Institute for Pure & Applied Mathematics
Optimal Transport in the Human Body: Lungs & Blood
From the Lung to the Cells‘ Powerhouses:
Symmorphosis in the Design of the Pathway for Oxygen
Ewald R. Weibel, Institute of Anatomy, University of Berne
From the Lung to the Cells‘ Powerhouses:
Symmorphosis in the Design of the Pathway for Oxygen
1. Why do cells need O2 ?
2. O2 consumption is regulated up to limit
3. What sets the limit?
4. Hypothesis of Symmorphosis:
structural design is economic
5. Quantitative structure-function model
6. Test of hypothesis: comparative physiology
7. Conclusions:
the special case of the lung
1. Why cells need Oxygen? — energy transduction
Liver cell
ATP = adenosine tri-phosphate
Mitochondria
Synthesis
Muscle cell
A-P-P~P
Liver, glands — brain, kidney
—
muscle
Muscle work
1. Energy transduction in cells: regenerate ATP
Liver cell
ATP = adenosine tri-phosphate
Mitochondria
Synthesis
Muscle cell
A-P-P ~P
Energy transfer
Liver, glands — brain, kidney
—
muscle
A-P-P + P
Muscle work
1. Energy transduction in cells: regenerate ATP
Liver cell
ATP
Phosphorylation in mitochondria
O2
Muscle cell
A-P-P~P
fuel
A-P-P + P
Energy input:
Depends on fuel & O2 supply
2. Energy demands vary with activity level
2. Energy demands vary with activity level
Rest to Work
O2 consumption
10x
2. Energy demands vary with activity level
Rest to Work
O2 consumption
10x
2. Energy demands vary with activity level
O2 flow rate increases linearly with work rate
•
up to limit VO2max
•
VO2max
glucose
Margaria 1967
more energy <— from glycolysis
Aerobic energy limited by O2 supply
2. Energy demands vary with activity level
Work in athletes
O2 consumption
athletes 15x
10x
Conclude:
Oxidative metabolism is limited by
O2 supply from ambient air to mitochondria
Is limit set by
structural design of the
Pathway for Oxygen ?
3. Design of the Pathway for Oxygen
Design elements & function:
Distribution trees & networks:
Regulation of air & blood flow
3. Design of the Pathway for Oxygen
Design elements & function:
Distribution trees & networks:
Regulation of air & blood flow
Exchange interfaces:
Transfer conductance
Pulmonary gas exchange
Tissue gas exchange
3. Design of the Pathway for Oxygen
Design elements & function:
Distribution trees & networks:
Regulation of air & blood flow
Exchange interfaces:
Transfer conductance
Mitochondria:
Energy transduction
Energy transduction
3. Design of the Pathway for Oxygen
Critical factors potentially
limiting O2 flux:
Exchange surfaces
Blood flow & heart
Mitochondria
Energy transduction
O2 transport: from Lung to Mitochondria
Driving force for O2 flow :
cascade of partial O2 pressures
O2 transport: from Lung to Mitochondria
Driving force for O2 flow :
• cascade of partial O2 pressures
Determining factors:
• Ventilation
• lung diffusion conductance
• blood flow & red blood cells
• tissue diffusion conductance
• mitochondrial energy transduction
O2 transport: from Lung to Mitochondria
Driving force for O2 flow :
• cascade of partial O2 pressures
Determining factors:
• Ventilation
• lung diffusion conductance
• blood flow & red blood cells
• tissue diffusion conductance
• mitochondrial energy transduction
Oxidative phosphorylation limited:
which factor is limiting?
