Modelling Optimization and Control of Biomedical Systems E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 466

Veröffentlichungsjahr: 2017

Beschreibung

Shows the newest developments in the field of multi-parametric model predictive control and optimization and their application for drug delivery systems 

This book is based on the Modelling, Control and Optimization of Biomedical Systems (MOBILE) project, which was created to derive intelligent computer model-based systems for optimization of biomedical drug delivery systems in the cases of diabetes, anaesthesia, and blood cancer. These systems can ensure reliable and fast calculation of the optimal drug dosage without the need for an online computer—while taking into account the specifics and constraints of the patient model, flexibility to adapt to changing patient characteristics and incorporation of the physician’s performance criteria, and maintaining the safety of the patients.

Modelling Optimization and Control of Biomedical Systems covers: mathematical modelling of drug delivery systems; model analysis, parameter estimation, and approximation; optimization and control; sensitivity analysis & model reduction; multi-parametric programming and model predictive control; estimation techniques; physiologically-based patient model; control design for volatile anaesthesia; multiparametric model based approach to intravenous anaesthesia; hybrid model predictive control strategies; Type I Diabetes Mellitus; in vitro and in silico block of the integrated platform for the study of leukaemia; chemotherapy treatment as a process systems application; and more.

• Introduces readers to the Modelling, Control and Optimization of Biomedical Systems (MOBILE) project
• Presents in detail the theoretical background, computational tools, and methods that are used in all the different biomedical systems
• Teaches the theory for multi-parametric mixed-integer programming and explicit optimal control of volatile anaesthesia
• Provides an overview of the framework for modelling, optimization, and control of biomedical systems

This book will appeal to students, researchers, and scientists working on the modelling, control, and optimization of biomedical systems and to those involved in cancer treatment, anaesthsia, and drug delivery systems.

Leseprobe

Table of Contents

Cover

Title Page

List of Contributors

Preface

Part I

1 Framework and Tools: A Framework for Modelling, Optimization and Control of Biomedical Systems

1.1 Mathematical Modelling of Drug Delivery Systems

1.2 Model analysis, Parameter Estimation and Approximation

1.3 Optimization and Control

References

2 Draft Computational Tools and Methods

2.1 Introduction

2.2 Sensitivity Analysis and Model Reduction

2.3 Multiparametric Programming and Model Predictive Control

2.4 Estimation Techniques

2.5 Explicit Hybrid Control

References

3 Volatile Anaesthesia

3.1 Introduction

3.2 Physiologically Based Patient Model

3.3 Model Analysis

3.4 Control Design for Volatile Anaesthesia

Conclusions

Appendix

References

4 Intravenous Anaesthesia

4.1 A Multiparametric Model‐based Approach to Intravenous Anaesthesia

4.2 Simultaneous Estimation and Advanced Control

4.3 Hybrid Model Predictive Control Strategies

4.4 Conclusions

References

Part II

5 Part A: Type 1 Diabetes Mellitus: Modelling, Model Analysis and Optimization

5.a Type 1 Diabetes Mellitus: Modelling, Model Analysis and Optimization

Part B: Type 1 Diabetes Mellitus: Glucose Regulation

5.b Type 1 Diabetes Mellitus: Glucose Regulation

Appendix 5.1

Appendix 5.2

Appendix 5.3

References

Part III

6 An Integrated Platform for the Study of Leukaemia

6.1 Towards a Personalised Treatment for Leukaemia: From

in vivo

to

in vitro

and

in silico

6.2

In vitro

Block of the Integrated Platform for the Study of Leukaemia

6.3

In silico

Block of the Integrated Platform for the Study of Leukaemia

6.4 Bridging the Gap Between

in vitro

and

in silico

References

7

In vitro

Studies: Acute Myeloid Leukaemia

7.1 Description of Biomedical System

7.2 Experimental Part

7.3 Cellular Biomarkers for Monitoring Leukaemia

in vitro

7.4 From

in vitro

to

in silico

References

8

In silico

Acute Myeloid Leukaemia

8.1 Introduction

8.2 Chemotherapy Treatment as a Process Systems Application

8.3 Analysis of a Patient Case Study

8.4 Conclusions

Appendix 8A Mathematical Model

Appendix 8B Patient Data

References

Index

End User License Agreement

List of Tables

Chapter 01

Table 1.1 The most common types of empirical pharmacodynamic models.

Chapter 02

Table 2.1 Classification of the main order reduction techniques

Table 2.2 Summary of the literature on model order reduction for mp‐MPC applications.

Table 2.3 mp‐QP algorithm.

Table 2.4 Literature review on continuous parametric programming techniques for static problems.

Table 2.5 Literature review on mixed‐integer parametric programming techniques for static problems.

Table 2.6 Literature review on parametric programming/sensitivity analysis in dynamic optimization.

Table 2.7 Relevant methods for designing robust model‐based controllers.

Chapter 03

Table 3.1 Calculation of patient‐specific tissue mass.

Table 3.2 Range and default values for PK and PD parameters and variables (partition coefficients at 37 °C for isoflurane).

Table 3.3 Summary of the cases for sensitivity analysis.

Table 3.4 Sobol’s sensitivity indices using GUI‐HDMR for Cases 1–4 given in Table 3.3 after 3.5 min and 20 min.

Table 3.5 Percentage of change of

C

E

and BIS after 5, 20 and 60 min, compared to the output with default PK and PD variables and parameters.

Table 3.6 Correlation matrix

C

of

C

50

,

γ

,

k

e

0

and

V

L

for the parameter estimation problem

PK

u

and

PK

u

;

PK

u

above diagonal and

PK

l

below diagonal.

Table 3.7 Correlation matrix of the PD parameters, entries for

PK

u

above diagonal and

PK

l

below diagonal, and estimated PD parameters for

PK

l

and

PK

u

.

Table 3.8 Patients’ characteristics, calculated values of the lung volume and cardiac output and estimated PD parameters.

Table 3A.1 Denotation of variables and parameters of the PBPK/PD model.

Table 3A.2 Denotation of subscripts in the PBPK/PD model.

Chapter 04

Table 4.1 Comparison of GMDH and HDMR at

t

 = 14.

Table 4.2 Biometric values of the virtual patients.

