# Mathematical Modeling of RBC Count Dynamics after Blood Loss

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## Abstract

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## 1. Introduction

#### 1.1. Overview of Erythropoiesis

#### 1.2. History of Erythropoiesis Modeling and Comparison with Our Approach

#### 1.3. Motivating Practical Application: Polycythemia Vera

## 2. Model

#### 2.1. Assumptions on the Subject

**Healthy adult subject**: The aim of the proposed model is to describe the RBC dynamics of healthy adult subjects. Healthy in this case especially means that the subject has no disturbed erythropoiesis as for example in the case of PV.**Existence of a steady state**: It is assumed that environmental conditions that affect the blood do not dramatically change. If this is the case, a mean value for the erythrocyte count should exist in healthy individuals, which is valid over a longer time period. This will be the $Base$ value of the system and the steady state for the erythrocyte count in the model. Using this assumption especially means that the model will not be able to capture large environmental effects to the blood.**Sufficient iron concentration**: The iron concentration in considered individuals is assumed to be sufficient, such that erythropoiesis is not affected by iron shortage. Therefore, iron influence on erythropoiesis is not included in the model.**Reasonable blood loss**: It is assumed that the blood loss in the considered subjects does not exceed a standard blood donation by much. Therefore, emergency reactions of the body in case of severe anemia (for example the release of stress reticulocyte, compare [2]) should not arise and are therefore not included in this model.

#### 2.2. Assumptions on the Physiological Process

- 5.
**Average cell**: A compartment modeling approach is used to describe the system dynamics. Therefore, all cells in one compartment share the same properties given by the mean values of the corresponding properties. On the one hand, cell individual information like cell age is lost, which leads to errors especially in early phases where proliferation is very dependent on the state of the cell. On the other hand, often only summarized properties like for example amount of cells or overall amount of hemoglobin is interesting for our application. In addition, influences in early cell stages are often not measured in clinical practice, such that an error in early cell stages is not preventable. The assumption that given mean values reflect summarized properties is justified, as there is a very large number of cells in each compartment and the law of large numbers can be applied.- 6.
**Partition of progenitor cells**: Measurements only include information on the mature RBC and not on previous cell stages. Therefore, in the application oriented model, a too detailed differentiation of early stem cells should be prevented. As EPO is the main factor for compensation of a blood loss, a partition into two cell types is proposed: cells that are strongly proliferating in dependence on EPO and cells that are not or only slightly affected by EPO. Cells strongly affected by EPO here include CFU-E cells and early erythroblasts. Not affected or less affected by EPO are BFU-E, later erythroblasts and reticulocytes. As BFU-E are not affected by the dynamics, they are modeled by a constant inflow term ${X}_{0}$, whereas the EPO-depending and Non-EPO-depending cells are each described by one compartment. The partition of erythropoiesis can be examined in Figure 1.- 7.
**Instant blood plasma regeneration**: After a blood loss, both solid and liquid blood components are lost. It is known that blood plasma regenerates very fast after about two to four days [23]. To prevent the usage of delay components in the model, it is here assumed that the regeneration of blood plasma is instant after a reasonable blood loss. This error should be low, as the horizon of about three days is small compared to the average time after a blood donation of about 42 days.- 8.
**Instant feedback by EPO**: The EPO production in the human body is adapted to a changing demand for erythrocytes. This process does not instantaneously change the production rate of erythrocytes, as it takes a while for the local EPO production in, for example, the liver to change the overall concentration of EPO. For our modeling approach, it is assumed that this process is instantaneous. This error also should be low, as the observed time scale is very large.- 9.
**Implicit feedback by EPO**: As stated in Section 1.1, EPO is mainly responsible for the regeneration of RBCs after changing hemoglobin concentrations for example due to blood loss. Unfortunately, EPO concentrations fluctuate strongly on a small time scale and are difficult to measure in clinical practice. Therefore, no direct information about EPO concentration in the subject is available [24]. The influence of EPO on erythropoiesis is therefore indirectly described by a negative feedback term based on the fractional blood loss.We assume the feedback function to be linear, which is certainly valid for small variations in RBC counts. This assumption may however be violated for larger amounts of blood loss and pathological cases, as discussed in Section 5.- 10.
**Only most necessary processes**: To reduce the model complexity, only the most essential features of erythropoiesis after blood loss are included. Therefore, processes like neocytolysis [2], which has a minor influence on young erythrocytes, is not included.- 11.
**No self-renewal of progenitor cells**: It is known that stem cells, especially those that are not committed to a hematopoetic lineage, have the capability of self-renewal to maintain the population of cells. The capability of self-renewal decreases with increased maturation progress [2]. There exist mathematical models that consider self-renewal to be even present in progenitor cells (see for example [25]). In contrast to that, here it is assumed that in the considered progenitor cell compartments only differentiation and no self-renewal is present.- 12.
**Transport rates of cells**: In case of severe anemia, cells might have an increased differentiation rate induced by high EPO concentrations to compensate for the lack of cells [2]. Due to Assumption 4, those reactions do not have to be considered here. Therefore, it is assumed that cells have a constant differentiation rate concerning EPO. Based on [2], it can be assumed that the EPO-Proliferating phase and the Non-EPO-Proliferating phase have an average duration of eight and six days, respectively. The transport rates ${k}_{1}$ and ${k}_{2}$ are set to the respective reciprocal values. As the average lifespan of a mature erythrocyte can be assumed to be 120 days, the death rate for the erythrocyte compartment is set to $\alpha =\frac{1}{120}$.- 13.
**Individual features of erythropoiesis**: Between individuals, there might be a strong variation in the pace of erythropoiesis. For example, a large individual often has more blood than a small individual and therefore should have a faster erythropoiesis to cover for example for fractional cell loss due to cell senescence. This is covered by two amplification variables: $\beta $ for the overall cell maturation and $\gamma $ for the reaction to a blood loss.