4. Limitation of O2 transport: Lung to Mitochondria
(1) bottleneck hypothesis:
single step limiting,
all others excess capacity
(2) distributed control hypothesis:
all steps with similar capacity
4. Limitation of O2 transport: Lung to Mitochondria
(1) bottleneck hypothesis:
single step limiting,
all others excess capacity
(2) distributed control hypothesis:
all steps with similar capacity
Distributed control requires:
all steps designed commensurate to needs
“enough but not too much”
= Symmorphosis
5. Model of Oxygen Pathway
Test of hypothesis of symmorphosis
FUNCTION
VÝOLung
= (PA O 2 - PbO 2 ){tc,O2}
2
• DESIGN
• DLO 2 {S(A),S(c),V(c),VV(ec),ht}
. Pa O 2 - v. PvO 2 ).f H • V s {V(LV)} . V V(ec )
VÝOHeart
=
(
a
2
VÝOCapillaries
= (PbO 2 - Pc O 2 ){tc,O2}
2
• DTO 2 {S(c),V(c),VV(ec),(c,m)}
Mitochondria
Ý
V O2 =  vÝO2
• V( mt) {S V(im,m)}
Ý ATP }
{m
5. Model of Oxygen Pathway
Test of hypothesis of symmorphosis
FUNCTION
• DESIGN
VÝO2
= (PA O 2 - PbO 2 ){tc,O2}
VÝO2
= (a . Pa O 2 - v. PvO 2 ).f H • V s {V(LV)} . V V(ec )
VÝO2
= (PbO 2 - Pc O 2 ){tc,O2}
• DTO 2 {S(c),V(c),VV(ec),(c,m)}
VÝO2
=  vÝO2
• V( mt) {S V(im,m)}
Ý ATP }
{m
• DLO 2 {S(A),S(c),V(c),VV(ec),ht}
•
Symmorphosis predicts: Design variables adjusted to VO2max
5. Model of Oxygen Pathway
•
Test by varying VO2max
FUNCTION
• DESIGN
VÝO2
= (PA O 2 - PbO 2 ){tc,O2}
VÝO2
= (a . Pa O 2 - v. PvO 2 ).f H • V s {V(LV)} . V V(ec )
VÝO2
= (PbO 2 - Pc O 2 ){tc,O2}
• DTO 2 {S(c),V(c),VV(ec),(c,m)}
VÝO2
=  vÝO2
• V( mt) {S V(im,m)}
Ý ATP }
{m
• DLO 2 {S(A),S(c),V(c),VV(ec),ht}
5. Model of Oxygen Pathway
•
Test by varying VO2max
Measure design parameters
FUNCTION
• DESIGN
VÝO2
= (PA O 2 - PbO 2 ){tc,O2}
VÝO2
= (a . Pa O 2 - v. PvO 2 ).f H • V s {V(LV)} . V V(ec )
VÝO2
= (PbO 2 - Pc O 2 ){tc,O2}
• DTO 2 {S(c),V(c),VV(ec),(c,m)}
VÝO2
=  vÝO2
• V( mt) {S V(im,m)}
Ý ATP }
{m
• DLO 2 {S(A),S(c),V(c),VV(ec),ht}
•
Symmorphosis predicts: Design variables adjusted to VO2max
6. Testing the hypothesis of
symmorphosis by
comparative physiology
FUNCTION
ACTIVITY
• DESIGN
BODY SIZE
VÝO2
= (PA O 2 - PbO 2 ){tc,O2}
VÝO2
= (a . Pa O 2 - v. PvO 2 ).f H • V s {V(LV)} . V V(ec )
VÝO2
= (PbO 2 - Pc O 2 ){tc,O2}
• DTO 2 {S(c),V(c),VV(ec),(c,m)}
VÝO2
=  vÝO2
• V( mt) {S V(im,m)}
Ý ATP }
{m
• DLO 2 {S(A),S(c),V(c),VV(ec),ht}
•
(1) Adaptive Variation of VO2max:
athletic vs. sedentary species
Goat / Dog / Pronghorn = 1 / 2.5 / 5
•
(1) Adaptive Variation of VO2max:
Antilocarpa americana
the pronghorn is the top athlete
among mammals
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Adaptive Variation
•
Species
Mb
VO2max/Mb
Goat
27.7
57
Dog
28.2
137
Pronghorn 28.4
272
kg
Karas, Taylor, Hoppeler, Lindstedt & al.
