A vector is a DNA molecule used as a vehicle to carry foreign genetic material into another cell. Once there it can be expressed and replicated. The effect and detection of gene doping is highly dependent on the vectors that are being used. In the early stages of our project we spoke with prof. Hidde Haisma from Groningen University, a gene doping expert, who told us the main vectors he would expect athletes to use now are plasmids and adenoviruses. This is because of their relative safety compared to vectors that integrate the DNA into the genome of the cell. Integrating vectors, such as the retrovirus, present the threat of insertional mutagenesis which can lead to the development of cancer. Table 1 shows our analysis of possible gene doping vectors and some of their properties based on the work of Ratko et al. 2003.

Table 1: Gene doping vectors advantages and disadvantages.

Vector	Advantages	Disadvantages
Plasmid	Relatively safe Generally low immune response Low cost and easy large quantity production Variable transgene insertion upto ±20 kb (Lodish et al. 2000) Long storage (Munier et al. 2005, Kircheis et al. 2001, Li and Huang 2000)	Very low transfection efficiency (Bergen et al. 2008) No targeting Transient expression
Adenovirus	High transduction efficiency Transduces proliferating and nonproliferating cells Transduces many cell types Easy Production Very high titers (1012 pfu/mL)	No targeting Transient expression Limited insert size: 4–5 kb Immune-related toxicity with repeated administration Potential replication competence
Adeno-associated virus	Continued expression No viral genes	No targeting Difficult production Not characterized well Potential Insertional Mutagenesis Limited insert size: 5kb
Lentivirus	Transduces proliferating and nonproliferating cells Prolonged expression Relatively high titers (106–107 pfu/mL)	Integrating virus Clinical experience limited Difficult to manufacture and store Limited insert size: 8 kb
Retrovirus	Relatively high titers (106–107 pfu/mL) Prolonged stable expression Larger insert size: 9–12 kb	Inefficient transduction Integrating virus Insertional mutagenesis Broad cell tropism No targeting Potential replication competence

Each vector has its own benefits and drawbacks. Plasmid vectors, as non-viral DNA vectors, have several advantages over viral vectors. Virus production is expensive (Templeton et al. 2002, Nagasaki and Shinkai, 2007) and safety of viral transfection remains a concern after several deaths (McCormack et al. 2004, Hacein-Bey-Abina et al. 2008). However, plasmid vectors have a very low transfection efficiency (Murakami et al. 2011), especially compared to adenoviruses that have been shown to have a 95% transfection efficiency in hepatocytes (Huard et al. 1995, Sullivan et al 1997). Transfection efficiency of plasmids can be improved through methods such as in vivo electroporation (Ataka et al. 2003). However, this method requires the insertion of electrode needles into the athlete to increase transfection efficiency. This is more invasive to the athlete than a single injection of vector particles. Therefore, adenoviruses are seen as the most likely transfection method for gene doping at this stage and even more so in the near future. Hence, most of the numerical values were based on this type of vector.

A step-by-step overview of the algorithm used for determination and creation of gRNAs is given below:

1. Convert coding sequence into numerical sequence (A=1, T=2, C=3, G=4).


Figure 4: Algorithm to convert coding sequence into numerical sequence in an ‘for’-‘if’ loop.



2. Translate genetic sequence into amino acid sequence.


Figure 5: Algorithm used to translate numeric DNA sequence into amino acid sequence in gene doping.



3. Based on the amino acid sequence, determine the number of possible different codons that will code for the same amino acid.


Figure 6: Algorithm used to determine the possible number of synonymous codons for each amino acid position in the gene.



4. Determine positions of PAM sequence close to the exon-exon junctions.
1. Analyze the total number of gRNAs necessary for each PAM sequence to cover all possibilities of synonymous mutations (product of possible codons). Determined by the identity necessary between gRNA and target sequence.


Figure 7: Algorithm used to find possible PAM sequences on the gene.



5. Gather the number of all PAM positions in the gene and determine their total number of gRNAs necessary, based on the codon variations.