Table 4.3 Hybrid model for intravenous anaesthesia.

Chapter 05-1

Table 5.a.1 Mathematical models of glucose–insulin system.

Table 5.a.2 Variables of glucose metabolism model.

Table 5.a.3 Variable subscript denotation.

Table 5.a.4 Ratio of cardiac output at rest.

Table 5.a.5 Ratio of glucose uptake.

Table 5.a.6 Ratio of capillary volume.

Table 5.a.7 Density of muscles and adipose tissue.

Table 5.a.8 Model equations of three proposed insulin kinetics models and a reference model; schematic representation of the models.

Table 5.a.9 Variable and parameter definition of Models 1, 2 and 3.

Table 5.a.10 Goodness of fit of proposed models and model selection.

Table 5.a.11 Optimal mean parameter estimates and standard deviations reported in parentheses.

Table 5.a.12 Parameter estimation results.

Table 5.a.13 Model parameters’ default values and range, and SIs for all parameters and for those related to intra‐patient variability calculated with the GUI‐HDMR toolbox.

Table 5.a.14 Optimal parameter estimates, presented as mean (lower‐upper) value for the 10 patients.

Table 5.a.15 Area under the curve (outside the normal range).

Chapter 05-2

Table 5.b.1 Selected clinical studies that evaluate MPC as a control strategy to regulate BG concentration in T1DM.

Table 5.b.2 Estimated parameters of linearized model for 10 adults.

Table 5.b.3 Meal disturbance types.

Table 5.b.4 Control designs.

Table 5.b.5 Prediction horizon for the 10 patients.

Table 5.b.6 Inequality constraints.

Table 5.b.7 Specifications of MPC 2 and the Kalman filter.

Table 5.b.8 CD

3

(predefined meal plan).

Table 5.b.9 CD

4

(unmeasured).

Chapter 07

Table 7.1 The phases of the cell cycle.

Table 7.2 Stress biomarkers for normal and abnormal HSCs.

Chapter 08

Table 8.1 PK models of cancer drugs.

Table 8.2 Formulas of PD models.

Table 8.3 Brief guide to model equations.

Table 8.4 Chemotherapy process optimisation algorithm.

Table 8.5 PK, PD and cell cycle parameters and inter‐individual ranges used for model sensitivity analysis and sensitivity index results.

Table 8.6 Cell cycle times fitted for the clinical data of 6 patients under LD and DA protocol (Appendix 8B).

Table 8.7 Leukaemic population of patient P016 based on simulation model results.

Table 8.8 Optimal schedule of the first chemotherapy cycle for Patient P016.

Table 8.9 Optimal LDAC induction treatment protocol for Patient P016.

Table 8.10 Leukaemic and normal cell populations for P016, over the simulation and optimisation induction treatment protocols.

List of Illustrations

Chapter 01

Figure 1.1 Mathematical representation of a drug delivery system.

Figure 1.2 Schematic of a two‐compartment pharmacokinetic model.

Figure 1.3 Schematic of a physiological pharmacokinetic model.

Figure 1.4 Illustration of a pharmacodynamic dose–response curve.

Figure 1.5 Framework towards optimal drug delivery systems.

Chapter 02

Figure 2.1 A framework for explicit/multiparametric model predictive control and moving horizon estimation.

Figure 2.2 Schematic representation of the MOR approximation procedure.

Figure 2.3 Concept of MHE.

Figure 2.4 A schematic representation of a binary search tree used in branch‐and‐bound methods for the solution of certain mp‐MIP problems.

Figure 2.5 A schematic representation of the comparison procedure employed in Acevedo and Pistikopoulos (1997). According to Equation (3.10), in case (a),

; in case (b),

; and in case (c),

CR

int

is split into

CR

1

and

CR

2

.

Figure 2.6 A schematic representation of different scenarios for a comparison procedure of objective functions featuring bilinear and/or quadratic terms, an issue considered in Axehill

et al.

(2011, 2014) and Oberdieck

et al.

(2014).

Figure 2.7 A schematic representation of two comparison procedures presented for the solution of mp‐MIQP problems: in case (a), McCormick relaxations (McCormick 1976) are used to divide

CR

int

into three regions, one of which contains an envelope of solutions (grey area) (Oberdieck

et al.

2014), while in case (b) the entire

CR

int

is regarded as an envelope of solution (grey area) (Axehill

et al.

2011, 2014).

Figure 2.8 The general framework for the solution of mp‐MIQP problems.

Figure 2.9 The three classifications of overlap between

CR

k

and

PP

j

.

Figure 2.10 A schematic representation for the exact comparison procedure for the solution of mp‐MIQP problems. In part (a), the quadratic boundary resulting from the exact use of

Δ

z

(

θ

) is combined with

CR

int

to form a quadratically constrained critical region

CR

(part (b)). For this region, a polyhedral outer approximation

is calculated such that

(part [c]). In part (d), the corresponding mp‐QP problem is solved in

, resulting in a partition of the parameter space. Each of these critical regions,

CR

1 ‐ 3

, is compared against the current upper bound, thus resulting in a new set of

Δ

z

(

θ

) (part [e]). Lastly, in part (f), the original quadratic constraints from

CR

are reintroduced, thus closing the loop.

Chapter 03

Figure 3.1 Structure of the physiologically based patient model. (a) Patient body; (b) fluxes in the lungs.

Figure 3.2 Structure of one tissue compartment.

Figure 3.3 BIS for PK (left) and PD (right) variability in Table 3.2. The solid line marks the BIS for the model adjusted to patient 1. The grey dots mark the measured BIS.

Figure 3.4 Time‐varying sensitivity indices (SI) for Cases 1–4. The three bottom plots denote a zoomed‐in scope for Case 1, Case 2 and Case 4.

Figure 3.5 BIS output for estimated PD parameters in Table 3.7 capturing PK variability

Figure 3.6 Inspired and expired isoflurane concentrations and BIS for patient characteristics and parameters given in Table 3.8.

Figure 3.7 Closed‐loop control design for volatile anaesthesia.

Figure 3.8 Control design for algebraic Hill equation.

Figure 3.9 Control design for linearized Hill equation.

