Mathematical modeling of SCD: a literature review

Abstract

SCD is a family of genetic blood disorders that affects over 20 million people worldwide. SCD complications include pain, anemia, and early death. The hallmark cause of medical visits for people with SCD is pain, initially in the form of acute, severe, vaso-occlusive crises stemming from obstructed blood vessels and a plethora of underlying disordered biological mechanisms. Vaso-occlusive crises are unpredictable and are often associated with acute disability and/or hospitalization. Both vaso-occlusive crises and longer-term, chronic sickle cell pain can contribute to multi-system organ damage and eventually early death. Many of the disordered biological mechanisms of SCD, and how they relate to painful outcomes, are not well understood. Mathematical modeling can be a useful tool to study and analyze such disordered SCD biological phenomena: biodynamics, vaso-occlusion, and responses to SCD drug and gene therapy. In this review, we summarize the variety of mathematical modeling methods used to study SCD and provide specific examples of how mathematical modeling contributes new understandings of SCD.

INTRODUCTION

SCD is a family of genetic blood disorders caused by mutated Hb genes that are responsible for beta-globin production. A single gene mutation ultimately leads to an abnormal Hb, which distorts the erythrocyte, or red blood cell (RBC), from within. The Centers for Disease Control and Prevention (CDC) estimates that there are approximately 100 000 people living with SCD in the United States, with 1 of every 365 Black-American births and 1 of every 16 300 Hispanic births affected. Historically, there has been little data collected on SCD patients, with widespread clinical research efforts only being established in the 1970s. Even today, few treatment options for SCD exist. Population-based studies as well as in vitro studies reveal the dynamic, unpredictable nature of SCD complications. Trying to better predict these outcomes is at the core of SCD prevention and control.¹ The current age of computational power and “big data” analytics has brought new techniques to better predict outcomes of interest, using population-based studies of SCD, along with the traditional basic science studies of RBCs and tissues.

One available tool of both population-based and basic SCD research is the use of mathematical modeling techniques. Systems biology and mathematical tools applied to complex biological systems have been a valuable part of the research landscape for many decades. Personalized medicine is now a possibility. Advancements in treatment protocols for suppression of HIV/AIDS, for immunotherapy for cancer treatments, individualized surgery plans. We discuss modeling frameworks, which can expand the scope of biological SCD research in this review.

Biological background

Hb is a large, complex protein within RBCs that is responsible for oxygen transport. Adult Hb (normal hemoglobin A, HbA) is made up of 4 protein chains: 2 alpha-globin chains and 2 beta-globin chains. The beta chains in SCD are abnormal, caused by a genetic mutation occurring in the HBB gene.¹

The abnormal beta chains result in abnormal Hb molecules (hemoglobin S, HbS^) inside the RBCs, causing the RBCs to no longer be flexible and doughnut-shaped, but instead inflexible and crescent or sickle-shaped. These inflexible RBCs cannot traverse very small blood vessels, and the subsequent backup obstructs blood flow. The term vaso-occlusion has been applied mainly to this red-cell-caused obstruction. But the abnormal RBC obstruction is accompanied by a cascade of white cell, platelet, and blood vessel wall abnormalities and is more correctly termed vasculopathy. While the primary complication of SCD is severe, acute painful episodes called vaso-occlusive crises (^VOCs), the vasculopathy results in destructive anemia, ubiquitous and sometimes chronic pain, inflammation, widespread tissue destruction, organ failure, and ultimately early death.²

Genetically, SCD is transmitted via autosomal recessive inheritance. In autosomal recessive inheritance, the host is an asymptomatic carrier if only one SCD gene mutation is transmitted from the host's parents, but a symptomatic patient if a mutated gene is transmitted from both parents and at least one is a SCD gene mutation. Different mutation combinations of the HBB gene produce different alternative symptomatic forms of SCD, with the most severe and common mutation being HbSS (homozygous SCD, also called SCA). Other forms of SCD include HbSC, HbSB0, HBSC, HbSB+, HbSD, HbSO, and HbSE.¹ See Figure 1 where an HbSS child inherits an abnormal (hemoglobin S) HBB gene from each of their parents.

Figure 1.

SCD genetic inheritance.

Pathologically, sickle-shaped RBCs are less deformable. Their cellular membranes are weaker. The cells literally burst (hemolysis) easily and die prematurely, lasting only 10-20 days rather than the usual 120 days. The resulting shortage of total RBCs in the body causes anemia, or low blood counts, usually associated with weakness and easy fatigue. RBCs may not always be sickle-shaped under the microscope. This is because they only assume abnormal shapes in low oxygen conditions. The mutated hemoglobin S protein chains inside the red cell can adhere to each other more easily, making rigid strings or polymers, only when they are not carrying oxygen³ (see Figure 2).

Figure 2.

Blood cell polymerization and sickling in SCD.

Only under low oxygen states can Hb polymerize and sustain long chains, grow, stretch the red cell membrane, and alter the shape of the RBC from doughnut to sickle shaped. This is the essential process of Hemoglobin S gelation.⁴

The cells' new rigidity and stickiness causes them to clump together and stick to the walls of vessels in the body. These obstructions lead to reduced oxygen supply to various organs. This is the process of vaso-occlusion and is the cause of the painful, hallmark, unpredictable VOCs. Indeed, vaso-occlusion is the underlying genesis of most SCD complications throughout the body. See Figure 3 for an example of an obstructed vessel caused by a VOC. VOCs last from hours to weeks and may occur as few as once a year during childhood, but more often for adolescents and adults. The types of SCD complications are vast, including skin ulcers, pulmonary hypertension, retinopathy, cardiomegaly, delayed puberty, and erectile dysfunction.⁵ Acute complications, ones that require immediate medical attention, include rapid spleen swelling due to blood flow obstruction (splenic sequestration), ischemic stroke (occlusion of a large cerebral artery), and ACS (lung sickling; the leading cause of death in adults).

Figure 3.

SCD pain symptoms and treatment.

Because SCD complications occur all over the body, systemic treatment is needed to address the full extent of the disease. However, limited options are currently available. Bone marrow transplants and gene therapy can be used to replace diseased cells and are the only functional cure for SCD.⁶ Since successful bone transplant matches are a rarity in the disease demographic, and since gene therapy is also invasive and expensive, most SCD treatments aim to relieve pain and symptoms, avoid VOCs, and prevent organ complications.⁵ Opioids and other analgesics are often prescribed to relieve pain. Periodic blood transfusions are often given to reduce anemia. Preventative disease modifiers, including hydroxyurea and newer anti-sickling drugs that can be additive to hydroxyurea,⁷ are given to promote healthy red cells and suppress sickling inside the RBCs. Vaccinations and prophylactic antibiotics are provided to counteract the high rate of infections.

