**3. Thermophysical theory of WBC**

The WBC thermophysical theory was formulated at St. Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University) in order to overcome the uncertainty of the technological requirements for specialized devices for WBC. In developing the theory, information on the conduct of WBC procedures and their effectiveness was used [1, 2, 7, 13]. As a criterion for optimizing the WBC technology, it is reasonable to use the duration of the analgesic effect of cryotherapy. The duration of the analgesic effect or effective time (WBC ET) is easy to determine in practice. To carry out computational experiments at ITMO University, a method was developed for calculating the WBC ET [16], which made it possible to perform studies on the optimization of the WBC technology in the mode of numerical experiment.

To calculate the WBC ET, a formula is proposed that relates the positive effect with the degree of approaching the skin surface temperature (*Ts*) to the temperature of the cryogenic damage onset (*Tcr* = 270.5 K), as well as with the area (*fs*) and the duration (*τ*max) of body surface contact with cryogenic gas:

$$
\sigma\_E = f\_s \int\_{\tau=0}^{\tau\_{\text{max}}} \frac{A}{\left(T\_s - T\_{cr}\right)^2} d\tau,\tag{6}
$$

Layers 1 and 2 endure significant hypothermia without any harm; patient's safety is ensured when the violation of the normal temperature distribution does not extends beyond the inner boundary of the BS [13]. In the normal state *tE* = 32°C, within the fat layer, the BS temperature rises 32°C ≤ *tf* ≤ 37°C; the temperature of the muscle layer is equal to the human body core (ВС) temperature *tm* = *tВС* = 37°C. It has been assumed that the thickness of layers 1 and 2 for an average patient has the following

The simplified physical model of a human BS has become the basis of a mathe-

BS is a relatively thin surface layer. The calculated thickness of the BS of an average human is Δ*В<sup>S</sup>* = 16 mm. In this case, the effective diameter of the body is 280 mm [7]. This allows us to describe the processes of heat transfer through the BS

where *h* means the tissue enthalpy, kJ/kg; τ means the time, sec; *qx* means the

The energy equation allows you to simulate thermal processes associated with significant changes in the temperature of the study object. The energy equation describes well the processes of changing the state of aggregation; this provides the mathematical model with certain advantages compared to models built based on the

approximants, it is possible to obtain an algebraic expression suitable for automated

Solving Eq. (8) with respect to the value of enthalpy at the new time layer *h*<sup>0</sup>

where *Δh* ¼ *h*<sup>0</sup> � *h*, *h* means the substance enthalpy value on the current time layer; *Δx* means the step of area elements along the *x* coordinate, m; and Δτ means the time step. Eq. (9) allows you to calculate the enthalpy of the nodal points of the

The structure of the cooling object and temperature distribution in the BS layers are described above (**Figure 3**). Thermophysical properties of the human body shell

Due to the high water content (φ), all human BS tissues have a high heat capacity, which ensures the accumulation of a significant amount of heat. The heat accumulated in the tissues protects the organs of the body core (BC), heart, lungs, kidneys, and liver from frigorism at a sharp decrease in ambient temperature.

*<sup>Δ</sup>qx* <sup>þ</sup> *qvΔ<sup>x</sup> Δτ*

þ *qv,* (7)

þ *qv:* (8)

*<sup>Δ</sup>x<sup>ρ</sup> ,* (9)

,

; *x* means the coordinate along which heat is trans-

.

*ρ ∂h <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> *<sup>∂</sup>qx ∂x*

heat transfer equation [13]. When replacing derivatives with difference

*ρ Δh Δτ* <sup>¼</sup> *<sup>Δ</sup>qx Δx*

*h*<sup>0</sup> ¼ *h* þ

simulated object under known boundary and initial conditions.

values: Δ*<sup>E</sup>* = 2 mm; Δ*<sup>f</sup>* = 2 mm.

heat flux through BS, W/m<sup>2</sup>

tissues are shown in **Table 1**.

Thermal balance of the BS area element

calculations:

we get:

**139**

**4. Mathematical model of the human body shell**

*Technique and Technology of Whole-Body Cryotherapy (WBC)*

*DOI: http://dx.doi.org/10.5772/intechopen.83680*

ferred, m; and *qv* means the heat of metabolism, W/m3

tissues by one-dimensional energy equation:

matical model.

where *A* is an empirical constant, providing the calculation of the WBC ET in seconds, *A* = 1200.

The maximum duration of WBC (τmax) is determined taking into account the requirements of the patient's hypothermic safety, which limit the permissible changes in body temperature on the surface (*Ts*) and on the inner boundary of the subcutaneous fat layer (*Tf*) (**Figure 3**).

Compliance with the established limitations of changing the value of *Ts* and *Tf* protects the patient from the danger of frostbite and frigorism, respectively:

$$T\_\text{'} \ge 271 \text{ K (}-2^\circ \text{C)}; \ T\_f \text{'} \ge 309 \text{ K (}36^\circ \text{C)}.$$

**Figure 3** shows a graphical representation of the patient's body shell (BS). BS is the outer layer of the body, the mass of which is 30% of the total body mass. BS consists of three types of tissues: epithelium 1, adipose tissue 2, and muscle tissue 3.

**Figure 3.** *Physical model of the human body shell (BS).*

*Technique and Technology of Whole-Body Cryotherapy (WBC) DOI: http://dx.doi.org/10.5772/intechopen.83680*

**3. Thermophysical theory of WBC**

*Low-temperature Technologies*

technology in the mode of numerical experiment.

seconds, *A* = 1200.