Figure 3.10 Linearized Hill equation at BIS = 50. The dot marks the linearization point.

Figure 3.11 Piecewise linearization of the Hill equation. The dots mark the intersection of the linearization functions and the switching points of the controllers, respectively.

Figure 3.12 Closed‐loop control design for on‐line parameter estimation of C

50

.

Figure 3.13 Decision process of the on‐line parameter estimation bloc.

Figure 3.14 Estimated

of

for Patient 3.

Figure 3.15 Control input for Patient 3 of

CD

1

and

.

Figure 3.16

and actual

C

e

for Patient 3 of

CD

1

and

.

Figure 3.17

BIS

R

and actual BIS for Patient 3 of

CD

1

and

.

Figure 3.18 Control input for Patients 1–3 of

.

Figure 3.19

and actual C

e

for Patients 1–3 of

.

Figure 3.20

BIS

R

and actual BIS for Patients 1–3 of

.

Figure 3.21 Estimated

of CD for Patients 1–3.

Figure 3.22 Framework presented in this thesis for volatile anaesthesia.

Chapter 04

Figure 4.1 Compartmental model of the patient. PK = the pharmacokinetic model; PD = the pharmacodynamic model.

Figure 4.2 Schematic representation of the nonlinear SISO patient model for intravenous anaesthesia.

Figure 4.3 Evolution of the first‐order sensitivity indices.

Figure 4.4 Comparison GMDH‐HDMR for small data samples (

N

 = 40),

t

 = 14.

Figure 4.5 Controller design using local linearization.

Figure 4.6 Controller scheme using local linearization.

Figure 4.7 Controller scheme using exact linearization.

Figure 4.8 Controller scheme using exact linearization.

Figure 4.9 Control scheme development flowchart.

Figure 4.10 Case 1: EPSAC control scheme.

Figure 4.11 Case 2:mp‐MPC without nonlinearity compensation – control scheme.

Figure 4.12 Case 3:mp‐MPC with nonlinearity compensation – control scheme.

Figure 4.13 Case 4: mp‐MPC with nonlinearity compensation and estimator – control scheme.

Figure 4.14 BIS output for all 13 patients for Case 1.

Figure 4.15 Map of critical regions, Case 2.

Figure 4.16 BIS output for all 13 patients for Case 2.

Figure 4.17 Map of critical regions, Case 3 and Case 4.

Figure 4.18 BIS output for all 13 patients for Case 3.

Figure 4.19 BIS output for all 13 patients for Case 4.

Figure 4.20 BIS response for the four controllers for PaN.

Figure 4.21 Output for the four controllers for PaN.

Figure 4.22 BIS response for the four controllers for patient 9.

Figure 4.23 Output for the four controllers for patient 9.

Figure 4.24 The artificially generated disturbance signal.

Figure 4.25 BIS response for the four controllers for PaN with disturbance.

Figure 4.26 Output for the four controllers for PaN with disturbance.

Figure 4.27 BIS response for the four controllers for patient 9 with disturbance.

Figure 4.28 Output for the four controllers for patient 9 with disturbance.

Figure 4.29 Schematic of simultaneous mp‐MHE and mp‐MPC for intravenous anaesthesia.

Figure 4.30 Map of critical regions – mp‐MPC.

Figure 4.31 BIS response for all 13 patients in the induction phase – nominal mp‐MPC.

Figure 4.32 Propofol infusion rate for all 13 patients in the induction phase – nominal mp‐MPC.

Figure 4.33 BIS response for all 13 patients in the induction phase – simultaneous mp‐MPC and Kalman filter.

Figure 4.34 Propofol infusion rate for all 13 patients in the induction phase – simultaneous mp‐MPC and Kalman filter.

Figure 4.35 BIS response for all 13 patients in the induction phase – simultaneous mp‐MPC and mp‐MHE.

Figure 4.36 Propofol infusion rate for all 13 patients in the induction phase – simultaneous mp‐MHE and mp‐MPC.

Figure 4.37 BIS response of the three controllers for patient 9 in the induction phase without noise.

Figure 4.38 BIS response of the three controllers for patient 9 in the induction phase without noise – zoomed in.

Figure 4.39 Propofol infusion rate of the three controllers for patient 9 in the induction phase without noise.

Figure 4.40 BIS response of the three controllers for patient 9 in the maintenance phase without noise.

Figure 4.41 Propofol infusion rate of the three controllers for patient 9 in the maintenance phase without noise.

Figure 4.42 BIS response of the three controllers for patient 9 in the maintenance phase – the B–C–D–E interval – without noise.

Figure 4.43 Propofol infusion rate of the three controllers for patient 9 in the maintenance phase – the B–C–D–E interval – without noise.

Figure 4.44 The original Hill curve and a piecewise linearized version. The red dots denote the points around which the linearization was performed, while the purple arrows show the switching points

λ

1

and

λ

2

, respectively.

Figure 4.45 Map of critical regions – mp‐hMPC.

Figure 4.46 Robust hybrid mp‐MPC control scheme.

Figure 4.47 BIS output for all 13 patients without offset correction – induction phase.

Figure 4.48 Drug infusion for all 13 patients without offset correction – induction phase.

Figure 4.49 BIS output for all 13 patients without offset correction – maintenance phase.

Figure 4.50 Drug infusion for all 13 patients without offset correction – maintenance phase.

Figure 4.51 BIS output for all 13 patients – strategy 2 – induction phase.

Figure 4.52 Drug infusion for all 13 patients – strategy 2 – induction phase.

Figure 4.53 BIS output for all 13 patients – strategy 2 – maintenance phase.

Figure 4.54 Drug infusion for all 13 patients – strategy 2 – maintenance phase.

Chapter 05-1

Figure 5.a.1 Incidence of type 1 diabetes mellitus (T1DM) worldwide.

Figure 5.a.2 The framework of an automated insulin delivery system.

Figure 5.a.3 Structure of the physiologically based compartmental model of glucose metabolism in T1DM.

Figure 5.a.4 Detailed glucose uptake in the periphery.

Figure 5.a.5 Comparison of Models 1, 2 and 3 and a reference model with experimental data.

Figure 5.a.6 Effect of subcutaneous insulin injection on endogenous glucose production.

