Timothy C. Y. Chan

Identifying Locations for Public Access Defibrillators Using Mathematical Optimization

Background— Geospatial methods using mathematical optimization to identify clusters of cardiac arrests and prioritize public locations for defibrillator deployment have not been studied. Our objective was to develop such a method and test its performance against a population-guided approach. Methods and Results— All public location cardiac arrests in Toronto, Ontario, Canada, from December 16, 2005, to July 15, 2010, and all automated external defibrillator (AED) locations registered with Toronto Emergency Medical Services as of September 2009 were plotted geographically. Current AED coverage was quantified by determining the number of cardiac arrests occurring within 100 m of a registered AED. Clusters of cardiac arrests without a registered AED within 100 m were identified. With the use of mathematical optimization techniques, cardiac arrest coverage improvements were computed and shown to be superior to results from a population-guided deployment method. There were 1310 eligible public location cardiac arrests and 1669 registered AEDs. Of the eligible cardiac arrests, 304 were within 100 m of at least 1 registered AED (23% coverage). The average distance from a cardiac arrest to the closest AED was 281 m. With AEDs deployed in the top 30 locations, an additional 112 historical cardiac arrests would be covered (32% total coverage), and the average distance to the closest AED would be 262 m. Conclusions— Geographic clusters of cardiac arrests can be easily identified and prioritized with the use of mathematical modeling. Optimized AED deployment can increase cardiac arrest coverage and decrease the distance to the closest AED. Mathematical modeling can augment public AED deployment programs.

A robust-CVaR optimization approach with application to breast cancer therapy

We present a framework to optimize the conditional value-at-risk (CVaR) of a loss distribution under uncertainty. Our model assumes that the loss distribution is dependent on the state of some system and the fraction of time spent in each state is uncertain. We develop and compare two robust-CVaR formulations that take into account this type of uncertainty. We motivate and demonstrate our approach using radiation therapy treatment planning of breast cancer, where the uncertainty is in the patient’s breathing motion and the states of the system are the phases of the patient’s breathing cycle. We use a CVaR representation of the tails of the dose distribution to the points in the body and account for uncertainty in the patient’s breathing pattern that affects the overall dose distribution.

Determining Risk for Out-of-Hospital Cardiac Arrest by Location Type in a Canadian Urban Setting to Guide Future Public Access Defibrillator Placement

STUDY OBJECTIVE Automated external defibrillator use by lay bystanders during out-of-hospital cardiac arrest rarely occurs but can improve survival. We seek to estimate risk for out-of-hospital cardiac arrest by location type and evaluate current automated external defibrillator deployment in a Canadian urban setting to guide future automated external defibrillator deployment. METHODS This was a retrospective analysis of a population-based out-of-hospital cardiac arrest database. We included consecutive public location, nontraumatic, out-of-hospital cardiac arrests occurring in Toronto from January 1, 2006, to June 30, 2010, captured in the Resuscitation Outcomes Consortium Epistry database. Two investigators independently categorized each out-of-hospital cardiac arrest and automated external defibrillator location into one of 38 categories. Total site counts in each location category were used to estimate average annual per-site cardiac arrest incidence and determine the relative automated external defibrillator coverage for each location type. RESULTS There were 608 eligible out-of-hospital cardiac arrest cases. The top 5 location categories by average annual out-of-hospital cardiac arrests per site were race track/casino (0.67; 95% confidence interval [CI] 0 to 1.63), jail (0.62; 95% CI 0.3 to 1.06), hotel/motel (0.15; 95% CI 0.12 to 0.18), hostel/shelter (0.14; 95% CI 0.067 to 0.19), and convention center (0.11; 95% CI 0 to 0.43). Although schools were relatively lower risk for cardiac arrest, they represented 72.5% of automated external defibrillator-covered locations in the study region. Some higher-risk location types such as hotel/motel, hostel/shelter, and rail station were severely underrepresented with respect to automated external defibrillator coverage. CONCLUSION We have identified types of locations with higher per-site risk for cardiac arrest relative to others. We have also identified potential mismatches between cardiac arrest risk by location type and registered automated external defibrillator distribution in a Canadian urban setting.

Modeling the impact of public access defibrillator range on public location cardiac arrest coverage.