Figure 8: Algorithm used to determine the total number of possible variations of codons behind a specific PAM sequence.



6. Chose the PAM sequence that has the minimal number of gRNAs possible and generate gRNAs.


Figure 9: Algorithm used to choose the position with smallest variation possible behind the PAM sequence and generate the gRNA_array for this position.



7. Output the position of PAM sequence in the gene and all the gRNAs’ sequence.

Our model starts from the point of injection, which is mostly either intramuscularly (IM) or intravenously (IV). The advantage of intravenous injections is that the viral vectors immediately enter blood circulation. The means that more vectors reach the target kidney cells before they are degraded. However, intravenous injections require a qualified doctor to administer and can lead to vein damage such as phlebitis. The advantages of intramuscular injections are that they are easy to administer and do not rely on a qualified doctor for administration. On top of this, they can supply relatively large volumes of the gene doping DNA as the muscles have larger uptake capacity than the veins. Also, the gene doping vectors are not directly getting into the bloodstream, which can lead to sustained release. The big disadvantage of intramuscular injection in comparison with intravenous injections though is the poor absorption. Also, for the athlete, intramuscular injections can cause local swelling, drainage and severe pain at the site of injection. Nevertheless, we take both ways of administration into account.

Pharmacological compartment models are developed to understand the distribution of drugs administered to the human body by oral or, intramuscular, or intravenous routes (M.A. Khanday et al. 2017). They are formulated based on diffusion processes using Fick’s principle and law of mass action. The same diffusion processes affect the administered adenoviral vectors. We start with pharmacological compartment models for both injection types as displayed in Figure 10. Here it is assumed that the mixing of the vectors with blood is instantaneous, based on an article by Tarr et al. 1933). Prof. Beltman, Assistant Professor Biomedical Modelling at Leiden University, later agreed with this assumption.

There are multiple tissues producing EPO, including liver, brain and kidney tissue. The main cell population responsible for EPO production are the interstitial fibroblasts in the kidney, spanning an average population of approximately 100 million cells, which produce more than 80% of the EPO in blood (Weidemann & Johnson 2009).

From the compartment models in Figure 10, equations 1 and 2 can be derived for the intravenous administration.

$$\frac{d[c_{blood}]}{dt} = -(k_{blood}+kel_{blood})[c_{blood}]+k_{tissue}[c_{tissue}]\tag{1}$$

$$\frac{d[c_{tissue}]}{dt} = k_{blood}[c_{blood}]-(k_{bind\,uptake}+k_{tissue})[c_{tissue}] \tag{2}$$

Similarly, equations 3, 4, and 5 can be derived for the intramuscular administration. In intramuscular administration, the viral vectors must first diffuse out of the muscle and enter the bloodstream. While in the muscle, the muscle macrophages break down and eliminate the viral vectors.

$$\frac{d[c_{muscle}]}{dt} = -(k_{muscle}+kel_{muscle})[c_{muscle}]\tag{3}$$

$$\frac{d[c_{blood}]}{dt} = -(k_{blood}+kel_{blood})[c_{blood}]+k_{tissue}[c_{tissue}]+k_{muscle}[c_{muscle}] \tag{4}$$

$$\frac{d[c_{tissue}]}{dt} = k_{blood}[c_{blood}]-(k_{bind\,uptake}+k_{tissue})[c_{tissue}] \tag{5}$$

The initial values and constantsused are as specified in Table 2 and Table 3 respectively.

Table 2: Overview of the nitial values used in the human body model for an adenoviral vector.

Constants	Values	Meaning
$$[c_{blood} (t=0)]$$	94 billion [#/mL] for IV Single Dose 64 billion [#/mL] followed by smaller doses of 18 billion vectors every 20 days for IV Microdosing 0 [#/mL] for IM	Initial injection of vectors intravenously
$$[c_{tissue} (t=0)]$$	0 [#/mL]	Initial vector concentration in in tissue
$$[c_{muscle} (t=0)]$$	141 billion [#/mL] for IM 96 billion vectors followed by smaller doses of 27 billion vectors every 20 days for IM Microdosing 0 [#/mL] for IV	Initial injection of vectors intramuscularly

Table 3: Overview of the constants used in the human body model for an adenoviral vector.

