ABSTRACT (original article): A secondary function of the respiratory system of the mammals is, during inspiration, to heat the air to body temperature and to saturate it with water before it reaches the alveoli. Relying on a mathematical model, we propose a comprehensive analysis of this function, considering all the terrestrial mammals (spanning six orders of magnitude of the body mass,
M
) and focusing on the sole contribution of the lungs to this air conditioning. The results highlight significant differences between the small and the large mammals, as well as between rest and effort, regarding the spatial distribution of heat and water exchanges in the lungs, and also in terms of regime of mass transfer taking place in the lumen of the airways. Interestingly, the results show that the mammalian lungs appear to be designed just right to fully condition the air at maximal effort (and clearly over-designed at rest, except for the smallest mammals): all generations of the bronchial region of the lungs are mobilized for this purpose, with calculated values of the local evaporation rate of water from the bronchial mucosa that can be very close to the maximal ability of the serous cells to replenish this mucosa with water. For mammals with a mass above a certain threshold (
≃5
kg at rest and
≃50
g at maximal effort), it appears that the maximal value of this evaporation rate scales as
-1/8
M
at rest and
-1/16
M
at maximal effort and that around 40% (at rest) or 50% (at maximal effort) of the water/heat extracted from the lungs during inspiration is returned to the bronchial mucosa during expiration, independently of the mass, due to a subtle coupling between different phenomena. This last result implies that, above these thresholds, the amounts of water and heat extracted from the lungs by the ventilation scale with the mass such as the ventilation rate (i.e. as
3/4
M
at rest and
7/8
M
at maximal effort). Finally, it is worth to mention that these amounts appear to remain limited, but not negligible, when compared to relevant global quantities, even at maximal effort (4-6%). CITATION (original article): Haut, B., Karamaoun, C., Mauroy, B. et al. Water and heat exchanges in mammalian lungs. Sci Rep 13, 6636 (2023). DOI: https://doi.org/10.1038/s41598-023-33052-y
In this notebook, the different figures of the paper are presented, in chronological order. Please refer to the paper for the different legends.
Choices of different sets of colours for the figures
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,
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256
,
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256
,RGBColor[1,0,0],RGBColor
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128
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,RGBColor
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,RGBColor
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,
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89
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,
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191
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143
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​​colorl=Black​​styleapr=Dashed;
Out[]=
{
,
,
,
,
,
,
}
Out[]=
{
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Out[]=
Definition of the physico-chemical parameters
νg=1.7
-5
10
;​​Dg=2.7
-5
10
;​​ρt=993;​​MMw=18
-3
10
;​​cpt=4180;​​λt=0.615+
0.629-0.615
10
7;​​αt=
0.6
ρtcpt
;​​ρg=1.138;​​cpg=1006;​​λg=0.027;​​αg=
λg
ρgcpg
;​​Lm=43470;​​Tb=273.15+36.2;​​Csat=2.44;​​Csatp=0.12;​​Psat[T_]=6014Exp-
Lm
8.314
1
T
-
1
Tb
;​​Csaturation[T_]=
Psat[T]
8.314T
;​​Scref=
νg
Dg
;​​Shth[x_,y_]=1.3+facrac
x
2
2
y
;
Definition of the allometric laws and of the different key dimensionless numbers
Off[InterpolatingFunction::dmval];​​Off[Solve::ifun];​​h=
-1
3
2
;​​Dtr[M_]=1.5
-2
10
3
8
M
70
;​​dalv[M_]=200
-6
10
1
12
M
70
;​​Qin[M_,ψ_]=ψ250
-6
10
3
4
M
70
;
Increase, as a function of the mass, of the inspiration flow rate at maximal effort, when compared to rest
ventinc[M_]=18
1
8
M
70
;
Increase, as a function of the mass, of the cardiac output at maximal effort, when compared to rest
hrinc[M_]=4.5
1
8
M
70
;
Ratio ψ/ϕ at maximal effort, which is independent of the mass
rapϕψeff=
hrinc[M]
ventinc[M]
;​​β[M_]=7
-1
8
M
70
;​​tbl[M_,ϕ_]=
2000
ϕ
1
4
M
70
;
Number of generations in the bronchial tree
ngen[M_]:=Ceiling
Log
2dalv[M]
Dtr[M]

Log[h]
+1;​​Peclet[M_,index_,ψ_]=
1
index-1
(2h)
β[M]Qin[M,ψ]
πDg
Dtr[M]
2
;​​Sh[M_,index_,ψ_]=1.3+
Peclet[M,index,ψ]
2
2
β[M]
;​​Γ[M_,index_,ψ_]=Exp
2
2
β[M]
Sh[M,index,ψ]
Peclet[M,index,ψ]
-1;​​Λ[M_,index_,ψ_,ϕ_]=
λt
CsatpLm
Sh[M,index,ψ]
Dtr[M]
2
index-1
h
Dg
αttbl[M,ϕ]
;​​Θ=
3
2

