Modeling a healthy and a person with heart failure conditions using the object-oriented modeling environment Dymola

Abstract

Several mathematical models of different physiological systems are spread through literature. They serve as tools which improve the understanding of (patho-) physiological processes, may help to meet clinical decisions and can even enhance medical therapies. These models are typically implemented in a signal-flow-oriented simulation environment and focus on the behavior of one specific subsystem. Neglecting other physiological subsystems and using a technical description of the physiology hinders the exchange with and acceptance of clinicians. By contrast, this paper presents a new model implemented in a physical, object-oriented modeling environment which includes the cardiovascular, respiratory and thermoregulatory system. Simulation results for a healthy subject at rest and at the onset of exercise are given, showing the validity of the model. Finally, simulation results showing the interaction of the cardiovascular system with a ventricular assist device in case of heart failure are presented showing the flexibility and mightiness of the model and the simulation environment. Thus, we present a new model including three important physiological systems and one medical device implemented in an innovative simulation environment.

Appendix

Only equations which are not given in the paper are presented.

List of variables

A :

area in $m^{2}$

C :

compliance in ml/mmHg

HR:

heart rate in bpm

K :

body

L :

inertance in kg/$\mathrm {m{^4}}$

M :

metabolic rate in watt

m :

amount of a gas in mmol

N :

amount of segments

Q :

flow in ml/s

$\dot{\mathrm {{q}}}$ :

heat flow in watts

P(t):

work load in watts

P :

pressure in mmHg

p :

pressure in mmHg

$P_{\mathrm {Pcd}}$ :

pressure in the Perciard in mmHg

$P_{\mathrm {Peri}}$ :

transmural pressure across the pericard in mmHg

PSNA:

parasympathetic nerve activity

SNA:

sympathetic nerve activity

$\theta$ :

actual temperature in $^\circ \hbox {C}$

V :

volume in ml

List of indices

a:

ambient

b:

basal

bl:

blood

Diff:

diffusion

C:

convection

$\mathrm {E_{ES}}$ :

elastance end-systolic

ED:

end-diastolic

env:

environmental

ES:

end-systolic

evap:

evaporation

F:

free wall, e.g.,

LAF:

left atrial free wall

f(t):

frequency in Hz

in:

inlet

K :

conduction

korr:

correction

LA:

left atrium

LV:

left ventricle

mus:

muscle

out:

outlet

Para:

parasympathetic

PC:

pulmonary capillary

Peri:

pericard

Pcd:

pericard

Prod:

production

r:

radiation

RA:

right atrium

RS:

airways

RV:

right ventricle

sat:

saturated

sh:

shivering

sk:

skin

sp:

set point

Spt:

septum

SR:

system resistance (arterial)

SS:

steady state

SW:

sweating

Sympa:

sympathetic

Tc:

tissue capillary

th:

thorax

tiss:

tissue

u:

unstressed

Vol:

volume

1.1 Model of the cardiovascular system

The following volume equations are given for the heart:

$$\begin{aligned} V_{\mathrm {LA}}= \,& V_{\mathrm {LAF}} \end{aligned}$$

(23)

$$\begin{aligned} V_{\mathrm {RA}}= \,& V_{\mathrm {RAF}}, \end{aligned}$$

(24)

$$\begin{aligned} V_{\mathrm {LV}}= \,& V_{\mathrm {LVF}} + V_{\mathrm {Spt}} \end{aligned}$$

(25)

$$\begin{aligned} V_{\mathrm {RV}}= \,& V_{\mathrm {RVF}} - V_{\mathrm {Spt}}, \end{aligned}$$

(26)

$$\begin{aligned} V_{Pcd}= \,& V_{\mathrm {LV}}+ V_{\mathrm {RV}} + V_{\mathrm {LA}} + V_{\mathrm {RA}} \end{aligned}$$

(27)

$$\begin{aligned}= \,& V_{\mathrm {LVF}}+ V_{\mathrm {RVF}}+V_{\mathrm {LAF}}+V_{\mathrm {RAF}} \end{aligned}$$

