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Phys. Med. Biol. 42 (1997) 2027–2039. Printed in the UK PII: S0031-9155(97)75049-8
A new ozone-based method for virus inactivation:
preliminary study
M M Kekez† and S A Sattar‡
† National Research Council of Canada, Ottawa, Ontario, K1A OR6, Canada
‡ The Faculty of Medicine, University of Ottawa, Ottawa, Ontario, K1H 8M5, Canada
Received 31 May 1996, in final form 10 July 1997
Abstract. The nebulization technique reported here could be used to inactivate viruses with
ozone in large volumes of body fluids, such as plasma, partial blood and perhaps whole blood
in a short time. Coliphage MS2 was used as a model because it is safe, easy to handle and
more resistant to chemical disinfections than viruses such as HIV. The theoretical curves and
experimental points, describing ozone inactivation of MS2, form a semi-sigmoid of congruent
data. There was a >7log
10
reduction in MS2 viability and the possibilities of minimizing the
ozone concentration required to kill viruses are indicated.
The analysis was expanded to account for the interaction of ozone with a virus suspension
in the shape of a thin film from the experimental findings of Bolton et al. We again find a semi-
sigmoid of congruent data for their case, i.e. describing ozone inactivation of the influenza A
virus (WSN strain) and the vesicular stomatitis virus versus time. For the method of nebulization,
the exposure time of droplets with ozone is a few seconds, whereas for the thin film method the
exposure time is measured in hours.
1. Introduction
A consortium involving the Canadian Department of National Defence, Agriculture Canada
and the Canadian Red Cross Society has conducted preliminary studies on ozone inactivation
of viruses in blood. One of the authors (MMK) was a member of this consortium and was
asked to develop an ozone delivery system to better facilitate interactions between the gas
(ozone) and the body fluids, because the two existing techniques used by the consortium,
(i) ‘hollow fibre’ (Wells et al 1991) (US patent of the date 911205) and (ii) ‘conventional
bubbling’ by M
¨
uller Medical Inc. (West German patent 1068428), convey inconsistent ozone
transfer to test fluids. M
¨
uller’s method also produces excessive bubbling and with both
methods the consistency of the treated fluid is not satisfactory. In M
¨
uller’s system ozone
is administered extracorporeally, i.e. ozone is foamed through a small sample of venous
blood (10 ml) removed from a patient by phlebotomy, exposed for 3 to 30 min to ozone,
and immediately injected intramuscularly back into the patient. This technique is referred
to as autoheamotherapy.
In the proposed method, the fluids are thoroughly nebulized or atomized, i.e. dispersed
into minute droplets to create a fine ‘rain’ to fall through a controlled atmosphere of O
3
/O
2
and/or O
3
/inert gas mixture. Electric and magnetic fields can be superimposed over the
space through which the droplets are passed. The preliminary experimental work shows
that this method yields consistent results. By comparison with M
¨
uller’s method, the system
described in this paper would ozonate test fluids at a rate of at least 20 ml min
−1
.
0031-9155/97/112027+13$19.50
c
1997 IOP Publishing Ltd 2027
2028 M M Kekez and S A Sattar
Many industrial applications (e.g. powdered milk, powdered eggs, instant coffee and
the carburation of liquid fuels in cars and jet planes) are based on the nebulization principle
described in this paper.
Our work is grouped into five parts: (i) theoretical analysis of the ozone penetration
into the biological fluids, (ii) comparison between the nebulization and the thin film method
of Bolton et al (1982), (iii) experimental examination of the nebulization method by
inactivating MS2 coliphage and Bacillus subtilis in the selected reactor configuration as
a function of ozone dosage, (iv) theoretical examination of the findings by Bolton et al
(1982) and (v) formulation of the suitability of surrogate viruses for subsequent evaluation,
particularly the lentivirus group of retroviruses.
2. Theoretical formulation
2.1. Nebulization
In general, ozone is transferred to the interior of the fluid by diffusion. The rate of ozone
absorption in an aqueous solution of droplets must be compensated by the addition of ozone
from outside the sphere. Razumovskii and Zaikov (1982), have shown that in zero-order
approximation the kinetics of the ozone–liquid interaction follows an exponential law
∂q/∂t
q
≈−
W
H
.
