Biol. Pharm. Bull. 38, 1714–1721 (2015)
1714
Vol. 38, No. 11
Regular Article
New Approach for Setting a Management Criterion in Microbiological Monitoring Using Rapid Microbiological Methods Noe Miyashita,* Makoto Tanaka, and Ryusuke Gotoda Air Conditioning & Plant Engineering Systems Department, Matsudo Research Center, Hitachi, Ltd., Infrastructure Systems Company; 537 Kami-Hongo, Matsudo, Chiba 271–0064, Japan. Received April 15, 2015; accepted August 30, 2015 The application of rapid microbiological methods (RMM) to bacterial monitoring in pharmaceutical manufacturing processes is now a key topic, since timely microbiological data are critical for product release, continuous process improvement and quality control. An automated, highly sensitive detection system has been developed which can measure the amount of ATP in a sample in 2 h with one hundredfold more sensitive than the conventional ATP method. One of the major subjects for adoption and implementation of RMM is how to set the criterion value for practical microbial control. This value was conventionally been set by experimental rule and indicated as the number of colonies counted after incubation in a particular medium. We have adopted a new approach to set a criterion value which enables assessment in whether the status of the object is normal or not. By setting this criterion value, it is possible to conduct the microbiological control with the intended probability of false-positive and false-negative. In this approach the probability distribution model of the measurement value of each object in a normal status has been established by performing repetitive measurement of each object. We have suggested and verified the probability distribution form of the ATP measurement value using measurement data of the standard bacterial solution of Staphylococcus aureus. The theoretical value of the model was in good agreement with the actual measured value. The results suggest it is possible to set an applicable management criterion value using this model and to conduct new microbiological monitoring using RMM. Key words tribution model
rapid microbiological method; bacterial monitoring; management criterion value; probability dis-
In pharmaceutical manufacturing facilities bacterial monitoring is essential for preventing microbiological invasion to products and for maintaining stable supply of it with high quality.1–3) Conventionally, microbes are monitored by using culturing method in which the sample is placed on an appropriate agar-based medium and the results are reported as colony forming unit (CFU) counted after a few days of cultivation.1–3) Rapid microbiological methods (RMM) can provide more timely microbiological data valuable not only for product release, but also for continuous process improvement and more efficient quality control.4–8) ATP method is one of RMM based on bioluminescent assay using luciferin–luciferase reaction which originated from the firefly.9) Luciferase catalyzes the oxidation of D -luciferin to oxyluciferin in the presence of ATP and oxygen as a bioluminescent reaction. ATP is one of the bioactivity indicators found in all living-cells.10) Therefore, by measuring the amount of ATP bioluminescence, we can evaluate the existence of livingbacteria in the sample.11,12) The disadvantage of using ATP bioluminescence assay is the low sensitivity, whereas it has advantage that ATP indicates the presence of living-bacteria that directly leads to critical contamination of products. As for common procedure, detection sensitivity of ATP method is as high as several hundreds of bacteria cells due to inefficiency in luminescence– intensity–measurement. We developed an automated, highly sensitive detection system with one hundredfold sensitivity than the conventional ATP method, primarily attributable to complete automation of the sampling and measurement steps and enhancements to
measurement technologies which result in lower background contamination, and a significant improvement in signal detection.13,14) The system underwent validation testing and is ready to be used in pharmaceutical manufacturing quality control operations. Implementation of RMMs is being more and more vigorously discussed,4–8) and regulations are improved and expanded in association with it.15–18) However, despite many impressive improvements in the technologies for rapid and sensitive microbe detection, few have been put into practical use in the pharmaceutical manufacturing processes. One of the reasons is the difficulty in making new operation procedure and establishing alternative management approach using new measurement methods. One big subject is how to set the criterion value. The acceptable microbial amount in products, raw materials and environments are described in pharmacopoeia or guidelines.1–3) Criterion values are set up for the stable supply of high-quality products17); alert-level indicates possibility of contamination and action-level demands the stop of production line, causal analysis, and recovery. It was conventionally set up by experimental rule and given as CFU. The aim of this study is to suggest the new approach for setting criterion value for microbial monitoring and risk control based on probability distribution analysis. By using probability distribution model of the measurement value of each object, it is possible to set the criterion value and assess the status of the object with intended probability of falsepositive and false-negative. In this study, we have established two types of probability distribution model formulae for ATP measurement value. The model formulae were verified using
* To whom correspondence should be addressed. e-mail: [email protected] © 2015 The Pharmaceutical Society of Japan
Biol. Pharm. Bull. Vol. 38, No. 11 (2015)1715
actual measurement data of standard bacterial solution of Staphylococcus aureus.
