The brief comparison of the operational efficiency of pooltesting strategies for COVID19 mass testing in PCR laboratories
Kirill Vechera
LOMT project, Jetware
20200708, revision 3
Introduction
For a PCR laboratory, the major indicator of the efficiency of a pooltesting strategy is the reduction of the number of assays for a certain number of specimens. The analysis process takes up to 4 hours, consumes reagents and requires significant manual labor. Other related processes require much less time and cost, and we can assume that a tenfold reduction in the number of assays increases the throughput of the laboratory by 9 times and also decreases the cost of the testing of one person by 9 times.
The number of assays during pool testing primarily depends on prevalence — that is, the proportion of positives samples. The lower the prevalence, the larger the pool size that can be used, and ultimately, fewer assays will be required. For example, for 1% prevalence, the maximal effective pool size is 69, for 10% prevalence it is 7 and for prevalence above 25% the pool size becomes less than 4 and the efficiency becomes too low to be worth using.
For a PCR laboratory, two other factors are important: the limit of dilution of a specimen and the time it takes to get results for a full batch of specimens.
The limit of dilution here is the maximum number of specimens that are mixed in a single pool, or the maximum pool size. The limit of dilution depends on the sensitivity of a test and its limit of detection, since if the dilution of a positive specimen leads to the amount of its detecting material decreasing below the limit of detection, the specimen is erroneously defined as negative. Studies have shown that a pool size of up to 32 allows for the confident identification of asymptomatic people in RTPCR testing for SARSCoV2 RNA ^{1, 2}.
The processing time of a batch of specimens depends not only on the number of specimens and how fast the laboratory can perform analyses but also on the depth (the number of stages) of the pooling strategy. Any pooling strategy requires two or more stages to obtain results for all the specimens in a batch, and each stage can begin only after the previous stage has been completed. In most cases, the more stages the pooling strategy has, the more assays are saved, but also the more time is required to get the results of a full batch.
As analyses are performed, some specimens are identified as positive or negative from the analysis results, while other specimens remain for repeated analyses in the next stages. The number of samples identified after the first analysis can vary significantly with different strategies. Algorithms in which one sample goes into only one pool at each stage can identify positive samples only at the last stage, while all other stages identify negative samples only. Combinatorial algorithms in which one sample goes into several pools are capable of identifying both negative and positive samples at any stage and can also obtain the results of a full batch in many cases in the single stage, requiring the second stage for only some rare cases.
There are also nonadaptive algorithms that the result of all samples to be obtained in one stage, at the cost of reducing the accuracy of detection and producing a certain number of false positives. While it can be useful in some other tasks, we see no sense in using such algorithms for COVID19 testing of people, so these algorithms are not considered here.
The comparison
Different laboratories can operate with different values of prevalence, constraints of testing and priorities of optimization, and these values can change over time. LOMT supports several different pooling strategies and automatically searches for the optimal strategy for every specific condition. This comparison provides an assessment of the efficiency of the most used strategies in several typical scenarios.
The task consists of RTPCR testing of 1000 swabs on a 1 thermal cycler with 96 wells; full analysis time is 4 hours, the dilution limit is 32, and high, medium and low prevalence rates are 10%, 1% and 0.1%, respectively. The time to transfer fluid from specimen tubes to assay tubes is not taken into account, although it can slightly increase using combinatorial algorithms.
The efficiency indicators assessed are as follows:
 The number of assays — the smaller the number, the more laboratory throughput increases and the cost of testing one person decreases.
 First results — the number of specimens completely classified after the first analysis (after the shortest possible time) and the number of positive specimens found.
 Total time to result — the time (hours) required to obtain the results of all swabs in the batch; some swabs from the batch are mixed in pools and analyzed the entire time.
Individual
In the standard testing method, without pooling, all specimens are tested individually. We use this as a baseline to assess the efficiency of pooltesting strategies. A prevalence rate of 10% means there are 900 negative specimens and 100 positive ones; a prevalence rate of 1% means 990 negative specimens and 10 positive ones; and a prevalence rate of 0.1% means 999 negative specimens and 1 positive.
Prevalence  10%  1%  0.1% 