#### 2.3. Model Equations

#### 2.4. Sensitivity Analysis

#### 2.5. Local Error Analysis of the System near the Steady State

## 3. Methods

#### 3.1. Data

#### 3.2. Relationship of $tHb$ Values and Number of Erythrocytes

#### 3.3. Calculation of $Base$ Value and Blood Loss

#### 3.4. Parameter Estimation

- ${n}_{\eta}$ are the number of available measurements of the subjects $tHb$,
- ${\eta}_{i}$ is the measurement value of $tHb$ at time ${t}_{i}$,
- ${x}_{3}\left({t}_{i}\right)$ is the according model response at time ${t}_{i}$,
- ${\sigma}_{i}$ is the standard deviation of the measurement at time ${t}_{i}$,
- p is the chosen parameter vector,
- F contains the right hand sides of the model Equation (5),
- ${x}_{0}$ fixed start values, and
- $\varphi \left(p\right)$ is a term that can be used to incorporate a priori information, in our setting chosen as 0

## 4. Numerical Results

#### 4.1. Results for Generic Subject Data from Literature

#### 4.2. Nonlinear Mixed-Effects Estimation for $\beta $ and $\gamma $

#### 4.3. Point Estimation Results for $\beta $ and $\gamma $

#### 4.4. Results if $Base$ Is a Free Parameter

**$1.4\%\pm 0.8\%$**, respectively.

## 5. Outlook: Polycythemia Vera

#### 5.1. Obtaining Data for PV Patients

#### 5.2. Extending the Mathematical Model

#### 5.3. Formulating and Solving Optimization Problems

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ALL | Acute Lymphoblastic Leukemia |

