^{1}

^{2}

^{3}

^{4}

^{5}

The authors have declared that no competing interests exist.

Recent technical and strategical developments have increased the survival chances for avalanche victims. Still hundreds of people, primarily recreationists, get caught and buried by snow avalanches every year. About 100 die each year in the European Alps–and many more worldwide. Refining concepts for avalanche rescue means to optimize the procedures such that the survival chances are maximized in order to save the greatest possible number of lives. Avalanche rescue includes several parameters related to terrain, natural hazards, the people affected by the event, the rescuers, and the applied search and rescue equipment. The numerous parameters and their complex interaction make it unrealistic for a rescuer to take, in the urgency of the situation, the best possible decisions without clearly structured, easily applicable decision support systems. In order to analyse which measures lead to the best possible survival outcome in the complex environment of an avalanche accident, we present a numerical approach, namely a Monte Carlo simulation. We demonstrate the application of Monte Carlo simulations for two typical, yet tricky questions in avalanche rescue: (1) calculating how deep one should probe in the first passage of a probe line depending on search area, and (2) determining for how long resuscitation should be performed on a specific patient while others are still buried. In both cases, we demonstrate that optimized strategies can be calculated with the Monte Carlo method, provided that the necessary input data are available. Our Monte Carlo simulations also suggest that with a strict focus on the "greatest good for the greatest number", today's rescue strategies can be further optimized in the best interest of patients involved in an avalanche accident.

In many countries the total number of avalanche victims did not increase during the last decade (e.g., [

Apart from technical developments, e.g. improving the performance and user friendliness of search devices, it seems most promising to improve rescue efficiency by optimizing rescue procedures with a strict focus towards saving as many lives as possible, a concept which is also known as the “greatest good for the greatest number” theorem (Jeremy Bentham, 1748-1832).

Based on the “greatest good for the greatest number” consideration Bogle

A first technical attempt to apply numerical simulations based on the Monte Carlo principle on an avalanche accident scenario was presented by Genswein

As in the case of the search strip width, there is usually a trade-off between competing processes involving complimentary parameters, such as e.g. speed and accuracy. If the probability distributions between both the first parameter (e.g. speed) and survival chance as well as the second parameter (e.g. accuracy) and survival chance are known, it is possible to calculate the optimal parameter setting in order to maximize the survival chances with respect to the “greatest good for the greatest number”.

If the probability distributions are known only empirically, as it is the case for distributions stemming from the relatively small database of avalanche accidents (e.g. the distribution for avalanche burial depth, or avalanche deposit size), performing a Monte Carlo simulation [

The Monte Carlo (MC) method was originally developed by Metropolis

In this study, we aim to demonstrate how Monte Carlo simulations can be applied to optimize avalanche rescue strategies. We do so for two exemplary rescue situations where competing, incompatible requirements exist. In a first practical example, we calculate an optimal probing depth in the first passage of a probe line search for one buried subject. Here, there is a trade-off between the chances of finding the buried subject (accuracy) and the time it takes for finding the buried subject (speed), i.e. the burial duration, which strongly influences the probability of survival. As a second application, we used a Monte Carlo simulation for demonstrating how to calculate the optimal duration of resuscitation performed by a single rescuer on a not-breathing patient, i.e. already excavated patient, in case of another buried subject not yet excavated. The scenario of one rescuer and two buried subjects both needing urgent rescue and medical assistance is the simplest case of shortage of resources–a setting requiring a triage criterion–within avalanche rescue.

The Monte Carlo method is a numerical method to solve mathematical problems by random sampling [

In simplified form the Monte Carlo method consists of the steps listed below. As an example, we include the specifics steps for the probing depth simulation. Note that we use bold italic letters for random variables and quantities depending on random variables to distinguish them from all other variables or parameters (for which italic letters are used).

Define the distributions, either empirical or analytical, of the random variables of interest. In the considered example random variables are: i) the lateral position of the buried subjects: uniform distribution and ii) the burial depth: empirical data from the SLF avalanche data base.

Generate a large number of values for the random variables, each value is chosen randomly according to its probability distribution. In our example, by using a random number generator, we generate uniformly distributed random numbers i) from position 1 to position “end”–this position is used for placing the buried subject laterally and ii) from 1 to the maximum number of data points, we then take the value of this data point as our burial depth.