Until more accessible curative options are available, a critical research goal is early detection as well as prediction and prevention of symptoms and complications with drugs and blood.⁸ Unfortunately, clinicians often have incomplete risk information, or have difficulty risk-stratifying adults with SCD, in order to quickly intervene on those at greatest risk. Ideal risk stratifiers would be simple, feasible, easily accessible, and available early. They would predict frequent hospitalizations, pulmonary hypertension, or renal failure as harbingers of death itself.

Researchers are therefore developing computational models of various biological phenomena in SCD. Mathematical modeling can allow researchers to quantitatively represent multiple components and scales of a system and investigate the dynamic behavior of these components and their interactions over time under various conditions. These models enable the analysis of a system on scales that range from inter- and intra-cellular tissue, organ, host, and population. Many mathematical techniques can be used for prediction and simulation of biological systems and events that aid in the decision-making process for potential disease treatment protocols and policies.

Current research efforts include using mathematical modeling to understand the inheritance frequency of the HBB gene mutation amongst different populations. These investigations also aim to provide clarity on the existence of genetic markers for SCD and optimize gene therapy intervention. Other researchers have focused on the role RBC system dynamics play in SCD complications, primarily the sickling process and mechanics of vaso-occlusion. Beyond the mechanics of SCD, computational researchers are interested in understanding the presentation and frequency of pain in individuals with SCD. They are also interested in investigating the feasibility and efficacy of different treatment options.

Mathematical background

When studying models, it is important to understand the broad classifications they can fall into, since this characterizes the essentials of their structure. Mathematical models can be mechanistic or empirical. Models that account for the mechanisms through which changes occur are called mechanistic models. The mechanisms typically describe physical and/or chemical processes taking place in the system. In SCD research, these models typically investigate processes such as RBC sickling and vaso-occlusion. On the other hand, empirical models do not incorporate the mechanisms by which changes occur but instead use mathematical structures to mimic observed behavior patterns. Mathematical models like these are sometimes used to describe genetic mutation frequencies and SCD pain episode severity or give heuristic descriptions of outcomes.

Another major classification category of models indicates whether the equations account for stochastic (random) events. Deterministic models always predict the same outcome from a given starting point, disregarding any random variation. Models that do include variation can predict a distribution of possible outcomes even with a common starting point. These models are known as stochastic models. Of the 4 classifications above, models can fall into one of each kind, ie, empirical/mechanistic and deterministic/stochastic. Below we will briefly introduce how these common modeling techniques are used to study SCD at various levels, ranging from the molecule to the cell to the organ to populations.

QUANTITATIVE MODELING OF SCD

Statistical modeling

Since many researchers collect data to test their hypotheses, statistical methods are commonly used to gain an understanding of relationships between quantities measured in the data. Common examples of these include heat maps, line of best fit testing, and regression. Regression quantifies the impact that variables have on the model output. While researchers of various fields use regression and statistical tools, statistical methods can also be used to supplement other modeling approaches, with regression tools being used to conduct statistical analysis beforehand. A model is a mathematical expression that quantitatively represents the relationships and dynamics of a system based on the data provided. Once an initial model is created, statistical methods can be used to assess model accuracy and provide insight into variable relationships through parameter estimation. For example, Chalacheva et al.⁹ utilized multiple regression analysis as part of determining underlying physiological mechanisms in the vaso-occlusion process based on patient lab data and blood samples.

Regression is also especially useful for the identification of correlation between continuous variables, making it a common tool for investigating collinearity (if multiple predictor variables are correlated—skewing individual variable impact). When studying complex systems such as SCD, regression models often have multiple variables of interest, making multivariate regression a useful tool, for instance, in investigating measurable indicators for VOCs in SCD.

Machine learning models

The improved accuracy brought by combining computational modeling alongside statistical analysis has led to an increase in using machine learning (ML) techniques for data-informed research. ML is the computational study of statistical models and algorithms using only pattern recognition and inference to predict outcomes from sample data. ML and statistical techniques are commonly used in pain and disease progression research. Quantitative models built through ML do not require explicit instruction, but rather sample data and appropriate algorithmic application, allowing researchers to analyze data without needing to know too much about the dynamics of a system. Since pattern recognition is a primary component of classification, SCD researchers may use ML to classify/categorize treatment outcome probabilities and/or SCD pain severity. Xu et al.,¹⁰ for example, created a ML model for sickle RBC classification from patient specific microscopy image data. Since early symptom detection is a major concern in SCD, ML algorithms can be extremely useful in processing patient data and predicting future behavior, especially from image data.^11–14 Most recently, though, Darrin et al. used the “fact that the individual movement of an RBC in shear flow is an indicator of its deformability,” along with ML, to classify RBCs through cell motion.¹⁵ Other ML tasks include cluster analysis, association analysis, and anomaly detection.

Because of its predictive power, ML is also often used to predict pain patterns and pain levels. Khalaf et al. tested several ML algorithms to find the best drug dosage classification for a particular SCD pain treatment.¹⁶ Yang et al.^2,17 produced ML models that showed promising results for predicting pain scores based on physiological patient data. Although Yang et al. first used self-reported data¹⁷ and later used wearable data,² both models highlighted that the patient cohort plays a major role in the relationship between pain and physiological symptoms. Although ML has improved the ability to predict pain levels, studies of the relationship between pain and physiology are often inconclusive due to the limited size of the patient datasets.

Another example of the utility of ML is the use of physiological measurements for pain prediction. Johnson et al.’s ML model tested the feasibility of using Microsoft bands health data to predict patient reported pain scores¹⁸ while Panaggio et al.¹⁹ used patient vital sign data. Padhee et al.²⁰ also used vital sign data but focused on determining which ML method best predicts pain scores. Patel et al.²¹ used ML in their model but aimed instead to predict hospital readmission rates from raw hospital data, outperforming standard hospital scoring systems. Similarly, Padhee et al.²² recently designed a ML classification algorithm to build pain prediction models using electronic health record data.

Another mathematical technique to model systems is network modeling (NM). NM involves creating graphs out of qualitative networks for analysis, where graph nodes (or vertices) are the system elements/variables and the edges, or lines/links, represent the pairwise relationships and connect the nodes. A system can be thought of as functioning through the relationships and connections between variables. These connections and interactions can be represented through dynamic NM. These models are useful since techniques from graph theory can be used to glean new biological insights and make predictions from the results.

One application of NM in SCD is modeling RBC behavior. Since disease progression depends upon poor RBC flow, a network model with each node representative of an RBC and each link representative of the processes and interactions connecting each RBC can be useful. A biological system's underlying network organization may sometimes be straightforward (linear) but often is incredibly complex.

