23. Hematopoiesis Regulation

Abstract

This chapter reviews key concepts of hematopoiesis and develops a mathematical model of erythropoiesis regulation via negative feedback to gain insights into hematopoietic homeostasis. The model describes the dynamics of erythrocyte and erythropoietin levels over time using a system of ordinary differential equations that captures the essential feedback inhibition mechanism. It utilizes a quasi-steady state approximation to simplify the model formulation. Steady state and stability analyses reveal the equilibrium conditions and relaxation behavior, providing a quantitative framework to understand how modulating degradation rates, sensitivity thresholds, and other regulatory parameters impacts the system response time and red blood cell production requirements. While simplified, the model demonstrates how even basic representations can provide mechanistic insights.

Introduction¶

Hematopoiesis precisely regulates blood cell production through negative feedback control. In this chapter we develop a mathematical model of erythropoiesis dynamics using ordinary differential equations. The model represents interactions between circulating erythrocytes and erythropoietin levels, which provides a framework to explore homeostasis.

A key feature is applying a quasi-steady state approximation. This leverages the large difference in erythrocyte and erythropoietin degradation timescales, allowing elimination of the fast erythropoietin variable. This simplification produces a reduced model focusing on long-term erythrocyte behavior while retaining feedback dynamics.

Analyzing the reduced model yields insights into steady states, stability, and response time properties of negative feedback regulation. Comparisons with literature validate the model’s biological relevance.

This chapter demonstrates how even simple mathematical models augmented with techniques like quasi-steady state approximations can generate mechanistic understanding of physiological regulation beyond observation alone.

Hematopoiesis Physiology¶

Hematopoiesis is the process of blood cell production that continuously replenishes various circulating cell types. This ensures a healthy supply of erythrocytes, leukocytes, and platelets, which all play crucial roles in oxygen transport, immune defense, and hemostasis. The most common site of blood cell production is the spongy tissue inside bones called bone marrow. Hematopoiesis that occurs in the bone marrow is called medullary hematopoiesis. Less often, hematopoiesis takes place in other parts of the body, like the liver and spleen. Hematopoiesis that occurs outside of the bone marrow is called extramedullary hematopoiesis Jaffe et al. (2024).

Red blood cells, also called erythrocytes, carry oxygen from the lungs to tissues and carbon dioxide back to the lungs. They are the most abundant cell type in blood. The production of red blood cells is termed erythropoiesis. White blood cells, or leukocytes, fight infection and protect the body from pathogens. They also destroy abnormal cells. Leukocyte production is termed leukopoiesis. Platelets, or thrombocytes, are cell fragments that clump to form clots after injury and create seals in damaged tissue to prevent blood loss. Their production is called thrombopoiesis.

Hematopoiesis begins with an originator cell common to all blood cell types. It’s called a hematopoietic stem cell (HSC). An HSC develops into a precursor cell, or “blast” cell. A precursor cell is on track to become a specific type of blood cell, but it’s still in the early stages. A precursor cell goes through several cell divisions and changes before it becomes a fully mature blood cell.

Hematopoiesis is tightly regulated through negative feedback loops that help maintain optimal blood cell levels. When cell concentrations rise in circulation, signals are sent to slow or inhibit further hematopoietic production. This prevents overabundance of cells.

Specifically for erythropoiesis, elevated red blood cell counts or hemoglobin trigger the kidneys to decrease secretion of erythropoietin (EPO). Since EPO estimulates proliferation and differentiation of red blood cell precursors in the bone marrow, lowered EPO then downregulates these processes. This negative feedback mechanism helps stabilize red blood cell production.

A similar process occurs for thrombopoiesis. Higher platelet counts inhibit thrombopoietin secretion by the liver, which dampens megakaryocyte development and thrombopoiesis in the bone marrow.

Meanwhile, leukocyte regulation involves feedback suppression via certain cytokines. Immune cells secrete less of growth factors like G-CSF and GM-CSF when higher leukocyte levels are detected. This decreased cytokine stimulation, in turn, curbs further leukopoiesis.

