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Toy model for a shopping spree analysis with PSA & VOI
Application: schedule surgeries - cost: operation time - effect: the prevented DALY
To do
Make a better shopping spree function
Include skipping one -> select nest
Include number of patients we have on the waiting list
Include estimating the residual time
Add an overview of the parameters used in the model + other values like utility and survival
= needed for the VOI analysis to quantify those most needed
mybe we have to make a meta-model for the surgergies if we can not run the VOI/PSA fast enough
Idea for now
Focus on the application, make a toy code + link with the results from the model We make choices about planning. Planning depends on # patients vs # minutes? In the PSA we evaluate the effects of parameter uncertainty from the model on the selected strategies. In one and two-ways sensitivity analysis with the uncertainty with min and max # patients & n_minutes
Would it make sense to compare if you lose efficiency if you change time. * what is the gain when using the total budget vs applying a “normal feasible time schedule” * Add an extra function that checks the surgeries we to based on the planning, not only budget, but really plannen. ** And what is left if we can not perform all the surgeries based on this planning -> and do we maybe need to consider an alternative to fill the planning if we have to much non-used budget
** Plot idea Benjamin Tornadao plot -> patienten & capaciteit? _> lengthe wachtlijst en capaciteit -> hoeveel invloed heeft dat op de NMB die we kunnen halen NMB -> WTP DALY prevented? 50000 / DALY prevented -> 0.10 DALY -> WTP in minutes/DALY
Questions
Aim: 1) to schedule surgeries 2) to quantify uncertainty and parameters that influence the decision most ## - How can this be useful for? - The day of tomorrow? - The overall strategy of the hospital in planning surgeries - The link with the supply of patients.
Toy model
For the demonstration of the model we make use of 5 hypothetical surgeries, Surgery A -E. After running the model, we have estimated outcomes for each surgery. These outcomes are the time a surgery requires to perform, minutes of operation room, which we define as costs. And the effects, are the prevented DALY’s when performing the surgery. This are the effects.
We also have an estimation of the number of patients that are eligible for this surgery that week.
Make toy data set
kable(df_input_order)| strategy | costs | effect | ratio | n_patients | tc | |
|---|---|---|---|---|---|---|
| 4 | surgery D | 200 | 0.10 | 2000.000 | 9 | 1800 |
| 2 | surgery B | 70 | 0.03 | 2333.333 | 3 | 210 |
| 3 | surgery C | 100 | 0.04 | 2500.000 | 6 | 600 |
| 1 | surgery A | 50 | 0.01 | 5000.000 | 5 | 250 |
| 5 | surgery E | 280 | 0.05 | 5600.000 | 4 | 1120 |
Hospital setting
n_days <- 5 # number of days the OR complex is used
n_hours_day <- 8 # number of hours per day the OR complex is used
n_time_or <- n_days * n_hours_day * 60 # the OR time per week in minutes
n_budget <- n_time_or # this is our budgetnumber of patients per disease -> hoeveel mensen hebben elke operatie nodig? -> maak je minuten op -> hoeveel kan je er dan daar van behandelen? daarna is het “op” -> stel je kan het niet verdelen om je laatste minuten gebruiken -> sla C over en ga naar A -> dus dan kan je nog 1 patient met A opleveren. “ineffienlcy minutes” “Is iets verdeelbaar? -> Je kan waarschijnlijk niet de operatie”half" doen? -> gaat niet op hier.
Question: What to do when the budget is smaller than the one with the highest CE ratio? Do we search for one we can effort? Although not the best value for money? How to include the # patients? What do we do when we have budget left?