(28)

Pressure in the heart results from the following equations:

$$\begin{aligned} P_{\mathrm {LAF}}= \,& P_{\mathrm {LA}} - P_{\mathrm {Peri}}, \end{aligned}$$

(29)

$$\begin{aligned} P_{\mathrm {RAF}}= \,& P_{\mathrm {RA}} - P_{\mathrm {Peri}},\end{aligned}$$

(30)

$$\begin{aligned} P_{\mathrm {LVF}}= \,& P_{\mathrm {LV}} - P_{\mathrm {Peri}},\end{aligned}$$

(31)

$$\begin{aligned} P_{\mathrm {RVF}}= \,& P_{\mathrm {RV}} - P_{\mathrm {Peri}}. \end{aligned}$$

(32)

$P_{\mathrm {Peri}}$ corresponds to the pressure in the pericard, whereas $P_{\mathrm {Pcd}}$ is the transmural pressure across the pericard:

$$\begin{aligned} P_{\mathrm {Peri}}= \,& P_{\mathrm {Pcd}}+P_{\mathrm {th}},\end{aligned}$$

(33)

$$\begin{aligned} P_{\mathrm {Spt}}= \,& P_{\mathrm {LV}}-P_{\mathrm {RV}}\end{aligned}$$

(34)

$$\begin{aligned}= \,& P_{\mathrm {LVF}}-P_{\mathrm {RVF}}\end{aligned}$$

(35)

$$\begin{aligned} P_{\mathrm {ed}}(V)= \,& P_\mathrm {0} \times (e^{\lambda \times (V-V_\mathrm {0})}-1),\end{aligned}$$

(36)

$$\begin{aligned} P_{\mathrm {es}}(V)= \,& E_{\mathrm {es}}\times (V-V_\mathrm {u}). \end{aligned}$$

(37)

$V_\mathrm {0}$ is the volume in the ventricle when the pressure is zero. The driver function for the atria is given by:

$$\begin{aligned} e(t) = \displaystyle \sum _{i=1}^{1}A_\mathrm {i} \times {e}^{-\frac{0,5}{B_\mathrm {i}^2} \times \left( \frac{\hbox {HR}}{80}\right) ^2 \times (t - C_\mathrm {i} \times \frac{80}{\hbox {HR}})^ 2}. \end{aligned}$$

(38)

The driver function for the septum and the ventricles is given in Eq. 1.

The following equation describes the behavior of the cardiac valves:

$$\begin{aligned} \frac{\hbox {d}Q}{\hbox {d}t}=\frac{p_{\mathrm {in}}-p_{\mathrm {out}}-Q \times R}{L}. \end{aligned}$$

(39)

whereas the following equations are used to simulate the compliance behavior of the vessels:

$$\begin{aligned} \Delta p= \,& p_{\mathrm {in}} - p_{\mathrm {out}} \end{aligned}$$

(40)

$$\begin{aligned} \frac{dV_{\mathrm {j}}}{dt}= \,& Q_{\mathrm {in}}-Q_{\mathrm {out}} \end{aligned}$$

(41)

$$\begin{aligned} p_{\mathrm {in}}= \,& \frac{1}{C} \times (V_\mathrm {j}-V_\mathrm {u}). \end{aligned}$$

(42)

The resistances of the vessels as depicted in Fig. 3a are given by:

$$\begin{aligned} \Delta P= \,& Q\times R \end{aligned}$$

(43)

$$\begin{aligned} \Delta P= \,& p_{\mathrm {in}} - p_{\mathrm {out}} \end{aligned}$$

(44)

1.2 Control of the cardiovascular system

In order to implement the control of the cardiovascular system, mainly first-order systems are used.

