1. Tidal volume based on ideal body weight does not accurately represent lung strain which varies with severity of ARDS. 2. Plateau pressure is not a reliable surrogate for lung stress as its relationship to transpulmonary pressure depends on lung and chest wall compliance. 3. Respiratory frequency and flow influence pulmonary mechanics in a time-dependent manner, contributing to mechanical power delivered and ventilator-induced lung injury (VILI).

1. MATHEMATICS OF PULMONARY MECHANICS
Clinical Implications in VILI
Dr. Ubaidur Rahaman
M.D.
Internist and Critical Care Specialist

2. “The lungs of one man may bear, without injury,
as great a force as those of another man can exert;
which by the bellows cannot always be determined.”
Fothergill J.
Observation on a case published in the last volume of the medical essays, and c. of recovering a man dead
in appearance, by distending the lungs with air.
Philos Trans R Soc Lon 1745; 43:275-281.

3. SPONTANEOUS VENTILATION MECHANICAL VENTILATION
INDEPENDENT
VARIABLE
PLEURAL PRESSURE AIRWAY PRESSURE/ FLOW
DEPENDENT
VARIABLE
AIRWAY PRESSURE/ FLOW PLEURAL PRESSURE

4. Elastic force
LUNG
Elastic force
CHEST WALL
Functional Residual Capacity
(FRC)
At End Expiration
RESTING VOLUME OF RESPIRATORY SYSTEM

5. 0
25
50
75
100
0-20 20
Chest wall
Lung
Chest wall and Lung
( respiratory system)
FRC
RV
Pressure ( cm H2O)
Ppl, Pcw, Prs
Vitalcapacity%
TLC
P-V curve of Lung, Chest wall and Respiratory system
E= ∆P/∆V
ERS= Ecw + EL
C= 1/E
C= ∆V /∆P
1/CRS= 1/Ccw +1/CL
CRS = (CL* CCW)/ CL+ CCW

6. PPL-PATM
PPLAT-PPL
PPLAT-PATM CRS= VT/ PLAT
CL= VT/ (PPLAT- PPL)
CCW= VT/ PPL
PL
(TRANSPULMONARY PRESSURE)

7. PLEURAL PRESSURE ESTIMATION
ESOPHAGEAL CATHETER
Position of Patient
Position of Catheter tip
Esophageal muscle tone
Weight of heart and lung
Respiratory variation of PES correlates with PPL

8. PLEURAL PRESSURE ESTIMATION
CENTRAL VENOUS PRESSURE
Static and dynamic components of esophageal and central venous pressure during intra-abdominal hypertension.
Crit Care Med 2007, Valenza F, Gattinoni L.

9. Mechanical ventilators calculate and display RESPIRATORY SYSTEM MECHANICS
as LUNG MECHANICS requires invasive PPL estimation
PPLAT-PPL
PPLAT
CL= VT/ (PPLAT -PPL)
CRS= VT/ PLAT
Problem with PPLAT as surrogate of PL
lies in their relation to each other
PL

10. MATHEMATICS OF PL AND PPLAT
PL= VT/ CL
PL/ PPLAT= CRS/ CL
PL= PPLAT * CCW/ (CL+ CCW)
In normal respiratory system chest wall and lung compliances are equal, about 200 ml/ cmH2O.
therefore, for an applied airway pressure, generated PL will be half of the PPLAT.
∆PL= ∆PPLAT * ½
Consider CL decreased to half (100) and CCW remains normal (200)
∆ PL= ∆PPLAT * 200 / (100+200)
∆PL = ∆PPLAT * 2/3
Consider CL remains normal (200), but CCW decreases by half (100)
∆ PL = ∆PPLAT * 100 / (200+100)
∆PL= ∆PPLAT * 1/3

