysis shows that even under the simplifying assumptions leading to our very idealistic model given by Eq.  single-t11. The main feature of this simulation is the possibility of tracking each particle, its decays and the size of the resulting particles during the time evolution of the system individually. In contrast to the model in the previous section, we included the more realistic possibility that particles resulting from a decay could stay in their initial size bin if the size after decay required it.To analyze the influence of some of the simplifying assumptions underlying our model, we implemented a scheme to numerically solve Eqs.  [in its 11. The simulation ran until a time of i and the total number of particles at time step Implementation of the algorithm was done as follows: An initial number of particles with size 5 mm was set. We implemented two reactions: a split of particles in bin 1 and bin 2 with rate Table Table Figure mentclass2pt{minimmentclass2pt{minimmentclass2pt{minimIn conclusion, all used decay variants achieve a quite similar ratio between particles in each of the three chosen bins and the total amount of particles (which was the target parameter). The simulation method is thus able to reproduce the target parameter from field data using the previously calculated decay parameter entclass1pt{minimaAt last, influx of particles might have a strong effect on the time evolution of size distributions. So far, this effect is not taken into account in our description. However, it is clear that inclusion of influx terms in our model  leads toThe previous section showed, that we are able to model the decay of microplastic using a Gillespie-algorithm, and to reconstruct field data information under simplified conditions. However, the results were obtained with simplified models which neglected influxes and outfluxes, which is most likely not legitimate in most instances.15, who quantified microplastic particles at three different times  and as calculated from the solution of model (k=mod). Due to the large difference between the The equations  were soli), and by i).Tables 15. Nevertheless, this demonstrates that our simple model can be used to extract relevant parameters from time series, although the time resolution is not sufficient to obtain sufficient information about the decay rates.The low value of 16 that proceeded as follows: we considered three size categories , most studies only report particle counts in size categories. Furthermore, our model approach needs data from subsequent measurements reporting both particle numbers and particle weights for particles in size categories, as already proposed by Metz et al.18. This time series data would then include a size histogram with particle counts and weights. Any increase or decrease could then be associated with particle influx or outflux in order to constrain the influx rates One way to realize this, is to take into account the mass of the particles. On the model side this could be easily done but on the experimental side this requires a specific quantification method. While mass of particles is reported in some studies  that could potentially influence the particle decay and that provide additional information for the theoretical extraction of parameters.For all samples, number of particles and mass abundance should be measured simultaneously for all particle categories (such as shape and composition) and reported in size categories. All data should be provided as time series. In this way it is guaranteed that the corresponding theoretical models can be solved and used to extract the relevant decay rates and influx and outflux parameters. These theoretical models can be obtained from our model given by Eq. (Our theoretical findings support the following suggestions for an improved measurement protocol to obtain more meaningful data: n by Eq.  by generStarting from Eq.  we intromentclass2pt{minima and b. Since a and b are simply particle ratios that can be measured in the field, we extracted ten values for ten spots from the publication by Klein et al.11.We solved these equations for initial conditions16. We implemented this algorithm in Python v3.7 along the lines of Erban et al.22 and Gillespie16.Eqs.  are a co11, i.e. More specifically, we used the 3 size categories of Klein et al.x, particle 2 has the size of the initial particle multiplied with x was randomly chosen between 0 and 1 using a Gaussian distribution with mean value We implemented two reactions: A decay of particles in bin 1 with rate 11. To do so, the fractionsAn optimization routine was implemented, to minimize the differences between the calculated particle size distributions and the experimental data of Ref. The algorithm ran until a time entclass1pt{minima22 and Gillespie16. The following size categories for microplastic particles were used: To create artificial data, a Gillespie-algorithm was implemented in Python v3.7, along the lines of Erban et al.A, B and C correspond to a particle belonging to bin 1, 2 or 3, respectively. This means, for example, that reaction (An initial number of particles tx and x was randomly drawn from a Gaussian distribution between 0 and 1. We used two different Gaussian distributions with mean and standard deviation x an The time after which a reaction occurs is given by:A global minimization routine was used to identify those values of  of Eqs.  and the  of Eqs.  (see mai of Eqs.  was mini