The stroke volume depends on the stiffness of the ventricle and its elastance (see also Eq. 6). Both are influenced by the sympathetic system (SNA). Therefore, $E_\mathrm {ES}$ in Eq. 37 is multiplied by:

$$\begin{aligned} 5 \times K'_\mathrm {E_{ES}}(t)+K'_\mathrm {E_{ES}}(t)= K_\mathrm {E_{ES},SV} \end{aligned}$$

(45)

In the veins, the unstressed volume as well as the compliance is influenced by the sympathic system:

$$\begin{aligned} 34 \times K'_\mathrm {Vu}(t)+K_{\mathrm {Vu}} (t)= K_\mathrm {Veins,Vol} \end{aligned}$$

(46)

and

$$\begin{aligned} 10 \times K'_\mathrm {Comp}(t)+K_\mathrm {Comp}(t)= K_\mathrm {C,Vein} \end{aligned}$$

(47)

The variables $K_\mathrm {E_{ES},SV}, K_\mathrm {Veins,Vol}, K_\mathrm {C,Vein}$ are all calculated using a polynomial equation of fourth order with the $\hbox {SNA}(t)$ as input variable. The same applies to the gain $K_\mathrm {SR}$ used to adapt the systemic arterial resistance. Here, no additional delay element is applied. For the active skeletal muscles, local regulation mechanisms predominate the control of the vessels. Therefore, the systemic arterial resistance of the active skeletal muscles is not influenced by the baroreceptor reflex but is directly influenced by the intensity of exercise P(t):

$$\begin{aligned} K_\mathrm {ex}= a\times e^{b\times P(t)}+ c\times e^{d\times P(t)}, \end{aligned}$$

(48)

where a, b, c and d are constant parameters. $K_\mathrm {ex}$ is used as gain in a first-order element with a time constant of 1 s. Thus, in the model of the cardiovascular system, the work rate directly influences the systemic arterial resistance (Eq. 48) and the baroreceptors and the heart rate additionally via the central command. The activation of the muscle pump results in a mechanical influence presented by the environmental pressure of the active skeletal muscles.

The heart rate is calculated using the following equation:

$$\begin{aligned} \hbox {HR}= 53 + e \times K_\mathrm {Sympa}- f\times K_\mathrm {Para}, \end{aligned}$$

(49)

where additional delay elements are used to calculate $K_\mathrm {Sympa}$ and $K_\mathrm {Para}$. The influence of the sympathetic nervous system is also dependent on the current work rate due to the central command.

1.3 Model of the respiratory system

The calculation of the volume flow to the airways is given by:

$$\begin{aligned} \dot{V}_{\mathrm {RS}}(t) = \frac{1}{R_{\mathrm {RS}}} \times \left( \Delta P_{\mathrm {Lung}}(t) - p_{\mathrm {mus}}(t) \right) . \end{aligned}$$

(50)

and for excitation of the muscular driving force a sine function is assumed:

$$\begin{aligned} p_{\mathrm {mus}}(t) = p_{\mathrm {mus,0}}(t) \times \mathrm {sine}(2\pi f(t)\times t), \end{aligned}$$

(51)

where $p_{\mathrm {mus,0}}(t)$

$$\begin{aligned} p_{\mathrm {mus,0}}(t) = \frac{V_{\mathrm {T}}(t)}{2} \times \sqrt{1 + \left( 2 \times \pi \times f(t) \times C_{\mathrm {RS}} \times R_{\mathrm {RS}} \right) ^2}. \end{aligned}$$

(52)

depends on the tidal volume $V_{\mathrm {T}}(t)$.

The pressure drop across $C_{\mathrm {RS}}$ is given by:

$$\begin{aligned} \Delta P_{\mathrm {Lung}}(t) = \frac{1}{C_{\mathrm {RS}}} \times \Delta V_{\mathrm {Lung}}(t). \end{aligned}$$

(53)

Therefore, the pressure in the lung is the sum of $p_{\mathrm {Lung}}(t)$, $p_{\mathrm {mus}}(t)$ and $P_{\mathrm {env}}$:

$$\begin{aligned} P_{\mathrm {Lung}}(t) = \Delta P_{\mathrm {Lung}}(t) + p_{\mathrm {mus}}(t) + P_{\mathrm {env}}. \end{aligned}$$

(54)

