Project Page for Artificial Pancreas Verification Project
This page describes ongoing collaborative research funded by the US
National Science Foundation (NSF) under awards
CPS-1446900
and CPS-1446751. All opinions expressed here are those of the authors
and not necessarily of the US National Science Foundation (NSF).
Introduction
This project is a collaboration between engineers, computer
scientists, mathematicians, and clinical researchers centered around
the mathematical modeling, simulation and formal analysis of the
artificial pancreas. The artificial pancreas is a concept that
involves the development of a series of increasingly sophisticated
Cyber-Physical Systems that will automate the delivery of insulin to
people with Type-1 Diabetes.
Background on Type-1 Diabetes
Warning: This is a simple (possibly simplistic) explanation that omits many key details. It is suitable as an overview and not intended to be a clinical presentation. The interested reader should consult a book (we recommend the Pink Panther book by Drs. Chase and Maahs).
What is Type-1 Diabetes?
Type-1 Diabetes (T1D) is a serious condition caused by the inability
or the decreased ability of the pancreas to secrete insulin. This can
happen due to numerous reasons: for example, as a result of an
autoimmune response that destroys the beta cells of the pancreas; or
as a result of certain surgical procedures. Insulin is an important
hormone that is responsible for regulating the uptake of glucose by
various cells in the body causing the lowering of blood glucose
levels. Lack of insulin leads to increased blood glucose
levels. However, the glucose in the blood is not taken up by the cells
that convert them to energy or store them. As a result, the cells are
starved of much needed energy and this results in their breakdown.
T1D Treatment
Currently, T1D is treated through the external administration of
artificial insulin analogs either through daily injections or an
insulin infusion pump. The latter option is increasingly popular with
patients, since it provides substantial freedom in their choice of
meals, exercise and other activities that affect their metabolism. The
infusion pumps are operated manually by the patient based on their
anticipated mealtimes, meal carbohydrate amounts and exercise
times/intensities. The infusion pumps currently available in the
market can deliver short acting insulin analogs at programmable doses,
all day and night. This typically involves a background “basal”
delivery of insulin and numerous boluses delivered at
mealtimes. Manual control of pump has two main drawbacks:
It imposes the burden of managing the blood glucose levels on the
patients. Typically this involves roughly 6-9 blood glucose
measurements, basal adjustments and bolus infusions throughout the
day. Failure to maintain blood glucose levels inside the euglycemic
range [70,170] mg/dl can have serious consequences. If too much
insulin is infused, it can drive the blood glucose levels dangerously
low, causing hypoglycemia. If too little is infused, it can cause high
blood gluocse levels with short term (ketacidosis) and long term
consequences (damage to kidneys, heart, nerves and eyes).
The Artificial Pancreas
The artificial pancreas (AP) is a series of
increasingly sophisticated devices that will increasingly automate
insulin delivery to the patient. It has been enabled by insulin
infusion pumps that can deliver insulin automatically and be
programmed externally by a software-based controller. On the other
hand, glucose sensor technology has developed sufficiently to provide
accurate, near realtime subcutaneous glucose measurement. Many stages
have been envisioned for the AP, and the technological feasibility of
each stage has been demonstrated in clinical trials. The original
stages of the AP were proposed by the JDRF as follows:
Pump shutoff: this stage simply shuts down the pump and alarms when the blood glucose is too low. This technology has already been useful at nighttimes to alert patients and their care providers when the blood glucose level drops.
Predictive Pump shutoff: this device forecasts the near term future glucose trend form the current observed glucose values. If the forecast calls for a low in the next 30-70 minutes, the pump is shutoff in anticipation of the low and turned back on as soon as the glucose levels rise.
Hybrid Closed Loop: This stage controls the basal insulin delivery, relying on daytime boluses by the user.
Fully automatic closed loop: This stage controls both basal delivery and automatic meal boluses.
Multihormone closed loop: This stage uses multiple hormones instead of just insulin. Specifically, glucagon (the counterregulatory hormone to insulin) is also used to increase blood glucose levels and in some designs, amylin is added to control the digestion of food.
Recently, it has been noted that all stages have been implemented and
proven to be technologically viable in various stages of clinical
trials. A simpler classification has been proposed by the JDRF as (a)
daytime vs. nighttime-only control and (b) insulin-only
vs. multi-hormonal control.
Formal Verification
The formal verification project proposes to verify AP control systems to find possible issues with these systems before deployment.