Rate Constants	Values (days-1)	Meaning	Source
$$k_{tissue}$$	1440	Vector displacement from tissue to blood	Estimate based on mean blood circulation time
$$k_{muscle}$$	1440	Vector displacement from muscle to blood in IM injections	Estimate based on mean blood circulation time
$$k_{blood}$$	1440	Vector displacement from blood to tissue	Estimate based on mean blood circulation time
$$kel_{blood}$$	720	Elimination of viral vectors from the blood	Ganesan et al. 2011
$$kel_{muscle}$$	720	Elimination of viral vectors from the muscle	Ganesan et al. 2011
$$k_{bind\,uptake}$$	8.64	Endosomal uptake	Varga et al. 2005

Cellular uptake

After the uptake of the vectors in the tissue, the stage of cellular uptake ensues. In this process we modelled multiple steps as indicated in Figure 11 according to a model first developed by Varga et al. 2005. The cellular uptake of gene doping vectors as depicted in Figure 11 can be dissected into multiple steps described by a set of coupled differential equations, for which the constants are given in Table 3.

First, the complex is taken up by endocytosis after which it is either degraded or taken up, as represented by equations 6 and 7.

$$\frac{d[vesicle]}{dt} = k_{bind\,uptake}[c_{tissue}]-(k_{Escape}+k_{deg\,vesicle})[vesicle] \tag{6}$$

$$\frac{d[complex\,intracell]}{dt} = k_{escape}[vesicle]-k_{unpack}[complex\,intracell]-k_{bind\,vector}[complex\,intracell]\tag{7}$$

Second, vector dissociation and either degradation or nuclear target complex binding takes place in either dissociated or complexed form, as given by equations 8, 9 and 10.

$$\frac{d[plasmid]}{dt} = k_{unpack}[complex\,intracell]-k_{bind \,plasmid}[plasmid]-k_{deg}[plasmid]\tag{8}$$

$$\frac{d[plasmid\,bound]}{dt} = k_{bind \,plasmid}[plasmid]-k_{NPC}[plasmid\,bound] \tag{9}$$

$$\frac{d[complex\,bound]}{dt} = k_{bind \,vector}[complex\,intracell]-k_{NPC}[complex\,bound] \tag{10}$$

Subsequently, transport to the inner part of the nucleus is believed to take place through first binding to a nuclear pore complex (NPC) and finally inside the nucleus dissociation of the nuclear target complex takes place. This is represented by equations 11 till 16.

$$\frac{d[complex\,boundNPC]}{dt} = k_{NPC}[complex\,bound]-k{in}[complex\,boundNPC] \tag{11}$$

$$\frac{d[complex\,bound\,nucleus]}{dt} = k{in}[complex\,boundNPC]-k_{dissociation}[complex\,bound\,nucleus] \tag{12}$$

$$\frac{d[complex\,nucleus]}{dt} = k_{dissociation}[complex\,bound\,nucleus] - k_{unpack2}[complex\,nucleus] \tag{13}$$

$$\frac{d[plasmid\,boundNPC]}{dt} = k_{NPC}[plasmid\,bound] -k_{in2}[plasmid\,boundNPC] \tag{14}$$

$$\frac{d[plasmid\,bound\,nucleus]}{dt} =k_{in2}[plasmid\,boundNPC]-k{kissociation2}[plasmid\,bound\,nucleus] \tag{15}$$

$$\frac{d[plasmid\,nucleus]}{dt} =k{kissociation2}[plasmid\,bound\,nucleus] + k_{unpack2}[complex\,nucleus]$$

$$- k_{cell\,death}[plasmid\,nucleus] \tag{16}$$

Detection of cell free doping DNA

Apart from the effect of the gene doping EPO on the production of red blood cells, the purpose of the model is to determine the dynamics of the detectable cfDNA concentration in the blood. Cell free doping DNA is released from dying infected cells and circulates in the blood where it is assumed to degrade at the same rate as natural cfDNA.