AML | Acute Myeloid Leukemia |

BFU-E | Blast Forming Unit-Hematopoietic |

CFU-E | Colony Forming Unit-Hematopoietic |

DDE | Delay Differential Equation |

EPO | Erythropoietin |

Hct | Hematocrit |

IVP | Initial Value Problem |

MCH | Mean Corpuscular Hemoglobin |

ODE | Ordinary Differential Equation |

PDE | Partial Differential Equation |

PK/PD | Pharmacokinetic/Pharmacodynamic |

PV | Polycythemia vera |

RBC | Red Blood Cell |

SD | Standard Deviation |

tHb | Total hemoglobin |

WBC | White Blood Cell |

%SD | Percentage Standard Deviation |

## Appendix A. Computational Details

#### Appendix A.1. How to Calculate P

#### Appendix A.2. Analysis of the Local Error at the Steady State

**Figure A1.**Numerical evaluation of the real parts of ${\lambda}_{1},{\lambda}_{2}$ and ${\lambda}_{3}$ on a grid of $1000\times 1000$ data points for $\beta ,\gamma \in \left(\right)open="["\; close="]">0,10$. The top row shows the values of the three roots. In the second row, the sign of the values is presented (negative corresponds to blue, positive to yellow). The third row contains the contour lines of the three roots.

## Appendix B. Detailed Results

Subject ID | $\mathit{\gamma}$ | SD(${\mathit{\gamma}}^{\ast}$) [%] | $\mathit{\beta}$ | SD($\mathit{\beta}$) [%]) | ${\mathit{R}}^{2}$ |
---|---|---|---|---|---|

01 | 0.769 | 13.3 | 1.65 | 26.2 | 0.961 |

02 | 0.388 | 12 | 0.867 | 24.4 | 0.996 |

03 | 0.51 | 15.3 | 1.617 | 28.7 | 0.915 |

04 | 0.323 | 12.4 | 0.424 | 24.2 | 0.801 |

05 | 0.061 | 341.6 | 1.381 | 84.9 | 0.61 |

06 | 0.59 | 20.6 | 2.615 | 33.4 | 0.915 |

07 | 0.262 | 67.9 | 1.518 | 69.7 | 0.887 |

08 | 0.324 | 53.3 | 2.676 | 49.1 | 0.911 |

09 | 0.356 | 16.6 | 0.891 | 29.7 | 0.806 |

10 | 0.089 | 242.2 | 2.557 | 61 | 0.947 |

11 | 0.243 | 20.7 | 0.925 | 29 | 0.897 |

12 | 1.003 | 12.5 | 1.409 | 20.1 | 0.959 |

13 | 0.057 | 178.1 | 0.879 | 67.8 | 0.895 |

14 | 0.762 | 42.7 | 0.46 | 53.6 | 0.507 |

15 | 0.344 | 33.1 | 2.132 | 40.9 | 0.863 |

16 | 0.141 | 257.3 | 1.661 | 116.9 | 0.344 |

17 | 0.47 | 66.2 | 0.544 | 69.1 | 0.842 |

18 | 0.525 | 8.8 | 0.631 | 20.1 | 0.871 |

19 | 0.423 | 44.7 | 1.525 | 56.2 | 0.624 |

20 | 0.661 | 31.1 | 2.798 | 47.9 | 0.812 |

21 | 0.686 | 20.9 | 1.943 | 36.7 | 0.92 |

23 | 0.613 | 36.1 | 3.142 | 49 | 0.897 |

24 | 0.421 | 26.9 | 1.528 | 42.1 | 0.905 |

25 | 0.863 | 15.9 | 2.078 | 24.7 | 0.868 |

26 | 0.414 | 16.3 | 1.172 | 29.8 | 0.703 |

27 | 0.635 | 8.2 | 0.836 | 16.8 | 0.866 |

28 | 0.952 | 15.8 | 1.596 | 26.1 | 0.992 |

29 | 0.805 | 22.2 | 1.486 | 39.9 | 0.925 |

**Table A2.**Results with Base and $\beta $ as free parameters ($\gamma =0.5\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\beta $).