What is formal verification?
Engineered systems are often prone to design faults that arise due to
numerous reasons, including design and implementation mistakes. These
are particularly problematic with software systems that tend to be
designed by a large team of people. Verification is the process by
which, these devices are tested and known defects that arise are
fixed. However, manual verification is expensive and prone to errors,
as well. Therefore, formal verification techniques try to automate the
verification process to provide more guarantees at the end of the
verification. Ideally, these are exhaustive techniques that either
provide a mathematical proof that the system is correct or expose a
bug/defect in the design.
However, there are fundamental reasons why fully exhaustive and
automated verification of software systems is a really hard problem,
if not downright impossible! Nevertheless, we have been trying many
partial solutions that have yielded varying degrees of success: these
solutions focus on solving the problem for special classes of systems
or abstracting the system model into a special form before reasoning
about them.
Why verify the Artificial Pancreas?
The artificial pancreas is safety critical. Ultimately, it takes
autonomy away from patients with T1D in return for a reduced burden of
diabetes management. It is essential that the control software be free
of common software bugs such a buffer overflows, division by zero, and
memory leaks. However, the AP is not just software: it includes
physical components like sensors, pumps and an entire human body in
the loop! As such, it has to deal with a large set of uncertainties in
its operation. Examples include unannounced meals, exercise, sensor
noise, erroneous sensor readings, pressure induced sensor attenuation,
set failures, and patient physiological changes. Besides software
bugs, we need to exhaustively run our designs through billions of
scenarios to see if the system behaves “reasonably” in all
scenarios. Worst of all, we need a good definition of what
“reasonable” is.
Aren’t Clinical Trials Good Enough?
Note: Clinical trials differ widely in what they study, and how they
study. The answer below is not truly representative of all possible
studies that have been carried out for AP controllers
In some sense, clinical trials will be “good enough” when the
technology is deployed to a large population of users and tested for a
long time under varying conditions. Otherwise, most trials are not
meant to test for the “worst case”. Worst case effects are by
definition rare and only seen when a product is deployed.
What stops us from verifying AP controllers?
As mentioned earlier, verification is a very hard problem. We have
billions, if not trillions, of scenarios to run through and test. It
is hard to go through each one by one. As a result, mathematical
models should be used by engineers to capture processes like the human
insulin-glucose response, the patterns of sensor noise, set failures
and other disturbances. There has been a lot of work on simulating AP
controllers. In fact, some models like the Dalla-Man et al. model have
been used with FDA’s encouragement as a stand in for animal trials of
AP controllers. However, these models are nonlinear and as such, many
verification tools that exist cannot handle these models or can be
very slow. However, we are making progress on tools like S-Taliro,
Flow* and dReal, that can perform restricted forms of automated
verification for nonlinear dynamics.
To conclude, a significant gap needs to be bridged between what formal
tools can do currently, and what is needed to automate an exhaustive
analysis of AP controllers.
Project Goals
The basic goal of this project is to provide a focused effort around
developing formal tools for verifying artificial pancreas
controllers. The project has multiple thrusts including:
Developing disturbance models for meal, exercise, sensor noise, dropouts, set
failures and other disturbances.
Developing simpler, perhaps even nondeterminstic models, of insulin-glucose response. We are looking
into delay differential models and nondeterministic “minimal models”
built using the available data.
Developing formal verification tools
for nonlinear dynamics so that they can reason about controller
implementations and nonlinear dynamics models.
Developing a set of
properties that need to be verified. Case-studies on real-life
systems that have been deployed in clinical trials.
Last but not least, Develop ways to
present verification results to engineers and clinicians to better
understand the root cause of defects and judge their real-life
feasibility.
Project Team
The project involves a mutidisciplinary team of computer scientists, chemical engineers, mathematicians and clinical researchers from multiple institutions.
Principal Investigators
Faye Cameron (joint PI), Chemical Engg. RPI.
Sriram Sankaranarayanan (joint PI), Computer Science, CU Boulder.
David Maahs (joint PI), Pediatric Endocrinology, Barbara Davis Center for Childhood Diabetes, UC Denver Medical School.
B. Wayne Bequette (senior person), Chemical Engg., RPI.
David Bortz (co-PI), Applied Mathematics, CU Boulder.
Shalom Ruben (co-PI), Mechanical Engg., CU Boulder.