**Figure 3.**

**138**

*Physical model of the human body shell (BS).*

subcutaneous fat layer (*Tf*) (**Figure 3**).

the duration (*τ*max) of body surface contact with cryogenic gas:

*τ<sup>E</sup>* ¼ *fs*

ð*<sup>τ</sup>*max *τ*¼0

The WBC thermophysical theory was formulated at St. Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University) in order to overcome the uncertainty of the technological requirements for specialized devices for WBC. In developing the theory, information on the conduct of WBC procedures and their effectiveness was used [1, 2, 7, 13]. As a criterion for optimizing the WBC technology, it is reasonable to use the duration of the analgesic effect of cryotherapy. The duration of the analgesic effect or effective time (WBC ET) is easy to determine in practice. To carry out computational experiments at ITMO University, a method was developed for calculating the WBC ET [16], which made it possible to perform studies on the optimization of the WBC

To calculate the WBC ET, a formula is proposed that relates the positive effect with the degree of approaching the skin surface temperature (*Ts*) to the temperature of the cryogenic damage onset (*Tcr* = 270.5 K), as well as with the area (*fs*) and

where *A* is an empirical constant, providing the calculation of the WBC ET in

The maximum duration of WBC (τmax) is determined taking into account the requirements of the patient's hypothermic safety, which limit the permissible changes in body temperature on the surface (*Ts*) and on the inner boundary of the

Compliance with the established limitations of changing the value of *Ts* and *Tf*

*Ts* ≥ 271 K (�2°C); *Tf* > 309 K (36°C).

**Figure 3** shows a graphical representation of the patient's body shell (BS). BS is the outer layer of the body, the mass of which is 30% of the total body mass. BS consists of three types of tissues: epithelium 1, adipose tissue 2, and muscle tissue 3.

protects the patient from the danger of frostbite and frigorism, respectively:

*A*

ð Þ *Ts* � *Tcr* <sup>2</sup> *<sup>∂</sup>τ,* (6)

Layers 1 and 2 endure significant hypothermia without any harm; patient's safety is ensured when the violation of the normal temperature distribution does not extends beyond the inner boundary of the BS [13]. In the normal state *tE* = 32°C, within the fat layer, the BS temperature rises 32°C ≤ *tf* ≤ 37°C; the temperature of the muscle layer is equal to the human body core (ВС) temperature *tm* = *tВС* = 37°C. It has been assumed that the thickness of layers 1 and 2 for an average patient has the following values: Δ*<sup>E</sup>* = 2 mm; Δ*<sup>f</sup>* = 2 mm.

The simplified physical model of a human BS has become the basis of a mathematical model.

### **4. Mathematical model of the human body shell**

BS is a relatively thin surface layer. The calculated thickness of the BS of an average human is Δ*В<sup>S</sup>* = 16 mm. In this case, the effective diameter of the body is 280 mm [7]. This allows us to describe the processes of heat transfer through the BS tissues by one-dimensional energy equation:

$$
\rho \frac{\partial h}{\partial \pi} = \frac{\partial q\_{\chi}}{\partial \chi} + q\_{v^{\bullet}} \tag{7}
$$

where *h* means the tissue enthalpy, kJ/kg; τ means the time, sec; *qx* means the heat flux through BS, W/m<sup>2</sup> ; *x* means the coordinate along which heat is transferred, m; and *qv* means the heat of metabolism, W/m3 .

The energy equation allows you to simulate thermal processes associated with significant changes in the temperature of the study object. The energy equation describes well the processes of changing the state of aggregation; this provides the mathematical model with certain advantages compared to models built based on the heat transfer equation [13]. When replacing derivatives with difference approximants, it is possible to obtain an algebraic expression suitable for automated calculations:

$$
\rho \frac{\Delta h}{\Delta \tau} = \frac{\Delta q\_{\infty}}{\Delta \infty} + q\_{v} \,\tag{8}
$$

Solving Eq. (8) with respect to the value of enthalpy at the new time layer *h*<sup>0</sup> , we get:

$$h' = h + \frac{\left(\Delta q\_x + q\_v \Delta \pi\right) \Delta \pi}{\Delta \pi \rho},\tag{9}$$

where *Δh* ¼ *h*<sup>0</sup> � *h*, *h* means the substance enthalpy value on the current time layer; *Δx* means the step of area elements along the *x* coordinate, m; and Δτ means the time step. Eq. (9) allows you to calculate the enthalpy of the nodal points of the simulated object under known boundary and initial conditions.

The structure of the cooling object and temperature distribution in the BS layers are described above (**Figure 3**). Thermophysical properties of the human body shell tissues are shown in **Table 1**.

Due to the high water content (φ), all human BS tissues have a high heat capacity, which ensures the accumulation of a significant amount of heat. The heat accumulated in the tissues protects the organs of the body core (BC), heart, lungs, kidneys, and liver from frigorism at a sharp decrease in ambient temperature. Thermal balance of the BS area element


**Table 1.**

*Properties of the human BS tissues [17].*

$$
\Delta h = h' - h = \frac{(\Delta q\_x + q\_v \Delta \pi) \Delta \pi}{\Delta \pi \rho},
\tag{10}
$$

replace WBC with cheaper water procedures have been ongoing for 40 years [18]. Proponents of such a replacement do not take into account the fact that the WBC ET is more than 360 minutes, and water treatments provide pain relief for a maximum of 30 minutes. Such a difference in efficiency should be based on the fundamental differences between the results of heat removal by liquid and gaseous

*Technique and Technology of Whole-Body Cryotherapy (WBC)*

*DOI: http://dx.doi.org/10.5772/intechopen.83680*

Simulation of the BS surface cooling process with water with a temperature of 273 K and a cryogenic gas with a temperature of 140 K has allowed revealing such a difference (**Figure 4**), which reduces to the level of the minimum surface temperature of the cooling object. In cold water, the minimum surface temperature of the cooling object is at least 5.5°C. As the temperature difference between the water and the cooled surface decreases, the intensity of convective heat removal to water decreases to less than 950 W/m2 [19]. The intensity of heat supply from the inner

