Patrick W. Nelson

Mathematical analysis of delay differential equation models of HIV-1 infection

Models of HIV-1 infection that include intracellular delays are more accurate representations of the biology and change the estimated values of kinetic parameters when compared to models without delays. We develop and analyze a set of models that include intracellular delays, combination antiretroviral therapy, and the dynamics of both infected and uninfected T cells. We show that when the drug efficacy is less than perfect the estimated value of the loss rate of productively infected T cells, delta, is increased when data is fit with delay models compared to the values estimated with a non-delay model. We provide a mathematical justification for this increased value of delta. We also provide some general results on the stability of non-linear delay differential equation infection models.

The role of cells refractory to productive infection in acute hepatitis B viral dynamics.

During acute hepatitis B virus (HBV) infection viral loads reach high levels (≈1010 HBV DNA per ml), and nearly every hepatocyte becomes infected. Nonetheless, ≈85–95% of infected adults clear the infection. Although the immune response has been implicated in mediating clearance, the precise mechanisms remain to be elucidated. As infection clears, infected cells are replaced by uninfected ones. During much of this process the virus remains plentiful but nonetheless does not rekindle infection. Here, we analyze data from a set of individuals identified during acute HBV infection and develop mathematical models to test the role of immune responses in various stages of early HBV infection. Fitting the models to data we are able to separate the kinetics of the noncytolytic and the cytolytic immune responses, thus explaining the relative contribution of these two processes. We further show that we need to hypothesize that newly generated uninfected cells are refractory to productive infection. Without this assumption, viral resurgence is observed as uninfected cells are regenerated. Such protection, possibly mediated by cytokines, may also be important in resolving other acute viral infections.

Time-Delay Systems: Analysis and Control using the Lambert W Function.

Time-Delay Systems Delay Differential Equations Lambert W Function Stability Controllability Observability Feedback Control Eigenvalue Assignment State Observer Chatter in Machining Human Immunodeficiency Virus Diesel Engine Control Robust Control Time-Domain Specification

Controllability and Observability of Systems of Linear Delay Differential Equations Via the Matrix Lambert W Function

Controllability and observability of linear time delayed systems have been studied, and various definitions and criteria have been presented since the 1960s. However, the lack of an analytical solution approach has limited the applicability of the existing theories. Recently, the solution to systems of linear delay differential equations has been derived using the matrix Lambert W function, in a form similar to the transition matrix in ordinary differential equations. The criteria for controllability and observability, and their Gramians, for systems of delay differential equations using the solution in terms of the matrix Lambert W function are presented for the first time and illustrated with examples.

Effect of drug efficacy and the eclipse phase of the viral life cycle on estimates of HIV viral dynamic parameters.

Summary: Fits of mathematic models to the decline in HIV‐1 RNA after antiretroviral therapies have yielded estimates for the life span of productively infected cells of 1 to 2 days. In a previous report, we described the mathematic properties of an extended model that accounts for imperfect viral suppression and the eclipse phase of the viral life cycle (the intracellular delay between initial infection and release of progeny virions). In this article, we fit this extended model to detailed data on the decline of plasma HIV‐1 RNA after treatment with the protease inhibitor ritonavir. Because the therapy in this study was most likely not completely suppressive, we allowed the drug efficacy parameter to vary from 70% to 100%. Estimates for the clearance rate of free virus, c, increased with the addition of the intracellular delay (as reported previously) but were not appreciably affected by changes in the drug efficacy parameter. By contrast, the estimated death rate of virus‐producing cells, &dgr;, increased from an average of 0.49 day‐1 to 0.90 day‐1 (an increase of 84%) because the drug efficacy parameter was reduced from 100% to 70%. Neglecting the intracellular delay, the comparable increase in &dgr; was only about 55%. The inferred increases in &dgr; doubled when the model was extended to account for possible increases in target cell densities after treatment initiation. This work suggests that estimates for &dgr; may be greater than previously reported and that the half‐life of a cell in vivo that is producing virus, on average, may be 1 day.

Design of observer-based feedback control for time-delay systems with application to automotive powertrain control

A new approach for observer-based feedback control of time-delay systems is developed. Time-delays in systems lead to characteristic equations of infinite dimension, making the systems difficult to control with classical control methods. In this paper, a recently developed approach, based on the Lambert W function, is used to address this difficulty by designing an observer-based state feedback controller via assignment of eigenvalues. The designed observer provides estimation of the state, which converges asymptotically to the actual state, and is then used for state feedback control. The feedback controller and the observer take simple linear forms and, thus, are easy to implement when compared to nonlinear methods. This new approach is applied, for illustration, to the control of a diesel engine to achieve improvement in fuel efficiency and reduction in emissions. The simulation results show excellent closed-loop performance.