Students
Alumni
Suhas Akshar Kumar
Ram Das Diwakaran
Alexandra Okeson
Dr. Xin Chen
Collaborators
Dr. Greg Forlenza, Pediatric Endocrinology, Barbara Davis Center for Childhood Diabetes, UC Denver Medical School.
Publications
Souradeep Dutta and Taisa Kushner and Sriram Sankaranarayanan, Robust Data-Driven Control of Artificial Pancreas Systems using Neural Networks In Computational Methods in Systems Biology, Volume tbd of Lecture Notes In Computer Science pp. tbd (2018).
URL Abstract Bib Topics
In this paper, we provide an approach to data-driven
control for artificial pancreas system by learning
neural network models of human insulin-glucose
physiology from available patient data and using a
mixed integer optimization approach to control blood
glucose levels in real-time using the inferred
models. First, our approach learns neural networks
to predict the future blood glucose values from
given data on insulin infusion and their resulting
effects on blood glucose levels. However, to provide
guarantees on the resulting model, we use quantile
regression to fit multiple neural networks that
predict upper and lower quantiles of the future
blood glucose levels, in addition to the mean.
Using the inferred set of neural networks, we
formulate a model-predictive control scheme that
adjusts both basal and bolus insulin delivery to
ensure that the risk of harmful hypoglycemia and
hyperglycemia are bounded using the quantile models
while the mean prediction stays as close as possible
to the desired target. We discuss how this scheme
can handle disturbances from large unannounced meals
as well as infeasibilities that result from
situations where the uncertainties in future glucose
predictions are too high. We experimentally
evaluate this approach on data obtained from a set
of 17 patients over a course of 40 nights per
patient. Furthermore, we also test our approach
using neural networks obtained from virtual patient
models available through the UVA-Padova simulator
for type-1 diabetes.
@inproceedings{Dutta_Others__2018__Robust,
author = { Souradeep Dutta and Taisa Kushner and Sriram Sankaranarayanan },
title = { Robust Data-Driven Control of Artificial Pancreas Systems using Neural Networks },
booktitle = { Computational Methods in Systems Biology },
year = { 2018 },
pages = { tbd },
publisher = { Springer-Verlag },
series = { Lecture Notes In Computer Science },
volume = { tbd }}
Taisa Kushner and David Bortz and David Maahs and Sriram Sankaranarayanan, A Data-Driven Approach to Artificial Pancreas Verification and Synthesis. In Intl. Conference on Cyber-Physical Systems (ICCPS'18), pp. 242-252 (2018).
URL Abstract Bib Topics
This paper presents a case study of a data driven approach to
verification and parameter synthesis for artificial pancreas control
systems which deliver insulin to patients with type-1 diabetes
(T1D). We present a new approach to tuning parameters using
non-deterministic data-driven models for human insulin-glucose
regulation, which are inferred from patient data using multiple time
scales. Taking these equations as constraints, we model the behavior
of the entire closed loop system over a five-hour time horizon cast
as an optimization problem. Next, we demonstrate this approach using
patient data gathered from a previously conducted outpatient
clinical study involving insulin and glucose data collected from
50 patients with T1D and 40 nights per patient. We use the
resulting data-driven models to predict how the patients would
perform under a PID-based closed loop system which forms the basis
for the first commercially available hybrid closed loop
device. Futhermore, we provide a re-tuning methodology which can
potentially improve control for 82\% of patients, based on the
results of an exhaustive reachability analysis. Our results
demonstrate that simple nondeterministic models allow us to
efficiently tune key controller parameters, thus paving the way for
interesting clinical translational applications.
@inproceedings{Kushner_Others__2018__Data,
author = { Taisa Kushner and David Bortz and David Maahs and Sriram Sankaranarayanan },
title = { A Data-Driven Approach to Artificial Pancreas Verification and Synthesis. },
booktitle = { Intl. Conference on Cyber-Physical Systems (ICCPS'18) },
year = { 2018 },
pages = { 242-252 },
publisher = { IEEE Press }}
Xin Chen and Sriram Sankaranarayanan, Model-Predictive Real-Time Monitoring of Linear Systems In IEEE Real-Time Systems Symposium (RTSS), pp. 297-306 (2017).