One example can be seen in Sebastiani et al.’s²³ work developing a predictive model of SCD severity using a Bayesian NM approach—a probability technique that quantifies uncertainty and incorporates prior knowledge into model analysis. Mehraei et al.’s²⁴ multi-scale network model functions similarly to Lu et al.'s 2019 treatment model for drug efficacy prediction. In these networks, the quantification of variable relationships is often incorporated through the graph edge definitions used. More recently, Balamanikandan and Bharathi²⁵ utilized a quantum graph theory mathematical model to extract elasticity properties and distinguish unsickled RBCs from sickled RBCs with DNA sequence data as input.

While machine learning’s ability for prediction with limited knowledge of system dynamics can be a great benefit, the lack of explicit instruction or explanation of network structure often makes them hard to interpret. Model complexity and data size are additional features of ML models that may decrease their interpretability.

Discrete time modeling: difference equations

While statistical and network-based models are great tools for understanding static variable relationships and prediction, discrete dynamical models may incorporate system patterns based on the differences in variable measurements over time. Discrete data (data taken at distinct time points) can be modeled by a difference equation as

$$x_t=f\left(x_{t-1}\right).$$

Difference equations are known as recursively defined equations, since to determine the next value of a variable, it is necessary to know the previous state of the variable.

For example, Chalacheva et al.²⁶ used experimental lab data and created a discrete dynamical system (DDS) model to identify key mechanisms that influence observed autonomic responses triggered from stimulation known to aggravate VOC. Different control groups' data were used to highlight mechanistic differences among individuals with SCD. Equations for heart rate variability and peripheral resistance variability were calculated using blood flow related mechanisms for each group. Statistical regression was then used to see which response variable could be a potential contributing factor to VOC.²⁶

In 2017, Chalacheva et al.²⁷ employed a discrete dynamical systems (DDS) to elucidate the relationships between physiological responses of interest and the experimentally applied stimulus. While SCD severity is commonly measured by genotype, the model equations based on underlying physiology enabled Chalacheva et al. to decompose the total vaso-occlusive response to pain into 4 measurable mechanisms of interest. The model showed that the biophysical marker neurogenic-vascular interaction ${BM}_{n-v}$ enhanced pain-induced vasoconstriction in SCD patients. A similar approach to finding biomarkers associated with VOC also used difference equations and statistical regression,⁹ with head-up tilt being the stimulation applied to induce VOC.

Dynamic discrete models have also been used to elucidate sickle RBC behavior. Li et al.'s²⁸ multi-scale DDS model equations represented the network of vertices along the cell membrane to understand sickle RBC membrane behavior under gelation. Bazzi et al.²⁹ also used a discrete modeling approach to locate systemic causes of changes in SCD blood during high oxygen tensions with a system of finite difference equations. Their combined experimental and computational approach enabled the quantification of sickle blood velocity.

Ordinary differential equations

Discrete models work well for distinct time events. In contrast, differential equations are mathematical expressions that equate the change in an independent (or state) variable $x$ over continuous time $t$ to the dynamics of the rates and relationships between the system variables. Each state variable represents a vital component of the system such as a cell, a chemical concentration, etc.

$$\frac{dx}{dt}=f(x,t).$$

For example, an equation of this form could be something like $\frac{dx}{dt}=-ax\left(t\right)+b$, where $\frac{dx}{dt}$ is measuring the change in x over the change in time, $t$, where the rate of change of x is proportional to its current size along with a constant source b. This is just a simplistic example, though, as these equations can become extensive and complex. Differential equations allow researchers to relate/connect biologically significant components in the system with appropriate relative rates of interactions. Ordinary differential equations (ODEs) and partial differential equations (PDEs, discussed later) allow the predictive outcome to be queried for any time point. Certain model analysis is also facilitated in these continuous models. In complex systems, interdependent differential equations form a dynamical system. In a SCD model, in the example equation above, $\frac{dx}{dt}$ could represent the change in the number of sickled RBCs $x$ over time $t$, where the number of sickle RBCs at any given time $x\left(t\right)$is impacted by a natural cell death rate $a$ and there is a constant source of new made sickle RBCs $b$ from the bone marrow.

While this equation is a simplistic example, ODEs can be used to represent multiple variable interactions over time that involve varying scales.

Zheng et al.³⁰ used multi-scale ODEs, (ie, models with multiple biological scales represented such as micromolecules, organelles, cells, and organism populations) used to model RBC production and Hb assembly. Zheng et al.’s model used ODEs to simulate transplantation of autologous stem cells that produce an anti-sickling Hb to understand how varying treatment parameters impacts the efficacy of different gene therapies. Their model findings reported the minimal dose of LT-HSC (stem cells) that were needed for a stable, long-term source of functional RBCs.

These types of ODE simulations based on real world data are especially useful because of their ability to be analyzed using several computational techniques. For example, sensitivity analysis can allow researchers to identify the most impactful model parameters and assess the associated changes in model outcomes (like above). Bifurcation analysis can elucidate the stability and intrinsic behavior of the system variables in the model. Optimal control methods can be used to hypothesize treatment options.

An early model of the RBC fluid system was created by Dong et al. in 1992³¹ using an ODE model and lubrication theory to describe the pulsatile flow in the gap between a cell and the vessel wall. Modelers can also use ODEs to investigate the intracellular and intercellular dynamics of sickle RBCs. For example, in their models, Lei et al.³ and Lu et al.³² used multi-scale ODEs to quantitatively simulate HbS polymerization inside RBCs by investigating cell morphology and fiber orientation, respectively.

Compartmental ODE methods, like the well-known SIR model, were originally developed to model disease transmission in epidemiology as members of a population move from one population compartment to another: susceptible (S) to infected (I) to resistant (R). Using this compartmental based framework, researchers have adapted it to investigate cell interactions in disease systems. In SCD research, models by Altrock et al.³³ and Zheng et al.³⁰ computationally determine the effect of potential treatment protocols on RBCs at different stages. Altrock et al.³³ created a mathematical model used to investigate RBC populations impacted by a gene therapy tool and then used the chimerism level found to inform future mouse model experimentation. The model was able to identify the number of HSCs required for successful RBC count/level by quantifying the relationship between HSCs and RBCs populations. Similarly, Zheng et al.'s model used multiscale ODEs to predict dynamics of stem cell engraftment on RBCs during gene therapy by modeling RBC production and Hb assembly during treatment.³⁰ Their model findings reported the minimal dose of stem cells needed for a stable, long-term source of RBCs.

Compartmental models have also been applied to studying genetic inheritance and frequency of individuals with SCD. Liddell et al.'s box population model,³⁴ for example, tracked the dynamics of genotypes within populations to investigate the impact of malaria selection pressure on sickle cell carrier population size. ODE models have also been used for treatment models, since parameters representing biological processes can be manipulated to reflect the impact of treatment (eg, death rate/growth rate × population size) within the systems. One ODE treatment model was used in Lu et al.³⁵ to monitor the impact of a potential anti-sickling drug on RBCs over time.

