MATHEMATICAL MODELLING OF NON-INVASIVE PRESSURE SUPPORT VENTILATION: INVESTIGATION OF TIDAL VOLUME INSTABILITY - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 281

P set during inspiration

P vent =

P peep during expiration

During expiration, the applied airway pressure to the mask and airway
pressure is xed and is denoted by the symbol P peep . The PEEP pressure
may be zero. In [3], it was assumed that the pressure drop due to the leak
in the face mask is

P mask = R m Q mask

where R m is a resistance factor and Q mask is the ow through the leak.
Using conservation of mass, the total ow from the ventilator at any
time during inspiration (Q vent ) is equal to the sum of the ow into the lung
and ow out through the mask leak (to the atmosphere):

Q vent = Q mask + Q lung :

The ow into the lung, Q lung , was assumed to be equal to the instantaneous
rate of change of the volume of the lung,

dV i

dt .
The ow from the ventilator Q vent (t) is determined by the equation:

!

ex V (n)

P set
R m

1
R i

(t)

Q (n) P set P (n1)

i

vent (t) =

+ (3)

C

for (n1)

tot t (n1)

+ t (n)
i

. To determine the end-expiratory pressure,

tot

we set V (n)
i

( (n1)
tot

+ t (n)
i

) = V (n) ( (n1)

tot

+ t (n)
i

) and solve for P (n)

ex . e

3. The Linear Lung Model

We now consider a ventilator connected to a patient. The ventilator applies
a constant pressure P set until the ow Q into the patient drops to specied
fraction of the initial ow. At this point inspiration ends and the patient
starts expiration. The dierential equations for the relative lung volume
during nth breath with = 1 are:
Inspiration

dV (n)
i
dt

+ V (n)

+ P (n1) = P set ; (n1)

tot t (n1)

+ t (n)
i

R i

i
C

ex

tot

(4)

V (n)
i

(0) = 0