Dtr[70]
2

λt
CsatpLm
αt
πβ[70]2
DgQin[70,1]tbl[70,1]
;
Limit value of Λ, for Pe  ∞, at rest
Λlimrep=Θ;
Limit value of Λ, for Pe  ∞, at maximal effort
Λlimeff=Θ
rapϕψeff
;​​Shconst=1.5;
Analytical expression of the yield
ηanal[x_]=-2x+2
x(1+x)
;
Definition of the colours and the legends
datapoids={{"Mouse",0.005},{"Weasel",0.05},{"Rat",0.5},{"Cat",5},{"Human",50},{"Horse",500},{"Elephant",5000}};​​listcolor=panel2;​​listMass=Table[datapoids[[i,2]],{i,1,Length[datapoids]}];​​listlegend=Table[StringJoin["M (kg) = ",ToString[listMass[[i]]]],{i,1,Length[listMass]}];​​listlegendp=Table[datapoids[[i,1]],{i,1,Length[listMass]}];
Function characterizing the exchanges, based on the resolution of the model equations, as a function of different parameters. Note: M is the mass, T0C is the temperature at the inlet of the trachea during inspiration (°C), and HR0 is the relative humidity at the inlet of the trachea during inspiration
Watex[M_,T0c_,HR0_,ψ_,ϕ_]:=​​thresh=0.1;​​T0=273.15+T0c;​​C0=HR0Csaturation[T0];​​​​nconsidered=ngen[M];​​rlistlung={Cin[0]0,Cex[nconsidered+1]1,Tin[0]0,Tex[nconsidered+1]1};​​inclung=Table[Cin[i],{i,1,nconsidered}];​​inclung=Join[inclung,Table[Cex[i],{i,1,nconsidered}]];​​inclung=Join[inclung,Table[Cμ[i],{i,1,nconsidered}]];​​eqlung=TableCin[i]
Cin[i-1]+Γ[M,i,ψ]Cμ[i]
1+Γ[M,i,ψ]
,{i,1,nconsidered};​​eqlung=Joineqlung,TableCex[i]
Cex[i+1]+Γ[M,i,ψ]Cμ[i]
1+Γ[M,i,ψ]
,{i,1,nconsidered};​​eqlung=Joineqlung,TableΛ[M,i,ψ,ϕ](1-Cμ[i])
1
2
Cμ[i]-
Cin[i]+Cin[i-1]
2
+
1
2
Cμ[i]-
Cex[i]+Cex[i+1]
2
,{i,1,nconsidered};​​buffer1=Table[Λ[M,i,ψ,ϕ],{i,1,nconsidered}];​​eqlung=FullSimplify[eqlung/.rlistlung];​​sollung=NSolve[eqlung,inclung][[1]];​​​​isign=Position[Table[If[1-Cin[i]>thresh,1,0]/.sollung,{i,1,nconsidered}],0][[1,1]];​​llocalrend=Table
(Cin[i]-Cin[i-1]+Cex[i]-Cex[i+1])
Cin[i]-Cin[i-1]
,{i,1,isign}/.sollung/.rlistlung;​​llocalrendbis=Table
(Cin[i]-Cin[i-1]+Cex[i]-Cex[i+1])
Cin[i]-Cin[i-1]
,{i,1,nconsidered}/.sollung/.rlistlung;​​llocalpuis=Table100
Cin[i]-Cin[i-1]+Cex[i]-Cex[i+1]
Cex[1]
,{i,1,nconsidered}/.sollung/.rlistlung;​​yield=Cex[1]/.sollung;​​imax=Position[llocalpuis,Max[llocalpuis]][[1,1]];​​​​listCex=Join[{{0,Cex[1]/.sollung}},Table[{i,(Cex[i+1]/.sollung)},{i,1,nconsidered-1}]];​​listCin=Join[{{0,0}},Table[{i,(Cin[i]/.sollung)},{i,1,nconsidered}]];​​listCμ=Tablei-
1
2
,(Cμ[i]/.sollung),{i,1,nconsidered};​​​​Wreal=
Qin[M,ψ]
2
(Csat-C0)yield
MMw
ρt
6
10
24×3600;(*enmlperday*)​​Preal=
Qin[M,ψ]
2
(Csat-C0)yieldLm;(*enW*)​​listevap=Table
Wreal
6
10
24×3600
Cin[i]-Cin[i-1]+Cex[i]-Cex[i+1]
Cex[1]
1
i-1
2
β[M]
Dtr[M]
i-1
h
2
(πDtr[M]
i-1
h
)
6
10
60,{i,1,nconsidered}/.sollung/.rlistlung;(*enμm/min*)​​Emax=Max[listevap];;
Plot of the dimensionless numbers, at rest (Fig 2a and Fig 2c)
Plot of the dimensionless numbers, at maximal effort (Fig 2b and Fig 2b)
Plot of the dimensionless concentration profiles, at rest and at maximal effort (Fig3)
Plot of imax as a function of M, at rest and at maximal effort, + analytical solution (Fig5)
Plot of dimensional parameters (Fig7)

CITE THIS NOTEBOOK

Water and heat exchanges in mammalian lungs​
by Benoit Haut​
Wolfram Community, STAFF PICKS, May 15 2023
​https://community.wolfram.com/groups/-/m/t/2920036