BACKGROUND Public access defibrillation with automated external defibrillators (AEDs) can improve survival from out-of-hospital cardiac arrests (OHCA) occurring in public. Increasing the effective range of AEDs may improve coverage for public location OHCAs. OBJECTIVE To quantify the relationship between AED effective range and public location cardiac arrest coverage. METHODS This was a retrospective cohort study using the Resuscitation Outcomes Consortium Epistry database. We included all public-location, atraumatic, EMS-attended OHCAs in Toronto, Canada between December 16, 2005 and July 15, 2010. We ran a mathematical model for AED placement that maximizes coverage of historical public OHCAs given pre-specified values of AED effective range and the number of locations to place AEDs. Locations of all non-residential buildings were obtained from the City of Toronto and used as candidate sites for AED placement. Coverage was evaluated for range values from 10 to 300 m and number of AED locations from 10 to 200, both in increments of 10, for a total of 600 unique scenarios. Coverage from placing AEDs in all public buildings was also measured. RESULTS There were 1310 public location OHCAs during the study period, with 25,851 non-residential buildings identified as candidate sites for AED placement. Cardiac arrest coverage increased with AED effective range, with improvements in coverage diminishing at higher ranges. For example, for a deployment of 200 AED locations, increasing effective range from 100 m to 200 m covered an additional 15% of cardiac arrests, whereas increasing range further from 200 m to 300 m covered an additional 10%. Placing an AED in each of the 25,851 public buildings resulted in coverage of 50% and 95% under assumed effective ranges of 50 m and 300 m, respectively. CONCLUSION Increasing AED effective range can improve cardiac arrest coverage. Mathematical models can help evaluate the potential impact of initiatives which increase AED range.

Generalized Inverse Multiobjective Optimization with Application to Cancer Therapy

We generalize the standard method of solving inverse optimization problems to allow for the solution of inverse problems that would otherwise be ill posed or infeasible. In multiobjective linear optimization, given a solution that is not a weakly efficient solution to the forward problem, our method generates objective function weights that make the given solution a near-weakly efficient solution. Our generalized inverse optimization model specializes to the standard model when the given solution is weakly efficient and retains the complexity of the underlying forward problem. We provide a novel interpretation of our inverse formulation as the dual of the well-known Bensons method and by doing so develop a new connection between inverse optimization and Pareto surface approximation techniques. We apply our method to prostate cancer data obtained from Princess Margaret Cancer Centre in Toronto, Canada. We demonstrate that clinically acceptable treatments can be generated using a small number of objective ...

Adaptive and robust radiation therapy optimization for lung cancer

A previous approach to robust intensity-modulated radiation therapy (IMRT) treatment planning for moving tumors in the lung involves solving a single planning problem before the start of treatment and using the resulting solution in all of the subsequent treatment sessions. In this paper, we develop an adaptive robust optimization approach to IMRT treatment planning for lung cancer, where information gathered in prior treatment sessions is used to update the uncertainty set and guide the reoptimization of the treatment for the next session. Such an approach allows for the estimate of the uncertain effect to improve as the treatment goes on and represents a generalization of existing robust optimization and adaptive radiation therapy methodologies. Our method is computationally tractable, as it involves solving a sequence of linear optimization problems. We present computational results for a lung cancer patient case and show that using our adaptive robust method, it is possible to attain an improvement over the traditional robust approach in both tumor coverage and organ sparing simultaneously. We also prove that under certain conditions our adaptive robust method is asymptotically optimal, which provides insight into the performance observed in our computational study. The essence of our method – solving a sequence of single-stage robust optimization problems, with the uncertainty set updated each time – can potentially be applied to other problems that involve multi-stage decisions to be made under uncertainty.

Optimizing a Drone Network to Deliver Automated External Defibrillators

Background: Public access defibrillation programs can improve survival after out-of-hospital cardiac arrest, but automated external defibrillators (AEDs) are rarely available for bystander use at the scene. Drones are an emerging technology that can deliver an AED to the scene of an out-of-hospital cardiac arrest for bystander use. We hypothesize that a drone network designed with the aid of a mathematical model combining both optimization and queuing can reduce the time to AED arrival. Methods: We applied our model to 53 702 out-of-hospital cardiac arrests that occurred in the 8 regions of the Toronto Regional RescuNET between January 1, 2006, and December 31, 2014. Our primary analysis quantified the drone network size required to deliver an AED 1, 2, or 3 minutes faster than historical median 911 response times for each region independently. A secondary analysis quantified the reduction in drone resources required if RescuNET was treated as a large coordinated region. Results: The region-specific analysis determined that 81 bases and 100 drones would be required to deliver an AED ahead of median 911 response times by 3 minutes. In the most urban region, the 90th percentile of the AED arrival time was reduced by 6 minutes and 43 seconds relative to historical 911 response times in the region. In the most rural region, the 90th percentile was reduced by 10 minutes and 34 seconds. A single coordinated drone network across all regions required 39.5% fewer bases and 30.0% fewer drones to achieve similar AED delivery times. Conclusions: An optimized drone network designed with the aid of a novel mathematical model can substantially reduce the AED delivery time to an out-of-hospital cardiac arrest event.

