EXPONENTIAL FUNCTIONS

1. MOHAMED ANWER
RIFKY
5-NATURAL
EXPONINTIAL
FUNCTONS:
A- WHAT
ARE THE
METHODS TO
MEASURE
THE CARDIAC
OUTPUT ?,
EXPLAING
ONE OF THEM
IN FULL
DETAILS.

2. 1-THE CONCEPT OF AN EXPONENTIAL
PROCESS:
A-Non-linear
change>>
filling of a bath(The height of the water
>>volume>>
graph of volume against time is
linear).
B-The emptying of the bath.
(High head of water >>high
pressure
-head of water gets less, the rate of
emptiness>>slower.
>>volume against time is a
curve.
>>getting less
steep.
-volume never reaches zero
>>level as the
plughole-
If the flow (laminar)>>rate of flow V is directly proportional to the pressure
head>>
- -
In an exponential process the rate of change of a quantity
at any
time is proportional to the quantity at that time.
A-The rate of filling of the lungs ( if constant) >>
graph of inspired volume against time is
linear>>>>
Relevace to
anaesthesia:
(linear)
B-Analogy to the emptying of the bath (
Elastic tissues
of the lungs>>
with a
constant
Compliance
and (process is
exponential).
Exponential
.
2-WASHOUT CURVES: (An exponential processes)
Drugs washed out of the tissues by the blood
flow.
Muddiness which must be
washed out
-
Plot of the concentration of mud in a bath against time during its
washout from the bath. when the concentration of mud is high, the mud is washed out at a high rate. The process is exponential.
A-Washout of this type occurs when indocyanine green is used to measure the
cardiac output (
( amount of green dye is injected via a catheter into the right heart
t)>>washed out
(C.O)
Concentration of green dye is measured in a peripheral artery using a spectrophotometric
technique-
concentration of dye is plotted against time >>
curve-
downstroke >>exponential and the extra hump>>
recirculation.
Straight line>>prolonged downwards and from it the stippled area
>>measured. The cardiac output is the amount of dye injected,divided by the
area,
or by microcomputers to make the
calculations.
B-Thermal dilution techniques: (Two catheters or the Swan-Ganz catheter-detecting catheter has a thermistor-Ten
millilitres of 5% dextrose at approximately 0°C is injected-temperature of this blood is measured(pulmonary artery)-plot
of temperature against time gives a washout type curve-no recirculation-repetition as an advantage).
C-The nitrogen washout
curve:
Breathe oxygen through a non-rebreathing valve, and the expired gas is
analysed.
It should fall below 2.5% after seven
minutes>>
>Failure>>uneven distribution of gas in the
lungs.
D-Radioactive decay (exponential change). The rate of decay >>proportional >>quantity of isotope
remaining.
.
3-DURATION OF EXPONENTIAL PROCESS:
(Negative exponential functions
and
length of time taken by the exponential process is infinite).
Two alternative systems : 1-The half-life ( Q to fall to half its initial value>>used to describe the properties of radioactive isotopes) 2-The time-
constant is the time at which the process would have been complete had the initial rate of change continued-Greek letter tau ( ) -after one time-constant the Q
has fallen to 37% of its value-the time-constant is longer than the half-life-A typical time-constant as (expiration) might be 0.3 s so after 0.3 s (one time-constant)
only 37% of the tidal volume is left to be expired, after 0.6 s (two time-constants) 13.5% is left,
and after 0.9 s, only 5% of the tidal volume is left to be expired. Thus, expiration is 95% complete after three time-constants .
**Although the quantity is falling, it
never actually reaches
zero. total time cannot be used
The log.y axis does't extend to zero !!

3. **In low compliance, the time-constant
is reduced >>short period of expiration suffices with a more
rapid
respiratory rate.
**In a normal person after about one second almost all the tidal volume
has been expired ,time-constant can be calculated if the compliance C and
resistance R of
the lung are known. It is obtained by the product of the two factors: = C x R
**A high airway resistance leads to a high time constant
and necessitates a prolonged period to allow expiration of
the
tidal volume.
4-THE EXPONENTIAL FORMULA:
A-Fixed exponent
y =
B-Variable exponent
y =
(Exponent =index)
(2 in the
example)
**(variable exponent
x.)
**k = 2.718, which is given the symbol
e.--
V0 = volume at zero
seconds
V = the volumes of air in the lungs in excess of the
FRC.
Exponent has a negative value >> time increases>>value of Vt steadily
decreases.
5-THE POSITIVE EXPONENTIAL PROCESS:
Bacterial growth in the
presence of unlimited nutrient,
6-THE BUILD-UP EXPONENTIAL PROCESS:
(Not to be confused with another type
of exponential, the build-up exponential)
>>Inverted
negative
exponential.
A-A build-up exponential
process>>bath on the right is being filled
>> through its plug hole
by water in a large reservoir (left). The
rate of filling of the bath at any time
>>difference in
Ah, = the pressure required to drive the
water through the connecting tube.
THE BUILD-UP EXPONENTIAL
PROCESS:(continuation)
B-Build-up exponential in
anaesthesia,
(inflation of the lungs with a constant-pressure
ventilator).
C-Different rates of exponential: Uptake of
the anaesthetic. Elimination of a volatile
anaesthetic is in the reverse direction
(several
exponential processes).
D-Buildup exponential>> washin' curves-The uptake of an anaesthetic>>not a simple
exponential process, but >>many exponentials>>inspired anaesthetic to the lungs
,exponential
the circulationto ,and to the tissue>>
plotted on semi-logarithmic
paper
>>Three exponential processes A, B and C, each giving one
straight line.
E-Orally administered drug from the gut into the plasma followed by the
excretion of the drug from the blood stream.
{more complex in the blood (metabolization or
excretion)}.
Frequency at which the drug is taken must be chosen to avoid the
plasma concentration falling below the therapeutic level or rising above the toxic
level.
In the case of a washout curve, the time-constant can be shown mathematically
to be equal to the volume **The volume undergoing
washout can be calculated if the time-constant and flow of the perfusing fluid
are known.
Radioactive isotopes are often used for such
measurements.
(N.B) ABOUT TIME CONSTANT:
and multiplication of cancer cells.
(The value
achieved and the eventual theoretical final
value).
Exponential
component
The emptying of a lung-V >>volumes of air in the lungs
in
undergoing washout divided by
the
flow of the perfusing fluid.
A less variable blood level can be achieved by infusing
the drug
>>infusion or syringe pump.