MATHEMATICAL MODELLING OF NON-INVASIVE PRESSURE SUPPORT VENTILATION: INVESTIGATION OF TIDAL VOLUME INSTABILITY - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 282

Expiration

dV (n)
e
dt

+ V (n) + P (n)

ex = P peep ; (n1)

+ t (n i t (n)

R e

e
C

tot tot

(5)

V (n) ( (n)

tot ) = 0

e

for n = 1; 2; 3; ::: .
In our study, we restrict our analysis to settings where there is no leakage
by the mask i.e., R m !1. In this case, equation (3) becomes

!

ex V (n)

1
R i

(t)

Q (n) ; (n1)

tot t (n1)

+ t (n)
i

P set P (n1)

i

vent (t) =

:

tot

C

(6)
The inspiratory time, t (n i , is determined by the ventilator cut-o. If we
assume that the ventilator cuts o at some predetermined fraction of the
initial ow, Q (n)

vent ( (n1)

), which we denote by , 0 < < 1, then, since

tot

we assume that V (n)
i

( (n1)
tot

) = 0, the inspiratory time during the nth cycle

can be computed as

!

P set P (n1)

t (n)
i

ex
K (n) R i

= CR i log n = 1; 2; 3; ::: (7)

to which there is a possible solution of Q (n)

vent (t (n)

) = Q (n)

vent ( (n1)

)

i tot

K (n) .

4. Model Simulations
The inspiratory initial value problem (4) can be solved for V (n)
i

(t), yielding

1e k i (t (n 1)

V (n)
i

)

P set P (n1)

(t) = C (8)

tot

ex

1

where k i =

CR i . We may also solve the expiratory initial value problem (5)

for V (n)

(t), obtaining

e

1e k e ( (n) V (n)

e

tot t)

P peep P (n) (t) = C (9)

ex

CR e . Finally, we determine P (n)

by setting V (n)
i

( (n)
tot

+ t (n)
i

1

where k e = ) = ex

V (n) ( (n)

tot

+ t (n)
i

) and obtain:

e