Cancers of the blood forming system are a very heterogeneous group of diseases. An important example is acute myeloid leukemia. It is driven by leukemic stem cells that resist treatment and trigger relapse. Mathematical models can help to characterize leukemic stem cell properties and to assign patients to groups that differ with respect to the clinical course of the disease.

**Cooperation partners:** A. D. Ho (Department of Hematology and Oncology, Heidelberg University Hospital), C. Lutz (Department of Hematology and Oncology, Heidelberg University Hospital), W. Wang (Fudan University, Shanghai Cancer Center), A. Marciniak-Czochra (Institute for Applied Mathematics, Heidelberg University)

**Acute leukemias** form a heterogeneous group of malignant diseases of the blood forming (hematopoietic) system. In many leukemia subtypes the malignant cell bulk is derived from a small and heterogeneous population of so called **leukemic stem cells** (LSC). Upon expansion, the leukemic cells out-compete healthy blood production which results in severe clinical symptoms.

The **inter-individual heterogeneity** of the disease dynamics is considerably high, even among patients suffering from the same disease subtype. Mathematical models provide a **mechanistic understanding** for this heterogeneity by linking experimentally inaccessible leukemic (stem) cell properties to measurable disease dynamics. This helps to predict **disease progression** including **maligant transformation, risk of relapse, response to therapy** and ** patient prognosis** based on individual clinical data.

Depending on the available data and on the question of interest the mathematical models rely on different framworks including **nonlinear ordinary differential equations, integro-differential equations and stochastic simulations**.

We focus on the following aspects of **human acute myeloid leukemia (AML)**:

** 1. Model-based quantification of leukemic stem cell properties:** Leukemic stem cells give rise to the malignant cell bulk. They play a key role in disease evolution and treatment failure. Due to experimental limitations leukemic stem cells cannot be observed directly. The combination of mathematical models, patient data and parameter estimation tools help to **estimate leukemic stem cell properties** such as **division rate** (quantifying the number of cell divisions per unit of time) or **self-renewal probability ** (quantifying the probability that progeny of LSC are again LSC and not non-stem cells). The models suggest that leukemic stem cell properties have a significant impact on the **speed of disease progression** in comparison to leukemic non-stem cells. The higher LSC self-renewal and proliferation, the poorer the prognosis.

** 2. Dependence of LSC on growth signals:** To adapt its output to the requirements of the organism the blood forming (hematopoietic) system is tightly regulated by feedback signals, so-called **cytokines**. To a variable extent malignant cells require such feedback signals to expand and survive. Mathematical models help to understand how the course of the disease differs between patients with **cytokine-dependent** and patients with **cytokine-independent** leukemic cells. The models suggest that **disease progression** is faster if leukemic cells can expand in absence of cytokines. Computer simultions help to understand in which cases external administration of cytokines can reduce the maligant cell load and in which cases it does the opposite.

**3. Impact of the stem cell niche on disease dynamics:** To maintain their stemness hematopietic (HSC) and leukemic stem cells (LSC) have to reside in a protective bone marrow microenvironment, referred to as ** stem cell niche**. There is evidence that LSC out-compete HSC from their niche and thus compromise healthy blood cell formation. Mathematical models suggest that dynamics in the stem cell niche have a relevant impact on the clinical course and patient survival. One possibilty to obatin insights into stem cell niche dynamics is the measurement of HSC in leukemia patients. We developed a mathematical model that supports the clinical concept that HSC counts can be used to monitor disease progression and serve as a marker for **minimal residual disease (MRD) ** in a subset of AML patients.

**4. Clonal evolution: ** A clone is defined as a set of genetically identical cells. The leukemic cell mass of an individual patient consists of different clones that evolve over time. During the course of the disease clones with unfavorable properties are out-competed and new clones arise due to mutations. This is referred to as **clonal evolution** or **clonal selection**. Mathematical models help to systematically study how leukemic (stem) cell properties, such as proliferation rate and self-renewal probability, drive clonal evolution and affect the **course of the disease**. The models suggest that at the time of diagnosis clones with high self-renewal and high proliferation (compared to healthy cells) dominate. Relapse is triggered by clones with reduced proliferation and higher self-renewal compared to the cells at diganosis. In computer simulations high LSC self-renewal is linked to a fast **disease progression** and a poor **survival**. Mathematical analysis implies that clones with high LSC self-renewal out-compete clones with non-maximal self-renewal.

**5. Therapy resistance: **The clinical management of acute myeloid leukemia is challenging since a large number of patients eventually relapse. Relapse is triggered by leukemic stem cells that survive therapy and subsequently expand. Computer simulations help to understand the change of leukemic cell properties during the course of the disease. They suggest that the combination of high LSC self-renewal and slow LSC proliferation leads to fast growing therapy resistant clones.

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