Subject ID | Base | SD(Base) [%] | $\mathit{\beta}$ | SD($\mathit{\beta}$) [%] | ${\mathit{R}}^{2}$ |
---|---|---|---|---|---|

01 | 861.014 | 1.3 | 1.685 | 21.2 | 0.963 |

02 | 882.041 | 1.2 | 0.838 | 17.6 | 0.996 |

03 | 856.779 | 0.7 | 1.296 | 14.4 | 0.907 |

04 | 881.885 | 3.3 | 0.431 | 22.9 | 0.806 |

05 | 930.774 | 1.2 | 1.306 | 44.4 | 0.595 |

06 | 986.155 | 0.5 | 1.985 | 12.2 | 0.899 |

07 | 954.28 | 2.1 | 0.968 | 43 | 0.78 |

08 | 692.458 | 0.8 | 1.286 | 14.1 | 0.861 |

09 | 956.621 | 1.4 | 0.787 | 21.2 | 0.803 |

10 | 815.247 | 0.9 | 1.345 | 19.1 | 0.897 |

11 | 980.574 | 1.1 | 0.851 | 17.7 | 0.902 |

12 | 936.771 | 0.7 | 1.665 | 13.9 | 0.949 |

13 | 611.74 | 2.5 | 0.64 | 45.3 | 0.876 |

14 | 1554.83 | 69 | 0.324 | 75.7 | 0.495 |

15 | 922.955 | 0.5 | 1.466 | 13.1 | 0.833 |

16 | 726.472 | 1.1 | 2.015 | 54 | 0.426 |

17 | 973.967 | 29.8 | 0.484 | 100.7 | 0.842 |

18 | 874.356 | 2.9 | 0.661 | 19.9 | 0.876 |

19 | 766.033 | 2.4 | 1.569 | 47 | 0.622 |

20 | 758.275 | 0.6 | 2.055 | 16 | 0.803 |

21 | 896.036 | 1 | 2.001 | 23.9 | 0.928 |

23 | 880.509 | 0.6 | 2.245 | 17.5 | 0.904 |

24 | 680.758 | 2.3 | 1.304 | 33.2 | 0.895 |

25 | 772.443 | 0.7 | 1.675 | 12.6 | 0.87 |

26 | 686.724 | 1.8 | 0.925 | 21.1 | 0.695 |

27 | 947.998 | 1.5 | 0.859 | 16.1 | 0.884 |

28 | 872.556 | 1.1 | 1.684 | 21.1 | 0.991 |

29 | 812.567 | 3 | 1.478 | 41.6 | 0.925 |

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**Figure 1.**Illustration of erythropoiesis with erythropoietin (EPO) dependency of the certain states. The stages (over the age of cell in days) are the stem cells, the precursor cells blast forming unit-hematopoietic (BFU-E), colony forming unit-hematopoietic (CFU-E), several stages of erythroblasts and reticulocytes in the marrow and blood reticulocytes and erythrocytes in the blood stream. Also included is a partition of the erythropoiesis scheme into three compartments, which reflect the dependency of the cells with respect to EPO. This partition will be used for the modeling process. ${X}_{0}$ describes the constant number of cells entering the red blood cell line. The more mature cells in the marrow are divided into proliferating and non-proliferating cells with respect to EPO. In the blood stream, again blood reticulocytes and erythrocytes do reside.

**Figure 2.**Monte Carlo simulation of 20 different values of parameter $\beta $ for an exemplary subject setting $\gamma =0.3$. ${x}_{1}$ denotes the proliferating cells, ${x}_{2}$ the non-proliferating cells and ${x}_{3}$ the erythrocytes. $Fb$ is the value of the feedback function at each time point. The simulation starts in the steady state and the simulated blood donation takes place at time 0. The dashed line marks the respective steady state.

**Figure 3.**As Figure 2, but with 20 different values of parameter $\gamma $ for an exemplary subject setting $\beta =1.0$.

