Mathematical modelling of immune condition dynamics : a clinical perspective

This thesis describes the use of mathematical modelling to analyse the treatment of patients with immune disorders; namely, Multiple Myeloma, a cancer of plasma cells that create excess monoclonal antibody; and kidney transplants, where the immune system produces polygonal antibodies against the implanted organ. Linear and nonlinear compartmental models play an important role in the analysis of biomedical systems; in this thesis several models are developed to describe the in vivo kinetics of the antibodies that are prevalent for the two disorders studied. These models are validated against patient data supplied by clinical collaborators. Through this validation process important information regarding the dynamic properties of the clinical treatment can be gathered. In order to treat patients with excess immune antibodies the clinical staff wish to reduce these high levels in the patient to near healthy concentrations. To achieve this they have two possible treatment modalities: either using artificial methods to clear the material, a process known as apheresis, or drug therapy to reduce the production of the antibody in question. Apheresis techniques differ in their ability to clear different immune complexes; the effectiveness of a range of apheresis techniques is categorised for several antibody types and antibody fragments. The models developed are then used to predict the patient response to alternative treatment methods, and schedules, to find optimal combinations. In addition, improved measurement techniques that may offer an improved diagnosis are suggested. Whilst the overall effect of drug therapy is known, through measuring the concentration of antibodies in the patient’s blood, the short-term relationship between drug application and reduction in antibody synthesis is still not well defined; therefore, methods to estimate the generation rate of the immune complex, without the need for invasive procedures, are also presented.

[1]  I. Lipkovich,et al.  Identifying patterns in treatment response profiles in acute bipolar mania: a cluster analysis approach , 2008, BMC psychiatry.

[2]  E. Gehan,et al.  Combination therapy for multiple myeloma , 1977, Cancer.

[3]  K R Godfrey,et al.  CODE: a deconvolution program implementing a regularization method of deconvolution constrained to non‐negative values. Description and pilot evaluation , 1998, Biopharmaceutics & drug disposition.

[4]  T. Hostetter,et al.  Chronic kidney disease awareness, prevalence, and trends among U.S. adults, 1999 to 2000. , 2004, Journal of the American Society of Nephrology : JASN.

[5]  Ian Guymer,et al.  Diameter and Surcharge Effects on Solute Transport across Surcharged Manholes , 2005 .

[6]  Reginald F. Brown.  Biomedical systems analysis via compartmental concept , 1985 .

[7]  Bernard Gallez,et al.  Cluster analysis of BOLD fMRI time series in tumors to study the heterogeneity of hemodynamic response to treatment , 2003, Magnetic resonance in medicine.

[8]  Keith R. Godfrey,et al.  Structural indistinguishability between uncontrolled (autonomous) nonlinear analytic systems , 2004, Autom..

[9]  T. Waldmann,et al.  Metabolism of human immunoglobulin D (IgD). , 1966, The Journal of clinical investigation.

[10]  J. Winchester,et al.  Beta-2 microglobulin in ESRD: an in-depth review. , 2003, Advances in renal replacement therapy.

[11]  E. Tuddenham,et al.  Plasma volume expansion and increased serum viscosity in myeloma and macroglobulinaemia. , 1974, Clinical and experimental immunology.

[12]  L. Racusen,et al.  Solid organ transplant rejection : mechanisms, pathology, and diagnosis , 1996 .

[13]  Arthur R. Bradwell.  Serum Free Light Chain Analysis , 2004 .

[14]  M. Fliess,et al.  Nonlinear observability, identifiability, and persistent trajectories , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[15]  J. J. Abel,et al.  ON THE REMOVAL OF DIFFUSIBLE SUBSTANCES FROM THE CIRCULATING BLOOD OF LIVING ANIMALS BY DIALYSIS II. SOME CONSTITUENTS OF THE BLOOD , 1914 .

[16]  W R Clark,et al.  Membrane adsorption of beta 2-microglobulin: equilibrium and kinetic characterization. , 1994, Kidney international.

[17]  Keith R. Godfrey,et al.  MODELLING OF HAEMODIALYSIS IN LIMITING SERUM FREE LIGHT CHAINS IN PATIENTS WITH RENAL FAILURE , 2006 .

[18]  Axel Wismüller,et al.  Cluster Analysis of Biomedical Image Time-Series , 2002, International Journal of Computer Vision.

[19]  J. Ritt.  Partial differential algebra , 1950 .

[20]  D. King.  Applications And Engineering Of Monoclonal Antibodies , 1998 .

[21]  C. Anderson,et al.  The protection receptor for IgG catabolism is the beta2-microglobulin-containing neonatal intestinal transport receptor. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[22]  Robert Tibshirani,et al.  An Introduction to the Bootstrap , 1994 .