MATERIALS AND METHODS Overview of BIOMAYTECTOR, the Highly Sensitive ATP Measurement System BIOMAYTECTOR (“BIOMAYTECTOR” is a registered trademark of Hitachi, Ltd. in Japan and overseas.) consists of two components: an air sampler and a measuring unit. This system could measure both air and water sample.13) As for air sampling, the process started with air sampling by the air sampler including inertial impactor. Microbes in 1000 L of air were collected in 10 min. Airborne microbes were trapped on the surface of a special sol–gel transition carrier in the BIOMAYTECTOR sampling cartridge. Then, sampling cartridge was set on the measuring unit after engaged with the BIOMAYTECTOR collection cartridge containing membrane filters for filtration and concentration. The BIOMAYTECTOR reagent kit is intended to remove ATP released from dead cells and then extract ATP contained in living cells or released from spores. The ATP luminescent reaction were set in to the measuring unit along with the BIOMAYTECTOR standard ATP solution. From this point, all processes were performed automatically in the measuring unit. First, sol–gel carrier was converted to liquid by heating and microbes were collected in small scale of solution on the surface of the filter by suction pump. After ATP reagent reactions were processed on the surface of the filter, sample solutions were moved to measurement tube by nozzle and the each luminescent intensity was measured by the optical system. Six cartridges could be set on the measuring unit and they were measured in 2 h at the same time.14) The system provided the calibration system for the correction of enzyme activity and residual ATP remained in reagents and also for the survey of contamination during postprocessing and measurement processes. Each sample measurement was done with a blank measurement and standard ATP measurement using same reagents derived from same reagent tubes, at the same time. ATP amount of sample was calculated using calibration data and the result was expressed in the unit of attomole (amol)/sample. Water sample was applied directly into the BIOMAYTECTOR collection cartridge and set on the measuring unit and following processes were the same as air sample measurement. Model Formulae ATP measurement value was characterized by two stochastically distributed quantities; microbial count and ATP extract amount per each cell. Distribution model of ATP measurement value was considered in two cases. The sample containing a large amount of microbes was expected to follow normal distribution model. The compound Poisson distribution model was developed for the sample that contains a small amount of microbes. Microbial count, which changes because of their random distribution in the environment, was approximated to Poisson distribution. ATP extract amount, which is dependent on the bacterial strain, component ratio and bioactivity of the bacteria, was approximated to gamma distribution. ATP measurement value with a microbial count of zero, which corresponded to dispersion of the blank
reading, was approximated to normal distribution. As the microbial count increases, the compound Poisson distribution model approaches to the normal distribution model. The boundary value of the microbe count that the normal distribution model can be adopted for the ATP measurement value was verified based on the skewness and the kurtosis of the compound distribution model. Reagents and Bacteria Phosphate-buffered saline (PBS) (pH 7.2) and UltraPure™ DNase⁄ RNase-Free Distilled Water were purchased from Life Technologies (Carlsbad, Canada). All consumables for BIOMAYTECTOR ZERO1-SX were purchased from Hitachi Plant Service Co., Ltd. (Tokyo, Japan). Tryptic soy agar culture media was purchased from ASONE Corp. (Osaka, Japan). Staphylococcus aureus (ATC C6538) was purchased from bioMérieux s.a. (Lion, France) as a freeze-dried sample which contains known amount of CFU. Microbe Solution PBS was autoclaved at 120°C for 2 h before use. Microbes were stored at temperature of −20±5°C. All microbes were solved and diluted by ten-fold serial dilution in autoclaved PBS just before each examination. ATP Measurement A hundred microliters of microbe solution was applied to the surface of the gel-phase carrier which is contained in the BIOMAYTECTOR sampling cartridge. The cartridges were set on the measuring unit of BIOMAYTECTOR after engaged with the BIOMAYTECTOR collection cartridge. ATP amount of each sample was measured in the airborne microbe measurement mode. As the blank test, ATP amount of the autoclaved PBS was measured in the same way. All ATP measurements were conducted in set with CFU measurement of the same sample. Colony Counting A hundred microliters of microbe solution was applied to the surface of the tryptic soy agar culture media. After incubation at 35±5°C for 5 d, the colonies on each sample were counted. As the blank test, microbial count of autoclaved PBS was checked in the same way. As the control test of the dilution process of the microbe solution, microbial count of each residual solutions of the serial dilution which contains 10–103 CFU/100 µL was also checked in the same way. Model Verification As a sample in the normal status, microbe solutions of 30 CFU/100 µL were prepared and of which ATP amount and CFU counting were measured repeatedly. The parameters of the compound distribution model were estimated by the repetitive measurement data. To evaluate the validity of the model formula and its parameter, microbe solutions of 5, 15, and 30 CFU/100 µL were prepared and measured repeatedly. The observed value was compared with the expected value and G tests were performed to evaluate the model.
RESULTS AND DISCUSSION Over All Procedure for Setting Management Criterion Value by Statistical Approach Figure 1 shows the flow chart for setting management criterion value based on probability distribution analysis. First, in order to develop the distribution model for the object in the normal status, repetitive colony counting and ATP measurement were necessary (S1 in Fig. 1). The type of the model was chosen from two types discussed in detail later based on the mean value of microbe
1716
Biol. Pharm. Bull.