W is the specific rate of gas-fluid feed (litre gas per litre solution) and H is the coefficient
of Henry’s law. Let us now apply their experimental observation to a droplet: the ozone
penetrates the surface area of the droplet with a diffusion velocity, v, and in time, 1t,it
will occupy the volume of 4πR
2
v1t. R is the radius of the droplet and the droplet has the
volume of (4/3)π R
3
. Since W is the rate of feed, we see that W = 3αv/R. By definition,
the diffusion velocity, v, is approximately D/1x
d
, where 1x
d
is the characteristic diffusion
length (= 2(Dt )
1/2
). If t →∞and O
3
becomes Q
0
, the total amount of ozone absorbed
according to the above equation, q(X),is
q(X)
Q
0
= 1−e
−
√
X
(1)
where X is the normalized time equal to 9α
2
Dt/R
2
and α = 1/H . As a numerical
example, the diffusion constant, D, for water is 2.14 × 10
−5
cm
2
s
−1
and for glucose
0.52×10
−5
cm
2
s
−1
. For 1% ozone concentration in oxygen at 20
◦
C for water, α is 0.386.
We find that the observation of Razumovskii and Zaikov is the well known Fick’s
law, which satisfies the classical mass diffusion–heat conduction (transport) law. In one
dimension (of x), this is
∂q
∂t
=D
∂
2
q
∂x
2
. (2)
In zero-order approximation equation (2) implies that the absorption of ozone by the fluid
corresponds to the same law as the flow of heat from the surroundings into the biological
fluid.
For a droplet (sphere), equation (2) was solved (see the appendix). The total amount of
ozone absorbed, q(X),is
q(X)
Q
0
= 1−e
X
Erf
√
X. (3)
Ozone-based method for virus inactivation 2029
Erf is the complementary error function. In figure 1, equation (3) is given as a full line. If the
droplet before landing meets the requirement that X<1, equation (3) can be approximated
by equation (1) (see figure 1). If X 1, equation (1) can be reduced further to
q(X)
Q
0
=
√
X =
3α
√
Dt
R
(4)
giving
R =
3αQ
0
q
0
√
Dt ∼ Q
0
√
D ∼ constQ
0
. (5)
When the time of flight is fixed, the time, t, becomes a constant quantity, t
0
, equal to
the droplet’s time of flight. To inactivate a virus by ozone, we assume that the dosage
must exceed a certain minimum value. If a single species of viruses is used, q(t) becomes
q
0
= constant. Q
0
is the saturation concentration of ozone in the liquid in accordance with
Henry’s law. If Q is the concentration in the chamber, then Q
0
= αQ (for t →∞). In
equation (5), Q is the only variable. The biological fluid is characterized by the diffusion
constant, D, and the absorption coefficient, α. The virus is specified by q
0
.
Figure 1. The full curve is equation (3): q(X)/Q
0
= 1 − expX ErfX
1/2
. The broken curve is
equation (1): q(X)/Q
0
= 1 − exp(−X)
1/2
.
To relate this analysis to virus inactivation in the fluid’s droplets, the distribution of the
droplets in a spray can be described by the Gaussian function (represented by the left-hand
side of equation (6), having mean drop size, m, and standard deviation, σ . When a virus
titre of 2 × 10
7
plaque forming units (pfu) per ml is added, on average, each droplet of
a mean size of 30 to 40 µm contains a single virus. When the droplets are subjected to
the ozone atmosphere, the viruses present in the smaller droplets will be the first to be
2030 M M Kekez and S A Sattar
inactivated. The relative number of inactivated viruses is
1
σ
√
2π
Z
x
−∞
{exp[−(t − m)
2
/(2σ
2
)]}dt =
1
2
1 + Erf
x − m
σ
√
2
. (6)
From −∞ to +∞, the integral has the value of 1, and Erf is the error function (see
Abramowitz and Stegun (1968)). Therefore, the relative number of viruses surviving the
ozone treatment is
s =
1
2
1 − Erf
x − m
σ
√
2
=
1
2
1 − Erf
−m + constQ
0
σ
√
2
. (7)
Here x represents a large size droplet (containing a virus) in the distribution that is not
affected by the ozone treatment. R of equation (5) is substituted in the middle part of
equation (7) as x = 2R, and the theoretical survival curves are produced following the path
given by Kekez et al (1996).