In Eq. 10 the first two summands determine the baseline of breathing. Both depend on the current work load [P(t)] of the patient. In order to calculate the steady state, the equations by Saunders [28] are used:

$$\begin{aligned} V_{\mathrm {T,SS}}(P) =&-2.572 \times 10^{-5}\times P(t)^2 + 6.55 \times 10^{-3}\times P(t) + 0.495, \end{aligned}$$

(55)

$$\begin{aligned} \hbox {FRC}_{\mathrm {SS}}(P) =&-5.244 \times 10^{-5} \times P(t)^3 + 2.148 \times 10^{-5}\times P(t)^2 \nonumber \\&- 4.47 10^{-3}\times P(t) + 3.51) \times 0.68. \end{aligned}$$

(56)

For changes in the steady state, the time response as investigated by Miyamoto [23] is used:

$$\begin{aligned} \dot{V}_{\mathrm {T}}(t)= \,& \frac{V_\mathrm{T,SS}(P) - V(t)}{50~\mathrm {s}} \end{aligned}$$

(57)

$$\begin{aligned} \dot{\hbox {FRC}}(t)= \,& \frac{\hbox {FRC}_{\mathrm {SS}}(P) - \hbox {FRC}(t)}{50~\mathrm {s}}. \end{aligned}$$

(58)

$$\begin{aligned} f_{\mathrm {SS}}(P)= \,& -1.29 \times 10^{-5} \times P(t)^2 + 2.629 \times P(t) + 0.192. \end{aligned}$$

(59)

$$\begin{aligned} \dot{f}(t)= \,& \frac{f_{\mathrm {SS}}(P) - f(t)}{10~\mathrm {s}}. \end{aligned}$$

(60)

The ideal gas law relates pressure and amount of gas in the lung:

$$\begin{aligned} \left( p_{\mathrm {Lung}}(t)-p_{\mathrm {H_2O}} \right) \times V_{\mathrm {Lung}}(t) = m_{\mathrm {sum,Lung}}(t) \times R_{\mathrm {G}} \times T_{\mathrm {body}} \end{aligned}$$

(61)

with

$$\begin{aligned} m_{\mathrm {sum,Lung}}(t) = m_{\mathrm {CO_2,Lung}}(t) + m_{\mathrm {O_2,Lung}}(t) + m_{\mathrm {N_2,Lung}}(t). \end{aligned}$$

(62)

The mass of gas in the lung is a result of the mass of gas coming from the airways subtracted by the exchanged mass of gas with the pulmonary capillaries

$$\begin{aligned} \vec {m}_{\mathrm {Lung}}(t) = \int _{t'=0}^t \left( \dot{\vec {m}}_{\mathrm {RS}}(t') - \dot{\vec {m}}_{\mathrm {Diff}}(t') \right) \, \mathrm {d} t' + \vec {m}_{\mathrm {Lung,0}}. \end{aligned}$$

(63)

Partial pressures $(\vec {p}_{\mathrm {Lung}}(t))$ for each gas in the lung depend on the relationship of the amount of the respective gas to the complete amount of gas:

$$\begin{aligned} \vec {p}_{\mathrm {Lung}}(t) = (p_{\mathrm {Lung}}(t)-p_{\mathrm {H_2O}}) \times \frac{\vec {m}_{\mathrm {Lung}}(t)}{m_{\mathrm {sum,Lung}}(t)}. \end{aligned}$$

(64)

The sum of mass flows through the single pulmonary capillaries presents the complete flow of mass given by diffusion:

$$\begin{aligned} \dot{\vec {m}}_{\mathrm {Pc,Diff}}(t) = \sum \limits _{i=1}^N \dot{\vec {m}}_{\mathrm {Pc}_i,\mathrm {Diff}}(t). \end{aligned}$$

(65)