Vol. 38, No. 11 (2015)
ity manager (S6 in Fig. 1). Normal Distribution Model An ATP measurement is determined by the number of bacteria collected from a sample and by the amount of ATP contained in the collected bacteria. According to the central limit theorem, a random variable represented as the sum of a large number of independent, identically distributed random variables with a mean and a standard deviation follows a normal distribution. Therefore, if at least the number of microbes in a sample is sufficiently large, it is thought that ATP observations can be approximated by normal distribution. The normal distribution model of ATP measurements is represented by Eq. 1 shown below.
S ~ N ( μS , σS2 ) fS ( x ) =
1 2πσS2
( x − μS ) 2 exp − 2σS2
(1)
μS and σS are the average and the standard deviation of ATP measurements, respectively. μS and σS have a relationship with the expected value of bacterial count λ, the average ATP amount contained in one bacterium μ, and the standard deviation σ, as expressed by the following equations:
μS = λμ σS2 = λ( μ 2+ σ 2 ) Compound Poisson Distribution Model While an ATP measurement is a non-negative variable, a random variable that follows a normal distribution can be a negative value. Thus, it is not appropriate to use normal distribution for the approximation of ATP measurements particularly when the number of collected bacteria is small. Consequently, we constructed a compound Poisson model for use with small bacteria counts. The ATP measurement value S is the sum of the ATP amounts in the collected bacteria and is represented by the following Eq. 2: Fig. 1. Flow Chart for Setting Management Criterion Value Based on Probability Distribution Analysis
count in the repetitive measurement (S2 in Fig. 1). Then, the parameters of each model were estimated based on measurement data (S3 in Fig. 1). In order to evaluate the model formula, G-test was performed (J2 in Fig. 1). In general, it is desirable to perform G-test where the expected frequency in each class is more than 5 measurement should be repeated until each expected frequency would be more than 5 (J1 in Fig. 1). Alert level was set by one-sided test based on the model for the normal status and probability of false-positive which could be decided by quality manager (S4 in Fig. 1). To set the action level, the model for the aberrance status was developed based on the model for the normal status and presumption of microbe contamination. Even if the model for the normal status has been rejected by G-test, it is still possible to develop the probability distribution model for the aberrance status and set action-level. In this case, parameters were estimated based on average ATP content of common microbe which was often detected from the object (S5 in Fig. 1). Action level was set by one-sided test based on the model for the aberrance status and probability of false-negative which could be decided by qual-
S = X 1 + X 2 ++ X N =
N
X
i
( N ≥ 1)
(2)
i=1
S = X 0 ( N = 0) where N is the number of collected bacteria, Xi is the ATP amount per bacterium, and X0 is ATP measurement when no bacteria were collected. N, Xi and X0 are random variables that follow a certain probability distribution. Because the number of collected bacteria N is a natural phenomenon that occurs discretely, it can be approximated by Poisson distribution. The probability density function f N(k) is represented by Eq. 3 shown below.
N ~ Po( λ) f N (k ) =
λk e− λ k!
(3)
where λ is a parameter of a probability model and is determined by the expected count of bacteria collected in the environment. Because the amount of ATP included in one bacterium Xi depends on the bacterial species, composition ratio, and bioactivity, the probability distribution is expected to be a complex shape with a large number of parameters. In this study, we adopted Gamma distribution for approxi-
Biol. Pharm. Bull. Vol. 38, No. 11 (2015)1717
mation for the following reasons: (1) This is a non-negative probability distribution model. (2) The average and variance of this model are determined arbitrarily by two parameters. (3) Having reproducibility, the model facilitates the calculation of the probability distribution of the sum of ATP amounts for any number of bacteria from the probability distribution of ATP amount per bacterium. Reason (1): Because the ATP amount is a non-negative variable, a distribution model having probability variables that might take negative values, such as in normal distribution, is considered inappropriate. The distribution of body weight is known to be approximated by Gamma distribution model better than normal distribution.19) Reason (2): According to the central limit theorem, the mean and variance of the probability model can be made the same as in the actual ATP amount distribution by parameter adjustment. Even if the distribution of ATP amount per bacterium is not exactly the same, the mean and variance of ATP amount distribution in multiple bacteria collected are the same between the actual and the model as long as the mean and variance are the same as the actual values. From the central
Fig. 2.
limit theorem, both the actual and model ATP amount distributions converge to an identical normal distribution as the number of bacteria increases. Reason (3): In order to determine the probability distribution of the sum of ATP amounts in multiple bacteria, it is necessary to perform a convolution operation using a probability distribution of ATP amount per bacterium. For a nonreproducible probability density function, the convolution involves numerical integration, but this operation is realistically difficult because its computational load is very high. The probability density function f X(x) is expressed by the following equation:
X i ~ Γ (η, κ ) f X ( x) = η=
x η−1e− x / κ Γ (η)κ η
μ2 σ2
µ and σ are the parameters in the probability model, representing the mean and standard deviation of the ATP amount per bacterium. μ and σ vary depending on the measurement
Comparative Verification of the Normal Distribution Model and the Compound Poisson Distribution Model with λ=10, 20, 50, 100, 300
(A) Probability density curve of the normal distribution and the compound Poisson distribution. (B) Skewness and kurtosis of the compound Poisson distribution model.