[23]  W. V. Loscutoff,et al.  General sensitivity theory , 1972 .

[24]  A. Katchalsky,et al.  Thermodynamic analysis of the permeability of biological membranes to non-electrolytes. , 1958, Biochimica et biophysica acta.

[25]  I Guymer,et al.  The use of deconvolution techniques to identify the fundamental mixing characteristics of urban drainage structures. , 2010, Water science and technology : a journal of the International Association on Water Pollution Research.

[26]  E. Kamil,et al.  Intravenous immune globulin treatment inhibits crossmatch positivity and allows for successful transplantation of incompatible organs in living-donor and cadaver recipients1 , 2003, Transplantation.

[27]  P Zucchelli,et al.  Controlled plasma exchange trial in acute renal failure due to multiple myeloma. , 1988, Kidney international.

[28]  G. P. King,et al.  Using cluster analysis to classify time series , 1992 .

[29]  F. J. Richards.  A Flexible Growth Function for Empirical Use , 1959 .

[30]  Willem J. Kolff,et al.  The artificial kidney and its effect on the development of other artificial organs , 2002, Nature Medicine.

[31]  M. Mineshima,et al.  Clinical evaluation of dual-dialyzer hemodialysis (DDHD) , 2007, Journal of Artificial Organs.

[32]  Evelyne Hubert,et al.  Differential Algebra for Derivations with Nontrivial Commutation Rules , 2005 .

[33]  Arseniy S. Karkach,et al.  Trajectories and models of individual growth , 2006 .

[34]  H. Pohjanpalo.  System identifiability based on the power series expansion of the solution , 1978 .

[35]  T. Waldmann,et al.  Familial hypercatabolic hypoproteinemia. A disorder of endogenous catabolism of albumin and immunoglobulin. , 1990, The Journal of clinical investigation.

[36]  J. Leypoldt.  Kinetics of β2‐Microglobulin and Phosphate during Hemodialysis: Effects of Treatment Frequency and Duration , 2005, Seminars in dialysis.

[37]  H. Renz,et al.  Immunochemical quantification of free light chains in urine. , 2005, Clinical chemistry.

[38]  A. W. Kelman,et al.  Compartmental models and their application , 1985 .

[39]  Anthony C. Davison,et al.  Bootstrap Methods and Their Application , 1998 .

[40]  D. Cutler.  Numerical deconvolution by least squares: Use of polynomials to represent the input function , 1978, Journal of Pharmacokinetics and Biopharmaceutics.

[41]  W. Ritschel,et al.  Handbook of basic pharmacokinetics-- including clinical applications , 1992 .

[42]  Jeffrey S. Barrett,et al.  Pharmacokinetic/Pharmacodynamic Modeling in Drug Development , 2004 .

[43]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[44]  D. Liberati,et al.  Linear and nonlinear techniques for the deconvolution of hormone time-series , 1993, IEEE Transactions on Biomedical Engineering.

[45]  John W. Sammon,et al.  A Nonlinear Mapping for Data Structure Analysis , 1969, IEEE Transactions on Computers.

[46]  T. Greene,et al.  Resistance to intercompartmental mass transfer limits beta2-microglobulin removal by post-dilution hemodiafiltration. , 2006, Kidney international.

[47]  G. Cozzi,et al.  Transmembrane pressure modulation in high-volume mixed hemodiafiltration to optimize efficiency and minimize protein loss. , 2006, Kidney international.

[48]  C Cobelli,et al.  Insulin Secretion and Hepatic Extraction in Humans by Minimal Modeling of C-Peptide and Insulin Kinetics , 1988, Diabetes.

[49]  F. Gotch,et al.  Evolution of the Single‐Pool Urea Kinetic Model , 2001, Seminars in dialysis.

[50]  R. Fournier.  Basic Transport Phenomena In Biomedical Engineering , 1998 .

[51]  A. Lupo,et al.  Prognostic factors and effectiveness of treatment in acute renal failure due to multiple myeloma: a review of 50 cases. Report of the Italien Renal Immunopathology Group. , 1987, Clinical nephrology.

[52]  S. Salmon,et al.  Kinetics of tumor growth and regression in IgG multiple myeloma. , 1972, The Journal of clinical investigation.

[53]  Brian Mohan Gurbaxani,et al.  Development of new models for the analysis of Fc-FcRn interactions. , 2006, Molecular immunology.

[54]  Amit X Garg,et al.  Plasma Exchange When Myeloma Presents as Acute Renal Failure , 2005, Annals of Internal Medicine.

