Microbial loads of spacecraft components measured prior to heat sterilization to determine if maximum tolerance is exceede

Cornell, R. G.

English

NASA Technical Reports Server

Richard G. Cornell  
Department of S t a t i s t i c s  
F lor ida  S ta te  University 
February 16,  1966 
Technical Report Number 2 
NASA Grant No. NGR-10-004-029 
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https://ntrs.nasa.gov/search.jsp?R=19660015204 2020-03-16T19:37:44+00:00Z
Variation i n  Measurements of Microbial Loads 
Richard G. Cornel1 
February 16, 1966 
A maximum microbial burden i s  allowable on spacecraf t  i t e m s ,  whether 
P a r t s  o r  subassemblies, p r i o r  t o  f i n a l  hea t  s t e r i l i z a t i o n .  
concerned with t h e  d i s t r i b u t i o n  of measurements based on bioassays which 
would be taken t o  see i f  t h e  maximum to le rance  is exceeded. The sources 
of v a r i a t i o n  involved i n  such measurements are of p a r t i c u l a r  concern. 
This r epor t  i s  
A measurement of microbial  load, denoted by X, may be represented 
by t h e  model 
X = u+ a + K(B+y). (1) 
Actually,  X is a measurement of mean load f o r  the  population from which t h e  
i t e m  being t e s t e d  is drawn. I n  t h e  bioassay of spacecraf t  components i t  is 
not always poss ib le  t o  test a component which w i l l  go i n t o  the  spacecraft .  
For instance, i n  order t o  measure t h e  i n t e r n a l  contamination of a small 
electrical component, t he  component would be destroyed. 
items a c t u a l l y  used i n  the  spacecraft ,  i t e m s  which have been manufactured and 
handled i n  the  same manner are of ten  assayed. 
corresponding i t e m  i n  t h e  spacecraft  are assumed t o  have been randomly 
s e l e c t e d  from t h e  same population. 
by t h e  term p i n  equation (1). 
of t h e  true microbial load f o r  the i t e m  assayed from p. 
an assayed i t e m  is therefore  u + a. 
Instead of t e s t i n g  
The assayed items and the  
The mean of t h i s  population i s  represented 
The t e r m  a i n  (1) r e f e r s  t o  the  devia t ion  
The t o t a l  load f o r  
I n  addi t ion  t o  v a r i a t i o n  from one i t e m  t o  another of t h e  same type, 
t h e r e  is  v a r i a t i o n  i n  microbial density over t he  su r face  and i n t e r i o r  of 
an i t e m .  For some l a rge  i t e m s ,  only a small por t ion  would be t e s t ed  i n  
- 2 -  
a bioassay. 
of which is of the  same s i z e  and i s  s m a l l  enough t o  be assayed. 
L e t  us think of an item as cons is t ing  of K u n i t s ,  K 2 1 ,  each 
I f  only 
one u n i t  of an i t e m  were se l ec t ed  fo r  assay, i ts  t r u e  microbial  load would 
be l i k e l y  t o  devia te  from the average load p e r  u n i t  f o r  t h a t  i t e m ,  and t h a t  
devia t ion  is  represented i n  the  model given by equation (1) by t h e  symbol 8 .  
Moreover, t he  bioassay measurement of the  microbial load f o r  an i t e m  would 
l i k e l y  devia te  from the  t r u e  load. This devia t ion  is  denoted by y i n  (1). 
The t o t a l  of B and y is mul t ip l ied  by K s ince  w e  are i n t e r e s t e d  i n  the  t o t a l  
load per  i t e m ,  not j u s t  t h e  load f o r  a u n i t  of an i t e m  s e l ec t ed  f o r  assay. 
It is  reasonable t o  assume tha t  t h e  expected values,  t h a t  is, the  
means over t h e  e n t i r e  population, of the  t e r m s  i n  (1) are given by 
E(a) = E(B) = E(Y) = 0 .  The term M is thought of as a constant,  s ince  it 
is  a population parameter which does not vary from one i t e m  t o  another within 
the  population. Therefore, E(X) = uand t h e  measurement X w e  are dealing with 
i s  an unbiased estimate of the  population mean. L e t  us a l s o  make t h e  reasonable 
assumption t h a t  t he  sources of va r i a t ion  given i n  equation (1) are independent 
of each o ther .  I f  w e  l e t  V denote a variance,  w e  may then w r i t e  
V(X) = V ( a )  + K2 CV(f3) + V(y) l  . (2) 