Assays  1000  1000  1000 
First results  94  94  94 
First results, positives  9.4  0.94  0.094 
Total time to result, h  44  44  44 
The efficiency is the same for any prevalence from 0 to 100% — 1000 swabs need 1000 assays. First results are available in 4 hours — these are 94 swabs with the number of positive samples proportional to the prevalence of the batch. Total time to obtain results is 44 hours, with the manual labor and thermal cycler operations as the bottleneck.
Plain
This is a classic strategy of pool testing, which runs in two stages: in the first stage, all the specimens are split into several groups, and each group is mixed into a single pool and tested. A negative result means that all the specimens in the pool are negative. The pools with a positive result go to the second stage, in which all specimens are tested individually.
Prevalence  10% (pool 4)  1% (pool 10)  0.1% (pool 32) 

Assays  594 (1.68)  196 (5.1)  63 (15.9) 
First results  246 (2.62)  895 (9.5)  969 (10.3) 
First results, positives  0  0  0 
Total time to result, h  28 (1.57)  12 (3.67)  8 (5.5) 
With 10% prevalence, 1000 specimens require 594 assays (1.68x reduction compared to 1000 assays at baseline). First results are available in 4 hours; 250 specimens, only negative ones (2.64x increase compared to 94 at baseline). Total time to obtain results is 36 hours (1.22x reduction compared to 44 hours at baseline).
With 1% prevalence, 1000 specimens require 196 assays (5.1x reduction). First results: 846 specimens, only negative ones (9x increase). Total time to obtain results is 16 hours (2.75x reduction).
With 0.1% prevalence, 1000 specimens require 63 assays (15.9x reduction). First results: 970 specimens, only negative ones (10.3x increase). Total time to obtain results is 8 hours (5.5x reduction).
Plain Multilevel — 2 levels
This strategy is a classic pooltesting strategy with several pooling stages instead of a single one. The pools with positive results go to the next stage, and the pool size decreases with each stage. This approach significantly reduces the number of samples that are tested individually at the last stage compared to the singlelevel pooling. The higher the number of levels, the fewer assays needed. The values are for the strategy with two levels of pooling.
Prevalence  10% (pool 5)  1% (pool 22)  0.1% (pool 32) 

Assays  597 (1.68)  134 (7.5)  44 (22.7) 
First results  277 (2.95)  803 (8.54)  969 (10.3) 
First results, positives  0  0  0 
Total time to result, h  32 (1.38)  12 (3.67)  12 (3.67) 
Bisector Waiting
This strategy applies the parallelized version of Generalized Binary Splitting algorithm. It provides the maximum possible reduction of the number of assays but increases the pooling depth incrementing total time to result. For 10% prevalence it is 13 stages on average, for 1% prevalence it is 9 stages and for 0.01% it is 5 stages.
Prevalence  10% (pool 8)  1% (pool 32)  0.1% (pool 32) 

Assays  477 (2.1)  90 (11.1)  39 (25.6) 
First results  323 (3.43)  730 (7.77)  969 (10.3) 
First results, positives  0  0  0 
Total time to result, h  56 (0.79)  36 (1.22)  20 (2.2) 
Bisector Waiting with a depth limit
This strategy limits the pooling depth by executing individual testing of all specimens left after the depth limit is reached. The number of specimens in the last stages is small, so switching to the individual testing slightly decreases the assays savings but improves the batch processing time.
Prevalence  10% (pool 7, depth 7)  1% (pool 32, depth 7)  0.1% (pool 32, depth 4) 