PDE modeling

When spatial dynamics need to be considered, PDEs are used in place of ODEs. PDEs are multidimensional and can track different populations throughout a spatial domain over time, making them especially useful for modeling fluid dynamics. For example, Deonikar et al.³⁶ utilized PDEs to make a flow dependent 2D model of nitric oxide (NO) production and transport in an arteriole and found that even a low presence of cell free Hb can reduce the amount of smooth muscle cell NO produced—potentially playing a role in the development of pulmonary hypertension.

Computational fluid dynamic (CFD) models derive from the Navier-Stokes equations, a set of partial differential equations that describe how “the velocity, pressure, temperature, and density of a moving fluid are related.”³⁷ These equations allow researchers to simulate SCD blood conditions given the proper data.

To identify triggers and indicators of VOCs, a deeper understanding of the disease hemodynamics (blood flow) is necessary and CFD models are commonly used to model the flow of fluid through space, thus making PDEs especially useful for modeling RBC movement in the arteries. In 1980, Berger and King³⁸ used the Krogh cylinder fluid dynamics model to represent SCD blood flow through the capillaries. The results suggested that feedback interactions between oxygen concentration and flow velocity through the capillary were important in SCD blood. Later, Rivera et al.³⁹ used computational fluid dynamics to alter fluid and artery wall properties to simulate scenarios causative of significantly elevated arterial blood velocities.

In 2020, Bazzi et al.²⁹ made a CFD model to simulate microfluidic movement and investigate VOCs, highlighting the advantages of microfluidic modeling. Szafraniec et al.⁴⁰ used an experimental microfluidic system to separate the cell flow profile into a bulk component and a wall component and then computationally modeled the system to evaluate differential contributions of effective viscosity and wall friction to the overall blood resistance.

To investigate potential reasons for the inflammation of the endothelial cells lining blood vessel walls, Zhang et al. in 2020 implemented a suspension flow PDE model and simulated idealized blood flow in SCD. Similarly, a PDE model was developed by Chaturvedi et al.⁴¹ to model how an RBC (in his case a pellet) moves through a capillary (or narrow fluid filled cylindrical tube) to investigate the impact of viscosity on blood flow. Sawyer et al.⁴² used CFD modeling to reconstruct the intracranial portion of the internal carotid artery and branches and extract the geometry to analyze vascular architecture influence on cerebrovascular risk. Most recently, a CFD model of the flow of a single RBC with alterable rheological properties was implemented to test the feasibility of the lab-on-a-chip microfluidic diagnostic tool for SCD.⁴³

The use of this foundational system CFD along with new computational simulation power has led to a rise in dissipative particle dynamic (DPD) models. DPD is a simulation technique used for molecular or mesoscopic modeling and investigation, also commonly used in chemistry and pharmacology. Lei et al.^44,45 constructed a 3D model of 3 typical shapes of sickle cells using experimental SEM observations to observe and quantify adhesion behavior of SS-RBCs inside capillaries. The model was able to validate that vaso-occlusion is a “complex process triggered by the interactions between multiple density groups, where each group contributes differently to the occlusion crisis.”⁴⁴

PDEs have also been utilized in studying genetic inheritance (in place of ODEs) when the number of individuals is large enough. Tchuenche et al.,⁴⁶ for example, created a theoretical PDE population model to track genetic inheritance of SCD.

Hybrid modeling

Hybrid models are models that contain 2 or more model types. For example, Clifton et al.⁴⁷ utilized a hybrid statistical and mechanistic ODE model to predict pain levels in SCD patients using smartwatch data along with drug mechanics. The statistical component of the model estimated the best estimates of parameter rates from the data (elucidating variable relationships). These parameters are then utilized in an ODE model to capture the relationship between the data and those parameters and how their relationship impacts the system. Clifton et al.’s⁴⁷ hybrid model results along with the effectiveness of ML on outcome prediction elucidate the need for more models that can accurately assess the personalized data available today. The increased accessibility of patient monitoring data combined with the low-cost implementation of complex quantitative models will allow for research advancements that save error and injury. Table 1 lists different types of models, what systems they are often used to analyze, and examples of investigation of that system.

Table 1.

Math models summary.

Math model type	Example	System applications	References
Statistics (regression)	$Y_i=f\left(X_i,\beta \right)+e_i$	VOC mechanism investigation	9 , 26 , 27
Machine learning (ML) (algorithms)		RBC shape/severity classification Subjective pain score prediction Medication/treatment dosage/frequency analysis	8 , 10–14 , 16–19 , 21 , 22 , 47 , 48
ODEs (differential equations)	$\frac{d}{dt}=ax\left(t\right)+b$	RBC modeling (cell shape, cell behavior) Insight into polymerization process (Hemoglobin S molecule) Pain score prediction Pharmacology/medication dosage + interactions	3 , 31 , 32 , 35 , 43 , 47 , 49
PDEs (computational fluid dynamics)	$\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial x_j}\left[\rho u_j\right]=0$	VOC mechanism investigation Modeling sickle RBC blood flow (re: microfluidic channels) Cellular interaction impact on complications	23 , 36 , 38 , 41 , 42 , 45 , 48 , 49
Difference equations (discrete time modeling)	$x_t=a{x}_{t-1}+b$	VOC mechanism investigation RBC behavior	9 , 26 , 27 , 40
Networks (graph theory)		Biodynamic behavior of sickle RBCs Cell shape classification/diagnosis Pain severity prediction Genetic therapy modeling	15 , 24 , 25 , 28 , 50

Choosing a math model type

Mathematical techniques can be used to quantify biological system processes, simulate these systems, elucidate variable relationships, and predict future outcomes. The type of model applied is based on multiple factors that should be considered (see Figure 4).

Figure 4.

How to choose a math modeling technique.

• The research objective are as follows:

  • Classification: RBC shape, disease progression, pain class

  • Prediction: gene inheritance, pain score prediction

  • Investigating variable relationships: hemoglobin S molecule, RBC flow and behavior, VOC triggers, treatment efficacy

• The data collected/available are as follows:

  • Aggregate or individual

  • System or local

  • Sample size

  • Frequency of data collection

  • Blood image and/or video data

  • Self-reported survey data

  • Physiological data (vital signs + lab data)

• What is known and unknown about the system?

  • The relationships between the variables are known (eg, “oxygen concentration directly impacts gelation”)

  • There are unknown factors or missing variables in the analysis (eg, “some aggregate of these vital sign measurements influence pain expression”)

The same way many understand mathematics as a language in its own right, but mathematical modeling can be seen as an art form. It offers the unique ability to investigate and experiment with biological processes that may otherwise elude us due to their complexity. Because of this complexity, there is not necessarily a right way to model biological phenomena. The type of model chosen is entirely dependent upon the research question, the available data, and the dynamics of interest. Consulting with mathematicians/computational researchers, sharing the data to be analyzed, and collaborating on research questions may lead researchers and clinicians to revelations that save time, errors, and money.