$$\frac{d[Doping\,DNA]}{dt} =k_{cell\,death}[plasmid\,nucleus]-kel_{cfDNA}[Doping\,DNA] \tag{17}$$

Equation 17 provides us with a detection window for which we assume that we can detect both, DNA left in the tissue and bloodstream after injection, and DNA released after cell death (kcelldeath). Any other degradation terms or transient expression we incorporated in the constants used. Based on the above model we obtained the concentration developments of cfDNA in time for both intramuscular and intravenous injections and the estimated detection windows where we assumed an estimated detection limit of 100 copies DNA. The concentrations of DNA over time were used in the laboratory for our sample preparation to mimic real life detection potential. The cell death rate constant is directly linked to the average lifetime of renal interstitial fibroblasts, which we estimated to be around 57 days based upon measurements in chicks by Weissmanshomer et al. 1975.

According to Haller et al. 2018, the expected concentration of doping cell free DNA may be higher for athletes in endurance and intermittent sports. Haller et al. found a 22.7 fold increase in venous cfDNA concentrations in footballers after a professional football match. Given the high amount of training top level athletes endure, this finding leads us to believe that we might be able to have even longer detection windows than our model predicts.

The Protein Effect

Lastly, the uptaken DNA can be translated into protein according to equation 18, after which it can be exported to the extracellular environment according to equation 19.

$$\frac{d[protein]}{dt} =k_{protein}[plasmid\,nucleus]-k_{deg\,protein}[protein]-k_{export}[protein] \tag{18}$$

$$\frac{d[protein\,extracellular]}{dt} =k_{export}[protein]-k_{deg\,protein\,extracellular}[protein\,extracellular]\tag{19}$$

Table 4: Overview of the constants used in the human body model for an adenoviral vector.

Rate Constants (Ad5)	Values (days-1)	Meaning	Source
$$k_{bind\,uptake}$$	8.64	Endosomal uptake	Varga et al. 2005
$$k_{deg\,vesicle}$$	28.8	Degradation of complex within uptake vesicle	Varga et al. 2005
$$k_{escape}$$	23.0	Complex movement from endosome to intracellular	Varga et al. 2005
$$k_{bind\,vector}$$	144	Binding of gene delivery vector to compound targeting for the nucleus	Varga et al. 2005
$$k_{unpack}, k_{unpack2}$$	144	Plasmid detaches from vector either in cytoplasm(1) or in the nucleus(2)	Varga et al. 2005
$$k_{deg}$$	7.2	Degradation of unbound plasmid in the cytoplasm	Lechardeur et al. 1999
$$k_{bind\,plasmid}$$	2.88	Binding of plasmid to compound targeting for the nucleus	Varga et al. 2001
$$k_{NPC}$$	1.44*106	Binding formed complexes to Nuclear Pore Complex	Vacik et al. 1999 Wilson et al. 1999 Chan et al. 1999 Dean et al. 1997
$$k_{in}, k_{in2}$$	2.88	Uptake nucleus through Nuclear Pore Complex	Varga et al. 2001
$$k_{dissociation}, k_{dissociation2}$$	1.44*106	Dissociation from the NPC targeting compound	Moroianu et al. 1996
$$k_{protein}$$	14.4	Protein production from plasmid	Schaffer et al. 1998
$$k_{degprot}$$	1.04	Cytoplasmic degradation of the protein	Fuertinger et al. 2012
$$k_{export}$$	1.44*106	Export protein to extracellular environment	Estimate
$$k_{deg\,protein\,extracellular}$$	1.04	Degradation of EPO protein in blood	Fuertinger et al. 2012
$$k_{cell\,death}$$	0.0167	Average death rate of renal interstitial fibroblast	Estimate based on chicks; Weissmanshomer et al. 1975
$$kel_{cfDNA}$$	100	Clearance of cfDNA from the blood	Alegre et al. 2015

With the doping DNA degradation and doping EPO formation determined, the effect of EPO on erythropoiesis, the process which produces red blood cells, is determined. We developed a model using an anemia EPO treatment model by Fuertinger et al. 2012 as reference.