Humoral Autoimmunity against the Extracellular Domain of the Neuroendocrine Autoantigen IA-2 Heightens the Risk of Type 1 Diabetes

The objective of this study was to determine whether antigenic determinants localized within the extracellular domain of the neuroendocrine autoantigen tyrosine phosphatase-like protein IA-2 are targets of humoral responses in type 1 diabetes (T1DM). Previous studies indicated that the immunodominant region of IA-2 is localized within its intracellular domain (IA-2ic; amino acids 601-979). We analyzed 333 subjects from the Childrens Hospital of Pittsburgh study, 102 of whom progressed to insulin-requiring diabetes (prediabetics). Autoantibodies from these individuals were initially assayed for ICA512bdc (Barbara Davis Center amino acids 257-556; 630-979), IA-2ic (amino acids 601-979), and IA-2 full-length (amino acids 1-979) in addition to islet cell antibody (ICA), glutamic acid decarboxylase, 65-kDa isoform, and insulin autoantibodies. We identified an autoantibody response reactive with the extracellular domain of IA-2 that is associated with very high risk of T1DM progression. Relatives with no detectable autoantibodies against ICA512bdc (or IA-2ic) exhibited antibody responses against the IA-2 full-length peptide (log rank, P = 0.008). This effect was also observed in first-degree relatives who were positive for glutamic acid decarboxylase, 65-kDa isoform (log rank, P = 0.026) or at least two islet autoantibodies but were negative for ICA512bdc (log rank, P = 0.022). Competitive binding experiments and immunoprecipitation of the IA-2 extracellular domain (amino acid residues 26-577) further lend support for the presence of autoantibodies reactive with new antigenic determinants within the extracellular domain of IA-2. In summary, the addition of measurements of autoantibodies reactive with the IA-2 extracellular domain to assays geared to assess the progression of autoimmunity to clinical T1DM may more accurately characterize this risk. This has considerable implications not only for stratifying high diabetes risk but also facilitating the search for pathogenic epitopes to enable the design of peptide-based immunotherapies that may prevent the progression to overt T1DM at its preclinical stages.

Primer: Immunity and Autoimmunity

For nonimmunologists, a daunting and rapidly evolving immunologic vocabulary, our incomplete understanding of both normal and abnormal immune function, and multiple interrelated complex immune cellular pathways can be a barrier to using basic immunology to understand and improve care for patients with type 1 diabetes. Our task in this review is to introduce current immune concepts specifically relevant to type 1 diabetes. Because our understanding is not complete, as evidenced perhaps by our lack of standard immunotherapy to prevent type 1 diabetes, we can only provide a partial framework. Perhaps the simplest framework (which may be wrong) is to consider the development of type 1 diabetes as the balance between regulatory and effector T lymphocytes. We know that in the absence of a major portion of regulatory T lymphocytes in a rare syndrome caused by mutation of the FOXP3 gene, most infants (even neonates) develop type 1 diabetes. Central to the development of type 1 diabetes are T lymphocytes with specific T-cell receptors that recognize islet molecules. When a T-cell is activated through its receptor, it can orchestrate protection from infection or autoimmunity, depending on the target. Other T-cells can suppress or enhance autoimmunity (either in general or only for T lymphocytes in islets). The activation of a T-cell involves multiple different cell types and genes, as we will discuss. The modern era of immunology began with the clonal selection theory independently expressed by David W. Talmage and Sir Frank Macfarlane Burnet (1,2). The clonal selection theory postulates that a foreign antigen entering the body binds to one unique antibody selected from an unlimited repertoire of antibodies formed early in the organisms life. This explains how the immune system is able to recognize and respond to a virtually inestimable number of foreign antigens. The immune system is a …

Association Between Impaired Cardiovascular Autonomic Function and Hypoglycemia in Patients With Type 1 Diabetes

OBJECTIVE We studied the association between glycemic variability (GV) reflecting hypoglycemic stress and cardiovascular autonomic function in subjects with type 1 diabetes. RESEARCH DESIGN AND METHODS Forty-four type 1 diabetic patients (mean age 34 ± 13 years, 40% male, 86% Caucasian, mean diabetes duration 13 ± 6 years, mean hemoglobin A1c [HbA1c] 8.0 ± 1.2% [64 ± 5 mmol/mol]) without cardiovascular disease, dyslipidemia, or hypertension participated in this pilot study. Indices of GV reflective of hypoglycemic stress (low blood glucose index [LBGI] and area under the curve [AUC] for hypoglycemia) were computed using data obtained during 5-day continuous glucose monitoring. Cardiovascular autonomic neuropathy (CAN) was assessed using standardized cardiovascular reflex testing and measures of heart rate variability (HRV), which were analyzed as time and frequency domain measures. RESULTS Both LBGI and AUC hypoglycemia had a significant negative association with the low-frequency power of HRV (r = −0.47, P = 0.002; r = −0.43, P = 0.005, respectively) and with the high-frequency power of HRV (r = −0.37, P = 0.018; r = −0.38, P = 0.015, respectively). These inverse associations persisted after adjusting for HbA1c, although they were attenuated in multivariable analysis after adjustment for age, diabetes duration, and several other covariates. CONCLUSIONS Increased GV promoting hypoglycemic stress was associated with reduced HRV independent of glycemic control as assessed by HbA1c. These pilot data suggest that glucose variability may contribute to cardiovascular autonomic dysfunction among adults with type 1 diabetes.

Robust Control and Time-Domain Specifications for Systems of Delay Differential Equations via Eigenvalue Assignment

An approach for eigenvalue assignment for systems of delay differential equations (DDEs), based upon the Lambert W function, is applied to the problem of robust control design for perturbed systems of DDEs, and to the problem of time-domain specifications. The real stability radius, which measures the ability of a system to preserve its stability under a certain class of real perturbations, can be computed from known nominal coefficients of the DDE representing the system. In this paper, considering the stability radius, the real part of the eigenvalues is assigned. Also, time-domain specifications for the transient response of systems of DDEs are improved in a way similar to systems of ordinary differential equations using the eigenvalue assignment approach.