Computational Models for Diagnosing and Treating Endometriosis

Abstract

Endometriosis is a common but poorly understood disease. Symptoms can begin early in adolescence, with menarche, and can be debilitating. Despite this, people often suffer several years before being correctly diagnosed and adequately treated. Endometriosis involves the inappropriate growth of endometrial-like tissue (including epithelial cells, stromal fibroblasts, vascular cells, and immune cells) outside of the uterus. Computational models can aid in understanding the mechanisms by which immune, hormone, and vascular disruptions manifest in endometriosis and complicate treatment. In this review, we illustrate how three computational modeling approaches (regression, pharmacokinetics/pharmacodynamics, and quantitative systems pharmacology) have been used to improve the diagnosis and treatment of endometriosis. As we explore these approaches and their differing detail of biological mechanisms, we consider how each approach can answer different questions about endometriosis. We summarize the mathematics involved, and we use published examples of each approach to compare how researchers: (1) shape the scope of each model, (2) incorporate experimental and clinical data, and (3) generate clinically useful predictions and insight. Lastly, we discuss the benefits and limitations of each modeling approach and how we can combine these approaches to further understand, diagnose, and treat endometriosis.

Keywords: biomarker; computational; endometriosis; hormone therapy; machine learning; mechanism; pharmacokinetics; systems biology.

Figure 1

Structure of logistic regression models for diagnosing endometriosis. Logistic regressions calculate the odds and probability of a binary outcome (e.g., positive endometriosis diagnosis) using measurements taken across several predictor variables (e.g., patient observations). The model parameters, the β coefficients, are identified by applying the logistic regression model to a many-patient data set for whom the outcomes are known, and these coefficients can then be used with new patient data to predict the likelihood of endometriosis in that patient.

Figure 2

Overview of model development and validation by Nnoaham et al. (25). (A) The authors created their models using a set of 771 patients. (B) They then evaluated the performance of this model using a ROC curve to identify probability thresholds for classification that produce a specificity and sensitivity within the optimal range. (C) They further validated the model by first updating it with new predictor variable values for a separate set of 625 patients (leaving the β coefficients as they were) and then creating a new ROC curve.

Figure 3

Relationship between pharmacokinetic (PK), pharmacodynamic (PD), and population PK-PD modeling. These three modeling modalities can be used to make predictions about treatment from drug dosing to resulting effects, on an individual and population scale.

Figure 4

Structure of basic two-compartment pharmacokinetic model. (A) Schematic of continuous processes represented in a two-compartment model. (B) The two ordinary differential equations (ODEs) used here describe the rate of change in concentration of the drug in the central and peripheral compartments over time as a result of these processes occurring. PK models can be more or less complex, with different compartments and processes included as needed to fully describe the drug being investigated in the simplest reasonable form.

Figure 5

Structure of Reinecke et al.'s (26) population pharmacokinetic models for an intravaginal ring that delivers anastrozole (ATZ) and levonorgestrel (LNG). These models include the influence of estradiol (E₂) and LNG on sex hormone binding globulin (SHBG), and vice versa. Solid lines represent a mass flow; dashed lines represent an indirect influence – as described in Reinecke et al. (26).

Figure 6

Agreement between pharmacokinetic simulation predictions and subsequent clinical trial results. X-axis: Simulation predictions for mean plasma ATZ concentrations 28 days following ring placement (C₂₉) in low-dose (290 μg/day), medium-dose (630 μg/day), and high-dose (1,080 μg/day) treatment groups. Horizontal error bars represent the 5th and 95th percentiles. Adapted from Reinecke et al. (26). Y-axis: Observed median plasma ATZ concentration as average of measurements taken 28, 56, and 84 days following first ring placement (C_ss) in low-dose (300 μg/day), medium-dose (600 μg/day), and high-dose (1,050 μg/day) treatment groups from a phase 2b clinical trial. Vertical error bars represent the 10th to 90th percentile. Adapted from Nave et al. (45).

Figure 7

Overview of Röblitz et al.'s (27) quantitative systems pharmacology model of the menstrual cycle and gonadotropin-releasing hormone (GnRH) therapies. (A) This schematic shows where molecules are produced and whether they stimulate (green line/arrow head), inhibit (red line/flat head), or have a mixed effect (orange line/both) on the production of other molecules in this model. The dotted circles labeled “Cellular Model(s)” represent processes affecting pituitary GnRH receptors and ovarian LH/FSH receptors that have been modeled in detail. The delivery of GnRH agonist and antagonist are modeled using PK models that feed into the pituitary cellular model. (B) The pituitary cellular model is summarized here. Each reaction has a unique reaction rate constant (k) that can depend on the receptor state (e.g., whether it's internalized or the specific molecule it's bound to). For simplicity, reactions involving an active complex have just been shown once; however, the rates of these processes do depend on the receptors' states, as described in Röblitz et al. (27).