Seriliaion D Values Seriliaion by seam consis of he simple observaion ha baceria die over ime during exposure o hea. They do no all live for a finie period of hea exposure and hen suddenly die a once, bu raher a consan hea inpu, hey cease o be alive in direc proporion he saring number of colony forming d unis (CFU). Where N is he number of CFU a any ime, we have he relaionship d N = k N. Somewha obviously, he rae consan, k, is a funcion of he amoun of hea pu ino he populaion, and hus i is a rue consan only a consan hea inpu. In a seam auoclave, hea inpu is indirecly measured as emperaure and pressue under known condiions of seam sauraion (no o imply "sauraed seam"). Shown below are he classical mahemaics o quanify suscepibiliy of each species and srain of microorganism o hea. d d N () = k N ()...Eqn ) The primary differenial equaion. N N dn() = k d (Eqn 2) Soluion by separaion of variables for inegraion N () ln N () N = k (Eqn 3) The inegraed resul N () = e k (Eqn 4) A simplificaion for undersanding N() N N () = N e k Obviously, if he logarihmic base were changed from e o, he effec would be o have a differen value for he consan k. [k' = k/ln()] Saring wih 9 organisms, and a k' value beween.5 and 2, he graphs are shown below.
N := 9 N (, k' ) := N k' (Eqn 5) k' :=.5,.. 2 :=,.5.. 5 Nk' (, ) N.5 (, ) N (, ) N(,.5) N2 (, ). LETHALITY AT DIFFERENT TEMPERATURES. 9. 8. 7. 6. 5. 4. 3 2 3 4 5 The consan, k, has unis of min-, since k* mus be uniless. Biologis have radiionally repored a value "D", which is he ime required for a fold reducion of CFU. D = So, in he graph k' shown, D=2 is he uppermos line wih he leas slope, i.e. he mos difficul srains o serilie, while D=.5 is he lowermos line. D Puing D ino equaion 5 in place of k resuls in ND (, ):= N Solving for D, and remembering ha D (a rae consan) is a funcion of emperaure, T, we obain he following. DT ( ) = log N D = k N D has unis of ime (usually minues), and i is he ime required o reduce he microbiological populaon by 9%. From equaion (3) above, we can obain a raio represening he survival of % of he organisms, i.e. 9% lehaliy. log N () N log(.) k ' = k' = log(.) = = D = 9 The D Value is he ime for 9% of he organisms o be killed. k'
An explanaion of Experience has shown ha 25 deg F is a suiable emperaure for he seam seriliaion of many microorganisms. However, auoclaves may run hoer or colder. The quesion arises as o how o relae ime a an acual auoclave emperaure o ime a a reference emperaure. How many degrees of emperaure would be required o make D vary by fold? For any given microorganism he quesion can be answered and he answer will be generically called "". Tha is, represens a fold change in D. D( Temp) D( Temp) = DT ( ) = <= Noe he log() = Simulaneously, represens he emperaure difference beween T and T. = T T log DT ( ) = = T T Where is he D value deermined a T. log DT ( ) = T T DT ( ) = = DT ( ) T T T T <= his expression would have unis of inverse minues, since D has unis of minues.
:= T := 2. :=. D( Temp) := D is he ime required, calculaed for any emperaure, T, o Temp T annihilae 9% of he populaion. When = min, he reference emperaure (here 2. deg C) is needed for 9. D( Temp) 4 6 2. 3.2434 D( Temp) 2.3832 6 8 2 22 24 Temp Inerpreaion: Using a value of. for he enire graph, when he emperaure is 6, here will be required 3.24 minues o reduce he populaion by 9%. Conversly (or inversely) in minue a 6 C, only.38 minues of equivalen lehaliy are accumulaed. Bu now we have a funcion ha relaes lehaliy(9% annihilaion),, and Temp, and can T T T T be inegraed o obain ime. FT ( ) = ( ) d = This inegraion resul assumes a consan emperaure T for he enire period from o. If T() is a funcion of ime, i.e. if Temperaure varies wih ime, as i almos cerainly will, hen a specific inegral mus be wrien afer he emperaure funion is known. In mos cases, here is no mah funcion for he variaion of Temperaure wih ime, and hus a numerical inegraion will be needed.
:= := T := 2. T := 22 FD (, ) := ( ) T T 3 Increasing D Values for F F' (, ) F( ',.4) F( ',.8) F( ', 2.2) F( ', 2.6) 2 2 ' The inegral funcion is used by microbiologis and referred o as F, ofen wihou he / erm which is assumed o have a value of. min. When used wih, F is uniless since ime in minues (d) over, also in minues, cancel each oher. F = T T d Is primary use is in relaing seriliaion emperaure and ime wih one microorganism back o a sandard seriliaion emperaure and ime. F is usually used as he equivalen number of minues a 2. T, ha are couned as a resul of he Temperaure a which he sysem is measured.
Example: := min, 6 2 min.. 6min N := 7 <= Saring number of microorganisms..8 k' := N () N k' := =.25 min min k'. 7 N () 5. 6 N () 2 3 6. 7. 6. 5 N (). 4. 3 5 min 5 := Temperaure T () 5 Temperaure ime race from a seam serilier, using simulaed daa, T(). 2 3 4 5 6
:= 4 T ( ) 2. Tj ( ) 2. d = 28.68 L := j 2 F values accumulaed for each minue 2 T ( ) 2. 2 3 4 5 6 For nearly fla Temperaure ime races such as he one shown, beween 2 and 4 minues, he inegral is someimes evaluaed using a sum over evenly imed incremens. T ( ) 2. 4 = 29.567 = 2 This is a calculaion mehod for inegraion. There are many differen mehods of numerically obaining he inegral, hence here will be small differences in he values. These differences should be insignifican in all cases.
I is also possible o inegrae over he enire ime period, hus geing Lehaliy accumulaion for he up and down imes, geing o and from he seriliing emperaure plaeau. 6 T ( ) 2. d = 3.25 6 = T ( ) 2. = 3.25 When he emperaure,t, is close o 2., influence from he value of beween and 2 is very limied. Clearly, when T maches he reference Temperaure, hen = no maer wha value may assume. T 2. F( T,, ) := d :=,.. 2 2 F( 22,, ) 2 Indeed, when he Temperaure is greaer han he reference emperaure (2. above), hen accumulaed F decreases when increases, bu when he Temperaure is less han he reference, accumulaed F increases when increases.
T 2. F (,, T) = d F (,, T) := ( T 2.) Effec of a change in value on Accumulaed Lehaliy, F. 25 8 F2 (,, 22) 24 F2 (,, 2) 7 23 6 22 5 2 T=22 > TRef = 2. 5 5 2 T=2 < TRef = 2.