11. MATHEMATICS OF PL AND PPLAT

12. MATHEMATICS OF PPL AND PPLAT
PPL= VT/ CCW
PPL = PPLAT * CRS/ CCW
PP=PPLAT * CL/ (CCW+CL)
In normal respiratory system, CCW= CL, each about 200 ml/ cm2O.
Therefore, in mechanical ventilation, for an applied airway pressure, rise in PPL will be half of the PPLAT.
∆PPL= ∆PPLAT * 200/ (200+200) = ½
Assuming that CL decreases to half (100) and CCW remains normal (200),
∆ PPL = ∆PPLAT * 100 /100+200
∆PPL = ∆PPLAT * 1/3
Now Consider that CL remains normal (200), but CCW decreases by half (100)
∆ PPL= ∆PPLAT * 200/200+100
∆PPL= ∆PPLAT * 2/3

13. BARO……
TRAUMA
VOLU……
TRAUMA
STRESS = Y * STRAIN

14. MATHEMATICS OF STRESS AND STRAIN
FORCE
STRESS
FORCE/AREA
FORCE
STRAIN
∆L/LI
STRESS = Y * STRAIN

15. Stress (PL)= K * strain (VT/ FRC)
K is specific lung elastance, proportionality constant equivalent in pulmonary physiology
Assume VT = FRC
strain (VT/ FRC)= 1
Stress= K
Specific lung elastance is the PL which doubles the lung volume
K is animal species specific
In humans K= 13.5
In early ARDS, baby lung K is unaltered: not stiff but small healthy lung
MATHEMATICS OF STRESS AND STRAIN

16. Human lung FRC= 35 ml/kg , TLC= 80 ml/kg
MATHEMATICS OF STRESS AND STRAIN
One K (PL of 13.5 cmH2O) will increase lung
volume equal to FRC (35 ml/kg)
2.2 K will inflate lung to TLC (80/35)
PL of 30 cmH2O (13.5* 2.2) will increase lung
volume to TLC
TLC=complete unfolding of collagen
fibers=structural damage
1K(13.5cmH2O)
2.2K(30cmH2O)

17. Lung Stress and Strain during Mechanical Ventilation
Any Safe Threshold?
Alessandro Protti1, Massimo Cressoni1, Alessandro Santini1, Thomas Langer1, Cristina Mietto1, Daniela Febres1,
Monica Chierichetti1, Silvia Coppola1, Grazia Conte2, Stefano Gatti2, Orazio Leopardi1, Serge Masson3,
Luciano Lombardi4, Marco Lazzerini4, Erica Rampoldi5, Paolo Cadringher1, and Luciano Gattinoni1,6
Am J Respir Crit Care Med Vol 183. pp 1354–1362, 2011
Strain > 2.1 VILE and died
Strain≤ 1.5 survived with healthy lung
Strain 1.5-2 Gray zone
STRAIN 2
THRESHOLD FOR
ALVEOLAR INURY
HEALTHY LUNG

18. STRESS RAISERS
STRESSED
BUT NOT
STRAINED
INCREASED
STRESS AND
STRAIN
ALVEOLAR INHOMOGENEITY
MULTIPIES STRAIN OF 1.5-2 TO
HIGHER THAN 2

19. Stress =
4.5*PL
C1/10
N
Applied PL of 30 cmH2O
Generated strain of 132 cmH2O

20. SAFE LIMIT OF STRESS- STRAIN
BABY LUNG VOLUME 20% = 7 ml/kg (20% OF 35 ml/kg)
VENTILATED WITH 6 ml/kg
STRAIN= 6/7= 0.85, STRESS= 0.85*13.5= 11.6 cmH2O

21. Similar VT produced different strain and stress,
Different tidal volume generated similar stress
and strain
depending upon difference in FRC
VT based on IBW is poor surrogate of lung strain
as it generates variable strain (safe to injurious)
depending upon the baby lung volume,
which varies with severity of ARDS.