Biol. Pharm. Bull.
1718
environment, such as the types of bacteria collected. Given the collected bacteria count k, the sum of ATP amounts in k bacteria X k is expressed by Eq. 4 shown below. X k = X 1 + X 2 ++ X k X k ~ Γ (kη, κ ) f Xk ( x ) =
x kη−1e − x / κ Γ (kη) ⋅ ( κ ) kη
( k ≥ 1)
(4)
When the number for collected bacteria is 0, the ATP measurement X 0 is considered to be equivalent to variations in blank measurements of the device and is approximated by a normal distribution shown in Eq. 5. X 0 ~ N ( μb , σ b2 ) f X0 ( x ) =
1 2πσ b2
( x − μb ) 2 exp − 2σ b2
Vol. 38, No. 11 (2015)
facilities, we find that the skewness and kurtosis are 0.32 and 0.15, respectively, or practically 0 (zero). Therefore, the compound Poisson distribution can be substantially regarded as a normal distribution. Consequently, we applied the normal distribution model for the average bacteria count of 100 CFU or more in prior repetitive measurements and applied the compound Poisson distribution model for the count of less than 100 CFU. Parameter Estimation for the Normal Status In the normal distribution model, λ, μ and σ are defined by Eq. 7. Here, c¯ and qt are the average bacteria count in prior repetitive measurements and the sample amount collected in the culture method, respectively. Similarly, qb, x¯ and u 2 are the sample amounts collected for the ATP measurement in prior repetitive measurements, their average, and unbiased variance, respectively.
(5)
where μb and σb are the mean and standard deviation in blank measurements, and are determined by the equipment specifications. Using Eqs. 3–5, the compound Poisson distribution model is expressed by Eq. 6.
fS ( x ) =
∞
f
k N (k ) f X ( x)
(6)
k=0
Model Selection The boundary value of the collected bacteria count that allows ATP measurements to be approximated also by the normal distribution model was verified based on the skewness and kurtosis of the compound Poisson distribution model. Figure 2 compares the normal distribution model with the compound Poisson distribution model in terms of the average count of collected bacteria. With our attention focused on the Grade-B management criterion value of 100 CFU for pharmaceutical manufacturing
Fig. 3. Conceptual Diagram Which Expresses the Relationship between Criterion Values and Probability of False-Positive and False-Negative To determine the alert level xc1, the normal status is subjected to the one-sided test with the significance level ε1, where ε1 is a probability of false-negative. To determine the action level xc2, the aberrance status was subjected to the one-sided test with the significance level ε2, where ε2 is a probability of false-positive.
Fig. 4. Comparison of Observed Value with the Estimated in the Compound Poisson Distribution Model (A) Cumulative distribution curve for microbe counts in sample. (B) Cumulative distribution curve for ATP extract amount per cell.
Biol. Pharm. Bull. Vol. 38, No. 11 (2015)1719
Fig. 5.
G-Tests for the Compound Poisson Distribution Model
(A) For microbe counts in sample. (B) For ATP extract amount per cell.
Biol. Pharm. Bull.
1720
qb qt
λ=c x λ
μ=
(7)
u2 x 2 − 2 λ λ
σ=
For the complex Poisson distribution, the Poisson distribution parameter λ is calculated from Eq. 8. Here, the average number of bacteria collected in prior repetitive measurements is c¯, the sample amount collected in the culture method is qt, and the sample amount collected for the ATP measurement is qb.
λ=c
qb qt
(8)
Given the pre-measured ATP amount values x1, x2, …, xm, the parameters μ and σ related to ATP amount distribution in bacteria are adjusted so that the log likelihood function l(λ, μ, σ) in Eq. 9 becomes the maximum. l ( λ, μ, σ ) = log =
m
∏ f ( x ; λ, μ, σ ) i
S
i=1
m
∑ log( f ( x ; λ, μ, σ )) S
i
(9)
i=1
Parameter Estimation for the Aberrance Status In order to set an action level, it is necessary to construct a probability distribution model for the aberrance status. For the parameter λe, we applied the management criterion value determined in the current culture method. μe and σe were determined to be the same as in the normal status. When the probability distribution model of the normal status was rejected in G-test, the average ATP amount and standard deviation of typical bacteria that frequently occur in the measured object were applied to μe and σe. Criterion Value Setting Figure 3 is a conceptual diagram showing the relationship between the management criterion value and the false-positive or false-negative probability. In order to determine the alert level xc1, the normal status was subjected to the one-sided test with the significance level ε1. xc1 was calculated from Eq. 10 shown below. Here, ε1 is a false-positive probability. The rejection region of the normal status was assumed to be S>xc1.