Perform deterministic calculations of the quantity of interest, using the randomly chosen set of values for the random variables as input. Each calculation with a different set of values represents a simulation run. In our probing depth example, we calculate the survival probability using randomly chosen lateral position and burial depth from above.

The mean value for the quantity of interest can be estimated by repeating the simulation a predefined number of times, thus generating the distribution for the quantity of interest from which respective values such as mean or variance can be calculated. In our example this means, calculating the mean survival probability from the results of all simulation runs for a given probing depth.

We applied the Monte Carlo method in two scenarios within avalanche rescue–probe line search and resuscitation strategy in case of an additional buried subject–that are described in detail below.

A probe line search is a search method, which is applied when buried subjects cannot be located by electronic search means (e.g. transceivers or Recco) or an avalanche dog. It involves penetrating the avalanche debris with long probes [

As the survival chance of a buried subject decreases dramatically with increasing burial time [

The strategy of limiting probing depth to improve the odds of finding those victims that are less deeply buried more quickly is not new [

Following the approach of the “greatest good for the greatest number” we performed a Monte Carlo simulation to find an optimal probing depth for maximizing the survival chance of a buried subject. We focused on the Monte Carlo method for determining an optimal probing depth, other important considerations such as the search strategy are described by Genswein

The Monte Carlo simulation for calculating an optimal probing depth was performed as follows. We drew random values for both the position of the buried subject–using a uniform distribution–and its burial depth–using the avalanche database. This was repeated for six different areas of avalanche debris (50, 100, 500, 1000, 5000 and 10,000 m^{2};

(a) Burial depth for 1490 buried subjects, and (b) avalanche deposit area for 541 avalanches. The vertical lines mark the first, second (median, red line), and third quartile. Both data sets are from SLF’s avalanche data base.

This sampling procedure was repeated 10,000 times for each size of avalanche debris and 21 equally spaced probing depths ranging from 0.5 to 2.5 m. The convergence of the simulation was checked empirically, and 10,000 simulation runs appeared sufficient. Each simulation run proceeds as described below.

The randomly sampled burial depth was compared to the probing depth. If the victim was buried deeper than the probing depth, the victim was missed and the probability of survival set to zero.

If the probing depth was larger than the burial depth, the victim was found. We did not consider lateral misses during probing. The total burial time was the sum of search and excavation time. The search time was _{search} = _{search} _{rescuers}), and the excavation time was _{dig} = _{burial} / _{dig}. The variables denote time (^{2}, we used a fixed number of rescuers, _{rescuers} = 5. The scenarios for deposits sized between 5000 and 100,000 m^{2} were also calculated but assuming that 20 rescuers were available to probe. We assumed the slalom probing strategy [

The total burial time was calculated as _{search} + _{dig}. Given the burial time the probability of survival was calculated. For the probability of survival _{survival} we used a smooth interpolation of the survival curve based on the Swiss avalanche survival data presented by Haegeli

The length (in meters) of search area (width 1.5 m) on avalanche debris covered by one probing rescuer per minute as a function of probing depth using the slalom probing technique [

For the simulation a smooth interpolation (blue line) of the avalanche survival curve based on Swiss accident data was used, adapted from [

Finally, the average probability of survival for a given probing depth was calculated as the average over the survival probabilities of each of the 10,000 simulation runs.

Our test scenario is simple, but demands for a triage decision due to shortage of resources. We assume two buried subjects, but only one rescuer. The simulation starts as soon as the rescuer has excavated the first buried subject. The excavated patient, from now on referred to as patient 1, has no obvious lethal injuries, is normothermic, but has no vital signs. The second buried subject, referred to as patient 2, is still buried. We assume that the rescuer has witnessed the avalanche, i.e. the duration of burial is known; it is assumed an upper estimate of the time when the (apparent) cardiac arrest of patient 1 has happened.

The goal of every rescue is to save as many lives as possible. As only one rescuer is available, this person can either engage on resuscitation of patient 1, or excavate patient 2. The situation demonstrates that the shortage of resources leads to the tricky situation as we have two competing processes: the increase of survival chance of patient 1 as a function of resuscitation duration [

In the above situation, the avalanche resuscitation algorithm as defined by the European Resuscitation Council [

Performing CPR for a maximum of at least 20 min strongly favours the survival chances of patient 1 (_{1}) at the cost of the survival chances (_{2}) of patient 2. Proceeding to patient 2 immediately, on the other hand, while optimal for patient 2, reduces the survival chances of patient 1.