22. HOW TO ALLEVITE STRESS RISERS
REDUCE ALVEOLAR INHOMOGENEITY
PEEP
PRONE POSITION

23. SEVERE ARDS
(P/F 150)
16 HOURS
PER DAY

24. strainDYN
strainSTAT
strainGLOBA L= VT + VPEEP/ FRC
strainDYN = VT/ FRC
strainSTAT = VPEEP/ FRC
strainGLOBA L

25. Crit Care Med. 2013 Apr;41(4):1046-55
Lung stress and strain during mechanical ventilation: any difference between statics and
dynamics?
Protti A1, Andreis DT, Monti M, Santini A, Sparacino CC, Langer T, Votta E, Gatti S, Lombardi L, Leopardi O, Masson S, Cressoni
M, Gattinoni L

26. Viscoelastic Polyurethane
(memory foam)
Elastic rubber band
Viscous Paint
ELASTIC: : Stress = E * strain
VISCOUS: Stress= ή * Strain rate
VISCOELASTIC: Stress = E * strain + ή * strain rate
MATERIAL PROPERTY

27. STRESS RESPONSE DURING DYNAMIC AND STATIC STRAIN

28. CREEP: Gradual elongation (increasing strain) of material under constant stress,
STRESS RELAXATION: Gradual decrement in stress, under constant strain, due to dissipation of
stored energy in friction.
stress
strainstress
strain
time time
timetime
VISCOELASTIC MATERIAL EXHIBITS THREE PROPERTIES
CREEP, STRESS RELAXATION AND HYSTERESIS

29. HYSTERESIS: Area between loading and unloading stress-strain curve, which travel different paths due to stress
relaxation.
Duane V. Knudson. Fundamentals of Biomechanics 1st ed 2003, Springer
stress
strain
VISCOELASTIC MATERIAL EXHIBITS THREE PROPERTIES
CREEP, STRESS RELAXATION AND HYSTERESIS

30. VISCOELASTIC LUNG
DYNAMIC STRAIN
stress ᾱ strain and strain rate
STATIC STRAIN
stress relaxation

31. (low dynamic strain + high
static strain) + low strain rate
(low tidal volume + high PEEP)
+ low flow rate Less stress
Stress is
higher during dynamic
strain than in static
strain
Differential tolerance of
dynamic and static
strain in
mechanical ventilation
Static strain is better
tolerated than
dynamic strain
In ARDS DURING MECHANICAL VENTILATION,
FOR A GIVEN END INSPIRATORY LUNG VOLUME (GLOBAL STRAIN)

32. PRESSURE-TIME SCALAR IN VC-CMV
VISUAL REPRESENTATION OF PULMONARY MECHANICS
RAW= (PPEAK-P1)/ flow
RVE= (P1-PPLAT)/ flow
ERS STAT= (PPLAT- PEEP)/ VT
EVE= (P1-PPLAT)/ VT
ERS DYN= (P1-PEEP)/ VT
LUNG
RESISTANE
(RL)
ERSSTAT

33. For a constant tidal volume, if flow increases
VISCOELASTICITY MAKES PULMONARY MECHANICS TIME DEPENDENT
RL DECREASES
RAW increases
RVE decreases
ERS DYN INCREASES
EVE increases
FREQUENCY
FLOW
Ti

34. VISCOELASTIC MODEL OF MILIC-EMILI

35. TIME DEPENDENCY TO PULMONARY MECHANICS
• RVE falls, decreasing RL
• EVE rises, increasing ERS DYN
Increases flow
Dashpot does not get enough time to
dissipate energy by friction
• RVE rises, increasing RL
• EVE falls, decreasing ERS DYN
Decrease flow
Dashpot gets enough time to dissipate
energy by friction
FORACONSTANTTIDALVOLUME
For a
constant tidal volume and PEEP,
higher rate will increase stress