Figure 5.a.7 Time‐varying SIs when all parameters are considered.

Figure 5.a.8 Time‐varying SIs when intra‐patient variability‐related parameters are considered.

Figure 5.a.9 Comparison of blood glucose concentration (mg/dL) as predicted from the proposed model with the Simulator, for the 10 adults when a meal plan of 45 g, 70 g and 70 g of carbs are considered at 420 min, 720 min and 1080 min, respectively. The insulin infusion (U) is shown at the right axis for every patient.

Figure 5.a.10 Delayed insulin effect.

Figure 5.a.11 Patient‐dependent time delay.

Figure 5.a.12 Time delay dependent on patient and bolus.

Figure 5.a.13 Optimization (grey line) and simulation (black line) glucose profiles.

Figure 5.a.14 Optimal glucose profiles when insulin is given as a bolus and as a piecewise constant infusion.

Chapter 05-2

Figure 5.b.1 Model‐based control structure.

Figure 5.b.2 Framework for MPC controller design.

Figure 5.b.3 Comparison of full‐state and reduced linearized model for patient 2.

Figure 5.b.4 Comparison of original model and linearized model when 50 g of carbs are consumed and a 5 U bolus is given to patient 2.

Figure 5.b.5 Basic scheme of discrete MPC.

Figure 5.b.6 Proposed control strategy to compensate for unknown meal disturbances consisting of two controllers: the reference control that regulates glucose for a reference meal plan, and the correction control that regulates the difference of the glucose between a real and reference meal plan.

Figure 5.b.7 MPC control for 10 adults of UVa/Padova Simulator for predefined (

CD

1

) and announced meal disturbances (

CD

2

). Upper graphs: blood glucose concentration (mg/dL) profiles; lower graphs: control action, insulin (U/min).

Figure 5.b.8 Comparison of glucose regulation with control designs 3 and 4 for adult 6. The meals are given at 420, 720 and 1080 min and contain 75, 100 and 90 g of carbohydrates, respectively.

Figure 5.b.9 Evaluation of

CD

3

when a meal of 50 g is given 30 min in advance, 30 min after and simultaneous with the reference meal of 30 g.

Figure 5.b.10 Multiparametric MPC.

Figure 5.b.11 Comparison of the original and state‐space model.

Figure 5.b.12 Critical regions for mp‐MPC.

Figure 5.b.13 Closed‐loop control performance.

Chapter 06

Figure 6.1 Towards optimisation/personalisation of chemotherapy for leukaemia treatment.

Figure 6.2 (a): Geometry of the 3D scaffolds; (b–c) scanning electron microscopy (SEM) images of the highly porous 3D scaffolds, including seeded leukaemic cells.

Figure 6.3 Mathematical optimisation of chemotherapy treatment for AML.

Chapter 07

Figure 7.1 The human haematopoietic system.

Figure 7.2 The cell cycle.

Figure 7.3 Cyclin expression throughout the cell cycle.

Figure 7.4 K‐562 growth (a,b) the 2D and (c,d) the 3D systems, at different oxygen levels: (a,c) 20% O

2

, and (b,d) 5% O

2

. Different colours represent different glucose levels: () [CTR], () [HIGH] and () [LOW].

Figure 7.5 Evolution of glutamate [Glu] for all the environmental conditions under study. Different colours represent the two different culturing systems: () 2D cultures and () 3D scaffolds.

Figure 7.6 Evolution of lactate [Lac] for all the environmental conditions under study. Different colours represent the two different culturing systems: () 2D cultures and () 3D scaffolds.

Chapter 08

Figure 8.1 Schematic diagram of PK and PD: blue boxes are for the PK model, and they are connected to the red cycle that represents the PD part of drug action.

Figure 8.2 The process of drug delivery. Drug delivery is governed by four mechanisms: absorption, distribution, metabolism and excretion. Each of these mechanisms is deprived of further mechanisms. Inter‐ and intra‐patient variability in these mechanisms is the probable source of PK variability.

Figure 8.3 Framework for the derivation of an optimal personalised chemotherapy protocol.

Figure 8.4 Patient P016 behaviour over the first chemotherapy cycle (days 1–11) and the recovery period prior to the second chemotherapy cycle (days 11–67). The dashed line is for the leukaemic cell population over the optimised protocol; the straight black line is for the leukaemic cell over the simulation of the clinical applied protocol; the cycle signs are for the normal population at the start and end dates of the optimisation protocol; the

x

signs are for the normal population at the start and end dates of the simulation protocol; and the grey line represents BM hypoplasia objective.

Figure 8.5 Patient P016 behaviour over the second chemotherapy cycle (days 67–77) and the recovery period prior to the BM aspirate at treatment completion (days 77–100). The dashed line is for the leukaemic cell population over the optimised protocol; the straight black line is for the leukaemic cell over the simulation of the clinical applied protocol; the cycle signs are for the normal population at the start and end dates of the optimisation protocol; the

x

signs are for the normal population at the start and end dates of the simulation protocol; and the grey line represents BM hypoplasia objective.

Guide

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Modelling Optimization and Control of Biomedical Systems

Edited by

Efstratios N. Pistikopoulos

Texas A&M University, USA

Ioana Naşcu

Texas A&M University, USA

Eirini G. Velliou

Department of Chemical and Process Engineering University of Surrey, UK

This edition first published 2018© 2018 John Wiley & Sons Ltd

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List of Contributors

Dr. Maria Fuentes‐GariProcess Systems Enterprise (PSE)LondonUK

Professor Michael C. GeorgiadisLaboratory of Process Systems EngineeringSchool of Chemical EngineeringAristotle University of ThesalonikiGreece

Dr. Alexandra KriegerJacobs ConsultancyKreisfreie Stadt Aachen AreaGermany

Dr. Romain LambertDepartment of Chemical EngineeringImperial College LondonUK

Professor Athanasios MantalarisDepartment of Chemical EngineeringImperial College LondonUK

Dr. Ruth MisenerDepartment of ComputingImperial College LondonUK

Dr. Ioana NaşcuArtie McFerrin Department of Chemical EngineeringTexas A&M UniversityCollege StationUSA