What is the value of more patients that benefit over the overall health benefit? Make shopping spree -> optimal for effect make shopping spree -> optimize for # of patients treated -> see how much they differ 210 budget -> surgery D = 1 patient with 10, vs 3 patients with surgery B
Make shopping spree function
# Arguments
# df_input: data with PSA results input about the costs & effects of each surgery
# n_budget: the available OR time in minutes
# Optional planning constrains:
## n_days: optional - days of surgery per week
## n_hours_day: optional - max. number of hours per day
## NOTE: @Benjamin - if we have n_days & hours, shall we overwrite n_budget?
## where/what to spit out when we have contrains?
make_shopping_spree <- function(df_input, n_budget, n_days = NULL, n_hours_day = NULL, planning_pd=TRUE){
# IF a number of days and hours per day is specified, overwrite the budget (because that ís the budget)
if(!is.null(n_days) & !is.null(n_hours_day)){
n_budget_f <- n_days*n_hours_day*60
}else{
n_budget_f <- n_budget
}
df_input_f <- df_input[order(df_input$ratio), ] # order
df_input_f$cumsum <- cumsum(df_input_f$tc) # add column cumulative sum
v_names <- df_input_f$strategy
# make a matrix of the dataframe, with strategy being the rownames
m_input <- as.matrix(subset(df_input_f ,select = -(strategy)))
rownames(m_input) <- v_names
m_strategy_spree <- matrix(NA, nrow = 1, ncol = length(v_names))
colnames(m_strategy_spree) <- v_names
for (n in v_names){ # For every surgery (from urgent to least urgent)
#Check how many patients can be treated
n_max_patients <- floor(n_budget_f/m_input[n, "costs"])
if((n_max_patients - m_input[n, "n_patients"]) > 0){
# if the max patients is more than the eligible patients, we could the costs based on the eligible patients
# since we can only operate those we have
n_budget_left <- n_budget_f - m_input[n, "costs"] * m_input[n, "n_patients"]
m_strategy_spree[, n] <- m_input[n, "n_patients"]
} else {
# if we have more patients than we can we can max perform surgery on, we can only use the n_max patients
n_budget_left <- n_budget_f - m_input[n, "costs"] * n_max_patients
m_strategy_spree[, n] <- n_max_patients
}
n_budget_f <- n_budget_left # store the budget that is left to use for the next calculation
}
# Idea only report the selected strategies & the # of patients per strategy if the total number of strategies becomes to large
if(sum(m_strategy_spree > 0)){
# If we treat at least one patient (>0), select the strategies & numbers
v_strategy_selected <- v_names[which(m_strategy_spree > 0 )]
n_patients_selected_strategy <- m_strategy_spree[, which(m_strategy_spree > 0) ]
}
# Make a list for the results
l_out <- list(strategies = v_strategy_selected,
n_per_strategy = n_patients_selected_strategy,
budget = n_budget, # report total budget
budget_left = n_budget_left # report budget left
)
################ Planning-based selection of surgeries #############
if ((!is.null(n_days) & !is.null(n_hours_day))& # if it is possible
planning_pd # and if it is specified
){
v_strategy_spree <- rep(colnames(m_strategy_spree), m_strategy_spree)
v_match <- match(v_strategy_spree, rownames(m_input))
v_costs_selection <- m_input[v_match, "costs"]
n_budget_day <- n_hours_day * 60
bp <- binPack(v_costs_selection, capacity = n_budget_day)
l_xs <- split(v_costs_selection, bp)
#print(l_xs)
#print(sapply(l_xs, sum))
v_minutes_day <- sapply(l_xs, sum)
# If we have more combinations than number of days, we select those with the highest # minutes @benjamin this is not per se the best (we like I think highest CE ratio)
#@Eline, I think it would be easier to select the "most valuable" days, not per se best ce. I'll write some code below:
if(length(l_xs) > n_days){
effects_pd <- sapply(l_xs, FUN = function(x){sum(m_input[names(x),"effect"])})
effects_pd <- effects_pd[order(effects_pd, decreasing = TRUE)]
selected_days <- names(effects_pd)[1:n_days] # Best days possible
l_xs <- l_xs[selected_days]
v_minutes_selected_days <- sapply(l_xs, sum)
}
#If there is budget left per day, fill it in again with the most cost effective surgery