2.2. Thin film method
The thin film method of Bolton et al (1982) also produces consistent results. The biological
fluid is suspended in the shape of a thin film on the surface of a rotating culture bottle
and the ozone is drawn through the bottle at a constant rate. The total amount of ozone
absorption in the thin film is also given by equation (3) but here X = α
2
Dt/(1x
2
), where
1x is the thickness of the film. If X 1, equation (3) with new X becomes
q(X)
Q
0
= 1−
1
2
√
X
= 1−
1x
2α
√
Dt
(8)
giving
1x = 2α
1 −
q
0
Q
0
√
Dt. (9)
Equation (3) (approximation) and equation (8) (exact solution) are compared in figure 2. We
see that the approximation is valid only for X>1. In the Bolton et al (1982) experiments
the ozone concentration in the chamber, Q, is the parameter and the time, t, is the main
variable. Their time is measured in hours, hence X>1. For their experimental conditions,
the titre of viruses is large (10
8
pfu ml
−1
). It is useful to assume that the thickness of the
film approaches that of a monolayer; therefore, 1x is also proportional to the diameter of the
virus, x, having its own mean size, m, and standard deviation, σ . Substituting equation (9)
in the second bracket of equation (7), the relative number of the viruses, s, surviving the
ozone treatment is obtained when the ozone exposure time, t, is a variable quantity:
s =
1
2
1 − Erf
−m + const
√
t
σ
√
2
. (10)
2.3. Comparison between the nebulization method and the thin film method
To make comparison for a given (constant) volume, we consider that the cubic volume of
the thin film (1x)
3
is equal to the equivalent volume of the sphere (4/3)π R
3
. This gives
R = 1x(3/4π)
1/3
. However, the normalized time, X, for the sphere has the multiplier
of 9 in its definition in comparison to X used in the thin film case. This implies that the
nebulization method is faster by a factor of 9(3/4π)
1/3
= 5.58 in comparison to the thin
film method.
Ozone-based method for virus inactivation 2031
Figure 2. The full curve is equation (3): q(X)/Q
0
= 1 − expX ErfX
1/2
. The broken curve is
equation (8): q(X)/Q
0
= 1 − 1/2X
1/2
. The comparison applies when X>1.
3. Basic methodologies and rationales for their selection
3.1. Nebulization
In our experimental work, the fluid is atomized into small droplets by a nebulizer. The
droplets are injected into an ozonation chamber and allowed to fall through a controlled
atmosphere containing ozone as indicated in figure 3; therefore the exposure time is a
function of the height of the ozonation chamber. The properties and composition of the fluid
affect the ability of the nebulizer to atomize. The method proposed is fluid specific, requiring
an appropriate nozzle for each fluid. Since the absorption is an exponential function of the
radius (see equation (1)), the dosage of ozone received by any viral contaminants is related
to the droplet size and the reactor design. To treat plasma and/or serum it is necessary
that the droplets are as small as possible, in order for the ozone to eliminate effectively the
viruses inside the droplet. To make the apparatus compact, it is also advisable to minimize
the time of flight, i.e. to shorten the height of the inactivation chamber. Reduction of
the effective dosage will also minimize additional damage to desirable components of the
test fluids. The experiment used a commercial ozone generator (model GTC-0.5, Griffin
Technics, New Jersey).
3.2. Selection of test fluid
A major concern with the ozone treatment process is virus inactivation without unacceptable
damage to desirable components of the treated product(s). Since the likely damage to
whole blood is greater and more difficult to assess because of its cellular components, in
this study we propose to test only plasma, a major source for blood products. Although
2032 M M Kekez and S A Sattar
Figure 3. The ultrasonic nebulizer system.
it is recognized that the possibility of damage to proteinaceous material as well as the
possibility of generating toxins during the process exists, we consider that the assessment
of these factors is beyond the scope of this exploratory study.