Dr. Richard OberdieckDONG energy A/SGentofteDenmark

Dr. Nicki PanoskaltsisDepartment of MedicineImperial College LondonUK

Dr. Eleni PefaniClinical Pharmacology Modelling and SimulationGSKUK

Professor Efstratios N. PistikopoulosTexas A&M Energy InstituteArtie McFerrin Department of Chemical EngineeringTexas A&M UniversityUSA

Dr. Pedro RivottiDepartment of Chemical EngineeringImperial College LondonUK

Susana Brito dos SantosDepartment of Chemical EngineeringImperial College LondonUK

Dr. Eirini G. VelliouDepartment of Chemical and Process EngineeringFaculty of Engineering and Physical SciencesUniversity of SurreyUK

Dr. Stamatina ZavitsanouPaulson School of Engineering & Applied SciencesHarvard UniversityUSA

Preface

A great challenge when dealing with severe diseases, such as cancer or diabetes, is the implementation of an appropriate treatment. Design of treatment protocols is not a trivial issue, especially since nowadays there is significant evidence that the type of treatment depends on specific characteristics of individual patients.

In silico design of high‐fidelity mathematical models, which accurately describe a specific disease in terms of a well‐defined biomedical network, will allow the optimisation of treatment through an accurate control of drug dosage and delivery. Within this context, the aim of the Modelling, Control and Optimisation of Biomedical Systems (MOBILE) project is to derive intelligent computer model‐based systems for optimisation of biomedical drug delivery systems in the cases of diabetes, anaesthesia and blood cancer (i.e., leukaemia).

From a computational point of view, the newly developed algorithms will be able to be implemented on a single chip, which is ideal for biomedical applications that were previously off‐limits for model‐based control. Simpler hardware is adequate for the reduced on‐line computational requirements, which will lead to lower costs and almost eliminate the software costs (e.g., licensed numerical solvers). Additionally, there is increased control power, since the new MPC approach can accommodate much larger – and more accurate – biomedical system models (the computational burden is shifted off‐line).

From a practical point of view, the absence of complex software makes the implementation of the controller much easier, therefore allowing its usage as a diagnostic tool directly in the clinic by doctors, clinicians as well as patients without the requirement of specialised engineers, therefore progressively enhancing the confidence of medical teams and patients to use computer‐aided practices. Additionally, the designed biomedical controllers increase treatment safety and efficiency, by carefully applying a “what‐if” prior analysis that is tailored to the individual patient’s needs and characteristics, therefore reducing treatment side effects and optimising the drug infusion rates. Flexibility of the device to adapt to changing patient characteristics and incorporation of the physician’s performance criteria are additional great advantages.

There were several highly significant achievements of the project for all different diseases and biomedical cases under study (i.e., diabetes, leukaemia and anaesthesia). From a computational point of view, achievements include the construction of high‐fidelity mathematical models as well as novel algorithm derivations. The methodology followed for the model design includes the following steps: (a) the derivation of a high‐fidelity model, (b) the conduction of sensitivity analysis, (c) the application of parameter estimation techniques on the derived model in order to identify and estimate the sensitive model parameters and variables and (d) the conduction of extensive validation studies based on patient and clinical data. The validated model is then reduced to an approximate model suitable for optimisation and control via model reduction and/or system identification algorithms. The several theoretical (in silico) components are incorporated in a closed‐loop (in silico–in vitro) framework that will be evaluated with in vitro trials (i.e., through experimental evaluation of the control‐based optimised drug delivery). The outcome of the experiments will indicate the validity of the suggested closed‐loop delivery of anaesthetics, chemotherapy dosages for leukaemia and insulin delivery doses in diabetes. It should be mentioned that this is the first closed‐loop system including computational and experimental elements. The output of such a framework could be introduced, at a second step, in phase 1 clinical trials.

Chapter 1 is an overview of the framework for modelling, optimisation and control of biomedical systems. It describes the mathematical modelling of drug delivery systems that usually requires a pharmacokinetic part, a pharmacodynamic part and a link between the two. Model analysis, parameter estimation and approximation are used here in order to obtain an in‐depth understanding of the model. Mathematical optimisation and control of the biomedical system could lead to a better prediction of the optimal drug and/or therapy treatment for a specific disease.

Chapter 2 presents in detail the theoretical background, computational tools and methods that are used in all the different biomedical systems analysed within the book. More specifically, Chapter 2 focuses on describing the computational tools, part of the developed multiparametric model predictive control framework presented in Chapter 1. It also presents the theory for multiparametric mixed‐integer programming and explicit optimal control. This is part of the larger class of hybrid biomedical systems (i.e., biomedical systems featuring both discrete and continuous dynamics).

Chapters 3 and 4 aim at applying the presented framework to the process of anaesthesia: both volatile as well as intravenous. They present the procedure step by step from the model development to the design of a multiparametric model predictive controller for the control of depth of anaesthesia. Chapter 3 focuses on the process of volatile anaesthesia. A detailed physiologically based pharmacokinetic–pharmacodynamic patient model for volatile anaesthesia is presented where all relevant parameters and variables are analysed. A model predictive control (MPC) strategy is proposed to assure safe and robust control of anaesthesia by including an on‐line parameter estimation step that accounts for patient variability. A Kalman filter is implemented to obtain an estimate of the states based on the measurement of the end‐tidal concentration. An on‐line estimator is added to the closed control loop for the estimation of the PD parameter C50 during the course of surgery. Closed‐loop control simulations for the system for conventional MPC, explicit MPC and the on‐line parameter estimation are presented for induction and disturbances during maintenance of anaesthesia.

In Chapter 4, we describe the process of intravenous anaesthesia. The mathematical model for intravenous anaesthesia is presented in detail, and sensitivity analysis is performed. The main objective is to develop explicit MPC strategies for the control of depth of anaesthesia in the induction and maintenance phases. State estimation techniques are designed and implemented simultaneously with mp‐MPC strategies to estimate the state of each individual patient. Furthermore, a hybrid formulation of the patient model is performed, leading to a hybrid mp‐MPC that is further implemented using several robust techniques.