Predicting objective function weights from patient anatomy in prostate IMRT treatment planning.

PURPOSE Intensity-modulated radiation therapy (IMRT) treatment planning typically combines multiple criteria into a single objective function by taking a weighted sum. The authors propose a statistical model that predicts objective function weights from patient anatomy for prostate IMRT treatment planning. This study provides a proof of concept for geometry-driven weight determination. METHODS A previously developed inverse optimization method (IOM) was used to generate optimal objective function weights for 24 patients using their historical treatment plans (i.e., dose distributions). These IOM weights were around 1% for each of the femoral heads, while bladder and rectum weights varied greatly between patients. A regression model was developed to predict a patients rectum weight using the ratio of the overlap volume of the rectum and bladder with the planning target volume at a 1 cm expansion as the independent variable. The femoral head weights were fixed to 1% each and the bladder weight was calculated as one minus the rectum and femoral head weights. The model was validated using leave-one-out cross validation. Objective values and dose distributions generated through inverse planning using the predicted weights were compared to those generated using the original IOM weights, as well as an average of the IOM weights across all patients. RESULTS The IOM weight vectors were on average six times closer to the predicted weight vectors than to the average weight vector, using l2 distance. Likewise, the bladder and rectum objective values achieved by the predicted weights were more similar to the objective values achieved by the IOM weights. The difference in objective value performance between the predicted and average weights was statistically significant according to a one-sided sign test. For all patients, the difference in rectum V54.3 Gy, rectum V70.0 Gy, bladder V54.3 Gy, and bladder V70.0 Gy values between the dose distributions generated by the predicted weights and IOM weights was less than 5 percentage points. Similarly, the difference in femoral head V54.3 Gy values between the two dose distributions was less than 5 percentage points for all but one patient. CONCLUSIONS This study demonstrates a proof of concept that patient anatomy can be used to predict appropriate objective function weights for treatment planning. In the long term, such geometry-driven weights may serve as a starting point for iterative treatment plan design or may provide information about the most clinically relevant region of the Pareto surface to explore.

Optimal margin and edge-enhanced intensity maps in the presence of motion and uncertainty

In radiation therapy, intensity maps involving margins have long been used to counteract the effects of dose blurring arising from motion. More recently, intensity maps with increased intensity near the edge of the tumour (edge enhancements) have been studied to evaluate their ability to offset similar effects that affect tumour coverage. In this paper, we present a mathematical methodology to derive margin and edge-enhanced intensity maps that aim to provide tumour coverage while delivering minimum total dose. We show that if the tumour is at most about twice as large as the standard deviation of the blurring distribution, the optimal intensity map is a pure scaling increase of the static intensity map without any margins or edge enhancements. Otherwise, if the tumour size is roughly twice (or more) the standard deviation of motion, then margins and edge enhancements are preferred, and we present formulae to calculate the exact dimensions of these intensity maps. Furthermore, we extend our analysis to include scenarios where the parameters of the motion distribution are not known with certainty, but rather can take any value in some range. In these cases, we derive a similar threshold to determine the structure of an optimal margin intensity map.

Optimizing the Deployment of Public Access Defibrillators

Out-of-hospital cardiac arrest is a significant public health issue, and treatment, namely, cardiopulmonary resuscitation and defibrillation, is very time sensitive. Public access defibrillation programs, which deploy automated external defibrillators (AEDs) for bystander use in an emergency, reduce the time to defibrillation and improve survival rates. In this paper, we develop models to guide the deployment of public AEDs. Our models generalize existing location models and incorporate differences in bystander behavior. We formulate three mixed integer nonlinear models and derive equivalent integer linear reformulations or easily computable bounds. We use kernel density estimation to derive a spatial probability distribution of cardiac arrests that is used for optimization and model evaluation. Using data from Toronto, Canada, we show that optimizing AED deployment outperforms the existing approach by 40% in coverage, and substantial gains can be achieved through relocating existing AEDs. Our results suggest that improvements in survival and cost-effectiveness are possible with optimization.