3.3. Selection of nebulizer
In this work, the nebulizers considered were: a compressed gas atomizer (or pressure jet
atomizer) and twin fluid atomizer; an ultrasonic nebulizer and a rotary nebulizer. Of these,
the ultrasonic nebulizer is preferable due to the low velocity of the incoming spray, and the
experimental chamber with such a nebulizer appears to yield the most compact design for
an equivalent ozone dose. Very fine atomization is achieved at low velocity (1–3 m s
−1
),
making this device ideal for use with the flow rates of 10 to 400 ml min
−1
.
The nebulizer system in figure 3 consists of a piezoelectric resonant device (19), whose
transducer discs are energized by high-frequency electrical signals applied to the terminals
(20). This results in the propagation of pressure waves in both directions along the nozzle
(21). The liquid inlet is at point (22) and the spray originates at the atomizing surface (23).
The droplets are collected at a dish (24), and a peristaltic pump (5) removes the liquids
from this dish (24). The inactivation chamber is formed between the two flanges (25 and
26) and a glass cylinder (27). Ozone is injected into the chamber via an annular ring (28)
through 24 small-diameter openings to ensure that the ozone stream is slow-moving. The
ozone outlet (29) feeds the used gas to the vent decomposer (11).
Ozone-based method for virus inactivation 2033
3.4. Application of electric and magnetic fields
The flight characteristics of the droplets can be augmented and controlled electronically.
Each droplet is charged by an electrostatic field. A high voltage is applied to the dish (24)
and the nozzle (21) acts as a ground electrode, ensuring that the droplets do not fuse in
flight. To further control the speed of the droplets, another electrode in the form of a ring
could be placed in the space between the nozzle (21) and the dish (24) to create a classical
triode structure. If this process is used for whole blood, a magnetic field could be applied
to increase the flight time of the droplets by making them follow a helical path, due to the
iron in red blood cells. The magnetic field is produced by putting a coil around the glass
cylinder (27).
3.5. Separation of ozone from oxygen
When ozone is generated by electrical discharges, a gas mixture (O
3
–O
2
or O
3
–air) is
produced in which the ozone concentration is typically 1 to 5% by volume. Organic
matter is prone to oxidation and it may be important in some applications to minimize the
oxidative stress caused by oxygen radicals. Work is in progress to develop a low-temperature
distillation (adsorption–desorption) technique to attain an ozone-to-oxygen ratio of about 1
to 5 (i.e. 20 to 30% by volume over silica gel with a low impurity level of Fe) during
the adsorption cycle. Later, during the desorption phase (when ozone is separated from
oxygen), an inert gas (e.g. N
2
, He, etc) will carry ozone to the inactivation chamber.
3.6. Control of droplet size
The size and distribution of the droplets are determined by a Doppler velocity apparatus.
This apparatus can also help measure the time of flight, optimize the flow rate and determine
the droplet’s evaporation rate.
4. Experimental results
4.1. Whole blood
Preliminary experiments were conducted to demonstrate some aspects of the proposed
concept. A drop of blood was released from a 10 ml syringe with a needle (gauge 14–20)
by knocking the needle with a finger, and these drops were allowed to fall freely over a
distance of 5 to 10 cm until enough blood collected in a Petri dish for the analysis. There
was minimal (<2%) or no haemolysis of the red blood cells due to free fall and exposure
to air. With the first (compressed gas) atomizer built, 87% of the total number of the red
blood cells present in whole blood survived the nebulization treatment.
4.2. Tests with bacteriophage MS2
An ultrasonic nebulizer was used in this study. A bacteriophage, the tailless icosahedral
RNA-coliphage MS2, suspended in bovine serum, was used to optimize a ‘viral inactivator’.
MS2 was used as a model here because it is safe, easy to handle, and can grow to titres up
to 10
13
pfu ml
−1
in a suitable bacterial host. MS2 phage, being a small (27 nm in diameter),
icosahedral and non-enveloped virus, survives better in the environment and is generally
more resistant to inactivation by chemical agents when compared with larger, enveloped
viruses such as HIV. The titre of active MS2 in the serum/plasma sample can be accurately
2034 M M Kekez and S A Sattar
measured by counting the number of plaques produced in an Escherichia coli lawn. The
effectiveness of the ozone treatment for the inactivation of the test virus in treated serum
samples was indicated by the reduction in the number of plaques as a function of ozone
dosage.