# budget_left_planning_pd <- (n_hours_day*60)-v_minutes_selected_days
#
# for(d in 1:n_days){
# #Compare budget of that day, to all sugeries (do we have enough time this day?)
# test_budgetleft_pd <- budget_left_planning_pd[d]>m_input[,"costs"]
# if(sum(test_budgetleft_pd)>0){ #If there is time left for a surgery, fill in the planning with this surgery
# surg_extra <- v_names[first(which(test_budgetleft_pd))]
# l_xs[[d]] <- c(l_xs[[d]],surg_extra=m_input[surg_extra,"costs"])
# l_xs[[d]] <- l_xs
# #Test whether there is still time left, and if so: fill it again
# }
# }
# total budget
n_time_scheduled <- sum(v_minutes_selected_days)
budget_left_planning_pd <- (n_hours_day*60)-v_minutes_selected_days
v_selected_strategies <- substr(x = names(unlist(l_xs)), start=3,stop=1000)
selected_strategies_planning <- unique(sapply(l_xs, FUN= function(x){unique(names(x))}))
l_out <- list(planning = l_xs,
selected_strategies = unique(v_selected_strategies),
n_per_strategy = table(v_selected_strategies),
budget = n_budget, # report total budget
budget_left_pd = budget_left_planning_pd # report budget left per day
)
}
return(l_out)
}Test the function
# without scheduling constrains
make_shopping_spree(df_input, n_budget = 100) #check the logical statement when the budget is to limited
make_shopping_spree(df_input, n_budget = 1000) #check with higher budget
make_shopping_spree(df_input, n_budget = 1e5) # check with unlimited budget
# with scheduling constrains
make_shopping_spree(df_input, n_budget = 100, n_days = 5, n_hours_day = 8) #check the logical statement when the budget is to limited
make_shopping_spree(df_input, n_budget = 1000, n_days = 5, n_hours_day = 8) #check with high budget
make_shopping_spree(df_input, n_budget = 1e5, n_days = 5, n_hours_day = 8) # check with unlimited budgetMake grid for all possible options
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## [8400] 0Sensitivity analysis
Probabilistic sensitivity analysis
Data PSA
## Warning: strategy name 'surgery A' was converted to 'surgery.A' for
## compatibility. See ?make.names## Warning: strategy name 'surgery B' was converted to 'surgery.B' for
## compatibility. See ?make.names## Warning: strategy name 'surgery C' was converted to 'surgery.C' for
## compatibility. See ?make.names## Warning: strategy name 'surgery D' was converted to 'surgery.D' for
## compatibility. See ?make.names## Warning: strategy name 'surgery E' was converted to 'surgery.E' for
## compatibility. See ?make.namesShow and plot the psa restuls
These PSA results from the model inform the shopping spree function
summary(psa_obj) # Show a summary of the results## Strategy meanCost meanEffect
## 1 surgery.A 245.5570 0.009557777
## 2 surgery.B 206.2701 0.030051456
## 3 surgery.C 593.1204 0.039032800
## 4 surgery.D 1759.6914 0.092934446
## 5 surgery.E 1080.2456 0.048119737# plot the PSA results
plot(psa_obj, ellipse = TRUE, alpha = 1) +
ylab("Costs (duration surgery)") +
xlab("Effect (DALY/month delay)") +
ggtitle("The CE-plane of the PSA results",
subtitle = "with mean and confidence ellipse ") +
geom_vline(xintercept = 0, colour = "gray") +
geom_hline(yintercept = 0, colour = "gray") 