URL Abstract Bib Topics
The predictive monitoring problem asks whether a deployed system is
likely to fail over the next T seconds under some environmental
conditions. This problem is of the utmost importance in
cyber-physical systems and has inspired real-time architectures
capable of adapting to such failures upon forewarning. In this
paper, we present a linear model-predictive scheme for the real-time
monitoring of linear systems governed by time-triggered controllers
and time-varying disturbances. The scheme uses a combination of
offline (advance) and online computations to decide if a given plant
model has entered a state from which no matter what control is
applied, the disturbance has a strategy to drive the system to an
unsafe region. Our approach is independent of the control strategy
used: this allows us to deal with plants that are controlled using
model-predictive control techniques or even opaque machine-learning
based control algorithms that are hard to reason with using existing
reachable set estimation algorithms. Our online computation has a
simplified efficient framework which reuses the symbolic reachable sets
computed offline. The real-time monitor instantiates the reachable set
with a concrete state estimate, and repeatedly performs emptiness
checks with respect to a safety property. We classify the various alarms
raised by our approach in terms of what they imply about the system
as a whole.
We implement our real-time monitoring approach over numerous linear system
benchmarks and show that the computation can be performed rapidly in
practice. Furthermore, we also examine the alarms reported by our approach
and show how some of the alarms can be used to improve the controller in an
offline fashion.
@inproceedings{Chen_Sankaranarayanan__2017__Model,
author = { Xin Chen and Sriram Sankaranarayanan },
title = { Model-Predictive Real-Time Monitoring of Linear Systems },
booktitle = { IEEE Real-Time Systems Symposium (RTSS) },
year = { 2017 },
pages = { 297-306 },
publisher = { IEEE Press }}
Xin Chen and Sergio Mover and Sriram Sankaranarayanan, Compositional Relational Abstraction for Nonlinear Systems In ACM Transactions on Embedded Computing Systems (Special Issue for EMSOFT 2017) Vol. 16(5s), pp. 187 (2017). Note: EMSOFT 2017 Best Paper Award Nomination
URL Abstract Bib Topics
We propose techniques to construct abstractions for nonlinear
dynamics in terms of relations expressed in linear arithmetic. Such
relations are useful for translating the closed loop verification
problem of control software with continuous-time, nonlinear plant
models into discrete and linear models that can be handled by
efficient software verification approaches for discrete-time
systems. We construct relations using Taylor model based flowpipe
construction and the systematic composition of relational
abstractions for smaller components. We focus on developing
efficient schemes for the special case of composing abstractions for
linear and nonlinear components. We implement our ideas using a
relational abstraction system, using the resulting abstraction
inside the verification tool NuXMV, which implements numerous
SAT/SMT solver-based verification techniques for discrete
systems. Finally, we evaluate the application of relational
abstractions for verifying properties of time triggered controllers,
comparing with the Flow* tool. We conclude that relational
abstractions are a promising approach towards nonlinear hybrid
system verification, capable of proving properties that are beyond
the reach of tools such as Flow*. At the same time, we highlight
the need for improvements to existing linear arithmetic SAT/SMT
solvers to better support reasoning with large relational
abstractions.
@article{Chen_Mover_Sankaranarayanan__2017__Compositional,
author = { Xin Chen and Sergio Mover and Sriram Sankaranarayanan },
title = { Compositional Relational Abstraction for Nonlinear Systems },
journal = { ACM Transactions on Embedded Computing Systems (Special Issue for EMSOFT 2017) },
volume = { 16(5s) },
year = { 2017 },
pages = { 187 },
publisher = { ACM Press }}
Xin Chen and Souradeep Dutta and Sriram Sankaranarayanan, Formal Verification of a Multi-Basal Insulin Infusion Control Model. In Workshop on Applied Verification of Hybrid Systems (ARCH), pp. 16 (2017).
URL Abstract Bib Topics
Ram Das Diwakaran, Sriram Sankaranarayanan, and Ashutosh Trivedi, Analyzing Neighborhoods of Falsifying Traces in Cyber-Physical Systems In Intl. Conference on Cyber-Physical Systems (ICCPS), pp. 109-119 (2017).
URL Abstract Bib
We study the problem of analyzing falsifying traces of
cyber-physical systems.
Specifically, given a system model and an input which is a counterexample
to a property of interest, we wish to understand which parts of the inputs
are ``responsible'' for the counterexample as a whole.
Whereas this problem is well known to be hard to solve precisely, we provide an
approach based on learning from repeated simulations of the system under test.
Our approach generalizes the classic concept of ``one-at-a-time'' sensitivity
analysis used in the risk and decision analysis community to understand how
inputs to a system influence a property in question.