P ( S > xc1 ) ≤ ε1 FS ( xc1 ) =
xc1
0
fS ( x)dx =1 − ε1
(10)
In order to determine the action level xc2, the aberrance status was subjected to the one-sided test with the significance level ε2. xc2 was calculated from Eq. 11 below. Here, ε2 is a false-negative probability. The rejection region of the aberrance status was assumed to be S
1714
Vol. 38, No. 11
Regular Article
New Approach for Setting a Management Criterion in Microbiological Monitoring Using Rapid Microbiological Methods Noe Miyashita,* Makoto Tanaka, and Ryusuke Gotoda Air Conditioning & Plant Engineering Systems Department, Matsudo Research Center, Hitachi, Ltd., Infrastructure Systems Company; 537 Kami-Hongo, Matsudo, Chiba 271–0064, Japan. Received April 15, 2015; accepted August 30, 2015 The application of rapid microbiological methods (RMM) to bacterial monitoring in pharmaceutical manufacturing processes is now a key topic, since timely microbiological data are critical for product release, continuous process improvement and quality control. An automated, highly sensitive detection system has been developed which can measure the amount of ATP in a sample in 2 h with one hundredfold more sensitive than the conventional ATP method. One of the major subjects for adoption and implementation of RMM is how to set the criterion value for practical microbial control. This value was conventionally been set by experimental rule and indicated as the number of colonies counted after incubation in a particular medium. We have adopted a new approach to set a criterion value which enables assessment in whether the status of the object is normal or not. By setting this criterion value, it is possible to conduct the microbiological control with the intended probability of false-positive and false-negative. In this approach the probability distribution model of the measurement value of each object in a normal status has been established by performing repetitive measurement of each object. We have suggested and verified the probability distribution form of the ATP measurement value using measurement data of the standard bacterial solution of Staphylococcus aureus. The theoretical value of the model was in good agreement with the actual measured value. The results suggest it is possible to set an applicable management criterion value using this model and to conduct new microbiological monitoring using RMM. Key words tribution model
rapid microbiological method; bacterial monitoring; management criterion value; probability dis-
In pharmaceutical manufacturing facilities bacterial monitoring is essential for preventing microbiological invasion to products and for maintaining stable supply of it with high quality.1–3) Conventionally, microbes are monitored by using culturing method in which the sample is placed on an appropriate agar-based medium and the results are reported as colony forming unit (CFU) counted after a few days of cultivation.1–3) Rapid microbiological methods (RMM) can provide more timely microbiological data valuable not only for product release, but also for continuous process improvement and more efficient quality control.4–8) ATP method is one of RMM based on bioluminescent assay using luciferin–luciferase reaction which originated from the firefly.9) Luciferase catalyzes the oxidation of D -luciferin to oxyluciferin in the presence of ATP and oxygen as a bioluminescent reaction. ATP is one of the bioactivity indicators found in all living-cells.10) Therefore, by measuring the amount of ATP bioluminescence, we can evaluate the existence of livingbacteria in the sample.11,12) The disadvantage of using ATP bioluminescence assay is the low sensitivity, whereas it has advantage that ATP indicates the presence of living-bacteria that directly leads to critical contamination of products. As for common procedure, detection sensitivity of ATP method is as high as several hundreds of bacteria cells due to inefficiency in luminescence– intensity–measurement. We developed an automated, highly sensitive detection system with one hundredfold sensitivity than the conventional ATP method, primarily attributable to complete automation of the sampling and measurement steps and enhancements to
measurement technologies which result in lower background contamination, and a significant improvement in signal detection.13,14) The system underwent validation testing and is ready to be used in pharmaceutical manufacturing quality control operations. Implementation of RMMs is being more and more vigorously discussed,4–8) and regulations are improved and expanded in association with it.15–18) However, despite many impressive improvements in the technologies for rapid and sensitive microbe detection, few have been put into practical use in the pharmaceutical manufacturing processes. One of the reasons is the difficulty in making new operation procedure and establishing alternative management approach using new measurement methods. One big subject is how to set the criterion value. The acceptable microbial amount in products, raw materials and environments are described in pharmacopoeia or guidelines.1–3) Criterion values are set up for the stable supply of high-quality products17); alert-level indicates possibility of contamination and action-level demands the stop of production line, causal analysis, and recovery. It was conventionally set up by experimental rule and given as CFU. The aim of this study is to suggest the new approach for setting criterion value for microbial monitoring and risk control based on probability distribution analysis. By using probability distribution model of the measurement value of each object, it is possible to set the criterion value and assess the status of the object with intended probability of falsepositive and false-negative. In this study, we have established two types of probability distribution model formulae for ATP measurement value. The model formulae were verified using
* To whom correspondence should be addressed. e-mail: [email protected] © 2015 The Pharmaceutical Society of Japan
Biol. Pharm. Bull. Vol. 38, No. 11 (2015)1715
actual measurement data of standard bacterial solution of Staphylococcus aureus.