Red blood cells begin as stem cells and progress into different cell stages as they age. As progenitor and precursor cells age, they proliferate or undergo apoptosis at a rate dependent on the cell stage they are in. The resulting growth or decay rates may be constant or dependent on the concentration of EPO in the blood. Burst-Forming Unit-Erythroid (BFU-E) cells have a very small number of EPO receptors. EPO concentration has no effect on BFU-E proliferation, their proliferation is assumed to be constant. After leaving the stem cell stage, cells stay in the BFU-E stage for 7 days, after which they enter the Colony-Forming Unit-Erythroid (CFU-E) stage. In this stage, the cells divide at a faster rate than in the BFU-E cell stage. The CFU-E cells have a large number of EPO receptors and are strongly dependent on EPO for their survival. Their rate of apoptosis is inversely related to the concentration of EPO. Under normal conditions within the human body, a large number of CFU-E generated do not survive. As the concentration of EPO in the blood increases, the number of cells which survive increases.

After spending 6 days as CFU-E cells, the cells enter the erythroblasts stage. Here, the number of EPO receptors decline. During this stage, there is no evidence that additional divisions occur when production of EPO increases. For this reason we assume that the proliferation of erythroblasts is constant (Lichtman et al. 2005).

The cells stay in the erythroblasts stage for 5 days until they stop dividing, extrude their nuclei and mitochondria, and become marrow reticulocytes. Marrow reticulocytes no longer proliferate and their mortality rate is inversely dependent on iron concentration in the plasma. Since we assume that athletes have a sufficient iron supply, a constant apoptosis rate for marrow reticulocytes is assumed. The time cells stay in the reticulocytes stage is between 0.75-3 days. An increase in EPO concentration shortens the marrow transit time of reticulocytes.

Once reticulocytes are released from the bone marrow and enter the blood, they mature into erythrocytes (red blood cells) within 1-3 days. Reticulocytes have a hemoglobin content of around 27.5 ± 2.8 pg per cell and, red blood cells have a hemoglobin content of around 26.4 ± 2.4 pg per cell (Fishbane et al. 1997). Due to the similar ability of blood reticulocytes and red blood cells to carry oxygen, when red blood cells are discussed, we refer to both blood reticulocytes and mature red blood cells. The lifespan of red blood cells (RBCs) in healthy human adults is about 120 days before their components are recycled by microphages (Jandl 1987). Over the course of this time a small number of RBCs die due to random daily breakdown or, internal or external bleeding. This is taken into account with a small apoptosis rate for RBCs. Adults have a red blood cell count ranging from about 20 to 30 trillion. Women have a blood cell count range of 3.5-5.5 trillion cells per liter, while men have a range of 4.3-5.9 trillion cells per liter (Dean 2005). The average red blood cell count is estimated to be 24.98 trillion by Lichtman et al. 2005.The entire process, from stem cell to red blood cell recycling by microphages, takes 141 days.

The endogenous release of EPO is inversely related to the partial pressure of oxygen in the blood. The partial pressure of oxygen in the blood is proportionally related to the number of red blood cells circulating. An increase in the red blood cell population in blood will decrease endogenous EPO production. If the concentration of EPO in the blood falls below a certain level (9.8 mU/ml in the case of this model), neocytolysisis is triggered. Neocytolysisis the selective lysis of young red blood cells by the body to allow it to decrease its red blood count at a faster rate and reach the desired partial pressure of oxygen in the blood.