. Due to the small difference in heat fluxes at the boundary of the cooling

*<sup>∂</sup>Ti*¼<sup>1</sup>*=∂<sup>τ</sup>* � *const:* (15)

layers of the body to the surface, on the contrary, approaches the level of

object, the rate of temperature decrease *Ti =* <sup>1</sup> reduces to 0.01 K/s; therefore, an increase in the duration of water hypothermia leads only to overcooling the patient's body. The estimated duration of the analgesic effect after water hypothermia is

gives a completely different picture. The graph of surface temperature changes (**Figure 3**) is almost a straight line, which means that the temperature decreases

The use of a cryogenic gas with a temperature of 140 K for cooling the BS surface

The minimum temperature value *Ti =* <sup>1</sup> = 271 K was obtained due to the termination of the numerical experiment on the patient's safety conditions during WBC:

In the case of cooling with a cryogenic gas, the BS surface is supercooled to the minimum acceptable level. WBC technology is based on the use of this hypothermia to stimulate the cold receptors of the skin. Expression (6) for calculating the duration of positive effects contains a term that allows us to illustrate the intensity of stimulation of cold receptors by changing the temperature of the BS surface. This parameter of the WBC procedure is called the intensity of stimulating action (ISA):

*ISA* ¼ *A=*ð Þ *Ti*¼<sup>1</sup> � *Tkr*

2

*:* (16)

heat carrier.

880 W/m<sup>2</sup>

**Figure 4.**

**141**

*Dependence of skin temperature on time.*

31 minutes [16].

almost without the rate change:

*Ts* ≥ 271 K (�2°C); *Tf* > 309 K (36°C).

determined by the ratio of the intensity of the heat fluxes transferred by the thermal conductivity of the tissue along the *x* coordinate Δ*qx* and the heat released by internal sources *qv*. The intensity and direction of heat transfer by thermal conductivity depend on the temperature distribution along the *x* coordinate:

$$
\Delta q\_x = q\_{i+1} + q\_{i-1}; q\_{i+1} = \lambda \frac{T\_{i+1} - T\_i}{\Delta \varkappa}; q\_{i-1} = \lambda \frac{T\_{i-1} - T\_i}{\Delta \varkappa}.\tag{11}
$$

At BS boundaries, heat transfer is described by boundary conditions. For the outer boundary, the intensity of the convective heat removal is calculated:

$$i = \mathbf{1}; \quad q\_{i-1} = a(T\_{\mathbf{g}} - T\_i), \tag{12}$$

where α means the heat transfer coefficient at the natural convection of gas or liquid, *α* ¼ *f Tg; Ti* , and W/(m2 �K); *Tg* means the temperature of the heatremoving medium, K.

On the inner BS boundary, the temperature of the tissues during WBC does not change and is equal to the body core temperature (0):

$$i = n; \quad T\_{i+1} = T\_{BC} = \text{const.} \tag{13}$$

Eq. (10) describes the change in the heat content of tissues over time *hi* ¼ *f*ð Þ*τ* , and in order to form the trial of the WBC technology issues, a similar dependence for temperature should be obtained **Т***i*¼*f*ð Þ*τ* . At each time step, the temperature value is calculated from the known value of the enthalpy of the array elements:

$$T\_i = f(h\_i). \tag{14}$$

The described algorithm of computations forms the mathematical model of the human BS, which is suitable for studies of processes of the therapeutic effect of lowtemperature liquids and gases.

#### **5. Thermophysical bases of achievement of the WBC healthcare effect**

The mathematical model of the human BS allowed us to perform a numerical experiment to study the physical bases of the WBC healthcare efficacy. Quite often [4, 5] WBC is compared with the cold water immersion procedures. The basis for this comparison is that cryogenic gas and cold water remove a significant amount of heat from the body surface. Moreover, under conditions of natural convection, the coefficient of heat transfer from a source of heat to gas is usually 10 times lower than when heat is removed by water [7]. In reliance on this information, attempts to *Technique and Technology of Whole-Body Cryotherapy (WBC) DOI: http://dx.doi.org/10.5772/intechopen.83680*

replace WBC with cheaper water procedures have been ongoing for 40 years [18]. Proponents of such a replacement do not take into account the fact that the WBC ET is more than 360 minutes, and water treatments provide pain relief for a maximum of 30 minutes. Such a difference in efficiency should be based on the fundamental differences between the results of heat removal by liquid and gaseous heat carrier.

Simulation of the BS surface cooling process with water with a temperature of 273 K and a cryogenic gas with a temperature of 140 K has allowed revealing such a difference (**Figure 4**), which reduces to the level of the minimum surface temperature of the cooling object. In cold water, the minimum surface temperature of the cooling object is at least 5.5°C. As the temperature difference between the water and the cooled surface decreases, the intensity of convective heat removal to water decreases to less than 950 W/m2 [19]. The intensity of heat supply from the inner layers of the body to the surface, on the contrary, approaches the level of 880 W/m<sup>2</sup> . Due to the small difference in heat fluxes at the boundary of the cooling object, the rate of temperature decrease *Ti =* <sup>1</sup> reduces to 0.01 K/s; therefore, an increase in the duration of water hypothermia leads only to overcooling the patient's body. The estimated duration of the analgesic effect after water hypothermia is 31 minutes [16].

The use of a cryogenic gas with a temperature of 140 K for cooling the BS surface gives a completely different picture. The graph of surface temperature changes (**Figure 3**) is almost a straight line, which means that the temperature decreases almost without the rate change:

$$
\partial T\_{i=1}/\partial \mathfrak{r} \sim \text{const.} \tag{15}
$$

The minimum temperature value *Ti =* <sup>1</sup> = 271 K was obtained due to the termination of the numerical experiment on the patient's safety conditions during WBC: *Ts* ≥ 271 K (�2°C); *Tf* > 309 K (36°C).