36. TIME DEPENDENCY TO PULMONARY MECHANICS
(high tidal volume + low PEEP)
+
high frequency
high stress

37. Anesthesiology 5 2016, Vol.124, 1100-1108
Mechanical Power and Development of Ventilator-induced Lung Injury
Massimo Cressoni, M.D.; Miriam Gotti, M.D.; Chiara Chiurazzi, M.D.; Dario Massari, M.D.; Ilaria Algieri, M.D.; Martina Amini, M.D.; Antonio Cammaroto,
M.D.; Matteo Brioni, M.D.; Claudia Montaruli, M.D.; Klodiana Nikolla, M.D.;Mariateresa Guanziroli, M.D.; Daniele Dondossola, M.D.; Stefano Gatti, M.D.;Vincenza
Valerio, Ph.D.; Giordano Luca Vergani, M.D.; Paola Pugni, M.D.; Paolo Cadringher, M.Sc.; Nicoletta Gagliano, Ph.D.; Luciano Gattinoni, M.D., F.R.C.P.
MECHANICAL POWER
(Area between PL and VT in P-V loop)
F
PL
VT
Mechanical Power Threshold
12 J/min
Extensive pulmonary edema

38. MECHANICAL POWER
(Area between PL and VT in P-V loop)
F
PL
VT
Very
small VT
Very
high F
High
MECHANICAL
POWER
FAILURE OF HIGH FREQUENCY OSCILLATORY VENTILATION

39. They compared the measured and calculated mechanical power at different PEEP, and found it equivalent in
normal subjects and ARDS patients.
Mechanical power was measure from P-V loop,
Mechanical power was calculated from Mechanical Power Equation, derived from Equation of Motion,
Mechanical Power increases exponentially with TV, P, flow and RR and linearly with PEEP,
Though PEEP increases Mechanical Power linearly, it may decrease the other causes of VILI (inhomogeneity,
atelectotrauma. The final effect of PEEP in VILI would depend upon which two action prevails and in which
patients,
Mechanical Power Equation can be easily implemented in every mechanical ventilator software.

40. PPLAT
F
PEEP
RR
RAW
VT
MECHANICAL
POWER
• P= TV*ERS + RAW*F + PEEP
EQUATION
OF MOTION
• 1/2*VT*VT*ERS+ VT*RAW*F +
VT*PEEP
EBREATH
• 0.098*RR[VT
2 {1/2*ERS +
RR*(1+I:E)/(60*I:E)*RAW} + VT*PEEP]
POWERRS
MECHANICAL POWER

41. P-V LOOP
ZEEP VS PEEP
MECHANICAL POWER

42. MECHANICAL POWER

43. knowledge of mathematics of pulmonary mechanics is pertinent to understand the pathophysiology of VILI.
Aim of mechanical ventilation is to correct the deranged pulmonary mechanics (PEEP and prone position) and
optimize gas exchange with a strategy to prevent VILI by minimizing delivered mechanical power for a duration
when pathophysiology is reversed.
Mechanical power is a product of tidal volume, transpulmonary pressure and respiratory frequency.
Clinical studies are needed to identify safe threshold of mechanical power which may help in decision making
between low tidal volume mechanical ventilation and extracorporeal lung support.
CONCLUSION

44. 1. Tidal volume based on ideal body weight do not represent lung strain because of variability of baby lung with
severity of ARDS,
2. Plateau pressure (Airway pressure) is not as reliable surrogate of lung stress as relationship between
transpulmonary pressure and plateau pressure is determined by ratio of compliances of lung and respiratory
system. Transpulmonary pressure is inversely related to lung compliance.
3. Respiratory frequency and inspiratory flow influence pulmonary mechanics, making it time dependent, and
contribute to the mechanical power delivered to alveoli and resultant VILI.
4. Limiting the dynamic strain (VT) at the expense of static strain (PEEP) and reducing the effect of stress multipliers
by minimizing alveolar heterogeneity with optimal PEEP and prone position may reduce VILI.
5. High flow and respiratory frequency may increase the dynamic stress and delivered mechanical power resulting
in VILI.
6. Safety of low tidal ventilation can be assessed by estimating baby lung volume and mechanical power threshold,
which can become rationale indication for extracorporeal lung support.
SUMMARY