In the following sections, the authors give examples of the way multiple approaches can be used for each problem/research area in SCD. Section 3 illustrates models created to investigate VOCs and hemodynamics in SCD. Section 4 describes models made to represent SCD cellular dynamics and RBC behavior. Section 5 illustrates gene-based models and Section 6 explores pain prediction models. Lastly, Section 7 reviews models created to analyze treatment options in SCD. Each section is associated with an illustrative table. These tables, Tables 2-6, summarize models discussed in that section, which were published for the purpose being discussed in that section (investigate VOCs and hemodynamics, versus predict pain, analyze treatment options, etc).

Table 2.

Summary of VOC models.

References	Model type	Dataset	Key variables	Summary
Berger and King 198038	PDE model	n/a	Oxygen concentration, oxygen partial pressure, tissue region	Fluid model of blood flow in the capillaries
Lei and Karniadakis44	DPD model	Shapes observed in experiments by scanning electron microscopy (SEM)	RBC free energy, cell adhesion, reaction rates	3D model of morphology and dynamic properties of SS-RBC
Lei and Karniadakis45	DPD model	Ref Lei and Karniadakis 2012	RBC free energy, adhesive interaction rates	Adhesion behavior of SS-RBCs using balance of free energies inside capillaries
Chalacheva et al.26	Linear ODE system	28 patients age: 10-30 years old SS genotype (save 2)	Heart rate variability, peripheral resistance, variability	Functional mechanisms of blood pressure, respiration, and peripheral vascular resistance
Rivera et al.39	CFD	3 patients—SCA genotype	Arterial segments: distal ICA, MCA, ACA	Causes of elevated arterial blood velocities from fluid and artery wall properties
Chalacheva et al.27	Multiple regression analysis	45 subjects Age: 13-55 yrs old HbSS and HbS β0	Finger blood volume, temperature, blood pressure, respiration, blood pressure coupling, respiratory coupling, neurogenic thermal pain coupling, neurogenic-vascular interaction	Contribution of physiological components to subjects’ heat induced pain pulse with autonomic dysfunction
Chalacheva et al.9	DDS, regression	66 subjects 13-18 yrs old HbSS (21) HbS β0(1) HbSβ + (1)	Heart rate variability, peripheral vascular variability, diastolic and systolic blood pressure	Responses to head up tilt (HUT) stimulus
Blakely and Horton48	CFD	Cited Cho and Kensey 1991	Vascular branching, shear rate, drag force, particle-wall interactions, particle-particle interactions	Microfluidic movement to inform into vaso-occlusive crises
Chaturvedi and Shah43	DPD model	n/a	Pressure, shear viscosity, fluid film thickness	Models the flow of a single RBC with alterable rheological properties; lab-on-a chip microfluidic diagnostic tool for SCD diagnosis

Table 3.

Summary of RBC models.

References	Model type	Dataset	Key variables	Summary
Dong et al.31	Geometric ODEs	Chien et al. 1992	Cell radius, cell length, plasma velocity, vessel radius, distance between cells, distance between cell and wall	Models the flow in the gap between cell and the vessel wall
Deonikar and Kavdia36	PDE	DATASET	Steady-state SCD conditions, NO concentration, NO consumption rates, NO production rate	Role of Hb concentration in disturbing smooth muscle cells/pulmonary hypertension
Lei and Karniadakis3	Multi-scale DPD model	n/a	RBCs, leukocytes, cell endothelium interaction	Model of polymerization and VOC events for intervention/treatment
Altrock et al.33	Discrete time compartmental model	Berkeley sickle mice used	Hematopoietic stem cells, RBC populations, survival factor, age	Predicts the number of HSCs required for successful RBC count
Li et al.28	Multi-scale DDS	4 patient blood samples HbSS genotype	Elastic bond energy, bending energy, corresponding energy	Behavior of sickle RBCs under hypoxia due to sickle RBCs’ irregular geometry, decreased cell deformability, and elevated cell volume
Lu et al.32	ODE, MARS	n/a	Spring constant, instantaneous and equilibrium distance, equilibrium angle	Fiber orientation and interaction range between HbS fibers are key interactions during polymerization
Xu et al.10	Machine learning	SCD patient blood 7 patients	Patient-specific, microscopy image data	Classifies sickle RBCs into different types to identify RBC shape factors and parameters
Bazzi et al.29	Finite difference equations	Patient RBC samples	Viscosity, shear rate	Quantifies sickle blood velocity where results suggest sickle blood flow is altered systemically
Delgado-Font et al.13	Machine learning algorithms	Peripheral blood Smear sample images	n/a	Patient blood sample images and classified RBCs as normal, elongated, or deformed
Petrovic et al.12	Machine learning	825 blood smears images 2695 tagged cells	n/a	Defines how to select the most important features for sRBC classification
Zhang et al.49	CFD suspension flow model	n/a	Slow velocity, discoid radius, total energy, energy density	Investigates potential reasons for the inflammation of the endothelial cells lining the blood vessel walls
Chaturvedi et al.41	PDE model	n/a	Pellet radius, pellet curvature	Investigates the impact of viscosity on how an RBC moves through a narrow capillary
Praljak et al.14	Machine learning	Microfluidic blood, smear images	n/a	Morphology-based classification scheme to identify 2 naturally arising sRBC subpopulations that link to underlying cell biomechanical properties
Sawyer et al. 202242	CFD	10 participants Age 5-15 yrs old HbSS and HbS β0	3D model of internal carotid artery: ICA, MCA, ACA	Reconstructs the intracranial portion of the internal carotid artery and branches to analyze the vascular architecture influence on cerebrovascular risk in SCD
Szafraniec et al.40	Machine learning and difference equations	n/a	RBC flow rate, velocity, resistance	Models the microfluidic blood system to evaluate impact of effective viscosity and wall friction to the overall resistance in blood
Darrin et al.15	Machine learning CNN	4 patients Age 18+ yrs old HbSS, HbS β0	Flipping cells, tank treading cells, highly deformable cells	Machine learning algorithm to automatically classify cell motion in videos with high class imbalance

Table 4.

Summary of gene models.

References	Model type	Dataset	Key variables	Summary
Tchuenche46	PDE	n/a	Males, females, age, growth rate, birth rate, death rate, mating rate	The persistence of SCA is likely due to the selective advantage of the abnormal S gene over the normal hemoglobin A in tropical regions
Liddell et al.34	ODE	n/a	Carrier adults, noncarrier adults, carrier children, non-carrier children, afflicted adults, afflicted children	Reveals that selection pressure from malaria results in a higher fraction of sickle cell carriers and that malaria protection selectively favors the sickle cell gene
Balamanikandan and Bharathi25	Quantum graph theory	1387 patients SCD type: HbSS and HbS thalassemia	Input DNA sequence	Input DNA sequence information & used machine learning mining approaches to distinguish normal cells from sRBCs.

