Implementing Stepped Pooled Testing for Rapid COVID-19 Detection
Abhishek Srivastava, Anurag Mishra, Trusha Jayant Parekh, Sampreeti Jena
IImplementing Stepped Pooled Testing for Rapid COVID-19 Detection
Abhishek Srivastava, ∗ Anurag Mishra, Trusha Jayant Parekh, and Sampreeti Jena Minneapolis, MN 55414, USA Los Angeles, CA 90048, USA Mumbai, MH 400063, India Biochemistry, Molecular Biology, and Biophysics, University of Minnesota Twin Cities, Minneapolis, MN 55414, USA (Dated: 2020-07-21)Coronavirus Disease 2019 (COVID-19), a viral respiratory pandemic, has rapidly spread throughout the globe.Large scale and rapid testing of the population is required to contain the disease, but such testing is prohibitive interms of resources, cost and time. Recently RT-PCR based pooled testing has emerged as a promising way toboost testing efficiency. We introduce a stepped pooled testing strategy, a probability driven approach whichsignificantly reduces the number of tests required to identify infected individuals in a large population. Ourcomprehensive methodology incorporates the effect of false negative and positive rates to accurately determinenot only the efficiency of pooling but also it’s accuracy. Under various plausible scenarios, we show that thisapproach significantly reduces the cost of testing and also reduces the effective false positive rate of tests whencompared to a strategy of testing every individual of a population. We also outline an optimization strategy toobtain the pool size that maximizes the efficiency of pooling given the diagnostic protocol parameters and localinfection conditions.
I. INTRODUCTION
Coronavirus Disease 2019 (COVID-19), a viral infectious respiratory illness, has recently emerged as a major threat to public healthand economic stability in countries around the world. It has spread globally at an alarming pace and World Health Organization(WHO) has declared it a pandemic. In absence of a cure or a vaccine, large scale testing and quarantine is recognized as one of themost effective strategies for containing its spread. While there are various known diagnostic methods for COVID-19 includingnucleic acid testing, protein testing and computed tomography [1], they can be extremely prohibitive in terms of cost and time.Pooled testing is a promising strategy to boost testing efficiency. In pooled testing, several samples from each patient are dividedand grouped into various pools and the pool is then tested for the disease. If the pool tests negative, each sample of the pool mustbe negative too. This basic idea reduces the overall cost and time of testing large populations.Pooled testing was first proposed during World War II [2] and has been a part of diagnostic methodology ever since [3]. It hassince been employed several times to test for infections ranging including Malaria [4], Flu [5] and HIV [6, 7]. One of the firstimplementations of laboratory pooled testing for COVID-19 was demonstrated by Yelin et al [8] for pools as large as 32 or 64samples. Today, Physicians and Public Health Officials from India [9–15] and many other countries around the globe [16–20] areusing pooled testing for determine the spread of this pandemic in a rapid and cost efficient manner.A variety of different strategies have been proposed over the past several years to implement pooled testing [21, 22]. Theycan be broadly classified into two types: adaptive and non-adaptive. Adaptive methods [22–30] employ a sequential testingapproach, thus requiring fewer number of total tests but more time as each step of testing informs the next. On the other hand,non-adaptive pooled testing methods [29, 31, 32] usually involve a matrix type pooling that allows for simultaneous testing ofseveral pools whose results are then collated to pinpoint to infected samples. These methods are faster but can require a greaternumber of tests in total. While many of these methods might be mathematically efficient, their practical implementation is usuallychallenging [8, 29] and limits the complexity that can be incorporated, no matter the benefits. Hence, it is imperative to modifyand verify any proposed method according to clinical constraints.Here, we present a probability driven pooled testing approach that can significantly reduce the number of tests required to identifyinfected patients in large populations. The method divides and tests pools of samples in a hierarchical (stepped) manner. Thisapproach is general enough to not be limited to COVID-19 alone and can be applied to other infectious scenarios with minormodifications. The mathematical model used for implementing and optimizing this strategy is presented along with representativeresults for various probable real-life scenarios. Under various plausible scenarios, this strategy reduces the cost of testing between30% to 90% compared to a strategy of individually testing everyone in a population and cuts the false positive rate up to one-thirdof an individual test. a r X i v : . [ q - b i o . P E ] J u l Parameter Name Description M Pool (group) size β Fraction of the Population infected
Inputs f + False positives rate for a test f − False negatives rate for a test N max Maximum number of tests possible per patient
Outputs K Efficiency Amplification Factor compared to testing individually. (This is also the average effectivenumber of people that can be tested per test) F pool − False Negative rate for the complete stepped pooled testing (Different from f − )TABLE I. List of parameters used for Analysis and Results It can be used to rapidly determine the efficiency boost that can be obtained by pooling a desired number of samples together if weknow the accuracy of testing method and the rate of infection in the population being tested. It can also suggest optimal pool sizethat should be used to minimize the number of tests needed per 1000 people.