Chapter 5 is focused on type 1 diabetes mellitus, more specifically on modelling, model analysis, optimisation and glucose regulation. The basic idea is to develop an automated insulin delivery system that would mimic the endocrine functionality of a healthy pancreas. The first level is the development of a high‐fidelity mathematical model that represents in depth the complexity of the glucoregulatory system, presents adaptability to patient variability and demonstrates adequate capture of the dynamic response of the patient to various clinical conditions (normoglycaemia, hyperglycaemia and hypoglycaemia). This model is then used for detailed simulation and optimisation studies to gain a deep understanding of the system. The second level is the design of model‐based predictive controllers by incorporating techniques appropriate for the specific demands of this problem.

The last three chapters are focused on the development of a systematic framework for the personalised study and optimisation of leukaemia (i.e., a severe cancer of the blood): from in vivo to in vitro and in silico. More specifically, Chapter 6 is a general description of the independent building blocks of the integrated framework, which are further analysed in the next chapters. Chapter 7 focuses on the detailed description of the in vitro building block of the framework. More specifically, it includes analysis of the disease, analysis of the experimental platform and environmental (stress) stimuli that are monitored within the platform, and a description of cellular biomarkers for monitoring the evolution of leukaemia in vitro. Chapter 8 focuses on the in silico building block of the framework. It describes the pharmacokinetic and pharmacodynamic models developed for the optimisation of chemotherapy treatment for leukaemia. Finally, the simulation results and analysis of a patient case study are presented.

The main outcome of this work is to develop models and model‐based control and optimisation methods and tools for drug delivery systems, which would ensure: (a) reliable and fast calculation of the optimal drug dosage without the need for an on‐line computer, while taking into account the specifics and constraints of the patient model (personalised health care); (b) flexibility to adapt to changing patient characteristics, and incorporation of the physician’s performance criteria; and (c) safety of the patients, as optimisation of drug infusion rates would reduce the side effects of treatment. The major novelty introduced by mobile technology is that it is no longer necessary to trade off control performance against hardware and software costs in drug delivery systems. The parametric control technology will be able to offer state‐of‐the‐art model‐based optimal control performance in a wide range of drug delivery systems on the simplest of hardware. All of this will lead to some very important advantages, like: enhancing the confidence of medical teams to use computer‐aided practices, increasing the confidence of patients to use such practices, enhancing safety by carefully applying a “what‐if” prior analysis tailored made to patients’ needs, a simple “look‐up function,” an optimal closed‐loop response and cheap hardware implementation.

The book shows the newest developments in the field of multiparametric model predictive control and optimisation and their application for drug delivery systems.

This work was supported by the European Research Council (ERC), that is, by ERC‐Mobile Project (no. 226462), ERC‐BioBlood (no. 340719), the EU 7th Framework Programme (MULTIMOD Project FP7/2007‐2013, no. 238013), the Engineering and Physical Sciences Research Council (EPSRC: EP/G059071/1 and EP/I014640), the Richard Thomas Leukaemia Research Fund and the Royal Academy of Engineering Research Fellowship (to Dr. Ruth Misener).

Part I

1Framework and Tools: A Framework for Modelling, Optimization and Control of Biomedical Systems

Eirini G. Velliou1, Ioana Naşcu2, Stamatina Zavitsanou3, Eleni Pefani4, Alexandra Krieger5, Michael C. Georgiadis6, and Efstratios N. Pistikopoulos7

1 Department of Chemical and Process Engineering, Faculty of Engineering and Physical Sciences, University of Surrey, UK

2 Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, USA

3 Paulson School of Engineering & Applied Sciences, Harvard University, USA

4 Clinical Pharmacology Modelling and Simulation, GSK, UK

5 Jacobs Consultancy, Kreisfreie Stadt Aachen Area, Germany

6 Laboratory of Process Systems Engineering, School of Chemical Engineering, Aristotle University of Thesaloniki, Greece

7 Texas A&M Energy Institute, Artie McFerrin Department of Chemical Engineering, Texas A&M University, USA

1.1 Mathematical Modelling of Drug Delivery Systems

Drug delivery can be defined as the process of administering a pharmaceutical agent in the human body, including the consequent effects of this agent on the tissues and organs. Mathematical modelling of drug delivery can be divided into two different yet complementary approaches, the pharmacokinetic and pharmacodynamic approaches. Pharmacokinetics describes the effect of the drug in the body, by capturing absorption, distribution, diffusion and elimination of the drug. Pharmacodynamics describes the effects of a drug in the body, which are expressed mathematically by relations of drug dose–body responses. Usually, modelling of the drug delivery system requires a pharmacokinetic part, a pharmacodynamics part and a link between the two (Figure 1.1).

Figure 1.1 Mathematical representation of a drug delivery system.

Source: Ette and Willliams (2007). Reproduced with permission of John Wiley and Sons.

1.1.1 Pharmacokinetic Modelling

Two approaches for pharmacokinetic models dominate the literature, the compartmental models and the physiologically based pharmacokinetic models.

1.1.1.1 Compartmental Models

The basic idea of compartmental modelling is to group organs with similar properties, such as the well‐perfused organs, in one compartment and describe the uptake based on these tissues’ properties (e.g. drug solubility and perfusion). The basic assumptions of compartmental modelling are: (a) homogeneity: uniform distribution and instant mixing within the compartment; (b) conservation of mass; (c) the intrinsic properties are constant (e.g. temperature and volume); (d) there are no time delays between compartments; and (e) all exiting fluxes are linearly proportional to the drug concentration in the compartment.

The simplest approach is to consider the whole body as one single compartment in which the drug is administered and also eliminated. Usually, this mathematical approach is used for the description of drugs that are intravenously injected and well diffused, the elimination of which follows first‐order kinetics. Practically, within the human body, usually more than one compartment is considered due to the slow diffusion of the drug to the peripheral tissues (Figure 1.2).

Figure 1.2 Schematic of a two‐compartment pharmacokinetic model.

Source: Saltzman (2001). Reproduced with permission of Oxford University Press.