Table 5.

Summary of pain models.

References	Model type	Dataset	Key variables	Summary
Clifton et al.47	Hybrid ODE + statistics	47 patients Age: 18+ SCD type: HbSC, HbSS, HbSB+ thalassemia, HbB0	Patient pain level, amount of drugs within patient, pain level probability distribution, unmitigated pain level, drug relaxation rate, drug rate of decay, drug dosage times, number drug doses taken	Shows predictive value for adult patients by forecasting the probability distribution of pain for a patient at a point in the near future
Yang et al.17	Machine learning	5363 records from 40 in-patient participants	Pain score, vital signs (SpO2, systolic BP, diastolic BP, heart rate pulse, respiratory rate, temp)	Estimates pain score predictions based solely on physiological measurements without including patient medication information
Yang et al.2	Machine learning	29 patients	Heart rate, RR interval, galvanic skin response, skin temp, steps	Shows that subjective pain scores can be estimated using objective wearable sensor data with high precision
Padhee et al.20	Machine learning	67 927 records from 50 participants over 5 years	Pain score, vital signs (SpO2, systolic BP, diastolic BP, pulse, resp, temp)	Tests ML algorithms to predict pain scores, with the Decision Tree algorithm as the most promising model
Panaggio et al.19	Machine learning	46 patients over 3 years	Vital sign data including respiratory rate, heart rate, and blood pressure	Machine learning models that outperform baseline models in estimating subjective pain, distinguishing between typical and atypical pain levels, and detecting changes in pain
Padhee et al.22	Machine learning	5363 records from 40 participants	6 vital signs, patient self-reported pain, medication type, medication status, total medication dosage	Classification model using raw data and deep representational learning to predict subjective pain scores. that performed better when provided with medication data along with vital signs data

Table 6.

Summary of treatment models.

References	Model type	Dataset	Key variables	Summary
Mehraei et al.24	Hybrid functional petri network (HFPN) model	qPCR data	Therapy targets, protein, and multiprotein	Human fetal-to-adult Hb switch network model to test and compare 6 strategies for Hb drug treatment
Khalaf et al.16	Machine learning	Commissioned specifically for the purposes of this study and was collected within a 5-year period from the Alder Hey Children’s hospital	Weight, Hb, MCV, platelets, neutrophils, reticulocyte count, alanine aminotransferase, body bio blood, HbF, Bilirubin, LDH, AST	ML architectures tested to find best to classify dosage of medication required
Lu et al.35	Hybrid stochastic and mechanistic ODE	Microfluidic channel observations	Temperature, deoxygenation time, oxygen pressure, RBC volume, Hb concentration	Models HbS polymerization and growth against the RBC membrane and outputs numbers of nuclei generated, the lengths of HbS fibers, the configuration of fiber domains, and the sickling of RBCs.
Patel et al.21	Machine learning	3299 admissions data comprising of 446 adult SCD patients	n/a	Applies ML algorithms to predict 30-day unplanned hospital readmissions in SCD and identified most important variable predictors.
Zheng et al.30	Multi-scale ODE/quantitative systems pharmacology (QSP)	n/a	Endogenous LT-HSC, transduced LT-HSC into ST-HSC, fold-amplification, mean residence time	Varies specific treatment parameters to investigate short- and long-term measures of treatment efficacy

MODELING HEMODYNAMICS AND VASO-OCCLUSIVE CRISES IN SCD

For models of VOCs, findings indicate potential markers/mechanisms that contribute to the triggering of VOCs, including analysis of blood flow property contributions.

Of the hemodynamic models for VOC investigation, Lei et al.’s discrete dynamics model found that interactions between multiple density groups, where each group contributes differently to the occlusion crisis, is a trigger for VOCs.⁴⁵ Rivera et al.'s 3D fluid dynamics PDE model reconstructions revealed “an uneven, internal arterial wall surface in children with homozygous SCA and higher mean velocities in the Middle Cerebral Artery up to 145 cm/s compared to non-SCA reconstructions.” Identifying cellular causes of these microstructures of RBCs or how luminal narrowing due to endothelial hyperplasia is induced by disturbed flow would provide new targets to treat children with SCD.³⁹

During their investigation for potential triggers of VOC, Chalacheva et al.’s²⁶ ODE model system found that vascular disturbance when the baroflex is blunting could be a potential factor that contributes to VOCs. In a similar attempt to monitor and identify physiological symptoms that indicate pain induced VOCs, Chalacheva et al.²⁷ used multiple regression analysis and found that blood pressure markers showed strong differences between SCD and non-SCD individuals. Chalacheva et al.⁹ also found measurable factors that indicate the parasympathetic activity that appears when VOCs occur using their statistical regression model.

A common thread throughout these models is the emphasis on functional mechanisms. Since SCD is a blood disorder, most of the measured factors are related to cardiac activity and blood properties. In 2015, Chalacheva et al.²⁶ mathematically modeled heart rate variability and peripheral resistance variability to investigate vascular functionality. But Chalacheva’s later models^9,27 induced VOCs through stimuli and compared the functional differences in heart rate and blood pressure mechanisms in SCD and non-SCD individuals. While the findings from these models provide some insight into what measurements are important to track VOCs, many note that investigating the underlying physiological mechanisms that are responsible for these measurement differences is imperative. Table 2 lists the different mathematical models of VOCs in SCD, the key variables and data used in model formulation, and a quick summary of the model.

MODELING CELLULAR DYNAMICS + RHEOLOGY OF SCD

RBCs are at the center of all processes in SCD. HbS polymerization occurs inside RBCs and deforms RBCs from usual doughnut or biconcave disc-shaped cells into sickle-shaped and odd-shaped cells. Therefore, understanding the intercellular dynamics of RBCs and other blood elements is an imperative effort in SCD research. Since SCD is a multi-scale, multi-organ disease, RBC research in SCD has been approached through multiple mathematical methods.

The conclusions from RBC models generally fall under 4 broad categories: (1) insight into the complex processes in SCD; (2) recognition of the role of cell-cell and cell-endothelial wall interactions in SCD complications; (3) further understanding of sickle RBC behavior; and (4) classification of RBCs by shape/disease severity.