Specifically, we pose the problem as one of finding a neighborhood of inputs
that contains the falsifying counterexample in question, such that each point
in this neighborhood corresponds to a falsifying input with a high probability.
We use ideas from statistical hypothesis testing to infer and validate such
neighborhoods from repeated simulations of the system under test.
This approach not only helps to understand the sensitivity of
these counterexamples to various parts of the inputs, but also
generalizes or widens the given counterexample by returning a
neighborhood of counterexamples around it.
We demonstrate our approach on a series of increasingly complex examples
from automotive and closed loop medical device domains.
We also compare our
approach against related techniques based on regression and machine learning.
@inproceedings{Diwakaran_Sankaranarayanan_Trivedi__2017__Analyzing,
author = { Ram Das Diwakaran, Sriram Sankaranarayanan, and Ashutosh Trivedi },
title = { Analyzing Neighborhoods of Falsifying Traces in Cyber-Physical Systems },
booktitle = { Intl. Conference on Cyber-Physical Systems (ICCPS) },
year = { 2017 },
pages = { 109-119 },
publisher = { ACM Press }}
Xin Chen, and Sriram Sankaranarayanan, Decomposed Reachability Analysis for Nonlinear Systems In IEEE Real Time Systems Symposium (RTSS), pp. 13-24 (2016).
URL Abstract Bib Topics
We introduce an approach to conservatively abstract a nonlinear
continuous system by a hybrid automaton whose continuous dynamics are
given by a decomposition of the original dynamics. The decomposed
dynamics is in the form of a set of lower-dimensional ODEs with
time-varying uncertainties whose ranges are defined by the
hybridization domains. We propose several techniques in the paper to
effectively compute abstractions and flowpipe
overapproximations. First, a novel method is given to reduce the
overestimation accumulation in a Taylor model flowpipe construction
scheme. Then we present our decomposition method, as well as the
framework of on-the-fly hybridization. A combination of the two
techniques allows us to handle much larger, nonlinear systems
with comparatively large initial sets. Our prototype implementation
is compared with existing reachability tools for offline and online flowpipe construction
on challenging
benchmarks of dimensions ranging from 7 to 30, with favorable
results.
@inproceedings{Chen_Sankaranarayanan__2016__Decomposed,
author = { Xin Chen, and Sriram Sankaranarayanan },
title = { Decomposed Reachability Analysis for Nonlinear Systems },
booktitle = { IEEE Real Time Systems Symposium (RTSS) },
year = { 2016 },
pages = { 13-24},
publisher = { IEEE Press }}
Sriram Sankaranarayanan, Suhas Akshar Kumar, Faye Cameron, B. Wayne Bequette, Georgios Fainekos, and David M. Maahs, Model-Based Falsification of an Artificial Pancreas Control System. In ACM SIGBED Review (Special Issue on Medical Cyber Physical Systems) Vol. TBA, pp. TBA (2016). Note: Presented at MEDCPS Workshop 2016
URL Abstract Bib Topics
We present a model-based falsification scheme for artificial
pancreas controllers. Our approach performs a closed-loop simulation
of the control software using models of the human insulin-glucose
regulatory system. Our work focuses on testing properties of an
overnight control system for hypoglycemia/hyperglycemia minimization
in patients with type-1 diabetes. This control system is currently
the subject of extensive phase II clinical trials.
We describe how the overall closed loop simulator is constructed, and
formulate properties to be tested. Significantly, the closed loop
simulation incorporates the control software, as is, without any
abstractions. Next, we demonstrate the use of a simulation-based
falsification approach to find potential property violations in the
resulting control system. We formulate a series of properties about
the controller behavior and examine the violations obtained. Using
these violations, we propose modifications to the controller software
to improve its performance under these adverse (corner-case)
scenarios. We also illustrate the effectiveness of robustness as a
metric for identifying interesting property violations. Finally, we
identify important open problems for future work.
@article{Sankaranarayanan_Others__2016__Model,
author = { Sriram Sankaranarayanan, Suhas Akshar Kumar, Faye Cameron, B. Wayne Bequette, Georgios Fainekos, and David M. Maahs },
title = { Model-Based Falsification of an Artificial Pancreas Control System. },
journal = { ACM SIGBED Review (Special Issue on Medical Cyber Physical Systems) },
volume = { TBA },
year = { 2016 },
pages = { TBA },
publisher = { ACM Press }}
Gregory P. Forlenza, Sriram Sankaranarayanan, and David M. Maahs, Refining the Closed Loop in the Data Age: Research-to-Practice Transitions in Diabetes Technology (Editorial) In Diabetes Technology and Therapeutics Vol. 17(5), pp. 304-306 (2015).