For a single pulmonary capillary, the mass of O₂ and CO₂ results from:

$$\begin{aligned} \begin{aligned} \vec {m}_{\mathrm {Pc_i}}(t) =&\int _{t'=0}^t \dot{\vec {m}}_{\mathrm {Pc}_i,\mathrm {in}}(t') + \dot{\vec {m}}_{\mathrm {Pc}_i,\mathrm {Diff}}(t) - \dot{\vec {m}}_{\mathrm {Pc}_i,\mathrm {out}}(t') \, \mathrm {d} t'\\&+\dot{\vec {m}}_{\mathrm {Pc}_i,0}, \end{aligned} \end{aligned}$$

(66)

where $\dot{\vec {m}}_{\mathrm {Pc_i,in}}(t')$ presents the mass of a gas transported via the blood flow into the pulmonary capillary. It assumed that the volume of blood in each pulmonary capillary $V_{\mathrm {Pc}_i}$ is the same. Therefore, given the complete blood volume in the pulmonary capillaries $V_{\mathrm {Pc}}$, it can be calculated by:

$$\begin{aligned} V_{\mathrm {Pc}_i} = V_{\mathrm {Pc}} / N. \end{aligned}$$

(67)

Gas concentrations in the segments (e.g., pulmonary capillaries, tissue capillaries) are calculated using the corresponding equation to Eq. (11). All pulmonary capillaries are connected in series. Accordingly, the blood flow and masses of a gas going out of a pulmonary capillary are identical to the mass going into the next pulmonary capillary, e.g.,

$$\begin{aligned} Q_{\mathrm {Pc}_{i+1},\mathrm {in}}(t) = Q_{\mathrm {Pc}_i,\mathrm {out}}(t), \quad \text { for } 1<i<N-1. \end{aligned}$$

(68)

The same assumption is applied to the capillaries of the tissue. The amount of O₂ and CO₂ in the tissue is calculated by a balance equation:

$$\begin{aligned} \vec {m}_{\mathrm {tiss}}(t)&= \int _{t'=0}^t \left( \dot{\vec {m}}_\mathrm{GA}(t') - \dot{\vec {m}}_{\mathrm {Diff,tiss}}(t')\right) \, \mathrm {d} t' + \vec {m}_{\mathrm {tiss,0}}, \end{aligned}$$

(69)

where the vector $\dot{\vec {m}}_\mathrm{GA}(t')$ gives the production of CO₂ and consumption of O₂:

$$\begin{aligned} \dot{\vec {m}}_\mathrm{GA}(t) = \begin{pmatrix} \dot{m}_{\mathrm {CO_2,\quad Prod}}(t) \\ \dot{m}_{\mathrm {O_2,\quad Use}}(t) \end{pmatrix}. \end{aligned}$$

(70)

The amount of O₂ and CO₂ that diffuses in and out of the tissue results from:

$$\begin{aligned} \dot{\vec {m}}_{\mathrm {Diff,Tiss}}(t) = \dot{\vec {m}}_{\mathrm {Tc,out}}(t) - \dot{\vec {m}}_{\mathrm {Tc,in}}(t). \end{aligned}$$

(71)

Furthermore, it is assumed that a complete exchange of O₂ and CO₂ is given, and therefore, the partial pressure in tissue is identical with the partial pressure in the capillaries of the tissue:

$$\begin{aligned} \vec {p}_{\mathrm {Tc}}(t) = \vec {p}_{\mathrm {Tiss}}(t). \end{aligned}$$

(72)

Production of CO₂ and usage of O₂ also depend on the work load:

$$\begin{aligned} \dot{m}_{\mathrm {CO_2,Prod}}(t) = 7.81 \times 10^{-3} \times P(t) + 1.55 \times 10^{-2} \end{aligned}$$

(73)

and

$$\begin{aligned} \dot{m}_{\mathrm {O_2,Use}}(t) = - 0.01 \times P(t) - 0.245. \end{aligned}$$

(74)