MATERIALS AND METHODS Overview of BIOMAYTECTOR, the Highly Sensitive ATP Measurement System BIOMAYTECTOR (“BIOMAYTECTOR” is a registered trademark of Hitachi, Ltd. in Japan and overseas.) consists of two components: an air sampler and a measuring unit. This system could measure both air and water sample.13) As for air sampling, the process started with air sampling by the air sampler including inertial impactor. Microbes in 1000 L of air were collected in 10 min. Airborne microbes were trapped on the surface of a special sol–gel transition carrier in the BIOMAYTECTOR sampling cartridge. Then, sampling cartridge was set on the measuring unit after engaged with the BIOMAYTECTOR collection cartridge containing membrane filters for filtration and concentration. The BIOMAYTECTOR reagent kit is intended to remove ATP released from dead cells and then extract ATP contained in living cells or released from spores. The ATP luminescent reaction were set in to the measuring unit along with the BIOMAYTECTOR standard ATP solution. From this point, all processes were performed automatically in the measuring unit. First, sol–gel carrier was converted to liquid by heating and microbes were collected in small scale of solution on the surface of the filter by suction pump. After ATP reagent reactions were processed on the surface of the filter, sample solutions were moved to measurement tube by nozzle and the each luminescent intensity was measured by the optical system. Six cartridges could be set on the measuring unit and they were measured in 2 h at the same time.14) The system provided the calibration system for the correction of enzyme activity and residual ATP remained in reagents and also for the survey of contamination during postprocessing and measurement processes. Each sample measurement was done with a blank measurement and standard ATP measurement using same reagents derived from same reagent tubes, at the same time. ATP amount of sample was calculated using calibration data and the result was expressed in the unit of attomole (amol)/sample. Water sample was applied directly into the BIOMAYTECTOR collection cartridge and set on the measuring unit and following processes were the same as air sample measurement. Model Formulae ATP measurement value was characterized by two stochastically distributed quantities; microbial count and ATP extract amount per each cell. Distribution model of ATP measurement value was considered in two cases. The sample containing a large amount of microbes was expected to follow normal distribution model. The compound Poisson distribution model was developed for the sample that contains a small amount of microbes. Microbial count, which changes because of their random distribution in the environment, was approximated to Poisson distribution. ATP extract amount, which is dependent on the bacterial strain, component ratio and bioactivity of the bacteria, was approximated to gamma distribution. ATP measurement value with a microbial count of zero, which corresponded to dispersion of the blank
reading, was approximated to normal distribution. As the microbial count increases, the compound Poisson distribution model approaches to the normal distribution model. The boundary value of the microbe count that the normal distribution model can be adopted for the ATP measurement value was verified based on the skewness and the kurtosis of the compound distribution model. Reagents and Bacteria Phosphate-buffered saline (PBS) (pH 7.2) and UltraPure™ DNase⁄ RNase-Free Distilled Water were purchased from Life Technologies (Carlsbad, Canada). All consumables for BIOMAYTECTOR ZERO1-SX were purchased from Hitachi Plant Service Co., Ltd. (Tokyo, Japan). Tryptic soy agar culture media was purchased from ASONE Corp. (Osaka, Japan). Staphylococcus aureus (ATC C6538) was purchased from bioMérieux s.a. (Lion, France) as a freeze-dried sample which contains known amount of CFU. Microbe Solution PBS was autoclaved at 120°C for 2 h before use. Microbes were stored at temperature of −20±5°C. All microbes were solved and diluted by ten-fold serial dilution in autoclaved PBS just before each examination. ATP Measurement A hundred microliters of microbe solution was applied to the surface of the gel-phase carrier which is contained in the BIOMAYTECTOR sampling cartridge. The cartridges were set on the measuring unit of BIOMAYTECTOR after engaged with the BIOMAYTECTOR collection cartridge. ATP amount of each sample was measured in the airborne microbe measurement mode. As the blank test, ATP amount of the autoclaved PBS was measured in the same way. All ATP measurements were conducted in set with CFU measurement of the same sample. Colony Counting A hundred microliters of microbe solution was applied to the surface of the tryptic soy agar culture media. After incubation at 35±5°C for 5 d, the colonies on each sample were counted. As the blank test, microbial count of autoclaved PBS was checked in the same way. As the control test of the dilution process of the microbe solution, microbial count of each residual solutions of the serial dilution which contains 10–103 CFU/100 µL was also checked in the same way. Model Verification As a sample in the normal status, microbe solutions of 30 CFU/100 µL were prepared and of which ATP amount and CFU counting were measured repeatedly. The parameters of the compound distribution model were estimated by the repetitive measurement data. To evaluate the validity of the model formula and its parameter, microbe solutions of 5, 15, and 30 CFU/100 µL were prepared and measured repeatedly. The observed value was compared with the expected value and G tests were performed to evaluate the model.
RESULTS AND DISCUSSION Over All Procedure for Setting Management Criterion Value by Statistical Approach Figure 1 shows the flow chart for setting management criterion value based on probability distribution analysis. First, in order to develop the distribution model for the object in the normal status, repetitive colony counting and ATP measurement were necessary (S1 in Fig. 1). The type of the model was chosen from two types discussed in detail later based on the mean value of microbe
1716
Biol. Pharm. Bull.