[55]  W. A. Hemmings,et al.  A THEORETICAL MODEL OF GAMMA-GLOBULIN CATABOLISM. , 1964, Nature.

[56]  T. Petitclerc.  Estimation of mass transfer through a hemodialyzer: theoretical approach and clinical applications. , 1998, Artificial organs.

[57]  T. Waldmann,et al.  Immunoglobulin metabolism in ataxia telangiectasia. , 1968, The Journal of clinical investigation.

[58]  M. Denham,et al.  The mathematica book: Stephen Wolfram. Wolfram Media/Cambridge University Press, 3rd edition (1996) , 1997 .

[59]  Jon G. Rokne,et al.  New computer methods for global optimization , 1988 .

[60]  Elizabeth Ann Maharaj,et al.  Cluster of Time Series , 2000, J. Classif..

[61]  C. Janeway.  Immunobiology: The Immune System in Health and Disease , 1996 .

[62]  K. Stojanoski,et al.  Cluster analysis method in electrophoretic patterns classification , 2005 .

[63]  D. Levitt.  PKQuest: a general physiologically based pharmacokinetic model. Introduction and application to propranolol , 2002, BMC clinical pharmacology.

[64]  T J Holmes,et al.  Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach. , 1992, Journal of the Optical Society of America. A, Optics and image science.

[65]  Maria Pia Saccomani,et al.  DAISY: A new software tool to test global identifiability of biological and physiological systems , 2007, Comput. Methods Programs Biomed..

[66]  William H. Press,et al.  Numerical recipes , 1990 .

[67]  M. J. D. Powell,et al.  Direct search algorithms for optimization calculations , 1998, Acta Numerica.

[68]  Keith R. Godfrey,et al.  Identifiability of uncontrolled nonlinear rational systems , 2002, Autom..

[69]  Keith R. Godfrey,et al.  A comparison of six deconvolution techniques , 1996, Journal of Pharmacokinetics and Biopharmaceutics.

[70]  J. Jacquez.  Compartmental analysis in biology and medicine , 1985 .

[71]  Jonghan Kim,et al.  Kinetics of FcRn-mediated recycling of IgG and albumin in human: pathophysiology and therapeutic implications using a simplified mechanism-based model. , 2007, Clinical immunology.

[72]  Yasuaki Ogawa,et al.  Estimation of drug absorption rates using a deconvolution method with nonequal sampling times , 1986, Journal of Pharmacokinetics and Biopharmaceutics.

[73]  H P Wynn,et al.  Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences. , 2001, Mathematical biosciences.

[74]  J. Foreman,et al.  Textbook of Receptor Pharmacology , 2011 .

[75]  T. Waldmann,et al.  The metabolism of IgE. Studies in normal individuals and in a patient with IgE myeloma. , 1976, Journal of immunology.

[76]  E. Ward,et al.  Transcytosis and catabolism of antibody , 2002, Immunologic research.

[77]  D. Liberati,et al.  Deconvolution of infrequently sampled data for the estimation of growth hormone secretion , 1995, IEEE Transactions on Biomedical Engineering.

[78]  R. Bataille.  Management of myeloma with bisphosphonates. , 1996, The New England journal of medicine.

[79]  L. D'Angio,et al.  A new differential algebra algorithm to test identifiability of nonlinear systems with given initial conditions , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[80]  V. Phat,et al.  Global stabilization of linear periodically time-varying switched systems via matrix inequalities , 2006 .

[81]  J. Varga,et al.  Immune Recognition , 1977, The Yale Journal of Biology and Medicine.

[82]  Peter L. Bonate,et al.  Pharmacokinetic-Pharmacodynamic Modeling And Simulation , 2005 .

[83]  Jacek Waniewski,et al.  Impact of convective transport on dialyzer clearance , 2003, Journal of Artificial Organs.

[84]  A. George,et al.  Diagnostic and Therapeutic Antibodies , 2000 .

[85]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[86]  W. Ritschel.  Handbook of basic pharmacokinetics , 1976 .

[87]  Philip F Halloran,et al.  Immunosuppressive drugs for kidney transplantation. , 2004, The New England journal of medicine.

[88]  D. Levitt.  Physiologically based pharmacokinetic modeling of arterial – antecubital vein concentration difference , 2004, BMC clinical pharmacology.

[89]  Anna M. Wu,et al.  A Predictive Model of Therapeutic Monoclonal Antibody Dynamics and Regulation by the Neonatal Fc Receptor (FcRn) , 2005, Annals of Biomedical Engineering.

[90]  W. K. Bolton,et al.  Improved urea reduction ratio and Kt/V in large hemodialysis patients using two dialyzers in parallel. , 2000, American journal of kidney diseases : the official journal of the National Kidney Foundation.