Assays  503 (1.99)  100 (10)  47 (21.3) 
First results  314 (3.34)  730 (7.77)  969 (10.3) 
First results, positives  0  0  0 
Total time to result, h  36 (1.22)  28 (1.57)  12 (3.66) 
Enumerator Doublets
This strategy uses a combinatorial algorithm in which every specimen goes into two different pools. This approach allows for the classification of the maximum number of specimens at the first stage, and it detects both negative and positives swabs. The efficiency of combinatorial strategies increases with decreasing prevalence. A simplified variant of the strategy is called also Array testing or Matrix testing.
Prevalence  10% (pool 8, depth 3)  1% (pool 32, depth 3)  0.1% (pool 32, depth 2) 

Assays  562 (1.78)  118 (8.5)  65 (15.4) 
First results  247 (2.91)  920 (9.8)  998 (10.6) 
First results, positives  0.67  0.01  0.56 
Total time to result, h  28 (1.57)  12 (3.67)  8 (5.5) 
Enumerator Triplets
This is similar to the previous strategy with a combinatorial algorithm, in which every specimen goes into three pools. The strategy is able to identify more positive swabs in the first stages. For 0.1% prevalence, this strategy gives results for all specimens immediately after the first assay in 93% of cases, and requires a second stage of assays only in the remaining 7% cases.
Prevalence  10% (pool 8, depth 5)  1% (pool 32, depth 3)  0.1% (pool 32, depth 2) 

Assays  595 (1.68)  125 (8)  95 (10.5) 
First results  185 (1.96)  954 (10.1)  999 (10.6) 
First results, positives  5.8  3  0.9 
Total time to result, h  36 (1.22)  12 (3.67)  8 (5.5) 
Summary
After comparing these strategies, we can see which are best for the maximum increment of the throughput of a laboratory (by reducing the number of assays) and which are best for reducing total time to process a batch of swabs.
Prevalence  Strategy  Assays  Time to result 

10%  Bisector Waiting with a depth limit  1.99  1.22 
Enumerator Doublets  1.78  1.57  
1%  Bisector Waiting  11.1  1.22 
Enumerator Doublets  8.5  3.67  
0.1%  Bisector Waiting  25.6  2.2 
Plain  15.9  5.5 
For a high prevalence (10%), the Bisector Waiting strategy with a depth limit gives the best assay savings and throughput increment by 1.99 times, while the Enumerator Doublets strategy gives the best assay savings by 1.78 times with the least amount of time to obtain results — 1.57 times less compared to the individual testing.
For a medium prevalence (1%), the Bisector Waiting strategy gives the best assay savings and throughput increment by 11.1 times, and the Enumerator Doublets strategy gives the best assay savings by 8.5 times with the least amount of time to obtain results — 3.67 times less compared to the individual testing.
For a low prevalence (0.01%), the Bisector Waiting strategy again gives the best assay savings and throughput increment by 25.6 times, while the Plain strategy gives the best assay savings by 15.9 times with the least amount of time to obtain results — 5.5 times less compared to the individual testing.
See also
 Other algorithms and strategies, and the efficiency of the strategies with other conditions and prevalence, are addressed in “The detailed comparison of the operational efficiency of pooltesting strategies for COVID19 mass testing in PCR laboratories.”
 Tables of efficiency of the algorithms with different conditions and prevalence are compiled in “The handbook of operational efficiency of pooltesting strategies for COVID19 mass testing in PCR laboratories.”
References

Stefan Lohse, Thorsten Pfuhl, Barbara BerkóGöttel, Jürgen Rissland et al.
Pooling of samples for testing for SARSCoV2 in asymptomatic people
Lancet Infect Dis. April 28, 2020 https://doi.org/10.1016/S14733099(20)303625 
Idan Yelin, Noga Aharony, Einat Shaer Tamar, et al.
Evaluation of COVID19 RTqPCR test in multisample pools
Clinical Infectious Diseases 02 May 2020, ciaa531, https://doi.org/10.1093/cid/ciaa531