**Figure 4.**As Figure 2, but with fixed $\beta =1.0$ and $\gamma =0.5$ and 20 different values for $Base$. Note that the initial value ${x}_{3}\left(0\right)$ is constant to highlight the sensitivity of the dynamics with respect to $Base$.

**Figure 5.**As Figure 2, but with 20 different values of parameter $\beta $ for an exemplary subject, where $\gamma =0.5\times \beta $.

**Figure 6.**Comparison of hematocrit ($Hct$) and total hemoglobin ($tHb$) for randomly chosen subject 02.

**Figure 7.**Results of parameter estimation using data of the generic subject in Fuertinger et al. [8].

**Figure 8.**Goodness-of-fit plot for 276 observations from 29 subjects, using a nonlinear mixed-effects modeling approach. Shown are measured versus calculated total hemoglobin (tHb) values.

**Figure 12.**Results of the two methods described for subjects 10 (above) and 27 (below). The approach using $\beta $ and $\gamma $ as free parameters is depicted in red, the one using $\beta $ and $Base$ is depicted in blue.

**Figure 13.**Comparison of results of the two methods described for subject 14. The approach using $\beta $ and $\gamma $ as free parameters is depicted in red. The approach using $\beta $ and $Base$ is depicted in blue.

**Table 1.**List of model parameters and states with explanation and example steady state values and corresponding units.

State | Explanation | Value | Unit |
---|---|---|---|

${x}_{1}$ | Erythroid progenitor cells highly proliferating with respect to erythropoietin (EPO) | 59.03 | [1] |

${x}_{2}$ | Erythroid progenitor cells not proliferating with respect to EPO | 44.27 | [1] |

${x}_{3}$ | Mature erythrocytes and blood reticulocytes | 885.42 | [g] |

Parameter | |||

$Base$ | Mean erythrocyte count in steady state | 885.42 | [g] |

${X}_{0}$ | Committed stem cells transitioning into ${x}_{1}$ | 7.3785 | [${d}^{-1}$] |

${k}_{1}^{\ast}$ | Transition rate from ${x}_{1}$ to ${x}_{2}$ | 0.125 | [${d}^{-1}$] |

${k}_{2}^{\ast}$ | Transition rate from ${x}_{2}$ to ${x}_{3}$ | 0.1667 | [${d}^{-1}$] |

$\alpha $ * | Apoptosis rate of ${x}_{3}$ | −0.00833 | [${\left(gd\right)}^{-1}$] |

$\beta $ ** | Amplifying factor for individual blood regeneration, global factor independent of fraction of blood loss | 1.0 | [1] |

$\gamma $ ** | Amplifying factor for individual blood regeneration, depending on fractional blood loss | 0.3 | [${d}^{-1}$] |

$Fb\left({x}_{3}\right)$ | Feedback function depending on fractional blood loss | 0.0 | [1] |

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## Share and Cite

**MDPI and ACS Style**

Tetschke, M.; Lilienthal, P.; Pottgiesser, T.; Fischer, T.; Schalk, E.; Sager, S.
Mathematical Modeling of RBC Count Dynamics after Blood Loss. *Processes* **2018**, *6*, 157.
https://doi.org/10.3390/pr6090157

**AMA Style**

Tetschke M, Lilienthal P, Pottgiesser T, Fischer T, Schalk E, Sager S.
Mathematical Modeling of RBC Count Dynamics after Blood Loss. *Processes*. 2018; 6(9):157.
https://doi.org/10.3390/pr6090157

**Chicago/Turabian Style**

Tetschke, Manuel, Patrick Lilienthal, Torben Pottgiesser, Thomas Fischer, Enrico Schalk, and Sebastian Sager.
2018. "Mathematical Modeling of RBC Count Dynamics after Blood Loss" *Processes* 6, no. 9: 157.
https://doi.org/10.3390/pr6090157