[91]  A. Morse,et al.  Basic problems in stability and design of switched systems , 1999 .

[92]  Yonathan Bard,et al.  Nonlinear parameter estimation , 1974 .

[93]  Per Christian Hansen,et al.  Regularization Tools version 4.0 for Matlab 7.3 , 2007, Numerical Algorithms.

[94]  D G Altman,et al.  Statistics Notes: Measurement error proportional to the mean , 1996, BMJ.

[95]  Cevat Gökçek,et al.  Stability analysis of periodically switched linear systems using Floquet theory , 2004 .

[96]  G. Ayers,et al.  Interative blind deconvolution method and its applications. , 1988, Optics Letters.

[97]  A. Dispenzieri,et al.  Improvement of cast nephropathy with plasma exchange depends on the diagnosis and on reduction of serum free light chains. , 2008, Kidney international.

[98]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[99]  J. Skilling,et al.  Maximum entropy image reconstruction: general algorithm , 1984 .

[100]  O. Levenspiel.  Chemical reaction engineering : an introduction to the design of chemical reactors , 1962 .

[101]  P O Droz,et al.  Quantification of biological variability. , 1992, The Annals of occupational hygiene.

[102]  B. Jaber,et al.  Kinetics and dosing predictions for daily haemofiltration. , 2003, Nephrology, dialysis, transplantation : official publication of the European Dialysis and Transplant Association - European Renal Association.

[103]  L. Sklar,et al.  Competitive binding kinetics in ligand-receptor-competitor systems. Rate parameters for unlabeled ligands for the formyl peptide receptor. , 1985, Molecular pharmacology.

[104]  T. Cornwell,et al.  A simple maximum entropy deconvolution algorithm , 1985 .

[105]  N. Heyne,et al.  European trial of free light chain removal by extended haemodialysis in cast nephropathy (EuLITE): A randomised control trial , 2008, Trials.

[106]  T. Waldmann,et al.  METABOLISM OF HUMAN GAMMA MACROGLOBULINS. , 1964, The Journal of clinical investigation.

[107]  R. Montgomery,et al.  PLASMAPHERESIS AND INTRAVENOUS IMMUNE GLOBULIN PROVIDES EFFECTIVE RESCUE THERAPY FOR REFRACTORY HUMORAL REJECTION AND ALLOWS KIDNEYS TO BE SUCCESSFULLY TRANSPLANTED INTO CROSS-MATCH-POSITIVE RECIPIENTS , 2000, Transplantation.

[108]  H. Metzger.  Fc receptors and the action of antibodies , 1990 .

[109]  J T Daugirdas,et al.  Relationship between apparent (single-pool) and true (double-pool) urea distribution volume. , 1999, Kidney international.

[110]  J. Waniewski.  Linear approximations for the description of solute flux through permselective membranes , 1994 .

[111]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[112]  Akira Hasegawa,et al.  Excellent Long‐term Outcome of ABO‐Incompatible Living Donor Kidney Transplantation in Japan , 2004, American journal of transplantation : official journal of the American Society of Transplantation and the American Society of Transplant Surgeons.

[113]  D. Spiegel,et al.  Hemodialysis urea rebound: the effect of increasing dialysis efficiency. , 1995, American journal of kidney diseases : the official journal of the National Kidney Foundation.

[114]  N. Levin,et al.  Kinetics of beta-2-microglobulin in hemodialysis. , 1989, Contributions to nephrology.

[115]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[116]  R Vanholder,et al.  Two-pool versus single-pool models in the determination of urea kinetic parameters. , 1996, Blood purification.

[117]  Urszula Foryś,et al.  Logistic Equations in Tumour Growth Modelling , 2003 .

[118]  D. Altman,et al.  Missing data , 2007, BMJ : British Medical Journal.

[119]  J. Hiriart-Urruty,et al.  Comparison of public-domain software for black box global optimization , 2000 .

[120]  Keith R. Godfrey,et al.  The problem of model indistinguishability in pharmacokinetics , 1989, Journal of Pharmacokinetics and Biopharmaceutics.

[121]  P V Pedersen,et al.  Novel deconvolution method for linear pharmacokinetic systems with polyexponential impulse response. , 1980, Journal of pharmaceutical sciences.

[122]  P. Verdonck,et al.  Middle molecule removal in low‐flux polysulfone dialyzers: Impact of flows and surface area on whole‐body and dialyzer clearances , 2005, Hemodialysis international. International Symposium on Home Hemodialysis.

[123]  Gilbert G. Walter,et al.  Compartmental modeling with networks , 1999 .

[124]  Peter P. Valko,et al.  Principal component analysis of kinetic models , 1985 .