In the case of cooling with a cryogenic gas, the BS surface is supercooled to the minimum acceptable level. WBC technology is based on the use of this hypothermia to stimulate the cold receptors of the skin. Expression (6) for calculating the duration of positive effects contains a term that allows us to illustrate the intensity of stimulation of cold receptors by changing the temperature of the BS surface. This parameter of the WBC procedure is called the intensity of stimulating action (ISA):

$$I\_{\rm SA} = A / \left( T\_{i=1} - T\_{kr} \right)^2. \tag{16}$$

**Figure 4.** *Dependence of skin temperature on time.*

*<sup>Δ</sup><sup>h</sup>* <sup>¼</sup> *<sup>h</sup>*<sup>0</sup> � *<sup>h</sup>* <sup>¼</sup> *<sup>Δ</sup>qx* <sup>þ</sup> *qvΔ<sup>x</sup> Δτ*

**Tissue <sup>ρ</sup>, kg/m3 <sup>φ</sup>, %** *<sup>с</sup>***, J/(kg**�**K) <sup>λ</sup>, W/(m**�**K)** *qv***, W/m<sup>3</sup>** Skin 1093 72.0 3600 0.35 10,996 Muscles 1041 80.0 3458 0.475 7277 Adipose tissue 916 20 2250 0.21 —

determined by the ratio of the intensity of the heat fluxes transferred by the thermal conductivity of the tissue along the *x* coordinate Δ*qx* and the heat released by internal sources *qv*. The intensity and direction of heat transfer by thermal conductivity depend on the temperature distribution along the *x* coordinate:

At BS boundaries, heat transfer is described by boundary conditions. For the

*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*; qi*�<sup>1</sup> <sup>¼</sup> *<sup>α</sup> Tg* � *Ti*

where α means the heat transfer coefficient at the natural convection of gas or

On the inner BS boundary, the temperature of the tissues during WBC does not

Eq. (10) describes the change in the heat content of tissues over time *hi* ¼ *f*ð Þ*τ* , and in order to form the trial of the WBC technology issues, a similar dependence for temperature should be obtained **Т***i*¼*f*ð Þ*τ* . At each time step, the temperature value is calculated from the known value of the enthalpy of the array elements:

The described algorithm of computations forms the mathematical model of the human BS, which is suitable for studies of processes of the therapeutic effect of low-

**5. Thermophysical bases of achievement of the WBC healthcare effect**

The mathematical model of the human BS allowed us to perform a numerical experiment to study the physical bases of the WBC healthcare efficacy. Quite often [4, 5] WBC is compared with the cold water immersion procedures. The basis for this comparison is that cryogenic gas and cold water remove a significant amount of heat from the body surface. Moreover, under conditions of natural convection, the coefficient of heat transfer from a source of heat to gas is usually 10 times lower than when heat is removed by water [7]. In reliance on this information, attempts to

outer boundary, the intensity of the convective heat removal is calculated:

*Ti*þ<sup>1</sup> � *Ti*

*<sup>Δ</sup><sup>x</sup> ; qi*�<sup>1</sup> <sup>¼</sup> *<sup>λ</sup>*

�K); *Tg* means the temperature of the heat-

*i* ¼ *n; Ti*þ<sup>1</sup> ¼ *TBC* ¼ *const:* (13)

*Ti* ¼ *f h*ð Þ*<sup>i</sup> :* (14)

*<sup>Δ</sup>qx* <sup>¼</sup> *qi*þ<sup>1</sup> <sup>þ</sup> *qi*�<sup>1</sup>*; qi*þ<sup>1</sup> <sup>¼</sup> *<sup>λ</sup>*

, and W/(m2

change and is equal to the body core temperature (0):

liquid, *α* ¼ *f Tg; Ti*

**Table 1.**

*Properties of the human BS tissues [17].*

*Low-temperature Technologies*

removing medium, K.

temperature liquids and gases.

**140**

*<sup>Δ</sup>x<sup>ρ</sup> ,* (10)

*Ti*�<sup>1</sup> � *Ti*

*,* (12)

*<sup>Δ</sup><sup>x</sup> :* (11)

The dependence graph *ISA* = *f*(*Ti =* 1) (**Figure 5**) shows how the receptor signal increases with the skin surface supercooling. At the lowest possible temperature of the skin surface under conditions of water hypothermia (5°C), *ISA* = 21 s/s, while at WBC the maximum value is 225 times higher than *ISA* = 4800 s/s. Differences in the intensity of stimulation of cold receptors determine the therapeutic benefits of WBC.

Data on the amount of heat *Q HC* removed from a BS surface unit by a heat carrier (HC) and heat flux intensity has the fundamental importance for the development of WBC technology *qHC*. The total heat removal is determined by integrating the instantaneous values *qHCqHC*, which were calculated by Eq. (12):

$$Q\_{HC} = \int\_{\mathbf{r}=0}^{\tau\_{\text{max}}} q\_{HC} \, d\mathbf{r}.\tag{17}$$

values, at the beginning of the procedure, and the minimum values, at the time of completion of the cooling powers of the heat flux to the cold water and the gaseous heat carrier. For designing WBC devices, it is useful to know the mean value of the heat flux, which the heat carrier must remove in a single procedure, 2.9 kW/m2. This value is 29 times greater than the nominal calorific capacity of the human body; therefore, it is often challenged by manufacturers of WBC devices [13]. Estimation of the heat reserve in BS tissues before and after the procedure shows that the heat flux to the heat carrier is provided by the heat capacity of the body