ml·min-1·kg-1
5. Model of Oxygen Pathway
•
VO2max and Mitochondria
FUNCTION
• DESIGN
VÝO2
= (PA O 2 - PbO 2 ){tc,O2}
VÝO2
= (a . Pa O 2 - v. PvO 2 ).f H • V s {V(LV)} . V V(ec )
VÝO2
= (PbO 2 - Pc O 2 ){tc,O2}
• DTO 2 {S(c),V(c),VV(ec),(c,m)}
VÝO2
=  vÝO2
• V( mt) {S V(im,m)}
Ý ATP }
{m
• DLO 2 {S(A),S(c),V(c),VV(ec),ht}
Adaptive Variation
•
•
Species
Mb
VO2max/Mb V(mi)/Mb VO2max/V(mi)
Goat
27.7
57
10.0
5.7
Dog
28.2
137
29.7
4.6
Pronghorn 28.4
272
46.2
5.9
ml·min-1·kg-1
ml·kg-1
kg
Karas, Taylor, Hoppeler, Lindstedt & al.
mlO2 ·min-1/ml(mi)
•
(2) Allometric Variation of VO2max
0.002 kg
M1
500 kg +
M3/4
500 kg
•
VO2max = a · Mb
Etruscan shrew 2 g
at rest b = 3/4
at max b = 0.87
(M.Kleiber)
Allometric Variation
•
VO2max & V(mi) parallel
~ Mb 0.96
C.R.Taylor, H. Hoppeler, J.H. Jones & al.
Allometric Variation
•
VO2max & V(mi) parallel
~ Mb 0.96
Athletic species higher
C.R.Taylor, H. Hoppeler, J.H. Jones & al.
Conclude: in all variations:
•
VO2max ~ V(mi)
~ 5 mlO2/min/ml
Hoppeler, Lindstedt 1985
5. Model of Oxygen Pathway
•
VO2max and Capillaries
FUNCTION
• DESIGN
VÝO2
= (PA O 2 - PbO 2 ){tc,O2}
VÝO2
= (a . Pa O 2 - v. PvO 2 ).f H • V s {V(LV)} . V V(ec )
VÝO2
= (PbO 2 - Pc O 2 ){tc,O2}
• DTO 2 {S(c),V(c),VV(ec),(c,m)}
VÝO2
=  vÝO2
• V( mt) {S V(im,m)}
Ý ATP }
{m
• DLO 2 {S(A),S(c),V(c),VV(ec),ht}
O2 supply from capillaries
Hypothesis:
•
VO2max ~ V(c)
Allometric variation in capillary volume
Capillary volume V(c)
varies with same slope as
•
VO2max and V(mt)
Adaptive variation
Adaptive variation
Hct = VV(ec,c)
Adaptive variation
x
ec
V(ec)
x
2.5
Hct = VV(ec,c)
O2 supply from capillaries
Conclusion:
•
VO2max ~ V(c)xVV(ec)
~ V(ec)
Two structures share adaptive effort
5. Model of Oxygen Pathway
•
VO2max and Heart
FUNCTION
• DESIGN
VÝO2
= (PA O 2 - PbO 2 ){tc,O2}
VÝO2
= (a . Pa O 2 - v. PvO 2 ).f H • V s {V(LV)} . V V(ec )
VÝO2
= (PbO 2 - Pc O 2 ){tc,O2}
• DTO 2 {S(c),V(c),VV(ec),(c,m)}
VÝO2
=  vÝO2
• V( mt) {S V(im,m)}
Ý ATP }
{m
• DLO 2 {S(A),S(c),V(c),VV(ec),ht}
5. Model of Oxygen Pathway
•
VO2max and Heart
FUNCTION
• DESIGN
VÝO2
= (PA O 2 - PbO 2 ){tc,O2}
VÝO2
= (a . Pa O 2 - v. PvO 2 ).f H • V s {V(LV)} . V V(ec )
VÝO2
= (PbO 2 - Pc O 2 ){tc,O2}
• DTO 2 {S(c),V(c),VV(ec),(c,m)}
VÝO2
=  vÝO2
• V( mt) {S V(im,m)}
Ý ATP }
{m
• DLO 2 {S(A),S(c),V(c),VV(ec),ht}
Adjust in adaptive variation
5. Model of Oxygen Pathway
•
VO2max and Heart
FUNCTION
• DESIGN
VÝO2
= (PfAH O=2 150min
- PbO 2 ){t-1c,O2}
VÝO2
= (a . Pa O 2 - v. PvO 2 ).f H • V s {V(LV)} . V V(ec )
VÝO2
= (PbO 2 - Pc O 2 ){tc,O2}
VÝO2
=  vÝO2
fH = 1050min-1
Ý ATP }
{m
• DLO 2 {S(A),S(c),V(c),VV(ec),ht}
• DTO 2 {S(c),V(c),VV(ec),(c,m)}
• V( mt) {S V(im,m)}
Adjust in adaptive variation
Adjusts in allometric variation
fH ~ M-0.15
