Download Detailed derivation of the Stress-Strain Index SSI

Detailed derivation of the Stress-Strain Index (SSI)
Stress is defined as the distribution of force over an area. In other words, Stress has the
units of pressure and for the lung can be estimated from the change in transpulmonary
pressure (dPL).
Stress = dPL
[s1]
Strain is defined as a dimensionless measure of length of change. For the lung, strain
can be estimated as tidal volume (Vt) related to the End Expiratory Lung Volume
(EELV).
Strain = Vt / EELV
[s2]
Stiffness (also known as Young’s Modulus) is the change in Stress required for a
change in Strain or Stress/Strain ratio. For the lung, it is the change in pressure required
for a change in relative volume, in other word the specific Elastance (EL,sp).
EL,sp = EL * EELV
[s3]
Where EL is the Elastance of the lung defined as:
EL = dPL / Vt
[s4]
The [s3] equation can therefore be written like:
EL,sp = (dPL / Vt) * EELV
[s5]
From equation [s5] dPL can be expressed as:
dPL = (Vt * EL,sp) / EELV
The Stress-Strain Index (SSI) is defined as
[s6]
SSI = α * Stress * β * Strain * f
[s7]
Where the relative contribution of Stress and Strain is expressed by α and β coefficients,
both being between 0 and 1
Substituting Stress and Strain with the equations [s1] and [s2], the equation [s7]
becomes
SSI = α * dPL * β * (Vt / EELV) * f
[s8]
From equation [s6] the SSI can be expressed as:
SSI = α * [(Vt * EL,sp)/EELV] * β * (Vt / EELV) * f
[s9]
or
SSI = α * (Vt / EELV) * β * (Vt / EELV) * EL,sp * f
[s10]
To find the least injurious Vt – f combination, i.e. Vt and f at which SSI is minimal, the
first derivative of the equation [s10] with respect to f needs to be set to zero and be
solved for the following boundary conditions:
B1) alveolar ventilation (V’A) is constant
B2) exhalation time is long enough to avoid breath-stacking i.e. expiratory time longer
than 2 times the time constant of the respiratory system
B3) EL does not change with respiratory rate and is linear within the tidal pressurevolume ranges
B4) EELV and Stress and Strain are stable from breath to breath.
According to B1, alveolar ventilation V’A remains constant:
V’A = f * (Vt - Vd) = constant
[s11]
or
Vt = (V’A + f * Vd) / f = constant
[s12]
Vt = (V’A / f + Vd) = constant
[s13]
or
Where Vd is dead space (anatomical dead space or series dead space).
Substituting Vt in the equation [s10], the SSI becomes:
SSI = α * [(V’A / f + Vd) / EELV] * β * [(V’A / f + Vd) / EELV] * EL,sp * f
[s14]
SSI = α * β *[(V’A / f + Vd) / EELV] 2 * EL,sp * f
[s15]
or
The first derivative of this function with respect to f is:
dSSI / df = d[α * β * [(V’A / f +Vd) / EELV] 2 * EL,sp * f] / df
[s16]
or
dSSI / df = d[α * β / EELV 2 *(V’A / f + Vd) 2 * EL,sp * f] / df
[s17]
dSSI / df = d[α * β * EL,sp / EELV 2 *(V’A / f + Vd) 2 * f] / df
[s18]
or
or
dSSI / df = d[α * β * EL,sp / EELV 2 *(V’A2 / f2 + 2*V’A / f*Vd + Vd 2) * f] / df
[s19]
dSSI / df = d[α * β * EL,sp / EELV 2 *(V’A2 / f + 2 * V’A * Vd + f * Vd 2) ] / df
[s20]
or
or
dSSI / df = α * β * EL,sp /EELV 2 * d[V’A2 / f + 2 * V’A * Vd + f * Vd 2] / df [s21]
or
dSSI / df = α * β * EL,sp / EELV2 * (- V’A2 / f2 + Vd2)
[s22]
SSI is minimal when
dSSI / df = α * β * EL,sp / EELV2 * (-V’A2 / f2 + Vd2) = 0
or
[s23]
- V’A2 / f2 + Vd 2 = 0
[s24]
Vd 2 = V’A2 / f2
[s25]
f2 = V’A2 / Vd 2
[s26]
f = V’A / Vd
[s27]
or
or
or
The tidal volume Vt at which SSI is minimal follows from equation [s11]:
f = f * (Vt - Vd) / Vd
[s28]
1 = (Vt - Vd) / Vd
[s29]
Vd = V t - Vd
[s30]
- Vt + Vd = - Vd
[s31]
Vt - Vd = V d
[s32]
Vt = 2 * Vd
[s33]
or
or
or
or
or