*Technique and Technology of Whole-Body Cryotherapy (WBC)*

*DOI: http://dx.doi.org/10.5772/intechopen.83680*

Lowering the surface temperature of the BS creates conditions for increasing heat transfer with thermal conductivity from the deep to the periphery of the body. As a result, there is a change in the distribution of the tissue temperature throughout the entire thickness of the BS. The amount of heat removed from different BS tissues is determined by the enthalpy difference before and after the WBC procedure. Taking into account the constancy of the heat capacity of the tissues in the temperature range from �2 to 40°C [17], the amount of heat removed can be

*<sup>i</sup>* � *<sup>T</sup><sup>τ</sup>*¼*τ*max

The total amount of heat released due to supercooling of each of the three types of BS tissues is the sum of portions of heat released in the area elements of this tissue

where *nE*, *nF*, and *nBS* means the number of area elements located between the outer surface and the inner boundary of the epithelial layer, the fat layer, and the

*<sup>i</sup> :* (18)

*QAi* ¼ *ΔTiΔxρici:* (19)

*<sup>i</sup>*¼*nF*þ<sup>1</sup>*ΔTiΔxρici,*

(20)

*<sup>i</sup>*¼<sup>1</sup>*ΔTiΔxρici;*

*<sup>i</sup>*¼*nE*þ<sup>1</sup>*ΔTiΔxρici;*

*<sup>Δ</sup>Ti* <sup>¼</sup> *<sup>T</sup><sup>τ</sup>*¼<sup>0</sup>

The amount of accumulated heat removed from one area element:

calculated from the temperature difference:

*The temperature of the body shell before and after the WBC procedure.*

*epithelial layer,* <sup>1</sup><sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> *nE, QAE* <sup>¼</sup> <sup>∑</sup>*nE*

*fat layer, nE* <sup>þ</sup> <sup>1</sup><sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> *nF, QAF* <sup>¼</sup> <sup>∑</sup>*nF*

patient's body shell, respectively.

*muscular layer, nF* <sup>þ</sup> <sup>1</sup><sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> *nBS, QAM* <sup>¼</sup> <sup>∑</sup>*nBS*

layer:

**143**

**Figure 6.**

shell tissues (**Figure 6**).

The calculated values are given in **Table 2**. In cryogenic gas, heat removal was 440 kJ/m<sup>2</sup> , which is 10% more than in cold water. The result obtained is significantly less than that supposed by some WBC popularizers who estimate heat removal from the patient's body at 1250–2500 kJ/m<sup>2</sup> [20].

At the same time, the result obtained is significantly more than can be removed from a WBC device by using 2 kg of liquid nitrogen for one patient's cooling [6, 10].

The assessment of the power of the specific heat flux, which the BS surface gives to the heat carrier, has essential practical importance. **Table 1** shows the maximum

#### **Figure 5.**

*Dependence of ISA value on body surface temperature.*


#### **Table 2.**

*The results of a numerical experiment on the simulation of heat removal by water with a temperature of 273 K and gas with a temperature of 140 K [16].*

*Technique and Technology of Whole-Body Cryotherapy (WBC) DOI: http://dx.doi.org/10.5772/intechopen.83680*

The dependence graph *ISA* = *f*(*Ti =* 1) (**Figure 5**) shows how the receptor signal increases with the skin surface supercooling. At the lowest possible temperature of the skin surface under conditions of water hypothermia (5°C), *ISA* = 21 s/s, while at WBC the maximum value is 225 times higher than *ISA* = 4800 s/s. Differences in the intensity of stimulation of cold receptors determine the therapeutic benefits of

Data on the amount of heat *Q HC* removed from a BS surface unit by a heat carrier (HC) and heat flux intensity has the fundamental importance for the development of WBC technology *qHC*. The total heat removal is determined by integrat-

> ð*<sup>τ</sup>*max *τ*¼0

The calculated values are given in **Table 2**. In cryogenic gas, heat removal was

At the same time, the result obtained is significantly more than can be removed from a WBC device by using 2 kg of liquid nitrogen for one patient's cooling [6, 10]. The assessment of the power of the specific heat flux, which the BS surface gives to the heat carrier, has essential practical importance. **Table 1** shows the maximum

**Results Heat carrier**

Cooling time, τmax, sec 159 177 Minimum surface temperature of the object,*Ts*, °C �2.0 5.5 The minimum temperature at the inner boundary of the fat layer,*Tf*, °C 309.2 309.0 Heat removed by heat carrier from the body surface, *QHC*, kJ/m<sup>2</sup> 440 410 Heat removed through the inner boundary of the fat layer, *Qf*, kJ/m<sup>2</sup> 10.2 12.5

*The results of a numerical experiment on the simulation of heat removal by water with a temperature of 273 K*

Heat flux from the body surface at the beginning of the cooling process, *qmax*

Heat flux from the body surface at the end of the cooling process, *q*min

, which is 10% more than in cold water. The result obtained is signifi-

*qHC∂τ:* (17)

**gas water**

*HC* , kW/m<sup>2</sup> 3.5 11.3

*HC* , kW/m2 2.3 0.95

ing the instantaneous values *qHCqHC*, which were calculated by Eq. (12):

*QHC* ¼

removal from the patient's body at 1250–2500 kJ/m<sup>2</sup> [20].

cantly less than that supposed by some WBC popularizers who estimate heat

WBC.

*Low-temperature Technologies*

440 kJ/m<sup>2</sup>

**Figure 5.**

**Table 2.**

**142**

*Dependence of ISA value on body surface temperature.*

*and gas with a temperature of 140 K [16].*

values, at the beginning of the procedure, and the minimum values, at the time of completion of the cooling powers of the heat flux to the cold water and the gaseous heat carrier. For designing WBC devices, it is useful to know the mean value of the heat flux, which the heat carrier must remove in a single procedure, 2.9 kW/m2. This value is 29 times greater than the nominal calorific capacity of the human body; therefore, it is often challenged by manufacturers of WBC devices [13]. Estimation of the heat reserve in BS tissues before and after the procedure shows that the heat flux to the heat carrier is provided by the heat capacity of the body shell tissues (**Figure 6**).