There are several challenges related to compartmental model development, such as the correlation of the model parameters (e.g. transfer coefficients) to physiological parameters, as well as difficulties related to the determination of the appropriate number of compartments that should be used in order to represent the pharmacokinetics of a population. Furthermore, the ability of these models to give a valid estimation of the drug profile of a newly studied patient is rather questionable. The major source of model uncertainty is due to the fact that the values of the variables are based on the interpretation of the mean concentration profile of a group of patients. This mean concentration profile in most of the cases is not representative of the behaviour of patients in the group studied, let alone the whole patient population. These drawbacks are satisfied to a certain extent by the physiologically based pharmacokinetic models.

1.1.1.2 Physiologically Based Pharmacokinetic Models

Physiological models are high compartmental models that use existing knowledge of the physiological mechanisms which regulate the drug action. These models capture the administration, diffusion and elimination of a drug in body organs that react with the drug. The drug mass balance for each organ can be described by Equation 1.1 (Saltzman, 2001):

(1.1)

where i is each specific organ/compartment, Vi is the organ volume, Ci is the drug concentration in the organ/compartment i, qel,i is the rate of drug metabolism in the organ/compartment i, and Flowin and Flowout are the inflow and outflow of the drug in the organ/compartment i.

A schematic overview of a physiological pharmacokinetic model, where each body organ is considered an independent compartment, is shown in Figure 1.3.

Figure 1.3 Schematic of a physiological pharmacokinetic model.

Source: Saltzman (2001). Reproduced with permission of Oxford University Press.

This modelling approach requires an in‐depth understanding of the physiology, but it describes more accurately than empirical compartmental models the drug delivery system. The advantages of physiologically based models over empirical compartmental models lie in the ability to be extrapolated between different species and different drug dosages (Cashman et al., 1996; Saltzman, 2001). The main drawback of physiologically based models is that, sometimes, certain parameters cannot be measured, and their values are difficult to be accurately predicted.

The description of one compartment itself in either of the previously mentioned approaches can be described by complex interactions and flows between, for example, blood cells, plasma, intestinal fluid, a rapid interactive pool and a slow interactive pool.

Both compartmental and physiological models range from simple to more detailed models that are based on fewer assumptions. Simplifications in the previous scheme can be made, depending on the exact system which is studied. In Figure 1.3, organs which do not contain important amounts of the drug agent can be neglected (Saltzman, 2001). However, the level of detail added to the model depends on the data availability and the purpose of the model.

1.1.2 Pharmacodynamic Modelling

Pharmacodynamic models describe the effect of a drug in the body (i.e. the impact of a drug that enters the cell on the cellular function). Due to the high complexity of the drug mechanism of action that enables precise measurements of the drug effect, detailed pharmacodynamic models are not in use and empirical expressions which correlate the drug concentration with the drug effect are more preferable (Holford & Sheiner, 1982). Practically, the pharmacodynamic model is determined by testing potential models and estimating the parameters when a reference pharmacokinetic model is used, and the accuracy of the pharmacodynamic model is highly dependent on precision of the pharmacokinetic model. The usage of a pharmacokinetic model is essential for the valuable expression of a pharmacodynamic model, as the latter assumes that the concentration of the drug is in equilibrium with the effect site, which might be the case only in the steady state.

In general, pharmacodynamics is the study of dose–response relationships. For the development of pharmacodynamic models, target cells are exposed in vitro in different drug concentrations, and drug effect curves are obtained. These data are then used to fit empirical pharmacodynamic models (Table 1.1). An example of a common dose–response curve is presented in Figure 1.4. The drug effect curves are of crucial importance, especially for the early clinical trial phases, for the determination of maximal dose effect as well as for estimation of the effective drug dosing window.

Table 1.1 The most common types of empirical pharmacodynamic models.

Source: Holford and Sheiner (1982). Reproduced with permission of Elsevier.

Model

Model equations

Description

Fixed‐effect model

–

Effect: present (1) or absent (0), or degree of effect

Linear model

drug effect,

drug concentration,

slope parameter,

initial drug effect

Log‐linear model

drug effect,

drug concentration,

slope parameter,

constant

E

max

model

drug effect,

drug concentration,

maximum drug effect,

initial drug effect from previous application,

concentration producing half of the maximum drug effect

Sigmoid

E

max

model

drug effect,

drug concentration,

maximum drug effect,

initial drug effect from previous application,

concentration producing half of the maximum drug effect,

constant affecting the shape of the drug effect–concentration curve

Figure 1.4 Illustration of a pharmacodynamic dose–response curve.

1.2 Model analysis, Parameter Estimation and Approximation

Model analysis includes analysis of parameters and variables of the developed pharmacokinetic model, in order to define uncertainty of parameters. This uncertainty usually originates from inter‐patient or experimental variability. In a consecutive step, the model is analysed towards its most influential parameters and variables. The methods that are usually used in order to obtain in‐depth understanding of the model are global sensitivity analysis, variability analysis, parameter estimation and parameters correlation.

1.2.1 Global Sensitivity Analysis

Global sensitivity analysis allows the understanding and identification of crucial model parameters that affect the model output. In the case of mathematical models that describe biomedical systems, global sensitivity analysis enables the identification of the relative influence of parameters of the pharmacokinetic and/or pharmacodynamic part of the model, on the model output. Performance analysis is conducted in the graphical user interface/high‐dimensional model representation (GUI‐HDMR) software, which uses random sampling HDMR (RS‐HDMR) to construct an expression for the output as a function of the parameters with orthogonal polynomials. This expression accounts for up to second‐order interactions and corresponds to the ANOVA decomposition truncated to the second order. From the coefficients of the representation, the sensitivity index is derived. The sensitivity indices are calculated based on partial variances, which themselves are calculated from the approximation of the model by orthonormal polynomials (Li et al., 2002; Ziehn and Tomlin, 2009).

1.2.2 Variability Analysis

Variability analysis focuses on the identification of the influence of the individual parameters and variables on the model outputs. Global sensitivity analysis gives a measure of the relative influence of each parameter on the output. However, that approach does not incorporate whether a higher or lower value of the parameter or variable of interest is increasing or decreasing the model output. Variability analysis enables the detection of the influence of each parameter and variable on the output, therefore facilitating the understanding of the actual physical influence of the pharmacokinetic and pharmacodynamic variables and parameters. In particular, when performing variability analysis, an investigation of whether an increase in the pharmacokinetic and/or pharmacodynamic variable or parameter increases or decreases the model output, y, takes place (Equation 1.2):

(1.2)

where P%,i is the percentage of change due to an increase in variable or parameter i, ymax,i is the upper bound model output, ymin,i is the lower bound model output and ynom is the calculated nominal model output.