II. METHODOLOGY
The stepped pooled testing strategy is applicable to any testing method that involves sample collection such as the ReverseTranscription – Polymerase Chain Reaction (RT-PCR) [1] test which is being widely used for testing COVID-19. We begin byassuming that the sample(s) collected from the patients are enough for N max tests only (for instance if we are able to collect 3swabs per patient then N max = M samples.2. If the outcome is negative (not infected) we can surmise that all the M samples in the pool are infection free.3. If the pooled sample is tested positive (infected), we split the samples from these M patients into two sub pools of size M / N max − M is an integer multiple of M / N max − . The initial sizeof the pool M can be optimized to maximize the effective number of people tested per test or equivalently, minimize the number oftests needed per 1000 people. A flowchart for this strategy is shown in Fig. 1.Probabilistic calculations along this tree enable us to estimate the expected number of tests to be done for a pool of given size aswell as the overall chances of false negatives. The probability of a pool of M samples being infected (i.e. at least 1 out of M positive) is p ( M ) = − ( − β ) M . (1)The probability of the pooled testing positive is [33] G + ( M ) = p ( M )( − f − ) + ( − p ( M )) f + = p ( M )( − f − − f + ) + f + . (2)Note that we have assumed that the false negative and positive rate for pool of samples is the same as that for a single sample. Thiscan be justified based on the limits of detection for the commonly used RT-PCR protocols. Please refer to Appendix A for details.Following the flowchart in Fig. 1, we can deduce that T ( M ) , the expected number of tests for a pool of size M , is given by arecursive function ˆ Z ( M , s ) that terminates when we get to N max steps: T ( M ) = ˆ Z ( M , N max ) (3)ˆ Z ( m , s ) = (cid:40) + G + ˆ Z ( m / , s − ) for s > m for s = M M M M M M M M N max − M N max − M N max − Testindividually Testindividually TestindividuallyStep 1Step 2Step 3Step N max − + + + − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) ++ + FIG. 1. Flowchart (Decision Tree) representing the testing method for a pool of M samples. If a pool sample is tested negative (−) , the procedureis stopped for that pool sample. Here m denotes the subpool size and s denotes the step number.It follows that the number of persons per test, which we call the test efficiency amplification K , is given by K = MT . (5)Correspondingly, the number of tests needed per 1000 people is T = K = TM . (6)The total probability of showing a false positive at the end of all steps can also be calculated using a recursive formula. To betterunderstand the calculation for this step, it helps to write the probabilities at each step as shown in Fig. 5. The recursive formula forthe pooled test false negative F pool − can then be written as F pool − = ˆ U ( M , N max ) (7)ˆ U ( m , s ) = (cid:40) p ( m ) f − + p ( m )( − f − ) ˆ U ( m / , s − ) for s > f − / s = . (8)Here m denotes the subpool size and s denotes the step number. M T β = β =
5% 2 4 6 8 10 12 14 1651015 M F p oo l − ( % ) β = β = FIG. 2.
Effect of population infection rate β . Tests required per 1000 people (left) and pool test false negative percentage (right) as a functionof pool size. We assume number of steps N max =
2, a false positive rate of f + = .
12% and a false negative rate of f − = III. RESULTS
Indian Council of Medical Research (ICMR) recently published guidelines [34] for pool testing and suggested limiting the poolsize M to 5 to avoid dilution. ICMR also suggested a staggered approach to use of pooled testing: (a) for areas with infection ratein the population less than 2% pooled testing should be used, (b) For infection rate between 2 − M =
64, have beenreported in other studies [8]. These are also in agreement with our calculations regarding limits of detection (See Appendix A).In Figs. 2 to 4, we show the results for a representative set of parameters. We find that the number of tests per 1000 peopledecreases and the false negatives increases as we make the pool size larger. However, there is an optimum pool size that achievesmaximum efficiency (i.e. minimum T ).Figure 2 reveals that for the same pool size, a higher infection rate population requires more tests and will have an overall loweraccuracy (higher false negative rate). This is consistent with what we would expect clinically. In Fig. 3, we obtain the effect of falsenegative rate on stepped pooled testing. Interestingly, a diagnostic test with higher false negative would go through more samplesin a fewer number of tests but at the cost of overall higher pool test false negative making this trade-off possibly undesirable.In Fig. 4, we see the effect of the number of steps, N max (also the number of samples per patient) on the pooling strategy. Similarto the previous two parameter sweeps, we notice that the test required per 1000 people shows a non-monotonic behavior andhas an optimal pool size for which the pooling is most efficient (Note that for N max = T minimizes at M =
48 which isbeyond the visible horizontal axis). On the other hand, the false negative rate steadily increases but still remains below the falsenegative rate of a single test. It is obvious that using multiple samples significantly reduces the number of tests needed withoutcompromising the overall false negative of the pooling strategy.Table II summarizes the results for a broad set of plausible scenarios to demonstrate the efficiency of this strategy. In addition topredicting the efficiency and accuracy of different pooling strategies, we can also this method to calculate the optimal pool sizethat leads to the least number of tests (i.e. minimizes T ). Figures 2 to 4 clearly demonstrate the existence of such an optimum.In Table III, we show various possible testing scenarios and the corresponding optimal pool size. The results in this section showthat stepped pooled testing can reduce the overall pool false negative rate below the false negative rate of an individual test. IV. CONCLUSION
We propose a new stepped pooled testing strategy that can significantly reduce the cost of testing a large population. The strategyalso reduces the chances of false negative in almost all scenarios because an infected patient’s sample is likely to be tested multipletimes. Even in the simplest case with two samples per individual (i.e. two steps, also called
Dorfman
Pooling [2]) and an initialpool size of 2, we can significantly reduce the number of tests required per 1000 individuals, by up to 33 .
