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Summary

Final cortisol value calculations were conducted using three methods:

  • Standard Method (Method A): Calculates cortisol concentration without correction for spiked samples.

  • Spike-Corrected Method (Method B): Adjusts for spiked samples to account for addition of a known amount of cortisol, following Nist et al. 2020.

  • Sam’s Method (Method C): Adjusts for spiked samples using a different equation

Results: As we see below, the formula used by Nist et al. results in negative values, which would mean that there is no cortisol in original samples. This could be an artifact of an extremely high absorbance level caused by an excessive amount of spike. Non-spiked samples, on the other hand, result in values that are within the range found in similar studies of cortisol in human hair.

Summary Nist et al. (A) Standard (B) Spike-Corrected (C) Sam’s
Mean cort conc (pg/mg) — 16.376 -0.182 9.329
Median cort conc (pg/mg) — 10.531 4.328 9.806
Range cort conc (pg/mg) — 2.7 to 58.9 -29.933 to 11.763 2.716 to 22.002
Weight (mg) of my samples
Range 11 to 37.1
Mean 23.54
Median 22.4

Conclusions: After accounting for differences in dilution and weight, our results suggest future Assays should use the optimal parameters listed below:

  • Dilution of 250uL is preferable over 60uL
  • Non-spiked samples seem to generate expected results

Concerns

  • Spike results in unrealistic values
  • Could be explained by the higher weight of our samples
  • Dilution of 250uL results in values that are twice as big as with 60uL, but they should be very similar or at least overlap

Explanation of each variable used in calculations

  • Ave_Conc_pg/ml: average ELISA reading per sample in pg/mL

  • Weight_mg: hair weight in mg

  • Buffer_nl: assay buffer volume in nL → we convert to mL

  • Spike: binary indicator (1 = spiked sample)

  • SpikeVol_uL: volume of spike added in µL

  • Dilution: dilution factor (already present)

  • Vol_in_well.tube_uL: total volume in well/tube in µL (for spike correction)

  • std: standard reading value

  • extraction: methanol volume ratio = vol added / vol recovered (e.g. 1/0.75 ml)

Cortisol concentration calculations

Input is data with low quality samples flagged, but they get removed before continuing with calculations.

Parameters and unit transformations:

# Define volume of methanol used for cortisol extraction
# vol added / vol recovered (mL)
extraction <- 1.3 / 1

# Reading of spike standard and conversion to ug/dl
std <- (3133 + 3146) / 2 # test 3 backfit
std.r <- std / 10000 # std in ul/dl

# Creating variables in indicated units
# dilution (buffer)
df$Buffer_ml <- c(df$Buffer_nl/1000)

# Transform to μg/dl from assay output
df$Ave_Conc_ug.dl <- c(df$Ave_Conc_pg.ml/10000)
Wells Sample Category Binding.Perc Ave_Conc_pg.ml Ave_Conc_ug.dl Weight_mg Buffer_ml Spike SpikeVol_uL Dilution TotalVol_well_uL Failed_samples
E5 11 NoSpike 71.6 513.2 0.05132 17.5 0.25 0 0 1 50 NA
F5 12 YesSpike 30.0 2728.0 0.27280 24.1 0.25 1 25 1 50 NA
G5 13 YesSpike 32.1 2477.0 0.24770 16.8 0.25 1 25 1 50 NA

(A) Standard Calculation

Formula:

((A/B) * (C/D) * E * 10,000) = F

  • A = μg/dl from assay output;
  • B = weight (in mg) of hair subjected to extraction;
  • C = vol. (in ml) of methanol added to the powdered hair;
  • D = vol. (in ml) of methanol recovered from the extract and subsequently dried down;
  • E = vol. (in ml) of assay buffer used to reconstitute the dried extract;
  • F = final value of hair CORT Concentration in pg/mg.
##################################
##### Calculate final values #####
##################################