Vol. 38, No. 11 (2015)
ity manager (S6 in Fig. 1). Normal Distribution Model An ATP measurement is determined by the number of bacteria collected from a sample and by the amount of ATP contained in the collected bacteria. According to the central limit theorem, a random variable represented as the sum of a large number of independent, identically distributed random variables with a mean and a standard deviation follows a normal distribution. Therefore, if at least the number of microbes in a sample is sufficiently large, it is thought that ATP observations can be approximated by normal distribution. The normal distribution model of ATP measurements is represented by Eq. 1 shown below.
S ~ N ( μS , σS2 ) fS ( x ) =
1 2πσS2
( x − μS ) 2 exp − 2σS2
(1)
μS and σS are the average and the standard deviation of ATP measurements, respectively. μS and σS have a relationship with the expected value of bacterial count λ, the average ATP amount contained in one bacterium μ, and the standard deviation σ, as expressed by the following equations:
μS = λμ σS2 = λ( μ 2+ σ 2 ) Compound Poisson Distribution Model While an ATP measurement is a non-negative variable, a random variable that follows a normal distribution can be a negative value. Thus, it is not appropriate to use normal distribution for the approximation of ATP measurements particularly when the number of collected bacteria is small. Consequently, we constructed a compound Poisson model for use with small bacteria counts. The ATP measurement value S is the sum of the ATP amounts in the collected bacteria and is represented by the following Eq. 2: Fig. 1. Flow Chart for Setting Management Criterion Value Based on Probability Distribution Analysis
count in the repetitive measurement (S2 in Fig. 1). Then, the parameters of each model were estimated based on measurement data (S3 in Fig. 1). In order to evaluate the model formula, G-test was performed (J2 in Fig. 1). In general, it is desirable to perform G-test where the expected frequency in each class is more than 5 measurement should be repeated until each expected frequency would be more than 5 (J1 in Fig. 1). Alert level was set by one-sided test based on the model for the normal status and probability of false-positive which could be decided by quality manager (S4 in Fig. 1). To set the action level, the model for the aberrance status was developed based on the model for the normal status and presumption of microbe contamination. Even if the model for the normal status has been rejected by G-test, it is still possible to develop the probability distribution model for the aberrance status and set action-level. In this case, parameters were estimated based on average ATP content of common microbe which was often detected from the object (S5 in Fig. 1). Action level was set by one-sided test based on the model for the aberrance status and probability of false-negative which could be decided by qual-
S = X 1 + X 2 ++ X N =
N
X
i
( N ≥ 1)
(2)
i=1
S = X 0 ( N = 0) where N is the number of collected bacteria, Xi is the ATP amount per bacterium, and X0 is ATP measurement when no bacteria were collected. N, Xi and X0 are random variables that follow a certain probability distribution. Because the number of collected bacteria N is a natural phenomenon that occurs discretely, it can be approximated by Poisson distribution. The probability density function f N(k) is represented by Eq. 3 shown below.
N ~ Po( λ) f N (k ) =
λk e− λ k!
(3)
where λ is a parameter of a probability model and is determined by the expected count of bacteria collected in the environment. Because the amount of ATP included in one bacterium Xi depends on the bacterial species, composition ratio, and bioactivity, the probability distribution is expected to be a complex shape with a large number of parameters. In this study, we adopted Gamma distribution for approxi-
Biol. Pharm. Bull. Vol. 38, No. 11 (2015)1717
mation for the following reasons: (1) This is a non-negative probability distribution model. (2) The average and variance of this model are determined arbitrarily by two parameters. (3) Having reproducibility, the model facilitates the calculation of the probability distribution of the sum of ATP amounts for any number of bacteria from the probability distribution of ATP amount per bacterium. Reason (1): Because the ATP amount is a non-negative variable, a distribution model having probability variables that might take negative values, such as in normal distribution, is considered inappropriate. The distribution of body weight is known to be approximated by Gamma distribution model better than normal distribution.19) Reason (2): According to the central limit theorem, the mean and variance of the probability model can be made the same as in the actual ATP amount distribution by parameter adjustment. Even if the distribution of ATP amount per bacterium is not exactly the same, the mean and variance of ATP amount distribution in multiple bacteria collected are the same between the actual and the model as long as the mean and variance are the same as the actual values. From the central
Fig. 2.
limit theorem, both the actual and model ATP amount distributions converge to an identical normal distribution as the number of bacteria increases. Reason (3): In order to determine the probability distribution of the sum of ATP amounts in multiple bacteria, it is necessary to perform a convolution operation using a probability distribution of ATP amount per bacterium. For a nonreproducible probability density function, the convolution involves numerical integration, but this operation is realistically difficult because its computational load is very high. The probability density function f X(x) is expressed by the following equation:
X i ~ Γ (η, κ ) f X ( x) = η=
x η−1e− x / κ Γ (η)κ η
μ2 σ2
µ and σ are the parameters in the probability model, representing the mean and standard deviation of the ATP amount per bacterium. μ and σ vary depending on the measurement
Comparative Verification of the Normal Distribution Model and the Compound Poisson Distribution Model with λ=10, 20, 50, 100, 300
(A) Probability density curve of the normal distribution and the compound Poisson distribution. (B) Skewness and kurtosis of the compound Poisson distribution model.