We performed a Monte Carlo simulation to calculate the optimal time for performing CPR on patient 1 before proceeding to patient 2 in order to maximize the expected number of survivors. The number of expected survivors for _{1} and _{2} persons with respective survival probabilities _{1} and _{2} is calculated as _{1}∙_{1}+ _{2}∙_{2.} Since we have _{1} = _{2} = 1, the expected number of survivors simplifies to _{1} + _{2}. The probability, that both patients survive can be calculated as _{1}∙_{2}, and the probability that at least one of them survives is given by 1 − [(1 − _{1}) ∙ (1 − _{2})]. These calculations are valid since _{1} and _{2} are independent, i.e. do not intrinsically depend on each other.

We tested different times for performing CPR on patient 1 (_{CPR}) and calculated the average number of survivors for each _{CPR}, ranging from 0 to 30 min in one minute steps. Each simulation consisted of 10,000 runs. We empirically checked the simulation's convergence by varying the number of simulation runs. Once the simulation results did not change anymore with increasing number of runs, we considered the simulation as converged. The simulation steps of our Monte Carlo simulation are described in detail below.

The set of random variables and parameters we need for each simulation run were defined as follows:

Search time _{search} for patient 2. This parameter was drawn from a Gaussian distribution with mean of 2 min [

Excavation time _{dig} for patient 2. The excavation time depends on the burial depth. We used the data-based burial depth distribution shown in

We generated a large number of values for the random variables _{dig} and _{search} drawn from the respective probability distributions. Note that _{dig} and _{search} are assumed to be independent.

We deterministically calculate the expected number of survivors (_{1}+_{2}) with each set of values for the random variables. The survival probability of patient 1, _{1}, is estimated based on data presented in _{1} = _{CPR}^{1.3}]) with _{burial, patient1}) as a function of the initial burial time of patient 1 as well as the chosen values for the initial burial time are based on the data presented by Moroder _{2}, is calculated based on the curve in _{burial, patient1} _{CPR} + _{search} + _{burial} / _{dig}. The time _{CPR} was a fixed parameter ranging from 0 to 30 min.

We calculated the average expected number of survivors (_{1}+_{2}) over all simulation runs.

The magenta curves refer to the three scenarios of burial time for patient 1, namely 12, 20, and 35 min; adapted from Reynolds

A simulation as described above was done for each _{CPR} ranging from 0 to 30 min and four different assumed burial times for patient 1, namely 12, 20, and 35 min, and corresponding values of

An exemplary result for a debris size of 5000 m^{2} is shown in _{survival} initially increased. At a probing depth of 1.9 m, however, the probability of survival reached its maximum before decreasing again. This decrease of _{survival} at large probing depths was due to the decrease of the victim’s survival probability with longer burial time (^{2}.

Probability of survival (red curve) of an avalanche victim as a function of different probing depths for a search area (avalanche debris size) of 5000 m^{2}. Also shown (black curve) is the probability of missing the buried subject.

The value of the optimal probing depth increased for smaller search areas, since it takes less time to search the whole area, while the optimal probing depth decreased for larger search areas. The optimal probing depths for maximizing the survival chance of a buried subject as a function of avalanche debris size is shown in

Probing depth leading to the highest survival chance for a buried subject as a function of area to be probed for either five (blue diamonds) or 20 rescuers (magenta squares).

The median size of the avalanche deposit for the dataset shown in ^{2}. However, it is rare that a buried subject is only found after the entire deposit has been probed. Entrance tracks of the caught subjects, terrain shape, direction of flow, potential anchoring points, witnessed "last seen points" as well as visual clues on the surface allow to determine the "most likely burial areas", thus in many cases strongly reducing the total area which needs to be probed.

Hence, the scenario including five rescuers and 5000 m^{2} surface is typical in companion rescue within the first 20 to 30 min after an accident. Due to restrictions of resources, the companions will be forced to focus on a strongly limited, most likely burial area. Once organized rescue arrives on scene and buried subjects are still missing, the initial search area can be extended as the availability of resources strongly increases. A probing depth larger than 2.50 m might also be considered, but only once organized rescue with professional, long probes has arrived on scene.