The var iances  i n  equation (2) f o r  a given type of i t e m  could be estimated 
from experimental d a t a  f o r  each t y p e  of spacecraf t  i t e m .  
h i e ra rcha l ,  experiment design would be appropriate followed by t h e  ana lys i s  
of var iance  f o r  a random e f f e c t s  mode1,which gives the  information needed 
f o r  the es t imat ion  of variance components. Since da ta  from such experiments 
a r e n o t  present ly  ava i l ab le ,  w e  s h a l l  examine the V(X) as given by (2) on t h e  
b a s i s  of known information on microbial assays. 
A nested,  t h a t  is, 
- 3 -  
I f  t he  probabi l i ty  tha t  a microorganism w i l l  contaminate a u n i t  
is independent of the previous contamination of t h a t  u n i t  and is constant 
f o r  u n i t s  of equal s i z e ,  then t h e  d i s t r i b u t i o n  of counts of microorganisms 
per u n i t  would be Poisson. Since t h e  mean and variance of t h e  Poisson are 
equal and t h e  expected count per un i t  i s  p / K ,  we  may take V(B) = u / K .  I f  
these  assumptions are not s a t i s f i e d ,  clumping of organisms would tend t o  
occur, and t h e  V(B) would be grea te r  than p/K.  Furthermore, t he  sum of Poisson 
random va r i ab le s  i s  a l s o  Poisson, s o  l.i + a , t he  t o t a l  microbial load f o r  an 
i t e m ,  would a l s o  be expected t o  have a variance at least as l a r g e  as its 
expected value P . Since P i s  c o n s t m t ,  w e  may take  V(a) = v. The 
o the r  variance,  V(y), i s  f o r  many microbial assays propor t iona l  t o  t h e  
square of t h e  mean microbial count p e r  un i t .  This mean is  p/K f o r  our 
2 
model. So we s h a l l  represent  V(y) by C2u2/K2, where C is p o s i t i v e  and C 
is  a p ropor t iona l i t y  constant. Subs t i t u t ion  of these  variance expressions 
i n  (2) gives,  as a minimal variance,  
Equation (3) would be more e a s i l y  in te rpre ted  i n  w e  expressed it i n  terms 
of X = p/K,  s ince  i f  poss ib le  assay u n i t s  are se l ec t ed  s o  t h a t  t h e  mean 
count pe r  unit i s  small, c e r t a i n l y  less than 1000 and preferably less than 
300. This enables separa te  colonies t o  be more e a s i l y  i d e n t i f i e d .  Subs t i t u t ing  
K = P / A  i n  (3) and simplifying, we  have 
2 2 V(X) = kl + x [l + c A] . 
This may be rewr i t ten  as 
(4) 
( 5 )  1 1  2 
P 
V(X) = P 2  c - + 7 +  c I. 
I f  both 
o r  i n  terms of 
The constant C 
- 4 -  
P and are l a rge ,  equation (5) would lead t o  the  approximation 
2 2  V(X) h c P 
X' 
the  standard deviation of X, denoted by u 
0 g c p .  X 
would l i k e l y  be s m a l l ,  say less than 0.5 o r  even as small 
as 0.1. 
t o  NASA e n t i t l e d  "Sample S i z e  Considerations f o r  Voyager Capsule S t e r i l i z a t i o n  
Assays. However, i f  X w e r e  s m a l l ,  say 1 o r  2,  and p w e r e  l a rge ,  minimum 
values of V(X), based on our Poisson assumptions and t h e  range of C mentioned 
earlier, could be as high as 1 . 2 5 ~  , which would mean t h a t  u, could be 
g r e a t e r  than E(X) = p. That is, i f  an i t e m  has a l a r g e  microbial burden 
bu t  t h e  expected number of microorganisms per assay u n i t  wi th in  t h a t  i t e m  i s  
small, then u could be g rea t e r  than E(X), where X is based on a s i n g l e  
assay of one uni t .  
s idered  than those studied i n  t h e  Avco repor t .  The reason t h a t  they d id  not 
i n v e s t i g a t e  l a r g e r  values of u relative t o  P i s  t h a t  they d id  not s e r ious ly  
consider sources of v a r i a t i o n  i n  the model (1) o ther  than y ,  t he  e r r o r  in t ro-  
duced by the  bioassay, and they d id  not allow f o r  the  mul t ip l i ca t ion  i n  our 
model by K, t h e  number of u n i t s  per  i t e m .  
This i s  the  range considered i n  the  December 10, 1965 Avco r epor t  
11 
2 
A 
X 
This i m p l i e s  t h a t  l a r g e r  values of ax should be con- 
X 
I n  p rac t i ce ,  ins tead  of taking our minimum value of V(X) as being 
c o r r e c t ,  i t  would be b e t t e r  t o  estimate V(X) and i t s  square roo t ,  