The process of red blood cell formation, from stem cell to red blood cell can be modelled in time through a series of partial differential equations based on the age of the cells . Each of cell stage, BFU-E, CFU-E, erythroblasts, marrow reticulocytes and red blood cells, is represented by a partial differential equation.The general form is seen below.

Figure 13: Breakdown of the general PDE used to model the various cell stages that occur in the process of red blood cell production.

Starting at cell age 0 days, when stem cells become BFU-E cells, the age span up until cell age 141 days, when red blood cells are recycled by macrophages, the cells can be broken up into populations of cells at a certain age. For every cell stage, the population density of cells at a given maturity and time can modelled as a population mesh. A population mesh is a numerical estimation that breaks down the process of red blood cell production into populations at a given age. A mesh point is population of cells at a specific age. When modelled as a mesh, an assumption of the relation between a change in time and a change in maturity can be made.

Assumptions

1. For a population of cells u of maturity μ and at time t, a change in time Δt while result in an equal change in maturity Δμ, assuming that the maturation velocity \(vs(E(t))=1\).
2. The maturation velocity is 1 for every stage except the marrow reticulocytes, where the time cells stay in this stage is dependent on the concentration of EPO.

The commitment of stems cells to becoming red blood cells is an irreversible event (Fuertinger et al. 2012). Cells cannot regress back to a previous cell type or switch to a different differentiation pathway. For this reason, assumption 1 can be made. A change in time will lead to an equal change in the age of the cell. As a result, the finer you make the population mesh the smaller the time step is. This increases the accuracy of the model as the cells react to changes in EPO concentration in time faster. Once they reach the age at which they differentiated into another cell stage, the resulting growth and decay rates that govern them will change.

Figure 14: Transition of cells from one stage to another through differentiation. (a) Follows one populations of reticulocyte cells as they age over time and transition to marrow reticulocytes. (b) Shows the population density of cells every 0.1 days of maturity at a steady state EPO concentration of 9.8 mU/ml.

When a time step occurs, the cells in each mesh point (\(u_n\)) experience growth or decay based on the stage they are in. The new population of cells is then shifted to the next age mesh point. The general form of the equation that governs the change of population density at each time step is below.

$$ u_n = u_{n-1}\times e^{(\beta-\alpha(EPO(t)) \times \Delta t} \tag{20}$$

$$ u_{n+1} = u_{n}\times e^{(\beta-\alpha(EPO(t)) \times \Delta t} \tag{21}$$

$$ \Delta t = \mu_{n+1} - \mu_n = \Delta \mu \tag{22}$$

The dynamics of the mesh in response to a gene doping injection can be seen below in Figure 15.

Figure 15: A gif of the dynamic red blood cell formation model. The injection 141 billion copies of the EPO cDNA in adenoviruses occurs at day 0. The infected cells produce extra EPO leading to a maximum concentration of EPO of 31.3 mU/ml (3.2 times more that the steady state EPO concentration of 9.8 mU/ml).

To calculate the total population of each stage, the trapezoid rule was used to calculate the area under the population mesh and estimate the total population.

BFU-E Cells

10⁸ stem cells become BFU-E cells on day 0. BFU-E cells grow exponentially as they age. Their proliferation rate is not dependent on EPO. As result, the population of BFU-E cells grows at a constant exponential rate as they age.

$$ BFU{-}E_n = BFU{-}E_{n-1}\times e^{\beta_{BFU-E} \times \Delta t} \tag{23}$$

$$ BFU{-}E_{n+1} = BFU{-}E_{n}\times e^{\beta_{BFU-E} \times \Delta t} \tag{24}$$

Cells committed to become red blood cells continue to grow in this manner until they reach the age of 7 days. At this point, the BFU-E differentiate into CFU-E cells.