**Figure 6.** *The temperature of the body shell before and after the WBC procedure.*

Lowering the surface temperature of the BS creates conditions for increasing heat transfer with thermal conductivity from the deep to the periphery of the body. As a result, there is a change in the distribution of the tissue temperature throughout the entire thickness of the BS. The amount of heat removed from different BS tissues is determined by the enthalpy difference before and after the WBC procedure. Taking into account the constancy of the heat capacity of the tissues in the temperature range from �2 to 40°C [17], the amount of heat removed can be calculated from the temperature difference:

$$
\Delta T\_i = T\_i^{\tau=0} - T\_i^{\tau=\tau\_{\text{max}}}.\tag{18}
$$

The amount of accumulated heat removed from one area element:

$$Q\_{Ai} = \Delta T\_i \Delta \mathbf{x} \,\rho\_i \mathbf{c}\_i. \tag{19}$$

The total amount of heat released due to supercooling of each of the three types of BS tissues is the sum of portions of heat released in the area elements of this tissue layer:

$$\begin{array}{ll}\text{eta} & \text{iphelial layer}, \quad \mathbf{1} \le i \le n\_{\mathrm{E}}, \quad \mathbf{Q}\_{AE} = \sum\_{i=1}^{n\_{\mathrm{E}}} \Delta T\_{i} \Delta \mathbf{x} \, \rho\_{i} c\_{i};\\\text{fat løyer}, \quad n\_{\mathrm{E}} + \mathbf{1} \le i \le n\_{\mathrm{F}}, \quad \mathbf{Q}\_{AF} = \sum\_{i=n\_{\mathrm{E}}+1}^{n\_{\mathrm{F}}} \Delta T\_{i} \Delta \mathbf{x} \, \rho\_{i} c\_{i};\\\text{unsular løyer}, \quad n\_{\mathrm{F}} + \mathbf{1} \le i \le n\_{\mathrm{RS}}, \quad \mathbf{Q}\_{AM} = \sum\_{i=n\_{\mathrm{F}}+1}^{n\_{\mathrm{RS}}} \Delta T\_{i} \Delta \mathbf{x} \, \rho\_{i} c\_{i};\end{array} \tag{20}$$

where *nE*, *nF*, and *nBS* means the number of area elements located between the outer surface and the inner boundary of the epithelial layer, the fat layer, and the patient's body shell, respectively.

**Figure 7.** *Sources of the heat gained the WBC procedure.*

Part of the heat removed was obtained from internal sources in the epithelial and muscle layers, heat of metabolism *QMH*. This heat was calculated by a known value *qv* (**Table 1**):

$$Q\_{\rm MHE} = \tau\_{\rm max} \Delta \mathbf{x} (n\_E - \mathbf{1}) q\_{\rm VE}; \ Q\_{\rm MHM} == \tau\_{\rm max} \Delta \mathbf{x} (n\_{\rm RS} - n\_F - \mathbf{1}) q\_{\rm VM}, \tag{21}$$

where *qVE* and *qVMqVM* means the specific heat of metabolism of epithelium and muscles, respectively, W/m<sup>3</sup> .

Some of the heat removed came from the patient's body core; the amount of heat gained can be determined by numerical integration and instantaneous values of the heat flux transferred by thermal conductivity through the inner boundary of the body shell:

$$Q\_{BC} = \int\_{\mathbf{r}=0}^{\mathbf{r}\_{\text{max}}} q\_{n\_{i+1}} \, \text{d}\mathbf{r}.\tag{22}$$

differ significantly; therefore, the technology of group and individual WBC should

The GWBC technology was influenced by the design of the device for performing the procedures (**Figure 1**). Using a low-temperature food storage chamber for WBC procedures, Japanese engineers and doctors were forced to carry out WBC procedures in groups. The dimensions of the chamber were too large for individual procedures. This forced solution is contrary to the general practice of

regimes of GWBC and IWBC are fundamentally different.

*Technique and Technology of Whole-Body Cryotherapy (WBC)*

*DOI: http://dx.doi.org/10.5772/intechopen.83680*

physiotherapy; treatment is always carried out individually.

temperature of the gas in the WBC zone.

cooling gas during the procedure is different.

procedure takes place in isothermal conditions:

the general technological conditions of efficiency.

surface decreases, and the safe cooling time increases.

the WBC zone (**Figure 8**).

**145**

Contrary to the popular belief [7, 13], GWBC and IWBC provide effects on only a fraction of the skin area. In a group installation, the contact area of the cooling gas with the patient's body in a multi-seat cab is up to 70.5% of the total surface area of the body. In an individual cab, the contact area reaches 66% [7]. Temperature

Systems for implementing technology I, individual cryosaunas, were developed 20 years after multi-seat installations [7] with consideration of the experience of their operation. Modern installations for IWBC use a nitrogen cooling system (NCS), so they quickly reach a given temperature level and allow you to adjust the

It is impossible to develop universal recommendations on selecting the optimal temperature of the gaseous heat carrier for GWBC and IWBC, since in multi-seat and single-seat installations the algorithm for changing the temperature of the

To conduct a preliminary analysis of the effect of gas temperature in the WBC zone on the magnitude of the positive effect achieved, it can be assumed that the

It is impossible to implement WBC in the isothermal mode, since it takes some time for the patient to enter the low-temperature zone and exit from it. However, the study of WBC processes in ideal temperature conditions allows us to formulate