t h e  b a s i s  of experimental evidence i n  t h e  manner ind ica ted  earlier. 
us ing  estimated values of u and 
u l a t i o n  mean, P, i s  not  of primary i n t e r e s t .  
i n  t h e  spacecraf t  i s  not  t h e  one assayed, u i s  not of primary i n t e r e s t .  
ax, on 
I n  
ux, i t  should be kept i n  mind t h a t  t h e  pop- 
A t  least when the  i t e m  used 
. c  . 
- 5 -  
Instead estimates of t h e  population parameters 
an estimate of the probabi l i ty  tha t  another i t e m  se lec ted  randomly from t h a t  
and Q would be used t o  compute X 
same population would have a s a t i s f a c t o r i l y  s m a l l  microbial  load. This 
probabi l i ty  would be required to  be very c lose  t o  one. The Avco repor t  
i s  only concerned with the probabi l i ty  t h a t  t h e  estimate of p i t s e l f  i s  
less than the  maximum allowable microbial load. This would seem t o  be the  
appropriate  probabi l i ty  t o  study only i f  the i t e m  assayed i s  the one t o  be 
used i n  the spacecraf t .  This would sometimes be t h e  case. The equations 
worked out earlier i n  t h i s  pzper would s t i l l  apply i n  t h i s  s i t u a t i o n  i f  a were 
deleted from equation (l), V ( a )  were deleted from equation (2) and the  corres- 
ponding variance terms,including - i n s ide  t h e  parenthesis  i n  equation (5), 
were de le ted  from the  o ther  equations. 
would s t i l l  be per t inent  s ince  i n  tha t  discussion - w a s  assumed t o  be negl ig ib le ,  
t h a t  i s &  w a s  assumed t o  be large.  
1 
5.1 
Also, the discussion of equation (5) 
1 
1.I 
I f  the  V(X) were found t o  be unacceptably l a r g e ,  
i t  could be reduced by assaying several  u n i t s  within an i t e m  o r  by measuring 
seve ra l  i t e m s  of t he  same type. Then t h e  sample mean of such measurements 
would be used t o  estimate and the standard e r r o r  of t h i s  sample mean would 
be of interest ins tead  of 
(1) would be wr i t t en  as 
ax. In t h i s  s i t u a t i o n  the model given by equation 
= p + ai + K(Bij + y i j ) ,  i = 1 , 2  ,..., k; j 1 , 2 y 0 0 - , n  'i-j - 
t h  
where t h e  subscr ip t  i r e f e r s  t o  the  i- i t e m  measured and the double sub- 
t h  t h  
s c r i p t  i j  r e f e r s  t o  t h e  j- u n i t  assayed f o r  t he  i- i t e m .  It is 
assumed t h a t  an equal number of un i t s  would be assayed f o r  each i t e m .  
- 6 -  
The m e a n  which would be used t o  estimate 1-1 would be 
* 
- 1 k -  x = -  c x i  i=l 
where 
A 
Xi = - E X  . 
n i j  j =1 
The variance of % wwdd be given by 
assuming mutual independence of and equal d i s t r i b u t i o n s  f o r  a l l  of t h e  Bij, and of 
t h e y  
would be equal t o  t h e  square root of V(?). . 
*terms i n  equation ( 6 ) .  The standard e r r o r  of x, denoted by a- , 
i j  X 
I f  t h i s  assay procedure were followed, a decision would have t o  be 
made regarding the  number of items and u n i t s  wi th in  i t e m s  t o  be measured, 
t h a t  is, regarding t h e  s i z e s  of k and n. The relative magnitudes of k and 
n se l ec t ed  would depend on t h e  relative magnitude of V(a) as compared w i t h  
V ( B )  + V(Y).  Under t h e  assumptions which l ed  from equation (2) t o  equation 
(S ) ,  i t  would s e e m  reasonable t o  take n l a r g e  r e l a t i v e  t o  k. I n  the  Avco 
r epor t  t he  p o s s i b i l i t y  of taking n greater than one is  considered while k 
is held  equal t o  one s ince  consideration is not given t o  assaying items 
o the r  than those i n  t h e  spacecraf t .  However, f i n a l  recommendations on t h e  
approximate s izes  of k and n should be delayed u n t i l  experimental evidence 
is a v a i l a b l e  on the  magnitudes of V(a), V ( B )  and V ( Y ) .  This is contrary t o  
t h e  recommendation i n  t h e  Avco r e p o r t  t h a t  n be approximately equal to'2 
with  k equal t o  one. 


Variation in measurements of microbial loads

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