data$Final_conc_pg.mg <- c(
    ((data$Ave_Conc_ug.dl) / data$Weight_mg) * # A/B *
      extraction *                             # C/D  *     
      data$Buffer_ml * 10000 )                 # E * 10000
data <- data[order(data$Sample),]
write.csv(data, file.path(data_path, "Data_cort_values_methodA.csv"), row.names = F)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.716   7.820  10.531  16.376  15.336  58.971
Wells Sample Category Binding.Perc Ave_Conc_pg.ml Ave_Conc_ug.dl Weight_mg Buffer_ml Spike SpikeVol_uL Dilution TotalVol_well_uL Failed_samples Final_conc_pg.mg
27 H3 6 YesSpike 34.7 2204.0 0.22040 13.7 0.06 1 25 1 50 NA 12.54832
28 A5 7 YesSpike 30.5 2669.0 0.26690 16.4 0.06 1 25 1 50 NA 12.69402
29 B5 8 NoSpike 77.8 386.8 0.03868 15.3 0.25 0 0 1 50 NA 8.21634
30 C5 9 YesSpike 30.3 2693.0 0.26930 19.2 0.25 1 25 1 50 NA 45.58464

(B) Accounting for Spike

We followed the procedure described in Nist et al. 2020:

“Thus, after pipetting 25μL of standards and samples into the appropriate wells of the 96-well assay plate, we added 25μL of the 0.333ug/dL standard to all samples, resulting in a 1:2 dilution of samples. The remainder of the manufacturer’s protocol was unchanged. We analyzed the assay plate in a Powerwave plate reader (BioTek, Winooski, VT) at 450nm and subtracted background values from all assay wells. In the calculations, we subtracted the 0.333ug/dL standard reading from the sample readings. Samples that resulted in a negative number were considered nondetectable. We converted cortisol levels from ug/dL, as measured by the assay, to pg/mg—based on the mass of hair collected and analyzed using the following formula:

A/B * C/D * E * 10,000 * 2 = F

where - A = μg/dl from assay output; - B = weight (in mg) of collected hair; - C = vol. (in ml) of methanol added to the powdered hair; - D = vol. (in ml) of methanol recovered from the extract and subsequently dried down; - E = vol. (in ml) of assay buffer used to reconstitute the dried extract; 10,000 accounts for changes in metrics; 2 accounts for the dilution factor after addition of the spike; and - F = final value of hair cortisol concentration in pg/mg”

dSpike <- data

##################################
##### Calculate final values #####
##################################

dSpike$Final_conc_pg.mg <- 
  ifelse(
    dSpike$Spike == 1,    ## Only spiked samples
      ((dSpike$Ave_Conc_ug.dl - (std.r)) / # (A-spike) / B
        dSpike$Weight_mg) 
        * extraction *                  # C / D
        dSpike$Buffer_ml * 10000 * 2,    # E * 10000 * 2
        dSpike$Final_conc_pg.mg  
    )


write.csv(dSpike, file.path(data_path, "Data_cort_values_methodB.csv"), row.names = F)

# summary for all samples
summary(dSpike$Final_conc_pg.mg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
-29.933  -9.370   4.328  -0.182   9.944  11.763 
dSpikeSub <- data[c(data$Spike == 0), ]
summary(dSpikeSub$Final_conc_pg.mg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.716   5.087   8.874   7.908  10.486  11.763 
kable(tail(dSpike, 10))
Wells Sample Category Binding.Perc Ave_Conc_pg.ml Ave_Conc_ug.dl Weight_mg Buffer_ml Spike SpikeVol_uL Dilution TotalVol_well_uL Failed_samples Final_conc_pg.mg
21 B11 33 NoSpike 52.3 1100.0 0.11000 33.8 0.25 0 0 1 50 NA 10.576923
22 C11 34 NoSpike 51.7 1124.0 0.11240 35.5 0.25 0 0 1 50 NA 10.290141
23 E11 36 NoSpike 53.2 1062.0 0.10620 31.2 0.25 0 0 1 50 NA 11.062500
24 F11 37 NoSpike 36.1 2076.0 0.20760 30.8 0.06 0 0 1 50 NA 5.257403
25 G11 38 NoSpike 32.5 2444.0 0.24440 34.7 0.06 0 0 1 50 NA 5.493718
26 G3 5 NoSpike 72.1 501.4 0.05014 14.4 0.06 0 0 1 50 NA 2.715917
27 H3 6 YesSpike 34.7 2204.0 0.22040 13.7 0.06 1 25 1 50 NA -10.652409
28 A5 7 YesSpike 30.5 2669.0 0.26690 16.4 0.06 1 25 1 50 NA -4.475488
29 B5 8 NoSpike 77.8 386.8 0.03868 15.3 0.25 0 0 1 50 NA 8.216340
30 C5 9 YesSpike 30.3 2693.0 0.26930 19.2 0.25 1 25 1 50 NA -15.115885