To determine the optimal gas temperature in the WBC zone, the calculated values of the WBC ET obtained by Eq. (6) were used. Simulation of the BS cooling process under conditions of natural gas convection with a temperature from 90 to 190 K allowed us to plot the dependence of the ET value on the gas temperature in

When isothermally cooling the surface of the patient's body, the maximum value

of ET (325 min) is achieved at a temperature of 140 K. At temperatures below 140 K, the WBC efficiency gradually decreases. At a temperature of 100 K, the value of WBC ЕТ is almost three times lower than the maximum [7]; therefore, when conducting WBC procedures, it is advisable to use a gas with a temperature from 120 to 140 K [16]. At temperatures above 140 K, the WBC efficiency rapidly decreases. At a temperature of 160 K, the WBC ET of the procedures is 10 times lower than the maximum value and is close to the results achieved during water procedures. The results of the computational experiment on simulation of cooling the body surface with gas with a temperature of 160 K (�110°C) ideally coincide with the results of tests performed by doctors in sports medicine [8], which in comparing the therapeutic effect of WBC procedures at a temperature of �110°C and water baths with a temperature of 8°C, did not reveal any advantages of the WBC. The results obtained have clear thermophysical reasons. As the temperature of the gaseous heat carrier increases, the intensity of heat removal from the BS

0 < *τ* ≤ *τ*max*; T*<sup>1</sup> ¼ const*:* (23)

be developed separately.

The histogram on **Figure 7** gives an idea of what is the source of heat removed from the surface of the patient's body shell. The main share of the heat of 55.2% was gained due to supercooling the epithelial layer. The heat gained by supercooling the fat layer *QAF* is 39.8%. The supply of heat from the body core *QBC* and internal sources *QMH* in the body shell tissues is less than 2%. This supply of heat is gained by supercooling the muscle layer *Q<sup>А</sup>M*.

The calorific capacity of the body does not play any role in the formation of the heat load on the cooling system of the WBC device, which is determined by the heat storage capacity of the body shell tissues. The safety of the WBC procedures is ensured by the correct choice of the contact duration of the body surface with a cryogenic gas. The thermal control system of the body does not affect the safety of procedures.

#### **6. Selecting optimal gas temperature in the WBC zone**

In practice, there are two options for carrying out WBC procedures in multi-seat and single-seat installations [7, 21–23]. The cooling conditions in these installations

differ significantly; therefore, the technology of group and individual WBC should be developed separately.

Contrary to the popular belief [7, 13], GWBC and IWBC provide effects on only a fraction of the skin area. In a group installation, the contact area of the cooling gas with the patient's body in a multi-seat cab is up to 70.5% of the total surface area of the body. In an individual cab, the contact area reaches 66% [7]. Temperature regimes of GWBC and IWBC are fundamentally different.

The GWBC technology was influenced by the design of the device for performing the procedures (**Figure 1**). Using a low-temperature food storage chamber for WBC procedures, Japanese engineers and doctors were forced to carry out WBC procedures in groups. The dimensions of the chamber were too large for individual procedures. This forced solution is contrary to the general practice of physiotherapy; treatment is always carried out individually.

Systems for implementing technology I, individual cryosaunas, were developed 20 years after multi-seat installations [7] with consideration of the experience of their operation. Modern installations for IWBC use a nitrogen cooling system (NCS), so they quickly reach a given temperature level and allow you to adjust the temperature of the gas in the WBC zone.

It is impossible to develop universal recommendations on selecting the optimal temperature of the gaseous heat carrier for GWBC and IWBC, since in multi-seat and single-seat installations the algorithm for changing the temperature of the cooling gas during the procedure is different.

To conduct a preliminary analysis of the effect of gas temperature in the WBC zone on the magnitude of the positive effect achieved, it can be assumed that the procedure takes place in isothermal conditions:

$$0 < \tau \le \tau\_{\text{max}}; \ T\_1 = \text{const.} \tag{23}$$

It is impossible to implement WBC in the isothermal mode, since it takes some time for the patient to enter the low-temperature zone and exit from it. However, the study of WBC processes in ideal temperature conditions allows us to formulate the general technological conditions of efficiency.

To determine the optimal gas temperature in the WBC zone, the calculated values of the WBC ET obtained by Eq. (6) were used. Simulation of the BS cooling process under conditions of natural gas convection with a temperature from 90 to 190 K allowed us to plot the dependence of the ET value on the gas temperature in the WBC zone (**Figure 8**).

When isothermally cooling the surface of the patient's body, the maximum value of ET (325 min) is achieved at a temperature of 140 K. At temperatures below 140 K, the WBC efficiency gradually decreases. At a temperature of 100 K, the value of WBC ЕТ is almost three times lower than the maximum [7]; therefore, when conducting WBC procedures, it is advisable to use a gas with a temperature from 120 to 140 K [16]. At temperatures above 140 K, the WBC efficiency rapidly decreases. At a temperature of 160 K, the WBC ET of the procedures is 10 times lower than the maximum value and is close to the results achieved during water procedures. The results of the computational experiment on simulation of cooling the body surface with gas with a temperature of 160 K (�110°C) ideally coincide with the results of tests performed by doctors in sports medicine [8], which in comparing the therapeutic effect of WBC procedures at a temperature of �110°C and water baths with a temperature of 8°C, did not reveal any advantages of the WBC. The results obtained have clear thermophysical reasons. As the temperature of the gaseous heat carrier increases, the intensity of heat removal from the BS surface decreases, and the safe cooling time increases.