Pathway for Oxygen
Design parameters are
•
matched to VO2max
but not in simple fashion
Mitochondria
V(mt)
Capillaries
V(c) x VV(ec)
Heart
V(LV) x VV(ec) / fH
The adaptive effort in design parameters is shared
& accounts for constraints on functional parameters
compatible with Symmorphosis
5. Model of Oxygen Pathway
•
VO2max and the Lung
FUNCTION
• DESIGN
VÝO2
= (PA O 2 - PbO 2 ){tc,O2}
VÝO2
= (a . Pa O 2 - v. PvO 2 ).f H • V s {V(LV)} . V V(ec )
VÝO2
= (PbO 2 - Pc O 2 ){tc,O2}
• DTO 2 {S(c),V(c),VV(ec),(c,m)}
VÝO2
=  vÝO2
• V( mt) {S V(im,m)}
Ý ATP }
{m
• DLO 2 {S(A),S(c),V(c),VV(ec),ht}
Pulmonary Gas Exchanger is different
Lung at interface to environment
Complex Pathway for O2 from Air to Blood
Ventilation
Perfusion
Diffusion
Pulmonary Gas Exchange
•
VO2 = (PAO2 -PcO2)·DLO2
PIO2
Driving force
• PAO2 < PIO2
PAO
2
PcO2
PA O
PaO2
2
PvO2
• PvO2 < PcO2 < PaO2
PcO2
tc
Driving force is smaller than
input pressure difference (PIO2 – PvO2)
Pulmonary Gas Exchange
•
VO2 = (PAO2 -PcO2)·DLO2
PIO2
Determining conductance
Pulmonary Diffusing Capacity
PAO
DLO2
2
PcO2
PaO2
PvO2
PcO2
tc
DLO2-1 = DMO2-1 + DbO2-1
DLO2 = f {SA, Sc, Vc, Vvec, 1/hb}
Predict:
•
VO2max ~ DLO2
Pulmonary Gas Exchange
•
VO2 = (PAO2 -PcO2)·DLO2
PIO2
Determining conductance
Pulmonary Diffusing Capacity
PAO
DLO2
2
PaO2
PvO2
PcO2
DLO2-1 = DMO2-1 + DbO2-1
DLO2 = f {SA, Sc, Vc, Vvec, 1/hb}
... not what we find ...
tc
Predict:
•
VO2max ~ DLO2
Morphometry of Human Lung & DLO2
• Body mass
±
4
kg
• Alveolar surface
130 ±
12
m2
• Capillary volume
194 ±
30
ml
• Total barrier thickness
1.15 ±
0.01 µm
Gehr, Bachofen et al. 1978
74
Morphometry of Human Lung & DLO2
• Body mass
±
4
kg
• Alveolar surface
130 ±
12
m2
• Capillary volume
194 ±
30
ml
• Total barrier thickness
1.15 ±
0.01 µm
Gehr, Bachofen et al. 1978
74
Morphometry of Human Lung & DLO2
• Body mass
±
4
kg
• Alveolar surface
130 ±
12
m2
• Capillary volume
194 ±
30
ml
• Total barrier thickness
1.15 ±
0.01 µm
• Diffusing capacity
•
74
DLO2
158 mlO2.min-1.mmHg-1
VO2 of 3.5 L/min with PO2 difference of 20 mmHg
Gehr, Bachofen et al. 1978
Morphometry of Human Lung & DLO2
• Body mass
±
4
kg
• Alveolar surface
130 ±
12
m2
• Capillary volume
194 ±
30
ml
• Total barrier thickness
1.15 ±
0.01 µm
• Diffusing capacity
74
DLO2
158 mlO2.min-1.mmHg-1
• Physiological DLO2 exercise 100 mlO2.min-1.mmHg-1
Gehr, Bachofen et al. 1978
Diffusing capacity
DLO2
158 mlO2.min-1.mmHg-1
Physiological max
DLO2
100 mlO2.min-1.mmHg-1
Normal lung has excess capacity: not limiting O2 uptake
Diffusing capacity
DLO2
158 mlO2.min-1.mmHg-1
Physiological max
DLO2
100 mlO2.min-1.mmHg-1
Normal lung has excess capacity: not limiting O2 uptake
but can become limiting in athletes
1.3x
•
Is DLO2 limiting in athletes?