(C) Sam’s calculation

Spike contribution (pg/mL) = (Vol. spike (mL) x Conc. spike (pg/mL) ) / Vol. reconstitution (mL) or total vol. in well (50uL) (depending on where the spike was added)

# Calculate contribution of spike according to the different volumes in which it was added
# Consider that contribution of spike in serial dilutions gets smaller 

# Vol. of spike transformed to mL
data$SpikeVol_ml <- data$SpikeVol_uL/1000
# Concentration of the spike:
std
[1] 3139.5
# Vol. reconstitution (mL) is the total volume in tube or well (sample + spike), after adding spike.
# transform to mL
data$TotalVol_well_mL <- data$TotalVol_well_uL/1000


  ##( Spike vol. x Spike Conc.)
  ## ------------------------  / dilution = Spike contribution
  ##      Total vol. 
  
# Cortisol added by spike in wells: 0.0025 mL x 3200 pg/mL = 80 pg
# Calculate cort contribution of spike to each sample
data$Spike.cont_pg.mL <- (((data$SpikeVol_ml * std ) / # Volume of spike * Spike concentration
                            data$TotalVol_well_mL) / # divided by the total volume (spike + sample)
                              data$Dilution) # does not affect values because it is 1 for all
dSpiked <- data
std
[1] 3139.5
summary(dSpiked$Spike.cont_pg.mL)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    0.0     0.0     0.0   627.9  1569.8  1569.8 
summary(dSpiked$Weight_mg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  11.00   16.98   22.40   23.54   30.50   37.10 
summary(extraction)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    1.3     1.3     1.3     1.3     1.3     1.3 
summary(dSpiked$Buffer_ml)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0600  0.0600  0.2500  0.1677  0.2500  0.2500 
##################################
##### Calculate final values #####
##################################


dSpiked$Final_conc_pg.mg <- 
      ((dSpiked$Ave_Conc_pg.ml - dSpiked$Spike.cont_pg.mL) / # (A - spike) / B
      dSpiked$Weight_mg) *
      extraction *      # C / D
      dSpiked$Buffer_ml * dSpiked$Dilution  # E * dilution (does no affect results because it is 1 for all)

write.csv(dSpiked, file.path(data_path, "Data_cort_values_methodC.csv"), row.names = F)