Mature circulating blood cells provide another level of negative feedback regulation. As their numbers increase systemically, fully differentiated cells secrete signaling molecules that act to inhibit further proliferation and self-renewal of hematopoietic stem cells (HSCs) in the bone marrow niche. This feedback mechanism helps balance the production of new cells from the HSC pools with the clearance of older senescent cells from circulation.

As new blood cells are generated, aging cells are cleared through specialized elimination pathways. Mature red blood cells, with their biconcave disc morphology, are superbly adapted for oxygen transport as they flexibly navigate narrow vasculature. However, after approximately 120 days in circulation, they become increasingly fragile. Macrophages in the spleen and liver then break down senescent erythrocytes.

Upon fulfilling their immune duties, white blood cells undergo programmed cell death, or apoptosis. Lifespans vary considerably by subtype, from roughly one day for granulocytes like neutrophils, eosinophils and basophils, to months for B cells, and even decades for memory T cells.

Platelets, as small cytoplastic fragments released from megakaryocytes, play a vital role in initiating clot formation. Yet due to their inherent fragility, platelets are removed from circulation after about 10 days as they become progressively less sturdy.

Precise coordination between ongoing production and timely degradation of aging cells through specialized pathways ensures the balanced homeostasis of distinct blood cell populations essential for normal physiological functioning. Close regulation at both stages is critical for maintaining healthy hematopoiesis.

Mathematical modeling of erythropoiesis¶

We propose the following mathematical model to capture the dynamics of erythropoiesis regulation via negative feedback:

This mathematical model of erythropoiesis regulation via negative feedback captures the essential dynamics governing the process in a simplified yet biologically meaningful way. It describes the time-evolution of two key variables—circulating erythrocyte levels ($E$) and erythropoietin concentration ($P$)—using a system of ordinary differential equations under reasonable assumptions.

The underlying model assumptions include a stationary hematopoietic stem cell population, which focuses the model on erythrocyte production and turnover dynamics without explicitly modeling upstream hematopoiesis. Linear degradation terms represent natural removal of erythrocytes and erythropoietin from circulation at rates $\gamma_E$ and $\gamma_P$, respectively.

A key component is the negative feedback Michaelis-Menten function $K_E/(K_E + E)$, where the half-saturation constant $K_E$ determines the erythrocyte level at which erythropoietin production is reduced by half. This formalizes the process by which increased $E$ signals lower $P$, stabilizing erythropoiesis.

The production rates $k_E$ and $k_P$ provide a quantitative description of erythrocyte generation and erythropoietin secretion. Together with the other terms and parameters, this framework can be analyzed mathematically and simulations run to examine behaviors like equilibrium states and responsiveness.

Quasi-steady state approximation¶

We observe that erythrocytes have a mean half-life on the order of 120 days, while erythropoietin’s half-life is only a few hours Bélair et al. (1995). This implies that the degradation rate of EPO, $\gamma_P$, greatly exceeds that of red blood cells, $\gamma_P$. Mathematically, we can assume that $dP/dt \approx 0$, representing a quasi-steady state where EPO concentration adjusts much more rapidly than erythrocyte levels change.

With this approximation, the EPO dynamics can be reduced to:

Substituting back into the original equation for $dE/dt$ yields:

where the effective erythrocyte production rate is $k_e = k_E k_P / \gamma_P$.

This quasi-steady state approximation produces a simpler reduced model by eliminating the fast EPO timescale. While an idealization, it retains the negative feedback dynamics and steady states of the full system for analytically exploring long-term erythrocyte behavior.

Steady State Analysis¶

The steady state condition for the reduced dynamical system (3) is $dE/dt=0$, which transforms into

where $E^*$ represents the steady state value of $E$. The left-hand side of the above equation is a decreasing function of $E$, while the right-hand side is increasing. Thus, the dynamical system has a single positive steady state.