7% for populations with M T f − = f − =
25% 2 4 6 8 10 12 14 1624681012 M F p oo l − ( % ) f − = f − = FIG. 3.
Effect of false negative rate f − . Tests required per 1000 people (left) and pool test false negative percentage (right) as a function ofpool size. We assume number of steps N max =
2, an infection rate β =
2% and a false negative rate f − = M T N max = N max = M F p oo l − ( % ) N max = N max = FIG. 4.
Effect of number of steps N max . Tests required per 1000 people (left) and pool test false negative percentage (right) as a functionof pool size for N max = N max =
4. We assume an infection rate of β = f − =
15% and a false positive rate f + = . a high infection rate and up to 46 .
5% for populations with a low infection rate. As the number of steps and initial pool size isincreased, the testing efficiency progressively improves, albeit at the cost of slightly higher false negative rate. Never the less,barring the cases with very high infection rate, the pooled false negative rate is still below that of an individual test.Based on our results, we make several suggestions about the effective pool size and the number of samples that should be collectedfrom an individual. This methodology should be customized dynamically and regularly based on evolving local levels of infection.Most significant benefits of this strategy can be realized by collecting 2 or 3 samples from each individual and pooling them intogroups of 4 to 6. Increasing the number of steps N max means collecting more samples from each patient being tested. Hence, thevalue of N max should be chosen pragmatically based on consultation with the physician or health professional. Finally, we notethat machine learning methods may be implemented to utilize data collected on disease spread and dynamically adapt this strategyfor maximum efficiency. We leave this as a topic for future research. Infection Rate ( β ) Number of Steps N max Pool Size ( M ) Test Needed Per1000 People T Pool Test FalseNegative F pool − % Testing CostReduction (%) Testing cost reduction from stepped pool strategy.
We show the overall testing cost reduction for various plausible scenariosoutlined by ICMR. We assume a false negative rate f − =
15% and a false positive rate f + = . Optimal PoolSize ( M ) Test Needed Per1000 People T Pool Test FalseNegative F pool − % Testing CostReduction (%) β ) 5 % 6 393 7.35 60.710 % 4 544 9.54 45.65 % 8 268 1.46 73.2False negative 15 % 8 253 4.14 74.7rate ( f − ) 30 % 9 229 8.48 77.140 % 10 211 11.71 78.92 8 253 4.14 74.7Number of steps ( N max ) 3 18 122 6.96 87.84 32 81 12.07 91.9TABLE III. Optimal pool size M under various scenarios. Unless specified in the first column, we use number of steps N max =
2, an infectionrate of β = f − =
15% and a false positive rate f + = . ACKNOWLEDGMENTS
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Bustin and Tania Nolan, “Pitfalls of quantitative real-time reverse-transcription polymerase chain reaction,” Journalof Biomolecular Techniques , 155–166 (2004).[36] Yi Wei Tang, Jonathan E. Schmitz, David H. Persing, andCharles W. Stratton, “The Laboratory Diagnosis of COVID-19Infection: Current Issues and Challenges,” Journal of clinicalmicrobiology (2020), 10.1128/JCM.00512-20.[37] Jasper Fuk Woo Chan, Cyril Chik Yan Yip, Kelvin Kai Wang To,Tommy Hing Cheung Tang, Sally Cheuk Ying Wong, Kit HangLeung, Agnes Yim Fong Fung, Anthony Chin Ki Ng, Zijiao Zou,Hoi Wah Tsoi, Garnet Kwan Yue Choi, Anthony Raymond Tam,Vincent Chi Chung Cheng, Kwok Hung Chan, Owen Tak YinTsang, and Kwok Yung Yuen, “Improved molecular diagnosisof COVID-19 by the novel, highly sensitive and specific COVID-19-RdRp/Hel real-time reverse transcription-polymerase chainreaction assay validated in vitro and with clinical specimens,”Journal of clinical microbiology (2020), 10.1128/JCM.00310-20.[38] Linda J. Carter, Linda V. Garner, Jeffrey W. Smoot, Yingzhu Li,Qiongqiong Zhou, Catherine J. Saveson, Janet M. Sasso, Anne C.Gregg, Divya J. Soares, Tiffany R. Beskid, Susan R. Jervey,and Cynthia Liu, “Assay Techniques and Test Development forCOVID-19 Diagnosis,” ACS Central Science , 591–605 (2020). Specimen Type Mean (range) viral load (RNA copies/mL) in RdRp-P2-negativebut COVID-19-RdRp/Hel-positive specimens