# summary for all samples
summary(dSpiked$Final_conc_pg.mg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  2.716   5.235   9.806   9.329  11.212  22.002 
kable(tail(dSpiked[!is.na(dSpiked$Final_conc_pg.mg) , c("Sample", "Final_conc_pg.mg", "Ave_Conc_pg.ml", "Spike.cont_pg.mL", "Binding.Perc", "Weight_mg", "Buffer_ml", "SpikeVol_uL", "Dilution", "TotalVol_well_uL")],20))
Sample Final_conc_pg.mg Ave_Conc_pg.ml Spike.cont_pg.mL Binding.Perc Weight_mg Buffer_ml SpikeVol_uL Dilution TotalVol_well_uL
11 23 10.484553 793.6 0.00 60.9 24.6 0.25 0 1 50
12 24 10.836520 680.2 0.00 64.8 20.4 0.25 0 1 50
13 26 7.688020 1991.0 0.00 37.3 20.2 0.06 0 1 50
14 27 5.030278 1393.0 0.00 46.0 21.6 0.06 0 1 50
15 28 11.763039 839.7 0.00 59.5 23.2 0.25 0 1 50
16 29 4.427836 2072.0 0.00 36.2 36.5 0.06 0 1 50
17 3 5.085954 2287.0 1569.75 33.9 11.0 0.06 25 1 50
18 30A 14.474029 2888.0 1569.75 28.8 29.6 0.25 25 1 50
19 31 4.227453 1149.0 0.00 51.2 21.2 0.06 0 1 50
20 32 10.485849 1197.0 0.00 50.0 37.1 0.25 0 1 50
21 33 10.576923 1100.0 0.00 52.3 33.8 0.25 0 1 50
22 34 10.290141 1124.0 0.00 51.7 35.5 0.25 0 1 50
23 36 11.062500 1062.0 0.00 53.2 31.2 0.25 0 1 50
24 37 5.257403 2076.0 0.00 36.1 30.8 0.06 0 1 50
25 38 5.493718 2444.0 0.00 32.5 34.7 0.06 0 1 50
26 5 2.715917 501.4 0.00 72.1 14.4 0.06 0 1 50
27 6 3.611058 2204.0 1569.75 34.7 13.7 0.06 25 1 50
28 7 5.228140 2669.0 1569.75 30.5 16.4 0.06 25 1 50
29 8 8.216340 386.8 0.00 77.8 15.3 0.25 0 1 50
30 9 19.013346 2693.0 1569.75 30.3 19.2 0.25 25 1 50

Plots

(A) Standard Calculation

Version Author Date
82ad928 Paloma 2025-04-17
ccad031 Paloma 2025-04-09

(B) Accounting for Spike

Version Author Date
82ad928 Paloma 2025-04-17
16ce91c Paloma 2025-04-10
ccad031 Paloma 2025-04-09

(C) Sam’s calculation

Version Author Date
82ad928 Paloma 2025-04-17
16ce91c Paloma 2025-04-10
ccad031 Paloma 2025-04-09 
  Wells Sample Category Binding.Perc Ave_Conc_pg.ml Ave_Conc_ug.dl Weight_mg
1    E5     11  NoSpike         71.6          513.2        0.05132      17.5
2    F5     12 YesSpike         30.0         2728.0        0.27280      24.1
3    G5     13 YesSpike         32.1         2477.0        0.24770      16.8
4    H5     14 YesSpike         31.8         2504.0        0.25040      13.8
5    A7     15  NoSpike         66.2          643.6        0.06436      12.0
6    B7     16 YesSpike         26.8         3196.0        0.31960      23.4
  Buffer_ml Spike SpikeVol_uL Dilution TotalVol_well_uL Failed_samples
1      0.25    No           0        1               50           <NA>
2      0.25   Yes          25        1               50           <NA>
3      0.25   Yes          25        1               50           <NA>
4      0.25   Yes          25        1               50           <NA>
5      0.06    No           0        1               25           <NA>
6      0.06   Yes          25        1               50           <NA>
  Final_conc_pg.mg SpikeVol_ml TotalVol_well_mL Spike.cont_pg.mL Buffer
1         9.530857       0.000            0.050             0.00 250 uL
2        15.619554       0.025            0.050          1569.75 250 uL
3        17.550967       0.025            0.050          1569.75 250 uL
4        22.002264       0.025            0.050          1569.75 250 uL
5         4.183400       0.000            0.025             0.00  60 uL
6         5.420833       0.025            0.050          1569.75  60 uL

Evaluation Non-spiked Samples Only

Since all spiked samples produce negative values, we will continue our analyses using only non-spiked samples.

# non-spiked samples only
data2 <-data[data$Spike == "No", ]

#two datasets, separated by dilution
data2.06 <- data2[data2$Buffer == "60 uL", ]
data2.25 <- data2[data2$Buffer == "250 uL", ]

#### fit models ####

# model Buffer = 0.06
model06 <- lm(Final_conc_pg.mg ~ Weight_mg, 
              data = data2.06)
r_squared06 <- summary(model06)$r.squared

# model Buffer = 0.25
model25 <- lm(Final_conc_pg.mg ~ Weight_mg, 
              data = data2.25)
r_squared25 <- summary(model25)$r.squared