The single survival curves for patient 1 needing CPR and patient 2 waiting for excavation are shown in _{CPR}, the time used to perform CPR on patient 1. In addition, the probability, that both patients survive, as well as the probability, that at least one of the patients survives, are shown (_{1}∙_{1}: open orange stars and 1 − [(1 − _{1}) ∙ (1 − _{2})]: yellow hexagrams). For a starting time of 12 min (_{CPR}, the survival chances of patient 2 decrease, since increasing _{CPR} means increasing burial time for him or her. The probability of both patients surviving are highest for a _{CPR} of 12 min with a value of 0.2, i.e. 20%. This means that even for the short initial burial time of 12 min, the probability that at least one patient dies is about 80%. The probability that at least one patient survives initially increases with increasing _{CPR}, to reach a constant value of 72% between 17 and 30 min.

Single survival curves (blue/green/cyan crosses and stars) for patient 1 and patient 2 over _{CPR}, the time used to perform CPR on patient 1. The assumed burial time of patient 1 was (a) 12 min, (b) 20 min and (c) 35 min. Also shown is the probability of both patients surviving (open orange stars) as well as the probability that at least one of the patients survives (yellow hexagrams).

The probabilities of survival depend strongly on the time it takes to excavate patient 1 and start performing CPR. If this time is assumed to be 20 min (_{CPR} of 16 minutes. The maximum probability of at least one patient surviving again reaches a constant value at a _{CPR} between 17 and 30 min but has dropped to 47%.

In the case of a relatively long initial burial time–i.e. the burial time of patient 1 before CPR starts–of 35 min, the survival chances of patient 1 reduce to zero [

In the sense of the “greatest good for the greatest number” we are interested in maximizing the number of survivors. The expected number of survivors as a function of resuscitation time _{CPR} for different initial burial times of patient 1 is shown in

Expected number of survivors over resuscitation time _{CPR} for initial burial times for patient 1 of 12, 20, and 35 min.

The maximum expected number of survivors is achieved for a CPR time of 11 min, for an initial burial time of patient 1 of 12 minutes, as can be seen from _{CPR}.

Obviously, our simulations results depend on the input data. This is particularly true for the data on the probability of survival of patient 1. The data from Reynolds

Despite the above-mentioned limitations it seems likely that in our simple triage scenario, given the simulation results, performing CPR on patient 1 for less than the 20 minutes recommended by Truhlář

Our study shows that certain data is required to optimize avalanche rescue procedures. To calculate the optimal probing depth we specifically require data on the probing speed as a function of probing depth, preferably also as a function of the number of rescuers. We assume that the probing speed in reality increases less than linearly with increasing number of rescuers as more rescuers need more time to line up after each set of probing strokes. In addition, for all simulations concerning avalanche rescue we need to know the digging speed as a function of burial depth and the number of rescuers. Finally, for the simulation including CPR, the survival probabilities of buried subjects who require CPR as a function of burial time are crucial.

The application of numerical simulations based on the Monte Carlo principle allow finding optimal solutions in the realm of the "greatest good for the greatest number"–an approach previously not possible. For two optimization problems in avalanche rescue, probing depth and CPR duration in case of shortage of resources, we have exemplarily shown the potential of this simulation approach.

Our Monte Carlo simulations primarily provide exemplary results as we had to make several assumptions given that the underlying data are sparse or do not even exist. Nevertheless, we suggest to consider the optimized suggested values for probing depth in probe line searches and CPR duration while revisiting current search and rescue strategies as well as triage decision support systems. In particular, our simulations suggest that for the second triage scenario (i.e. resuscitation in case of an additional buried subject) a CPR duration shorter than the 20 min as recommended by Truhlář

In the future, we plan to extend our simulations, and in particular gather more medical data on avalanche victims to improve the underlying assumptions, e.g. on the success rate of CPR efforts.

Avalanche ID, burial depth (in cm) and avalanche deposit size (in m2) for fully buried subjects by avalanches in Switzerland recorded between 1973–1974 to 2012–2013 in the SLF avalanche database (1555 cases in total). Out of the 1555 cases either only the burial depth (1490 cases) or only the deposit size (541 cases) is known. For 477 cases both the burial depth and the deposit area were recorded.

(PDF)

Avalanche ID, avalanche burial depth, and avalanche deposit area.

(CSV)

We thank Frank Techel for extracting the required data from SLF’s avalanche database.