The phage was diluted to a final concentration of about 2 × 10
7
pfu ml
−1
for the data
shown in figure 4. Dose–response curves of MS2 inactivation by ozone were obtained
with the virus suspended in (i) Dulbecco’s phosphate buffered saline (PBS), (ii) a 10%
solution of bovine serum in PBS, and (iii) a 25% bovine serum in PBS. A comparison
of the experimental points and the theoretical curves for MS2 shows that the curves and
points form a semi-sigmoid of congruent data (figure 4). The increase in bovine serum
concentration only results in translating the curve to the right.
Figure 4. Dose–response curves of MS2 inactivation by ozone as a function of ozone
concentration using the nebulization method. MS2 was suspended in (i) Dulbecco’s phosphate
buffered saline (PBS), (ii) a 10% solution of bovine serum in PBS and (iii) a 25% bovine serum
in PBS. The experimental points are:
•, PBS; , 10% serum; , 25% serum. Data points are
the average of three experiments performed in triplicate. Theoretical curves are described by
the following expressions.
PBS: s = 0.5{1 − Erf[0.0241(−36.38 −0.7Q)]}
10% serum: s = 0.5{1 − Erf[0.0241(−36.38 −0.014Q)]}
25% serum: s = 0.5{1 − Erf[0.0241(−36.38 −0.008Q)]}
when the ozone exposure time used was about 1 s.
The theoretical curves were computed using equation (7) for the measured value of
m = 36 µm, σ = 29 (µm)
−1
. To get a good fit with the experimental points, a set of
values for the ‘constant’ of equation (7), named here β, was carried out. We find that β has
a value of 0.7, 0.014 and 0.008 for PBS, 10% serum and 25% serum respectively. From
equation (5) we see that β is proportional to the square root of the diffusion constant. The
Ozone-based method for virus inactivation 2035
ratio of β for 10% serum to β for 25% serum is 1.75, implying that the diffusion constant at
25% serum is decreased by a factor of 3 with respect to the diffusion constant at 10% serum
concentration. To get a more accurate evaluation of the change in the diffusion constant,
it is necessary to use equation (1) instead of equation (5) in determining the value of R.
Hence, we have estimated that q
0
(for an air-borne virus to be inactivated) has a value of
20 ppm, when the time of ozone exposure is a few seconds.
4.3. Tests with the spores of Bacillus subtilis
An ultrasonic nebulizer was used in this study. The system was examined for its ability to
inactivate the spores of Bacillus subtilis (ATCC 19659). Such spores are routinely used to
assess the sporicidal activity of liquid and gaseous chemicals and as biological indicators to
validate the performance of steam and gas sterilizers. The bacterium was grown in Columbia
Broth (Difco, Detroit, Michigan, USA) diluted ten-fold in deionized water. In this medium,
there was nearly a 100% sporulation after 48 h at 37
◦
C. The spores were washed three
times in deionized water and suspended in the test medium to a final concentration of 10
8
colony forming units/ml. When B. subtilis spores were suspended in PBS and nebulized in
the presence of ozone, there was >5log
10
reduction in their viability.
4.4. Thin film method of Bolton et al (1982)
4.4.1. Test with vesicular stomatitis virus. If equation (10) is to be applied to these results,
detailed knowledge of the virus distribution is required. From Wagner (1975) and Fields
et al (1996) we see that there is considerable heterogeneity in particle size. The typical
infectious B virion is a bullet-shaped (B particle) cylinder, 180 ± 10 nm in length and
65±10 nm in diameter at the blunt end. If this volume is approximated by a sphere, it will
have a radius of 50.65 nm. For the predominant defective truncated (T) virions having about
one-third the length (65 nm) of the infectious B virions, we find the equivalent radius to be
36 nm. The molecular weight of the infective VS B-virion was estimated to be 4.4 × 10
6
daltons at the upper extreme and 3.2 × 10
6
daltons at the lower end. For the density of
1.31 g cm
−3
, this corresponds to the equivalent radius of 9.9 nm at the lower end. Defective
VS virions have a molecular weight of (0.7–1.2) × 10
6
daltons which corresponds to the
equivalent radius of 5.9 to 7.1 nm.