Test: exercise in hypoxia
Athlete cannot
maintain O2 saturation
in hypoxia (altitude)
DLO2 is limiting
normal
athlete
athlete
Adaptive Variation of Pulmonary Gas Exchange
•
•
Species
Mb
VO2max/Mb DLO2 /Mb VO2max/DLO2
Goat
27.7
57
4.8
11.9
Dog
28.2
137
7.1
19.4
kg
Taylor, Karas, Weibel, Hoppeler 1987
ml·min-1·kg-1
ml· min-1·mmHg-1· kg-1
mmHg
1.6x
Varying driving force
••
at VO2max
excess capacity
sedentary
C.R. Taylor, R.H.Karas, H. Hoppeler, J.H. Jones et al.
athletic
Pulmonary Diffusing Capacity
•
VO2 = (PAO2 -PcO2)·DLO2
• normal sedentary species
•
VO2max
not ~ DLO2
excess capacity 1.5x
DLO2 with ± safety factor
PaO2
PvO2
PcO2
• athletic species
•
tc
VO2max
~ DLO2
no excess capacity
Allometric variation of Pulmonary Gas Exchange
•
VO2max ~ M0.86
DLO2 ~ M1.08
Similar scaling ?
Allometric variation of Pulmonary Gas Exchange
•
VO2max ~ M0.86
DLO2 ~ M1.08
•
VO2max / DLO2 = (PAO2 – PcO2)
somewhat smaller in large animals
Why is (PAO2 – PcO2) smaller in large mammals?
Consider geometry of acinar airways:
• Alveoli arranged along airways in acinus
• Capillaries perfuse surface of alveoli
Why is (PAO2 – PcO2) smaller in large mammals?
Consider geometry of acinar airways:
• Alveoli arranged along airways in acinus
• Capillaries perfuse surface of alveoli
—> serial ventilation – parallel perfusion
Why is (PAO2 – PcO2) smaller in large mammals?
Consider geometry of acinar airways:
• Alveoli arranged along airways in acinus
• Capillaries perfuse surface of alveoli
—> serial ventilation – parallel perfusion
O2 in alveolar air screened!
Why is (PAO2 – PcO2) smaller in large mammals?
Submit that screening effect is larger in large acini
—> lower average PAO2 in large lungs
Why is (PAO2 – PcO2) smaller in large mammals?
Size of acinus varies with body size
0.03 kg
0.3 kg
70 kg
3 kg
Diffusional “screening” : size matters!
Diameter of acinus
rest
exercise
inner acinar surface area
Diffusional “screening” : size matters!