# Calculate residuals
residuals06 <- residuals(model06)
residuals25 <- residuals(model25)

# Quantify residuals
# Mean Absolute Error
mae06 <- mean(abs(residuals06))          
# Standard Deviation of Residuals
std_dev06 <- sd(residuals06)   

# Mean Absolute Error
mae25 <- mean(abs(residuals25))      
# Standard Deviation of Residuals
std_dev25 <- sd(residuals25)                      



# scatterplot

ggplot(data2, aes(y = Final_conc_pg.mg, 
                  x = Weight_mg, 
                  color = Buffer, 
                  fill = Buffer)) +
  geom_point(size = 2.5) +  
  geom_text(label = c(data2$Sample), nudge_y = 0.75, nudge_x = -0.5) +
  geom_smooth(method = "lm", 
              color = "gold3", 
              se = TRUE,
              alpha = 0.1) + 
  geom_hline(yintercept = mean(data2$Final_conc_pg.mg), 
             color = "gray80",
             linetype = "dashed") +
  geom_hline(yintercept = mean(data2.06$Final_conc_pg.mg), 
             color = "lightblue3",
             linetype = "dashed") +
  geom_hline(yintercept = mean(data2.25$Final_conc_pg.mg), 
             color = "lightpink3",
             linetype = "dashed") +
  theme_minimal() +  
  xlim(5, max(data2$Weight_mg) + 4) +
  ylim(0, max(data2$Final_conc_pg.mg)+4) + 
  labs(
    title = "Final Cort Concentration and Weight
    (Non-spiked only), method A",
    y = "Final Concentration (pg/mg)",
    x = "Weight (mg)"  
  ) +
  theme(
    plot.title = element_text(hjust = 0.5, size = 16, face = "bold"),  
    axis.title = element_text(size = 14),  
    axis.text = element_text(size = 12) 
  ) +
  # Add R^2 annotation
  annotate("text", x = max(data2$Weight_mg) * 0.7, 
           y = min(data2$Final_conc_pg.mg) * 1.5,
           label = paste("R² =", round(r_squared06, 3)), 
           size = 5, color = "black") +
  annotate("text", x = max(data2$Weight_mg) * 0.7, 
           y = max(data2$Final_conc_pg.mg) * 0.84,
           label = paste("R² =", round(r_squared25, 3)), 
           size = 5, color = "black")

Version Author Date
16ce91c Paloma 2025-04-10

The previous figure shows that:

  • results are very stable across weights, particularly for the samples where a dilution of 250 uL was used
  • there is more error when using a dilution of 60 uL
  • dilution affects estimation of cortisol concentration in a significant way: even though final concentration numbers account for differences in the dilutions, the results we observe for each group do not overlap
  • the average value when using 250 uL of buffer is twice as big as when using 60 uL
# non-spiked samples only
data <- dSpiked
data2 <-data[data$Spike == "No", ]

#two datasets, separated by dilution
data2.06 <- data2[data2$Buffer == "60 uL", ]
data2.25 <- data2[data2$Buffer == "250 uL", ]

#### fit models ####

# model Buffer = 0.06
model06 <- lm(Final_conc_pg.mg ~ Weight_mg, 
              data = data2.06)
r_squared06 <- summary(model06)$r.squared

# model Buffer = 0.25
model25 <- lm(Final_conc_pg.mg ~ Weight_mg, 
              data = data2.25)
r_squared25 <- summary(model25)$r.squared

# Calculate residuals
residuals06 <- residuals(model06)
residuals25 <- residuals(model25)

# Quantify residuals
# Mean Absolute Error
mae06 <- mean(abs(residuals06))          
# Standard Deviation of Residuals
std_dev06 <- sd(residuals06)   

# Mean Absolute Error
mae25 <- mean(abs(residuals25))      
# Standard Deviation of Residuals
std_dev25 <- sd(residuals25)                      