HbS polymerization is a particular area of interest in the context of SCD as one of the main tasks inside RBCs that contribute to sickling/gelation. Using multi-scale modeling, Lu et al.³² was able to predict the efficacy of sickling inhibitors and identify additional effects of existing and future drugs. Differential equations and the MAR scheme allowed them to simulate the HbS polymerization process, measure the growth rate and bending stiffness of a single HbS fiber, and extract the interaction forces between HbS fibers that occur during polymerization.³² Multi-scale modeling also allowed for illustration of how VOCs begin and are sustained. Lei and Karniadakis's³ stochastic VOC model was used to represent the development of the intracellular aligned sickle Hb polymer domain. The insight gained from this model revealed that perhaps processes other than sickling such as vessel endothelium activation and cell-endothelium adhesion may be legitimate mechanistic targets of treatment and prevention of VOCs and treatment of SCD in general. Indeed, one SCD drug currently on the market utilizes one of these alternative targets.⁵¹

Other RBC models illustrated the merit of understanding RBC interactions in the formation of treatment protocols that impact sickle cell development. Altrock et al.’s³³ compartmental model revealed the number of stem cells necessary for a successful non-sickle RBC count/level that was found by quantifying the relationship between stem cell and RBC populations. Similarly, Zhang et al.⁴⁹ found the minimal dose of transduced stem cells necessarily engrafted for a stable, long-term source of functional RBCs after transfusion. Additionally, a model finding that even a low Hb concentration in the blood can negatively impact smooth muscle cells by Deonikar et al.³⁶ indicates a pathway for treating the development of pulmonary hypertension in individuals with SCD. All these models utilized differential equations (ODEs/PDEs) for simulation. PDEs were also used to create computational fluid dynamics models. Sawyer et al.'s⁴² computational fluid dynamics approach allowed for the evaluation of vascular topology and hemodynamics in SCD using MRA images. Bazzi et al.²⁹ made a model that can be used to identify patients who are at higher risk for cerebrovascular complications. The results suggest that sickle blood flow “is altered systemically from the arterial to the venous circulation.” Zhang et al.’s⁴⁹ findings from their idealized blood flow model may aid in understanding the pathophysiology of chronic endothelial inflammation in SCD from a biophysical perspective.

Other RBC models investigated the biodynamics of sickle RBC shape formation. Li et al.'s multi-scale network model²⁸ found that biodynamic behavior of sickle RBCs under hypoxia is heavily impacted by their irregular geometry, decreased cell deformability, and elevated cell volume. Dong et al.³¹ found a transition from membrane to internal polymer dominance of deformability as oxygen saturation was lowered. Chaturvedi et al.'s⁴¹ model showed that the higher viscosity of plasma exerted comparatively higher drag force through their PDE model. Modeling the microfluidic channel, Sawyer et al.⁴² showed that blood from patients with SCD exhibited elevated frictional and viscous resistances at all physiologic tensions. From another perspective, RBC shape was studied by classifying sickle RBCs from blood data. ML models by Wang et al.¹¹ and Petrovic et al.¹² allowed the classification of RBC types according to ML patterns and defined how to select the most important features for classification of blood smears.

The remaining models classified RBCs by shape and disease severity. Delgado-Font et al.¹³ analyzed blood sample images to classify RBCs as normal or elongated or having other deformations. Praljak et al.¹⁴ established a morphology based on classifying the RBC population into subgroups using novel visual markers that linked to underlying cell biomechanical properties. Xu et al.¹⁰ reported a ML model that created an automated algorithm for sickle RBC classification using a convolutional neural network (CNN). Similarly, Darrin et al.¹⁵ created a ML algorithm to automatically classify cell motion in videos with high motion imbalance. Table 3 lists the different mathematical models of RBC behavior in SCD, the key variables and data used in model formulation, and a quick summary of the model.

MODELING GENETIC INHERITANCE AND MUTATION IN SCD

The ODE models utilized to model gene populations investigated selection factors that impacted SCD inheritance. Liddell et al.’s³⁴ compartmental ODE model tracked the dynamics of genotypes within key SCD populations to understand SCD’s relationship with malaria. By using a compartmental model, similar to the SIR epidemiology framework,³⁴ the population size of children vs adult non-carriers, carriers, and afflicted populations was tracked. After simulations, stability and sensitivity analyses were done and supported the conclusion that selection pressure from malaria does result in a higher fraction of carriers of the HbS gene. This is likely because malaria protection selectively favors the sickle cell gene. Another supporting finding is that the selective advantage of the HbS gene over the HbA gene does allow SCD inheritance to persist in tropical populations as shown by Balamanikandan et al.'s theoretical model. Balamanikandan et al.’s²⁵ quantum network model also aimed to aid in early diagnosis and identification of SCD. The authors were able to input DNA sequence data into a quantum graph theory model and utilize ML mining approaches to distinguish normal RBCs from sickled RBCs. Table 4 lists the different mathematical models of the genetics in SCD, the key variables and data used in model formulation, and a quick summary of the model.

MODELING PAIN BEHAVIOR IN SCD

Pain is the primary symptom of SCD. As a symptom, information about pain may only come through observation—ie, using patient reports. Management of SCD related pain is particularly challenging due to its subjective nature. In clinical practice, medical providers often search for objective indicators, such as vital signs, non-verbal cues, and biomarkers, to guide their assessment and treatment of pain. This exercise often results in misdiagnosis and failure. Many envision instead employing a precision medicine, individualized, integrative treatment approach, based on a whole-person model of pain.⁴⁰ As such, the development of an objective automatic pain estimation method would lead to marginal improvements in pain assessment and management but multiple models might need to be applied to refine prediction. Since the goal of pain-related modeling is often to predict pain intensity, pain frequency, pain behavior, changes in pain over time, model conclusions may often fail to predict pain accurately over time. Instead, researchers, now with access to an ever-increasing amount of medical data, should think of more complex hybrid modeling and ML methods in order to design pain therapy. One example of this kind of thinking and ML modeling was Sebastiani's Bayesian network model.²³ It integrated individual disease complications and lab test results and was able to compute personalized severity scores. Severity scores were determined by a network of interactions of 14 individuals. Yang et al. and Padhee et al. experimented with ML methods/algorithms to predict patient pain scores.^2,17,22 In 2018, Yang et al. were able to predict pain scores based using vital signs, without needing medication information.¹⁷ Padhee et al. also used vital sign data along with self-reported pain data to predict pain scores.²² They found that the Decision Tree algorithm was the most promising approach for prediction. Consistent with the “big data” hypothesis of increasing accuracy with more complex models, all methods Padhee tested had higher accuracy when they had more training data to feed the model.²² In 2019, Yang et al. conducted a similar study, and used feature selection algorithms to identify which sensor data features were significant.² Sensor data were also used in Clifton's hybrid statistical and mechanistic ODE model, but Clifton et al.'s⁴⁷ data focused on pain scores and pain medication information.