Run PSA for shopping spree
Summarise the PSA results
# check if all values are stored somewhere?
sum(dt_combinations$count) == n_iter## [1] TRUE# select the combinations at are at least once the preferred strategy
df_selected_strategies_psa <- dt_combinations[dt_combinations$count > 0, ]
# How do we get from here to a CE-acceptability curve? / VOI analysis
df_selected_strategies_psa$probCE <- round(df_selected_strategies_psa$count / sum(df_selected_strategies_psa$count), 3)
# Make a bar-plot of the strategies
df_selected_strategies_psa <- df_selected_strategies_psa[order(df_selected_strategies_psa$probCE),]
df_selected_strategies_psa$strategy <- nrow(df_selected_strategies_psa):1
#heatmap (left)
m_hm <- matrix(unlist(df_selected_strategies_psa[,1:5]), ncol = 5, byrow = FALSE,
dimnames = list(df_selected_strategies_psa$strategy,colnames(df_selected_strategies_psa)[1:5]))
p1 <- as.grob(~heatmap(m_hm, Colv=NA, Rowv=NA,scale = "row",reorderfun = NA,
xlab = "Surgery", ylab="Strategy",
main="Number of patients treated",cexCol = 0.8))
#barchart (right)
p2 <- as.grob(~barplot(df_selected_strategies_psa$probCE,
main = "Probability of being best strategy",
xlab = "Probability",
names.arg = nrow(df_selected_strategies_psa):1,
ylab = "Strategy",
col = "orange",
horiz = TRUE))
plot_grid(p1, p2, ncol=2, rel_heights = c(0.2, 1.5))
# Make a legend with what these strategies are?
kable(df_selected_strategies_psa)| SurgeryA | SurgeryB | SurgeryC | SurgeryD | SurgeryE | count | probCE | strategy | |
|---|---|---|---|---|---|---|---|---|
| 68 | 1 | 3 | 2 | 0 | 0 | 1 | 0.01 | 17 |
| 115 | 0 | 3 | 4 | 0 | 0 | 1 | 0.01 | 16 |
| 190 | 3 | 3 | 0 | 1 | 0 | 1 | 0.01 | 15 |
| 92 | 1 | 3 | 3 | 0 | 0 | 2 | 0.02 | 14 |
| 355 | 0 | 3 | 0 | 2 | 0 | 2 | 0.02 | 13 |
| 69 | 2 | 3 | 2 | 0 | 0 | 3 | 0.03 | 12 |
| 349 | 0 | 2 | 0 | 2 | 0 | 3 | 0.03 | 11 |
| 91 | 0 | 3 | 3 | 0 | 0 | 4 | 0.04 | 10 |
| 337 | 0 | 0 | 0 | 2 | 0 | 4 | 0.04 | 9 |
| 189 | 2 | 3 | 0 | 1 | 0 | 5 | 0.05 | 8 |
| 211 | 0 | 3 | 1 | 1 | 0 | 5 | 0.05 | 7 |
| 343 | 0 | 1 | 0 | 2 | 0 | 6 | 0.06 | 6 |
| 169 | 0 | 0 | 0 | 1 | 0 | 7 | 0.07 | 5 |
| 175 | 0 | 1 | 0 | 1 | 0 | 10 | 0.10 | 4 |
| 188 | 1 | 3 | 0 | 1 | 0 | 11 | 0.11 | 3 |
| 181 | 0 | 2 | 0 | 1 | 0 | 17 | 0.17 | 2 |
| 187 | 0 | 3 | 0 | 1 | 0 | 18 | 0.18 | 1 |
Final
For questions contact: e.krijkamp@erasmusmc.nl This methodological code is written for the application of surgical priority described in: Gravesteijn B, Krijkamp E, Busschbach J, Geleijnse G, Retel Helmrich I, Bruinsma S, van Lint C, van Veen E, Steyerberg E, Verhoef K, van Saase J, Lingsma H, Baatenburg de Jong R. Minimizing Population Health Loss in Times of Scare Surgical Capacity During the Coronavirus Disease 2019 Crisis and Beyond: A Modeling Study. Value in Health, accepted for publication in May 2021 edition.
Erasmus MC, e.krijkamp@erasmusmc.nl↩︎
Erasmus MC, b.gravesteijn@erasmusmc.nl↩︎