Solving the last equation for $k_e$ gives:

Notice that, if $E^*$ remains constant, $k_e$ is an increasing function of $\gamma_E$ and a decreasing function of $K_E$.

Stability Analysis¶

Define the function corresponding to the right-hand side of the reduced model as

The stability of the steady state is determined by the sign of $f'(E^*)$. After taking the derivative, we obtain

Observe that $f'(E^*)$ is negative, implying that the steady state is stable.

Substituting the expression for $k_e$ in (5) yields:

The value of $f'(E^*)$ determines the relaxation time of the dynamical system; the more negative it is, the faster the system returns to steady state after a perturbation. Interestingly, increasing $\gamma_E$ and/or decreasing $K_E$ under constant $E^*$ accelerates this relaxation dynamics.

The regulatory mechanisms governing hematopoiesis appear designed to restore basal blood cell levels as quickly as possible following a perturbation such as hemorrhage. As shown in the stability analysis, decreasing the half-saturation constant $K_E$ or increasing the degradation rate $\gamma_E$ serves to accelerate the dynamical system’s relaxation back to the erythrocyte steady state $E^*$. However, holding $E^*$ constant requires compensating through higher erythrocyte production rate $k_E$—see (5). Therefore, while modulating parameters like $K_E$ and $\gamma_E$ can minimize the system’s response time, this comes at the cost of increased red blood cell generation via elevated $k_E$ under homeostasis.

It is remarkable that adult humans have the capacity to regenerate their entire complement of blood cells every 7 years through hematopoiesis Mackey (2001). This estimated production of an individual’s body weight in red blood cells, white blood cells, and platelets over such a short timeframe indicates the exquisitely optimized nature of blood cell generation and regulation in humans.

Overall, the model analysis suggests that hematopoietic regulatory mechanisms have evolved to prioritize rapid restoration of baseline blood cell populations through adjustable response kinetics. This analysis illustrates how quantitative mechanistic understanding of human blood cell dynamics and how perturbations impact the highly integrated negative feedback system sustaining blood homeostasis.

Discussion¶

This chapter developed a mathematical model of erythropoiesis regulation via negative feedback that provided insights into hematopoietic homeostasis. Steady state, stability, and response time analyses revealed key properties of the negative feedback system. However, opportunities remain to enhance the model’s scope and realism.

While the model illustrated how feedback control stabilizes erythron levels, a primary limitation was the omission of precursor cell dynamics. Explicitly modeling erythroid progenitor proliferation, differentiation, and apoptosis could provide a more mechanistic representation of how feedback cues impact red blood cell output over time.

Additionally, simplifying assumptions around exponential decay kinetics for cells and EPO ignored spatial and temporal variations inherent in biological systems. Incorporating distributed or age-structured terms may better represent the time-delayed responses between changes in populations, signaling factors, and feedback regulation.

The model also considered erythropoiesis independently from cross-talk with other cell lineages. In reality, erythropoiesis is integrated within broader networks regulating hematopoietic production. Accounting for cross-regulation could alter predictions, particularly regarding system recovery following pan-lineage perturbations.

Parameter values were estimated from literature rather than calibrated to experimental data, neglecting biological variability. Tailored calibration approaches may enhance the model’s clinical and predictive applicability.

Despite these limitations, the model provided a useful foundation for quantitative exploration of erythropoietic control mechanisms. Future efforts expanding on precursor dynamics, spatiotemporal details, and inter-lineage interactions have potential to deepen understanding of normal and dysregulated hematopoietic regulation. Overall, this work demonstrated how even simple models can provide mechanistic insights into negative feedback regulation sustaining blood cell homeostasis.

Exercises¶

1. Revisit exercise 4 of Chapter 2. Discuss the results from the perspective of quasi-steady-state approximation.

2. Revisit exercise 2 of Chapter 2. Discuss the results from the perspective of quasi-steady-state approximation.