1.2.3 Parameter Estimation and Correlation

Parameter estimation is the process of fitting the model parameters to clinical data. If the parameters are estimated with high precision, then the model’s response is closer to reality. The parameter estimation problem is evaluated by the correlation matrix C of the estimated parameters. An entry in the off‐diagonal elements of the correlation matrix C close to one indicates a high correlation of the corresponding parameters i and j, whereas an entry of zero indicates no correlation. The entries of the correlation matrix are calculated based on the variance–covariance matrix V, the variance of a parameter is given on the diagonal (Vii) and the covariance of two parameters i and j is given on the off‐diagonal elements (Vij).

(1.3)

(1.4)

1.3 Optimization and Control

Mathematical optimization and control of biomedical systems could lead to a better prediction of the optimal drug and/or therapy treatment for a specific disease. Advanced mathematical and computational techniques such as multiparametric predictive control, sensitivity analysis and model reduction are extensively discussed in Chapter 2. Moreover, those techniques are applied in a variety of diseases (i.e. anaesthesia, diabetes and leukaemia) that are further discussed in the following chapters.

Anaesthesia (see Chapters 3 and 4) is a process which provides hypnosis, analgesia and muscle relaxation while maintaining the vital functions of a living organism. For efficient prediction and control of this bio‐process, a model predictive controller (see Chapter 2) is required.

In type 1 diabetes (see Chapter 5), the goal is to maintain the blood’s glucose concentration within normal levels. From a mathematical point of view, this can be formulated as a model predictive control problem.

In acute myeloid leukaemia (see Chapters 6, 7, and 8), the ultimate goal is to determine the optimal chemotherapy dose that would lead to minimization of the cancerous population while maintaining the normal/healthy population above a minimum acceptable level. From a computational point of view, this is a scheduling problem.

For all of these diseases, the overall framework for the design of optimal drug delivery systems is presented in Figure 1.5.

Figure 1.5 Framework towards optimal drug delivery systems.

In order to move towards the design of optimal delivery systems, development of a high‐fidelity model able to describe the biomedical problem for an individual patient has to take place. Identification of parameters that crucially affect the model output enables the model reduction and, at a second step, the predictive control of the drug dose, therefore leading to process optimization.

Especially in the case of acute myeloid leukaemia, we have developed an appropriate in vitro system which allows ex vivo experimentation of leukaemic patient cells for the more efficient understanding and further identification of parameters that crucially affect the model output (i.e. the drug dose determination). Moreover, experimental data serve as an input for our mathematical model, allowing validation and improvement (Chapters 6, 7, and 8). Therefore, this in vivo–in vitro–in silico closed loop enables the accurate study and further determination of the optimal drug dose for an individual/specific patient (Velliou et al., 2014).

References

Cashman, J.R., Perotti, B.Y., Berkman, C.E., & Lin, J. (1996). Pharmacokinetics and molecular detoxication.

Environmental Health Perspectives

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Ette, E.I., & Williams, P.J. (2007).

Pharmacometrics: the science of quantitative pharmacology

. Hoboken, NJ: John Wiley & Sons.

Holford, N.H.G., & Sheiner, L B. (1982). Kinetics of pharmacologic response.

Pharmacology & Therapeutics

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Li, G., Wang, S.W., Rabitz, H., Wang, S., & Jaffe, F. (2002). Global un‐certainty assessments by high dimensional model representations (HDMR).

Chemical Engineering Science

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Saltzman, W.M. (2001).

Drug delivery: engineering principles for drug therapy

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Velliou, E., Fuentes‐Garí, M., Misener, R., Pefani, E., Rende, M., Panoskaltsis, N., Pistikopoulos, E.N., & Mantalaris, A. (2014). A framework for the design, modeling and optimization of biomedical systems. In M. Eden, J.D. Siirola, & G.P. Towler (Eds.),

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2Draft Computational Tools and Methods

Ioana Naşcu1, Richard Oberdieck2, Romain Lambert3, Pedro Rivotti3, and Efstratios N. Pistikopoulos4

1 Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, USA

2 DONG energy A/S, Gentofte, Denmark

3 Department of Chemical Engineering, Imperial College London, UK

4 Texas A&M Energy Institute, Artie McFerrin Department of Chemical Engineering, Texas A&M University, USA

2.1 Introduction

This chapter focuses on describing the computational tools that are part of the developed multiparametric model predictive control (MPC) framework presented in Chapter 1. The framework enables the solution of demanding optimization and control problems through a step‐by‐step procedure presented in this chapter. The key advantage of this is that it follows a multiparametric approach for the controller design that transfers the computational burden offline (Pistikopoulos 2000). Furthermore, the proposed procedure is not process dependent and can be adapted to any process at hand. All the steps included in the framework are realized through the developing software platform PAROC (PARametric Optimization and Control). PAROC is a user‐friendly software platform that utilizes the communication between gPROMS ModelBuilder and MATLAB. Through this software interoperability, the multiple steps are realized in a way that convenient for the user and, most importantly, tractable.

A comprehensive schematic representation of the framework is shown in Figure 2.1, and a thorough explanation of the computational tool required for the steps is provided within this chapter.

Figure 2.1 A framework for explicit/multiparametric model predictive control and moving horizon estimation.

Source: Naşcu et al. (2016). Reproduced with permission of Elsevier.

The high‐fidelity model developed in the modelling and design optimization step usually results in differential‐algebraic equation (DAE) systems of high complexity. The DAE systems are approximated by discrete time models in state‐space representation. In order to do that, complex model–order reduction techniques as well as identification methods and toolboxes are employed. The key objectives are to simplify the representation of the system without compromising the accuracy of the high‐fidelity model. Although there is a variety of model reduction and approximation techniques (Lambert et al. [2013] and references therein), the System Identification Toolbox of MATLAB is also commonly used. In this chapter, we will focus on model reduction techniques as a method of model approximation.

2.2 Sensitivity Analysis and Model Reduction

2.2.1 Sensitivity Analysis