It is important to note that nitrogen is an inert gas and is therefore only taken into consideration in the airways. The dissociation curves for O₂, CO₂ in the pulmonary capillaries are given by

$$\begin{aligned} c_{\mathrm {CO_2,Pc_i}}(t)= \,& - 2\times 7.89 \times 10^{-7} \times {p_{\mathrm {CO_2,Pc}}(t)}^4\nonumber \\&+ 1.93 \times 10^{-4}\times {p_{\mathrm {CO_2,Pc}}(t)}^3 \nonumber \\&- 1.75\times 10^{-2} \times {p_{\mathrm {CO_2,Pc}}(t)}^2\nonumber \\&+ 8.53\times 10^{-1} \times p_{\mathrm {CO_2,Pc}}(t) \nonumber \\&+ 5.908. \end{aligned}$$

(75)

$$\begin{aligned} c_{\mathrm {O_2,Pc_i}}(t)= \,& 1.26 \times 10^{-3} \times p_{\mathrm {O_2,Pc}}(t) \nonumber \\&+6.068 \times 1.55 \times \frac{1}{1+e^{(27.86-p_{\mathrm {O_2,Pc}}(t))/{10.73})}} \end{aligned}$$

(76)

and for the fluid in the tissues reduced dissociation curves due the lack of hemoglobin are used:

$$\begin{aligned} c_{\mathrm {O_2,tiss}}(t) = 0.141 \times p_{\mathrm {O_2,tiss}}(t) \end{aligned}$$

(77)

and

$$\begin{aligned} \begin{aligned} c_{\mathrm {CO_2,Pc}_i}(t) =\,&(- 2 \times 7.89 \times 10^{-7} \times {p_{\mathrm {CO_2,Tc}}(t)}^4\\&+ 1.93\times 10^{-4} \times {p_{\mathrm {CO_2,Tc}}(t)}^3\\&- 1.75 \times 10^{-2} \times {p_{\mathrm {CO_2,Tc}}(t)}^2\\&+ 0.853 \times p_{\mathrm {CO_2,Tc}}(t) \\&+ 5.908)\times 0.6. \end{aligned} \end{aligned}$$

(78)

Thus, in the respiratory model the work load is used as input variable in Eqs. (73) and (74) as well as the equation of the controller (Eq. 17). Finally, the current work load also influences equations describing the steady-state equations.

1.4 Model of the thermoregulatory system

For each layer, a heat balance equation is given. Thus for the core layer, the change in heat is calculated by:

$$\begin{aligned} \dot{q}_{\mathrm {stored}} = \dot{M}_\mathrm {i} + \dot{q}_{\mathrm {k, out}} + \dot{q}_{\mathrm {c, bl}}. \end{aligned}$$

(79)

For those layers located between two layers (muscle and fat), the heat balance equation is:

$$\begin{aligned} \dot{q}_{\mathrm {stored}} = \dot{M}_\mathrm {i} + \dot{q}_{\mathrm {k, in}} + \dot{q}_{\mathrm {k, out}} + \dot{q}_{\mathrm {c, bl}}, \end{aligned}$$

(80)

and for the skin additional heat losses, e.g., due to evaporation must be taken into account:

$$\begin{aligned} \dot{q}_{\mathrm {stored}} = \dot{M}_\mathrm {i} + \dot{q}_{\mathrm {k, in}} + \dot{q}_{\mathrm {r}} + \dot{q}_{\mathrm {c}} + \dot{q}_{\mathrm {evap}} + \dot{q}_{\mathrm {evap, res}} + \dot{q}_{\mathrm {c, bl}}. \end{aligned}$$

(81)

The amount of heat stored in a layer depends on its mass and specific heat capacity:

$$\begin{aligned} \dot{q}_{\mathrm {stored}}=m \times c \times \frac{\hbox {d}\theta }{\hbox {d}t}. \end{aligned}$$

(82)

Heat transfer by conduction between two adjacent layers results from

$$\begin{aligned} \dot{q}_\mathrm {k} = C_\mathrm {T} \times A \times (\theta _\mathrm {i} - \theta _\mathrm {j}) \end{aligned}$$

(83)

and heat transfer due to radiation:

$$\begin{aligned} \dot{q}_{\mathrm {r}} = h_\mathrm {r} \times A_\mathrm {K} \times (\theta _{\mathrm {sk}}-\theta _{\mathrm {a}}). \end{aligned}$$

(84)