Determined by:
rest
exercise
• Ratio (diffusivity in air) / (permeability) = 
• Ratio (surface) / (diameter acinus) = Lp
Screening depends on /Lp
Sapoval et al. PNAS 2002
Diffusional “screening”
• Ratio
(diffusivity in air) /
(permeability) = 
• Ratio
(surface) /
(diameter acinus) = Lp
Despite large size variation:
± invariant
Lp small variation
rest
exercise
Diffusional “screening”
• Ratio
(diffusivity in air) /
(permeability) = 
• Ratio
(surface) /
(diameter acinus) = Lp
... and yet:
Ratio /Lp
smaller in
large lungs
2–3 fold
rest
exercise
Species
Mb
V(ac) /Lp
kg
ml
Mouse
0.025 0.41 2.7
Rat
0.300 1.70 1.85
Rabbit
3.000 3.4
Human (1/8)
Sapoval et al. PNAS 2002
70.000 23.4
2.3
0.93
Diffusional “screening” stronger in large mammals
Lowers driving force (PAO2 – PcO2) more
than in small mammals
rest
exercise
(PAO2 – PcO2) / (PIO2 – PvO2) < 1
—> apparent acinus efficiency < 1
PI
PA
Pc
Pv
Diffusional “screening”
Small acini are more efficient:
rest
exercise
PI
PA
Pc
Pv
Diffusional “screening”
Small acini are more efficient:
rest
exercise
Low efficiency of larger acini in larger species
explains the reduced O2 uptake per DLO2 !
PI
PA
Pc
Pv
Diffusional “screening”
Small acini are more efficient:
rest
exercise
Low efficiency of larger acini in larger species
explains the reduced O2 uptake per DLO2 !
“Smaller is better — but not too small”
Sapoval et al. PNAS 2002
PI
PA
Pc
Pv
Pathway for Oxygen
Design parameters
are
•
matched to VO2max
but not in simple fashion
Mitochondria
Capillaries
Heart
Lung
V(mt)
V(c) x VV(ec)
V(LV) x VV(ec) / fH
~ DLO2 {S(a), V(c), VV(ec), b}
but with variable excess
capacity
Does structural design fit Symmorphosis principle:
„enough but not too much“ ?
Pathway for Oxygen
Design parameters
are
•
matched to VO2max
but not in simple fashion
Mitochondria
Capillaries
Heart
Lung
V(mt)
V(c) x VV(ec)
V(LV) x VV(ec) / fH
~ DLO2 {S(a), V(c), VV(ec), b}
but with variable excess
capacity
Lung apparently does not fit Symmorphosis principle:
„may have too much“ ?
Pathway for Oxygen
Why is the lung different ?
Constraints on lung design:
1) Plasticity & adaptability of adult lung is limited
2) Lung at interface to external PO2 = variable
3) Size constraints on diffusion in acinar airways
& on ventilation ?
Pathway for Oxygen
Constraints of body size on design of system:
1) Locomotor muscle: stride frequency
2) Heart muscle: heart frequency
3) Lung:
a) diffusion screening in acinus
b) respiratory frequency
fR 300 min-1
fH 1050
Pathway for Oxygen
Design parameters
are
•
matched to VO2max
but not in simple fashion
Mitochondria
Capillaries
Heart
Lung
V(mt)
V(c) x VV(ec)
V(LV) x VV(ec) / fH
~ DLO2 {S(a), V(c), VV(ec), b}
but variable
Symmorphosis in the pathway for oxygen:
yes for internal compartments —
perhaps for lung — constraints of size, environment etc.
Institute for Pure & Applied Mathematics
Optimal Transport in the Human Body: Lungs & Blood
From the Lung to the Cells‘ Powerhouses:
Symmorphosis in the Design of the Pathway for Oxygen
CONCLUSION
Is the design of this vital functional system favouring
„optimal transport in the (human) body“ ?
Yes — there is enough of everything but not too much !
The Pathway for Oxygen
Functional Limits & Structural Design
in the Respiratory System
University of Bern, Department of Anatomy
Harvard University, Museum of Comparative Zoology,
Concord Field Station
C. Richard Taylor†, Hans Hoppeler, Ewald R. Weibel
Richard Karas, James Jones, Stan Lindstedt, Eduardo Bicudo,
Kevin Conley, Odile Mathieu, Susan Kayar, Vilma Stalder-Navarro,
Jean-Michel Weber, Ruth Vock, Thomas Roberts, Connie Hsia,
Hans Howald, Arne Lindholm, Kim Longworth, Fabienne Doffey,
Bernard Sapoval, Marcel Filoche &&&&
Thanks
Hans Hoppeler
C. Richard Taylor†