If half of all the particles are occupied by the infectious B viruses in the Gaussian type
distribution curve we have from the definition of the error function that (x − m)/(σ
√
2) =
0.477 as Erf (0.477) = 0.50. For the mean radius of the infectious B virion, m, of 50.6 nm,
we get x −m = 14.8 nm. Here, x corresponds to the mean radius of the defective truncated
(T) virion of 36 nm. This gives 1/(σ
√
2) = 0.032 nm
−1
. Putting the values for m and
1/(σ
√
2) into equation (11), the theoretical curves for ozone inactivation versus time are
obtained. The results are given in figure 5, again showing a semi-sigmoid of congruent
data.
4.4.2. Test with influenza A virus (WSN strain). The information about the distribution of
this virus can be obtained from the work by Compans and Choppin (1975) and Fields et al
(1996). The majority of particles are spherical with a diameter of 80–100 nm, making the
average radius 45 nm. The purity, homogeneity and approximate composition of population
of virus particles were not conclusively established. However, to derive the theoretical
curves, we assume that the relative distribution for influenza A virus is the same as for
2036 M M Kekez and S A Sattar
Figure 5. Ozone inactivation of vesicular stomatitis virus versus time. Experimental points
obtained by Bolton et al (1982) were replotted on the log–log scale. The thin film method was
used. The virus suspension was exposed to: , 0.64 ppm of ozone; ◦, 0.16 ppm; M, 0.00 ppm.
Full curves are the theoretical curves for:
0.00 ppm: s = 0.5{1 − Erf[0.0323(−50.65 +20(time)
1/2
)]}
0.16 ppm: s = 0.5{1 − Erf[0.0323(−50.65 +26(time)
1/2
)]}
0.64 ppm: s = 0.5{1 − Erf[0.0323(−50.65 +33(time)
1/2
)]}.
vesicular stomatitis virus. This means that 1/(σ
√
2) equals 0.032 nm
−1
; the results are
given in figure 6. We suggest that the slightly higher resistance (manifested by higher value
of q
0
used in equation (9)) to inactivate influenza A viruses in comparison with vesicular
stomatitis viruses, may be attributed to the fact that they have a segmented genome. There
are eight separate segments which make up the full genetic complement of the influenza
virus.
5. Selection of test viruses for future study
To assess the ability of ozone applied by the nebulization method for inactivating viruses in
biological fluids, it is necessary to spike the test samples with relevant viruses and examine
their inactivation as a function of ozone dosage. Relevant human viruses may include the
hepatitis viruses, parvoviruses (including B19), the herpes viruses (including herpesvirus 2
and cytomegalovirus) and retroviruses (including type C and HIV). However, safety issues
as well as practical difficulties of working with some of these agents precluded their use at
the preliminary stages of such a study. It is also difficult at this stage to reliably extrapolate
to mammalian viruses from studies on bacteriophage(s). Therefore, experiments will be
performed using a panel of surrogate viruses in proper containment facilities.
Ozone-based method for virus inactivation 2037
Figure 6. As in figure 5, but for influenza A virus (WSN strain). Full curves are the theoretical
curves for:
0.00 ppm: s = 0.5{1 − Erf[0.0323(−45 +15(time)
1/2
)]}
0.16 ppm: s = 0.5{1 − Erf[0.0323(−45 +20(time)
1/2
)]}
0.64 ppm: s = 0.5{1 − Erf[0.0323(−45 +27(time)
1/2
)]}.
6. Conclusions
The technique reported could be used to inactivate viruses with ozone in large volumes of
body fluids such as plasma, partial blood and perhaps whole blood in a short period of time.
To enhance the absorption of ozone in the fluid, the method atomizes the fluid into small
droplets that are sprayed into an atmosphere of ozone which kills viruses. It is shown by
Carpendale and Freeberg (1991) that ozone inactivates HIV at non-cytotic concentrations.
The current work offers many opportunities to minimize the ozone concentration
required to kill viruses. It is necessary to decrease the droplets size as much as possible
in order that the virus inactivation is done at low ozone concentration; however, if a very
small droplet size is chosen, the use of hydrocyclone technology to collect droplets becomes
necessary.