Part of the heat removed was obtained from internal sources in the epithelial and muscle layers, heat of metabolism *QMH*. This heat was calculated by a known value

*QMHE* ¼ *τ*max*Δx n*ð Þ *<sup>E</sup>* � 1 *qVE; QMHM* ¼¼ *τ*max*Δx n*ð Þ *BS* � *nF* � 1 *qVM,* (21)

where *qVE* and *qVMqVM* means the specific heat of metabolism of epithelium and

Some of the heat removed came from the patient's body core; the amount of heat gained can be determined by numerical integration and instantaneous values of the heat flux transferred by thermal conductivity through the inner boundary of the

> ð*<sup>τ</sup>*max *τ*¼0 *qni*þ<sup>1</sup>

The histogram on **Figure 7** gives an idea of what is the source of heat removed from the surface of the patient's body shell. The main share of the heat of 55.2% was gained due to supercooling the epithelial layer. The heat gained by supercooling the fat layer *QAF* is 39.8%. The supply of heat from the body core *QBC* and internal sources *QMH* in the body shell tissues is less than 2%. This supply of heat is gained

The calorific capacity of the body does not play any role in the formation of the heat load on the cooling system of the WBC device, which is determined by the heat storage capacity of the body shell tissues. The safety of the WBC procedures is ensured by the correct choice of the contact duration of the body surface with a cryogenic gas. The thermal control system of the body does not affect the safety of

In practice, there are two options for carrying out WBC procedures in multi-seat and single-seat installations [7, 21–23]. The cooling conditions in these installations

*∂τ:* (22)

*qv* (**Table 1**):

**Figure 7.**

body shell:

procedures.

**144**

muscles, respectively, W/m<sup>3</sup>

*Sources of the heat gained the WBC procedure.*

*Low-temperature Technologies*

by supercooling the muscle layer *Q<sup>А</sup>M*.

.

*QBC* ¼

**6. Selecting optimal gas temperature in the WBC zone**

237 sec [3]. At a temperature of 140 K, the safe exposure time for cooling is 161 sec. The practice of using WBC has shown that, along with the maximum duration of cooling, it is necessary to limit the minimum duration of stay of patients in a

The reasons for this limitation are explained by the graph of dependence *ET = f*(τ) (**Figure 8**). The graph shows that there is a fairly long period in the WBC procedure when a positive effect is not formed. At a gas temperature of 140 K, this phase of the procedure accounts for almost 80%, but 93% of the positive effect is formed after its completion. The reason for the low efficiency of WBC at the beginning of the procedure is the relatively high skin temperature (*TS*) (**Figure 8**), which drops to 275 K (2°C) only by the end of the first phase of the procedure. As it can be seen from the graph of dependence *ISA = f*(*TS*) (**Figure 5**), at a skin temperature of *TS >* 275 K, the intensity of the WBC stimulating effect is negligible *ISA* < 60. The first phase of the WBC procedure reduces the surface temperature of the skin to a temperature of *TS* = 275 K, so it is called the cooling time (τcool) (**Figure 9**). It is obvious that the duration of the WBC procedure must be longer than the duration of the cooling time but less than the time of the violation of safety conditions τcool *<* τ *<* τmax*.* The second, effective, phase of the procedure ensures the formation of the main positive result, the longer the duration of the effective phase,

The calculated dependences of the WBC safe exposure (τmax) and the duration

of the cooling time (τcool) on the gas temperature (*Tg*) (**Figure 10**) show that increasing the gas temperature from 90 to 150 K increases the effective phase of the procedure. A further increase in temperature increases the duration of the cooling time. At temperatures above 160 K, the estimated duration of the cooling time exceeds the safe WBC exposure. Even with isothermal cooling, it is impossible to provide an effective WBC when using gas with a temperature *Tg* > 150 K; in real

Numerical experiments on a mathematical model of the human body shell allowed to formulate general ideas about the technological foundations of effective WBC. When developing technological recommendations on the design of installations for the implementation of GWBC or IWBC methods, it is necessary to take into account the algorithm for changing the temperature of the gas in contact with

*The change of the body temperature surface* Ts *and the value of* ET *during the WBC procedure with gas*

conditions the gas temperature should be no higher than 140 K.

*τEP* ¼ *τ*max � *τcool* (24)

cryotherapeutic installation [7].

*DOI: http://dx.doi.org/10.5772/intechopen.83680*

*Technique and Technology of Whole-Body Cryotherapy (WBC)*

the greater the effect of the procedure.

the patient's body surface.

**Figure 9.**

**147**

*temperature of 140 K.*

**Figure 8.** *The estimated duration of the effect of WBC at different gas temperature.*

When the gas temperature is above 150 K, the danger of supercooling of the body core (*Tf* ! 309 K) occurs before the surface of the body shell is supercooled. The reason for the termination of the WBC procedures becomes a violation of the condition *Tf* ≥ 309 K. At the same time, the temperature of the body shell surface remains at a sufficiently high level of *Ts* ≥ 275 K (2°C), due to which the cold receptors of the skin do not experience significant irritation and the accumulation of a positive WBC effect is extremely slow. The picture described is identical to what is observed at the time of completion of the water hypothermia procedure. Under conditions of isothermal cooling of the body with gas with a temperature of 160 K (�110°C), the estimated duration of the WBC procedure is 207 sec. During this time, the BS surface temperature drops only to 275 K. At this BS surface temperature, the ISA value is 80 times less than the maximum value (**Figure 5**). Under actual conditions, the WBC procedures in installations with a minimum temperature of 160 K (�110°C) do not ensure the constancy of the gas temperature, so the BS surface temperature after the procedure is much higher than the calculated one and is 15–20°C [24]. Such a temperature on the surface of the skin can be obtained using water baths with a temperature of 8°C; therefore, the doubts of some authors [11, 18, 25] on the advisability of using cryogenic technologies are fully justified.

According to the results of simulating the process of cooling the BS surface with a cryogenic gas, it can be argued that for effective procedures the gas temperature in the WBC zone should be not lower than 140 K.