URL Abstract Bib Topics
The editorial analyzes a recent clinical study reporting on
the impact of insulin pump shutoff on patients with
type-1 diabetes. In a broader context, it examines
the need for enhanced formal verification techniques
applied to more sophisticated artificial pancreas
controllers that remain under development.
@article{Forlenza_Others__2015__Refining,
author = { Gregory P. Forlenza, Sriram Sankaranarayanan, and David M. Maahs },
title = { Refining the Closed Loop in the Data Age: Research-to-Practice Transitions in Diabetes Technology (Editorial) },
journal = { Diabetes Technology and Therapeutics },
volume = { 17 },
year = { 2015 },
pages = { 304-306 },
publisher = { Mary Ann Libert Inc. }}
Fraser Cameron, Georgios Fainekos, David M. Maahs, and Sriram Sankaranarayanan, Towards a Verified Artificial Pancreas: Challenges and Solutions for Runtime Verification In Proceedings of Runtime Verification (RV'15), Volume 9333 of Lecture Notes in Computer Science pp. 3-17 (2015). Note: Invited Keynote Paper
URL Abstract Bib Topics
In this paper, we briefly examine the recent developments in artificial pancreas controllers, that automate the delivery of insulin to patients with type-1 diabetes. We argue the need for offline and online runtime verification for these devices, and discuss challenges that make verification hard. Next, we examine a promising simulation-based falsification approach based on robustness semantics of temporal logics. These ideas are implemented in the tool S-Taliro that automatically searches for violations of metric temporal logic (MTL) requirements for Simulink(tm)/Stateflow(tm) models. We illustrate the use of S-Taliro for finding interesting property violations in a PID-based hybrid closed loop control system.
@inproceedings{Cameron_Fainekos_Maahs_Sankaranarayanan__2015__Towards,
author = { Fraser Cameron, Georgios Fainekos, David M. Maahs, and Sriram Sankaranarayanan },
title = { Towards a Verified Artificial Pancreas: Challenges and Solutions for Runtime Verification },
booktitle = { Proceedings of Runtime Verification (RV'15) },
year = { 2015 },
pages = { 3-17 },
series = { Lecture Notes in Computer Science },
volume = { 9333 }}
Stephen M. Kissler, Cody Cichowitz, Sriram Sankaranarayanan, and David M. Bortz, Determination of personalized diabetes treatment plans using a two-delay model In J. Theoretical Biology Vol. 359(Oct), pp. 101-111 (2014).
URL Abstract Bib Topics
Diabetes cases worldwide have risen steadily over the past decades, lending urgency to the search for more efficient, effective, and personalized ways to treat the disease. Current treatment strategies, however, may fail to maintain ultradian oscillations in blood glucose concentration, an important element of a healthy alimentary system. Building upon recent successes in mathematical modeling of the human glucose-insulin system, we show that both food intake and insulin therapy likely demand increasingly precise control over insulin sensitivity if oscillations at a healthy average glucose concentration are to be maintained. We then suggest guidelines and personalized treatment options for diabetic patients that maintain these oscillations. We show that for a type II diabetic, both blood glucose levels can be controlled and healthy oscillations maintained when the patient gets an hour of daily exercise and is placed on a combination of Metformin and sulfonylurea drugs. We note that insulin therapy and an additional hour of exercise will reduce the patient's need for sulfonylureas. Results of a modeling analysis suggest that a typical type I diabetic's blood glucose levels can be properly controlled with a constant insulin infusion between 0.45 and 0.7 microU/(ml*min). Lastly, we note that all suggested strategies rely on existing clinical techniques and established treatment measures, and so could potentially be of immediate use in the design of an artificial pancreas.
@article{Kissler_Others__2014__Determination,
author = { Stephen M. Kissler, Cody Cichowitz, Sriram Sankaranarayanan, and David M. Bortz },
title = { Determination of personalized diabetes treatment plans using a two-delay model },
journal = { J. Theoretical Biology },
volume = { 359 },
year = { 2014 },
pages = { 101-111 },
publisher = { Elsevier Press }}
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Disclaimer
This page was authored and maintained by Sriram Sankaranarayanan, who is responsible for its content. Please direct all queries to Sriram at his email address.
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