An objection to our approach has been raised that we have neglected the chemical
reaction of ozone with liquid. For example, when ozone diffuses in a liquid and reacts
with it irreversibly, equation (2) must be modified according to a first-order reaction. This
means that, on the left hand side of equation (2) we must add the term βq to account for
this reaction. Here, β is a constant greater than zero. After solving this new expression, the
total amount of ozone absorbed versus normalized time again reaches the semi-saturation
for X>1 in a similar fashion as in figure 2, but at smaller value of q(X). The fundamental
relationship of both equations (4) and (8) is maintained and the validity of equation (4) is
expanded over the large domain of X.
2038 M M Kekez and S A Sattar
Acknowledgments
Interest shown by DrALVanKoughnett, NRC, and Dr G Adams, NRC, is appreciated.
The authors wish to thank Mrs V S Springthorpe of the Faculty of Medicine, University
of Ottawa for useful ideas and helpful discussions. The authors are indebted to Dr P
Savic, Researcher Emeritus, NRC, for his help in the theoretical formulation. The technical
assistance of Mrs H Rahman and Mr R Sansom is gratefully acknowledged.
Appendix
Carslaw and Jaeger (1989, p 349 (III)) quote the problem of a sphere of radius a and perfect
conductor of specific heat c
1
, surrounded by an infinite region of diffusivity D, specific heat
c and contact resistance (1/h). When the heat source Q is absent, the problem becomes
that given on p 350 (IV). With the conductor at the initial temperature T
0
, and with contact
resistance approaching zero, h →∞, the integral #21 on p 350, describing the cooling of
the sphere, reduces to
T =
2kT
0
π
Z
∞
0
[exp(−u
2
Dt/a
2
)]u
2
(u
2
− k)
2
+k
2
u
2
du
where k = 4πa
3
ρc/M
1
c
1
. Here, M
1
is the mass of the sphere. This integral can be evaluated
by partial fraction expansion of the polynomial part of the integrand. The denominator is a
quadratic in u
2
with two roots:
v = u
2
= k
1 −
k
2
±
k
2
2
r
1 −
4
k
.
For large values of k, the square root can be expanded to first order, giving the following
value for the two roots: v
1
= 0 and v
2
=−k
2
. Hence, the integral becomes
T =
2kT
0
π
Z
∞
0
exp(−u
2
Dt/a
2
)
u
2
+k
2
du. (A1)
From Gradshteyn and Ryzhik (1965, p 338, #3.466 (1)), we have
Z
∞
0
exp(−µ
2
x
2
)
x
2
+ β
2
dx = Erf(βµ)
π
2β
exp(β
2
µ
2
)
and the solution of equation (A1) is
T = T
0
e
X
Erf
√
X (A2)
where
√
X = βµ µ =
r
Dt
a
2
β = k =
3ρc
ρ
1
c
1
since
D
1
D
=
ρ
2
ρ
2
1
c
c
1
= α
we get
X =
9α
2
D
1
t
a
2
.
Ozone-based method for virus inactivation 2039
Here, D
1
is the diffusion constant inside the sphere. If the sphere were to be be heated,
equation (A2) would read
T
T
0
= 1 − e
X
Erf
√
X.
For the analogies between heat and mass transfer, we need to substitute T by Q. Hence,
the equation above becomes equation (3) given in the main text.
References
US patents
Item 2/5/3; date 911205; Patent Assignee: (Medi-) Medizone Inc; Inventors: Zee Y C; Bolton D C; Inactivating
lipid enveloped virus in blood or other tissues - by treatment with ozone without loss of physiological activity
(hollow fiber method)
Item 2/5/15 (also Item 2/5/4); US 4632980 Date 900626; Patent Assignee: (IMMU-) Immunologisics; (Medi-)
Medizone Inc; Inventors: Zee Y C; Bolton D C; Ozone treatment of blood and blood products to inactivate
viable enveloped viruses, e.g. AIDS virus (thin film method)
Item 2/5/20; WPI Acc. no: 83-742540/34; Patent Assignee: (REGC) Univ of California; Inventors: Zee Y C,
Bolton D C; Vaccines for immunisation of mammals containing ozone inactivated pathogenic microorganism
and carrier (thin film method; ozone concentrations: 0.1–10 ppm)
West German patent
Patent no 1068428; Dates: 21 May 1957; 5 Nov. 1959, 21 April 1960. Inventors: Jentjens H; M
¨
uller F W; Method
and design for production of oxygenated blood
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