Clifton et al. noted that their hybrid model would drastically improve in accuracy with the addition of blood pressure, heart rate, and activity level information via Fitbits. This assumption was supported by the conclusions from other works.^2,17,47,49 All the models mentioned here focused on physiological factors that often centered on heart-related measurements. The models from Padhee et al.²² and Yang et al.¹⁷ used vital sign data (SpO₂, systolic BP, diastolic BP, pulse, resp, temp) and pain scores for prediction. Similarly, Yang et al.'s feature selection algorithm revealed that galvanic skin temperature, heart rate variability, acceleration, and steps taken were physiological and body movement related features that had high significance in pain prediction.² These findings raised the perceived value of employing more ML and mechanistic ODEs for pain prediction in SCD.

Using ML, authors^18,19,22 used electronic health data to predict subjective pain scores from patient data. Electronic health data outperformed hospital prediction systems. Most recently, Padhee et al found that models performed better when provided with medication data along with vital signs data.²²  Table 5 lists the different mathematical models of pain behavior and dynamics in SCD, the key variables and data used in model formulation, and a quick summary of the model.

MODELING TREATMENT EFFICACY IN SCD

Mehraei et al.'s 2016 network model was created as an illustrative guide for drug discovery and prediction for $\beta$-globin disorders.²⁴ The authors hypothesized possible treatment of SCD and other $\beta$-globin disorders by inducing protective fetal Hb via modulating specific protein targets that induce $\beta$-globin. The authors validated the superiority of this treatment approach versus others, using simulation models based on the literature, and comparing the simulations for postulated efficacy at fetal Hb induction. Another model studying drug treatment impact, Lu et al.’s 2019 kinetic ODE model was able to reproduce different experimental results without tuning model parameters, indicating its capability to confirm and analyze laboratory and clinical findings.³⁵ Since a lot of treatments like hydroxyurea target multiple pathways like sickling, adhesion, and inflammation, Lu et al.’s 2019 model may be useful since it can predict the efficacy of sickling inhibitors.

Khalaf et al.'s ML classification algorithms found that the Random Forest Classifier worked best on their dataset to classify the dosage of medication required for the treatment of patients with SCD.¹⁶ Patel et al.²¹ applied ML algorithms to predict 30-day unplanned hospital readmission rates and outperformed the standard hospital readmission risk scoring systems (LACE and HOSPITAL) by a large margin. Zheng et al.'s³⁰ (QSP) model and dynamical approach to represent RBC production and treatment simulations found the minimal dose of LT-HSC needed for a stable long-term resource of functional RBCs. This will allow researchers to understand how varying specific treatment parameters affect short- and long-term measures of treatment efficacy. Table 6 lists the different mathematical models of SCD drug treatment efficacy, the key variables and data used in model formulation, and a quick summary of the model.

CONCLUSION: FUTURE OF MODELING

This review has demonstrated the utility of using mathematical modeling to gain a deeper understanding of SCD. By quantifying known dynamics and phenomena, fitting models to experimental data, and conducting parallel assays and simulations, researchers have been able to elucidate significant biomarkers, processes, and clinical findings that at least partially contribute to SCD progression.

However, the field of SCD research is relatively new, with official research efforts only starting in 1949¹ despite its discovery in 1910. There is much to still be understood about the disease’s dynamic behavior. Mathematical and computational efforts have progressed the field greatly, taking advantage of the new abundance of data on individuals with SCD. Like most research, however, these findings mark only the beginning of what can be understood.

The models investigating VOCs in SCD are still aim to uncover biomarkers and other physiological mechanisms that primarily impact the vascular and other system dysfunctions that together cause VOCs. There are also still many questions about RBCs, as everything from their size and shape to their rigidity and location have major implications on SCD presentation. Li et al. in 2017 demonstrated their interest in understanding the complex relationship between RBCs, morphological membrane distortion of RBCs, and deoxy HbS polymer chain formation. By classifying RBCs according to their shape, Xu et al. in 2017 have tried to build a gold standard RBC phenotype library within SCD. Petrovic et al. in 2020 have tried to optimize RBC classification by adding more features based on PCA and LDA for SCD diagnosis support. Zheng et al. 2021 and Altrock et al. 2016 have suggested extending their models by incorporating diffusion of chemicals and/or drug pharmacokinetics to help understand therapeutic effects on different RBC shape phenotypes and blood flow.

The common theme found in studies building cellular, genetic, behavioral, pain, and treatment SCD models is the need to increase model complexity. Parameters and variables have started with simplified models, in order to understand or explain the operation of small galaxies of a universe of SCD dynamics, and only at a basic level. Now, simple models must be either combined, expanded or layered properly, in a branching tree-like effect, to account for the impact of confounding variables and mathematical degrees of freedom at many levels-the molecule, cell, the momentary VOC, the organ, and the biopsychosocial milieu.

For example, Lu et al.’s 2019 model parameters must now be varied to incorporate the effect of anti-sickling agents. Similarly, researchers who have successfully predicted short-term SCD pain intensity, must now improve system prediction performance by predicting longer term SCD pain intensity, along with VOC visit behavior. These are the kinds of predictions needed to affect real-life events, modify the disease course, and prevent mortality. In order to accomplish that, researchers must now collect and incorporate into mathematical models more variables, and more layers of variables, simultaneously. Added to models of molecular and cellular behavior, must be simultaneous models of physiological measurements, pain reports, activity level, age, gender, medication usage, and psychosocial and environmental stressors—whole-body models. One investigator stated upon completing a model of red cell behavior: “This is a step towards continuous and non-invasive pain management for SCD patients during hospitalization and after they are discharged from the hospital. Doing so will allow us to create a remote pain management system that can hopefully reduce re-hospitalization and improve the quality of life for patients with SCD.”¹⁰ One hope is that mathematical modeling, among other benefits, will benefit the field by being the ideal platform for testing optimal dosage of medications and hypothesizing treatment effects in order to obviate conducting expensive and time-consuming randomized controlled trials.

Systems biology and mathematical modeling are having a resurgence with the rise in computational power along with an abundance of data. Computational modeling is going to play an important role in science, but models, to be used, must demonstrate reality. In most cases, life and the world are not simple. SCD is not simple.

AUTHOR CONTRIBUTIONS

Quindel Jones (Conceptualization, Visualization, Writing—original draft, Writing—review & editing), Reginald McGee (Funding acquisition, Writing—review & editing), Rebecca Segal (Conceptualization, Funding acquisition, Supervision, Writing—review & editing), Wally R. Smith (Conceptualization, Funding acquisition, Writing—review & editing), and Cecelia Valrie (Conceptualization, Funding acquisition, Writing—review & editing)

FUNDING

This work was supported by the National Institutes of Health Helping to End Addiction Long-term (HEAL) Initiative and the National Institute of Dental and Craniofacial Research and the National Institute of Neurological Disorders and Stroke (R21DE032583).

CONFLICTS OF INTEREST

W.R.S. is Executive Editor for the Journal of Sickle Cell Disease. Full peer-review for this manuscript was handled by Journal of Sickle Cell Disease Associate Editor Betty Pace. All (other) authors declare no conflict of interest.