Respiratory tract 4 . × ( . × − . × ) NPA/NPS/TS 1 . × ( . × − . × ) Saliva 5 . × ( . × − . × ) Sputum N/ANon-respiratory tract 7 . × ( . × − . × ) Plasma 7 . × ( . × − . × ) Urine N/AFeces/rectal swabs 4 . × ( . × − . × ) Total 3 . × ( . × − . × ) TABLE IV. Viral Load in respiratory and non-respiratory specimens. Reproduced from Chan et al. [37]
Appendix A: Diagnostic limit considerations for pooled testing with RT-PCR
One of the key advantages of real-time PCR assays utilizing target sequence specific primers (as is the case with all COVID-19 testkits) is their wide dynamic range. This enables the analysis of samples with widely varying levels of target RNA. The resolvingpower of RT-PCR is mostly limited by the efficiency of RNA-to-cDNA conversion, a real concern when the target RNA is scarce.Thus, determination of the Limit of Detection (LOD)– by performing serial dilutions of the positive control sample and obtainingstandard curves– is a critical step in the validation of any testing kit/protocol. The highest dilution of the standard curve, providedin the assay performance evaluation report of any RT-PCR assay kit, delineates the lowest concentration that can be quantifiedwith confidence. Thus, pooling patient samples as proposed by the current model is unlikely to influence the probability of a falsenegative prediction by the assay if the effective target concentration is maintained above the LOD. However, if the intensity valuesrecorded are comparable to that of the LOD, they should be recorded only as a qualitative (yes/no) prediction [35].Target RNA selection plays a big role in the assay sensitivity. These include RNA-dependent RNA polymerase (RdRp),hemagglutinin-esterase (HE), and open reading frames ORF1a and ORF1b. World Health Organization (WHO) recommends afirst line screening with the E gene assay followed by a confirmatory assay using the RdR p gene. Tang et al. [36] developed andcompared the performance of three novel real-time RT-PCR assays targeting the RdRp/Hel, S, and N genes of SARS-CoV-2.Among them, the COVID-19-RdRp/Hel assay had the lowest limit of detection in vitro and higher sensitivity and specificity.In this section, we will calculate the maximum possible pool size ( M ∗ ) that is consistent with the LOD of current COVID-19 tests. Calculation .– The LOD of the COVID-19-RdRp/Hel assay is 11 . µ L, this is equivalent 448 RNA copies in one mL sample. From Table IV, we find that the mean viral load fornasopharyngeal/nasal swabs is 1 . × RNA copies/mL. Assuming a pool size of M samples with only one infected sample, andthat samples are pooled first followed by RNA extraction, the net effective viral load in the pooled sample will be ( . / M ) × copies/mL. In the standardized protocol for RNA extraction and RT-PCR procedure, 200 µ L of pooled sample is diluted with 250 µ L of solvent and loaded for RNA extraction. Purified RNA is diluted into 50 µ L of solvent. 10 µ L of diluted solution is used perwell of PCR assay with a total reaction volume of 25 µ L [38].Thus, the net effective viral load per PCR well (in units of RNA copies/mL of solvent) is1 . × M × × × ≈ . × M . (A1)This quantity should be greater than the minimum LOD of the test, which is 448 RNA copies per one mL of solvent. Thus,2 . × M ≥
448 (A2)or, M ≤ . M ∗ =
62. This value is consistent with earlierliterature [8, 28].
Appendix B: Pool Test False Negative M M M M M M M M N max − M N max − M N max − Testindividually Testindividually TestindividuallyStep 1Step 2Step 3Step N max − + p ( M N max − )( − f − ) + ++ p ( M )( − f − ) + + + − p ( M ) f − (cid:131)(cid:216) − p ( M ) f − (cid:131)(cid:216) − (cid:131)(cid:216) − p ( M ) f − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) − (cid:131)(cid:216) + p ( M )( − f − ) + p ( M )( − f − ) ++