Biol. Pharm. Bull.
1718
environment, such as the types of bacteria collected. Given the collected bacteria count k, the sum of ATP amounts in k bacteria X k is expressed by Eq. 4 shown below. X k = X 1 + X 2 ++ X k X k ~ Γ (kη, κ ) f Xk ( x ) =
x kη−1e − x / κ Γ (kη) ⋅ ( κ ) kη
( k ≥ 1)
(4)
When the number for collected bacteria is 0, the ATP measurement X 0 is considered to be equivalent to variations in blank measurements of the device and is approximated by a normal distribution shown in Eq. 5. X 0 ~ N ( μb , σ b2 ) f X0 ( x ) =
1 2πσ b2
( x − μb ) 2 exp − 2σ b2
Vol. 38, No. 11 (2015)
facilities, we find that the skewness and kurtosis are 0.32 and 0.15, respectively, or practically 0 (zero). Therefore, the compound Poisson distribution can be substantially regarded as a normal distribution. Consequently, we applied the normal distribution model for the average bacteria count of 100 CFU or more in prior repetitive measurements and applied the compound Poisson distribution model for the count of less than 100 CFU. Parameter Estimation for the Normal Status In the normal distribution model, λ, μ and σ are defined by Eq. 7. Here, c¯ and qt are the average bacteria count in prior repetitive measurements and the sample amount collected in the culture method, respectively. Similarly, qb, x¯ and u 2 are the sample amounts collected for the ATP measurement in prior repetitive measurements, their average, and unbiased variance, respectively.
(5)
where μb and σb are the mean and standard deviation in blank measurements, and are determined by the equipment specifications. Using Eqs. 3–5, the compound Poisson distribution model is expressed by Eq. 6.
fS ( x ) =
∞
f
k N (k ) f X ( x)
(6)
k=0
Model Selection The boundary value of the collected bacteria count that allows ATP measurements to be approximated also by the normal distribution model was verified based on the skewness and kurtosis of the compound Poisson distribution model. Figure 2 compares the normal distribution model with the compound Poisson distribution model in terms of the average count of collected bacteria. With our attention focused on the Grade-B management criterion value of 100 CFU for pharmaceutical manufacturing
Fig. 3. Conceptual Diagram Which Expresses the Relationship between Criterion Values and Probability of False-Positive and False-Negative To determine the alert level xc1, the normal status is subjected to the one-sided test with the significance level ε1, where ε1 is a probability of false-negative. To determine the action level xc2, the aberrance status was subjected to the one-sided test with the significance level ε2, where ε2 is a probability of false-positive.
Fig. 4. Comparison of Observed Value with the Estimated in the Compound Poisson Distribution Model (A) Cumulative distribution curve for microbe counts in sample. (B) Cumulative distribution curve for ATP extract amount per cell.
Biol. Pharm. Bull. Vol. 38, No. 11 (2015)1719
Fig. 5.
G-Tests for the Compound Poisson Distribution Model
(A) For microbe counts in sample. (B) For ATP extract amount per cell.
Biol. Pharm. Bull.
1720
qb qt
λ=c x λ
μ=
(7)
u2 x 2 − 2 λ λ
σ=
For the complex Poisson distribution, the Poisson distribution parameter λ is calculated from Eq. 8. Here, the average number of bacteria collected in prior repetitive measurements is c¯, the sample amount collected in the culture method is qt, and the sample amount collected for the ATP measurement is qb.
λ=c
qb qt
(8)
Given the pre-measured ATP amount values x1, x2, …, xm, the parameters μ and σ related to ATP amount distribution in bacteria are adjusted so that the log likelihood function l(λ, μ, σ) in Eq. 9 becomes the maximum. l ( λ, μ, σ ) = log =
m
∏ f ( x ; λ, μ, σ ) i
S
i=1
m
∑ log( f ( x ; λ, μ, σ )) S
i
(9)
i=1
Parameter Estimation for the Aberrance Status In order to set an action level, it is necessary to construct a probability distribution model for the aberrance status. For the parameter λe, we applied the management criterion value determined in the current culture method. μe and σe were determined to be the same as in the normal status. When the probability distribution model of the normal status was rejected in G-test, the average ATP amount and standard deviation of typical bacteria that frequently occur in the measured object were applied to μe and σe. Criterion Value Setting Figure 3 is a conceptual diagram showing the relationship between the management criterion value and the false-positive or false-negative probability. In order to determine the alert level xc1, the normal status was subjected to the one-sided test with the significance level ε1. xc1 was calculated from Eq. 10 shown below. Here, ε1 is a false-positive probability. The rejection region of the normal status was assumed to be S>xc1.
P ( S > xc1 ) ≤ ε1 FS ( xc1 ) =
xc1
0
fS ( x)dx =1 − ε1
(10)
In order to determine the action level xc2, the aberrance status was subjected to the one-sided test with the significance level ε2. xc2 was calculated from Eq. 11 below. Here, ε2 is a false-negative probability. The rejection region of the aberrance status was assumed to be S