# scatterplot

ggplot(data2, aes(y = Final_conc_pg.mg, 
                  x = Weight_mg, 
                  color = Buffer, 
                  fill = Buffer)) +
  geom_point(size = 2.5) +  
  geom_text(label = c(data2$Sample), nudge_y = 0.75, nudge_x = -0.5) +
  geom_smooth(method = "lm", 
              color = "gold3", 
              se = TRUE,
              alpha = 0.1) + 
  geom_hline(yintercept = mean(data2$Final_conc_pg.mg), 
             color = "gray80",
             linetype = "dashed") +
  geom_hline(yintercept = mean(data2.06$Final_conc_pg.mg), 
             color = "lightblue3",
             linetype = "dashed") +
  geom_hline(yintercept = mean(data2.25$Final_conc_pg.mg), 
             color = "lightpink3",
             linetype = "dashed") +
  theme_minimal() +  
  xlim(5, max(data2$Weight_mg) + 4) +
  ylim(0, max(data2$Final_conc_pg.mg)+4) + 
  labs(
    title = "Final Cort Concentration and Weight
    (Non-spiked only) method D",
    y = "Final Concentration (pg/mg)",
    x = "Weight (mg)"  
  ) +
  theme(
    plot.title = element_text(hjust = 0.5, size = 16, face = "bold"),  
    axis.title = element_text(size = 14),  
    axis.text = element_text(size = 12) 
  ) +
  # Add R^2 annotation
  annotate("text", x = max(data2$Weight_mg) * 0.7, 
           y = min(data2$Final_conc_pg.mg) * 1.5,
           label = paste("R² =", round(r_squared06, 3)), 
           size = 5, color = "black") +
  annotate("text", x = max(data2$Weight_mg) * 0.7, 
           y = max(data2$Final_conc_pg.mg) * 0.84,
           label = paste("R² =", round(r_squared25, 3)), 
           size = 5, color = "black")

Optimal dilution

Error using 0.06 mL buffer

Mean Absolute Error (MAE) 0.06 mL: 0.873 
Standard Deviation of Residuals 0.06 mL: 1.377

Error using 0.25 mL buffer

Mean Absolute Error (MAE) 0.25 mL: 0.65 
Standard Deviation of Residuals 0.25 mL: 0.86

From this we conclude that using a 250 uL dilution provides more consistent results


sessionInfo()
R version 4.5.0 (2025-04-11)
Platform: aarch64-apple-darwin20
Running under: macOS Sequoia 15.4.1

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: America/Detroit
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] dplyr_1.1.4     paletteer_1.6.0 broom_1.0.8     ggplot2_3.5.2  
[5] knitr_1.50     

loaded via a namespace (and not attached):
 [1] sass_0.4.10       generics_0.1.3    tidyr_1.3.1       prismatic_1.1.2  
 [5] lattice_0.22-6    stringi_1.8.7     digest_0.6.37     magrittr_2.0.3   
 [9] evaluate_1.0.3    grid_4.5.0        fastmap_1.2.0     Matrix_1.7-3     
[13] rprojroot_2.0.4   workflowr_1.7.1   jsonlite_2.0.0    whisker_0.4.1    
[17] backports_1.5.0   rematch2_2.1.2    promises_1.3.2    mgcv_1.9-1       
[21] purrr_1.0.4       scales_1.3.0      jquerylib_0.1.4   cli_3.6.4        
[25] rlang_1.1.6       splines_4.5.0     munsell_0.5.1     withr_3.0.2      
[29] cachem_1.1.0      yaml_2.3.10       tools_4.5.0       colorspace_2.1-1 
[33] httpuv_1.6.16     vctrs_0.6.5       R6_2.6.1          lifecycle_1.0.4  
[37] git2r_0.36.2      stringr_1.5.1     fs_1.6.6          pkgconfig_2.0.3  
[41] pillar_1.10.2     bslib_0.9.0       later_1.4.2       gtable_0.3.6     
[45] glue_1.8.0        Rcpp_1.0.14       xfun_0.52         tibble_3.2.1     
[49] tidyselect_1.2.1  rstudioapi_0.17.1 farver_2.1.2      nlme_3.1-168     
[53] htmltools_0.5.8.1 rmarkdown_2.29    labeling_0.4.3    compiler_4.5.0