This document discusses exponential functions and their relevance to measuring cardiac output and anesthesia. It provides details on several methods to measure cardiac output, including using indocyanine green dye washout curves, thermal dilution techniques, and radioactive decay curves. It also discusses exponential processes in other contexts like bacterial growth, drug uptake and elimination curves during anesthesia, and the buildup of fluid in a bath. Exponential functions are characterized by a constant rate of change proportional to the quantity present, meaning the rate decreases over time as the quantity decreases towards zero, though it never actually reaches zero.

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.