Additionally, heat loss caused by evaporation due to breathing is present

$$\begin{aligned} \dot{q}_{\mathrm {evap, res}} = \dot{M} \times 1.725 \times 10^{-4} \times \left( 5866-p_{\mathrm {H_2O}}\right) . \end{aligned}$$

(85)

as well as by convection given by the transport of blood:

$$\begin{aligned} \dot{q_\mathrm {c}} = 3.6 \times 10^{6} \times \dot{v}_{\mathrm {bl}} \times (\theta _\mathrm {i} - \theta _{\mathrm {bl}}). \end{aligned}$$

(86)

The blood flow to the muscles is influenced by the current metabolic rate and the basal metabolic rate:

$$\begin{aligned} \dot{v}_{\mathrm {bl}} =\dot{v}_{\mathrm {bl, b}} + 3.6 \times 10^{-7} (\dot{M}_{\mathrm {mus}} - \hbox {BMR}). \end{aligned}$$

(87)

By contrast, the blood flow to the skin is strongly influenced by the thermoregulation as presented in Sect. 2.7:

$$\begin{aligned} \dot{v}_{\mathrm {bl}}= 2,78\times 10^{-7}\left( \frac{\dot{v}_{\mathrm {bl,b}}\times 3600000+F_{\mathrm {vc}}\times \hbox {VC}}{1+F_{\mathrm {vd}} \times \hbox {VD}}\right) 2^{\theta _{\mathrm {sk}}-\theta _{\mathrm {spsk}}/10}. \end{aligned}$$

(88)

The basal blood flow to the muscle and the skin layer is calculated by:

$$\begin{aligned} v_{\mathrm {bl,b,mus}}= \,& 0.4696\times v_{\mathrm {bl}\_i} + 0.8293 \end{aligned}$$

(89)

$$\begin{aligned} v_{\mathrm {bl,b,sk}}= \,& 0.5304\times v_{\mathrm {bl}\_i} - 0.8293. \end{aligned}$$

(90)

Heat loss by evaporation on the skin results from:

$$\begin{aligned} \dot{q}_{\mathrm {evap,sk}}=(\dot{q}_{\mathrm {evap,b}}+F_{\mathrm {sw}} \times \hbox {SW}) \times 2^{(\theta _{\mathrm {sk}}-\theta _{\mathrm {spsk}})/4} \end{aligned}$$

(91)

and is limited by:

$$\begin{aligned} \dot{q}_{\mathrm {evap, max}} = 0.0165 \times h_\mathrm {c} \times A \times (p_{\mathrm {sat}}-p_{\mathrm {H_2O}}), \end{aligned}$$

(92)

where

$$\begin{aligned} h_{\mathrm {c, korr}} = 3,16 \times h_\mathrm {c} \times v^{0,5} \end{aligned}$$

(93)

is the coefficient of convection in case of wind. The metabolic rate of the muscle is given by the basal metabolic rate and the current work load. In addition, heat can be generated by shivering:

$$\begin{aligned} \dot{M}_{\mathrm {mus}} = \hbox {BMR}+\dot{W}_{\mathrm {mus}} \times F_{\mathrm {mus}}+\hbox {SH} \times F_{\mathrm {sh}}. \end{aligned}$$

(94)

Therefore, in the thermoregulatory model, the current metabolic rate is the input variable in Eqs. (94) and (87). It is important to note that in the respiratory as well as in the cardiovascular model the intensity of exercise is used as input variables, whereas in the thermoregulatory the input variable is the current metabolic rate. Accordingly, a conversion factor is used to calculate the metabolic rate from the given intensity of exercise.

1.5 Model overview in Dymola

An overview on the first and second levels of the model is given in Figs. 10, 11, 12, 13, 14.

Fig. 10

Screenshot of the first model level of the Dymola model

Fig. 11

Screenshot of the second model level of the Dymola model

Fig. 12

Screenshot of the first model level of the lung model

Fig. 13

Screenshot of the second model level of the lung model

Fig. 14

Screenshot of the first model level of the thermoregulatory model