Lateral Flow Device sensitivity model

To compare the pooling model against lateral flow results, see the following document:

Preliminary report from the Joint PHE Porton Down & University of Oxford SARS-CoV-2 test development and validation cell

Figure 2 provides a relationship between Cq value from PCR tests of known positive samples, and LFD sensitivity. Through visual inspection, can obtain the following approximate points:

library(dplyr)

## Warning: replacing previous import 'vctrs::data_frame' by 'tibble::data_frame'
## when loading 'dplyr'

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

d <- tribble(
    ~cq, ~sensitivity,
    18.0, 1,
    19.0, 0.98,
    23.0, 0.96,
    26.5, 0.89,
    29.5, 0.65,
    32.5, 0.32,
    34.5, 0.22,
)
plot(d)

Fit a sigmoid model to this relationship, to be used to infer the probability of testing positive in LFD based on modelled Ct value

lfd_fit <- nls(sensitivity ~ SSlogis(cq, Asym, xmid, scal), data = d)
summary(lfd_fit)

## 
## Formula: sensitivity ~ SSlogis(cq, Asym, xmid, scal)
## 
## Parameters:
##      Estimate Std. Error t value Pr(>|t|)    
## Asym  0.99895    0.01683   59.35 4.83e-07 ***
## xmid 31.01328    0.19894  155.89 1.02e-08 ***
## scal -2.37948    0.18314  -12.99 0.000203 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.02523 on 4 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 5.256e-06

plot(seq(10, 50, length.out = 100), 
      predict(lfd_fit, newdata = data.frame(cq = seq(10, 50, length.out = 100))),
      xlab="RT-qPCR Cq value", ylab="LFD sensitivity", type="l")
points(d, col="red", pch=19)

This model is:

\[ sensitivity = Asym / (1 + exp((xmid - cq)/scal)) \]

Save model fit

lfd_fit <- summary(lfd_fit)
save(lfd_fit, file="../data/lfd_fit.rdata")