CFU-E Cells

CFU-E cells are strongly dependent on EPO for their survival.

$$ CFU{-}E_n = CFU{-}E_{n-1}\times e^{(\beta_{CFU{-}E}-\alpha_{CFU{-}E}(EPO(t))) \times \Delta t} \tag{25}$$

$$ CFU{-}E_{n+1} = CFU{-}E_{n}\times e^{(\beta_{CFU{-}E}-\alpha_{CFU{-}E}(EPO(t))) \times \Delta t} \tag{26}$$

The apoptosis rate of CFU-E cells is dependent on the concentration of EPO at a given time. As the concentration of EPO increases, the apoptosis rate of CFU-E cells decreases.

Figure 16: The effect of EPO on the apoptosis of CFU-E cells.

The logistic equation that governs the apoptosis rate of CFU-E is seen in equation 27 below.

$$\alpha_{CFU{-}E}(EPO(t))= \frac{(a_1 - b_1)}{1+e^{k_1 \times EPO(t) - c_1}}+b_1 \tag{27}$$

The CFU-E stage starts once cells reach the age of 7 days and continues until cells differentiate into Erythroblasts at an age of 13 days.

Erythroblasts

The proliferation of erythroblasts is not dependent on the concentration EPO. However, the population of cells entering the erythroblast stage at age 13 days is dependent on the concentration of EPO due to the CFU-E stage coming before.

$$ Erythroblasts_n = Erythroblasts_{n-1}\times e^{\beta_{Erythroblasts} \times \Delta t} \tag{28}$$

$$ Erythroblasts_{n+1} = Erythroblasts_{n}\times e^{\beta_{Erythroblasts} \times \Delta t} \tag{29}$$

Upon reaching the age of 18 days, erythroblast cells differentiate into marrow reticulocytes.

Marrow Reticulocytes

For the marrow reticulocytes, where the marrow transit time varies with EPO concentration, the same mesh method is used. Even with sufficient iron supply, a constant fraction of marrow reticulocytes is phagocytosed (Fuertinger et al. 2012). This is represented by a constant apoptosis rate independent of EPO concentration.

$$ Reticulocytes_n = Reticulocytes_{n-1}\times e^{-\alpha_{Reticulocytes} \times \Delta t} \tag{30}$$

$$ Reticulocytes_{n+1} = Reticulocytes_{n}\times e^{-\alpha_{Reticulocytes} \times \Delta t} \tag{31}$$

The amount time it takes for marrow reticulocytes to leave the bone marrow and enter the blood is dependent on the concentration of EPO. A faster marrow transit time means marrow reticulocytes spend more time as blood reticulocytes. The oldest marrow reticulocytes will become RBCs first when marrow transit time decreases. To account for the change in transit time, the stage boundary between reticulocytes and red blood cells is allowed to shift depending on the concentration of EPO.

Figure 17: The effect of EPO on reticulocyte marrow transit time.

The logistic equation that governs the marrow transit time can be seen below.

$$ \mu_{Reticulocytes,max}(EPO(t)) = a_2 - \frac{b_2}{1+e^{k_2 \times EPO(t) - c_2}} \tag{32}$$

Figure 18: The effect of EPO concentration on marrow transit time. The injection 141 billion copies of the EPO cDNA in adenoviruses occurs at day 0. The infected cells produce extra EPO leading to a maximum concentration of EPO of 31.3 mU/ml (3.2 times more that the steady state EPO concentration of 9.8 mU/ml). As EPO concentration in the blood increases, the marrow transit time will decrease, shifting the boundary between marrow reticulocytes and circulating red blood cells to the left. A decrease in EPO concentration increases the marrow transit time and shifts the boundary to the right.

Reticulocytes will stay in the bone marrow until age 18.75 to 21 days when they leave the bone marrow and circulate in the blood. There they become mature red blood cells.

Red Blood Cells and Blood Reticulocytes

There is a fixed rate of random daily breakdown of red blood cells due to random daily breakdown or, internal or external bleeding. This is represented by a constant apoptosis rate independent of EPO concentration.

$$ RBCs_n = RBCs_{n-1}\times e^{-\alpha_{RBCs}(EPO(t), \, \mu_{RBCs}) \times \Delta t} \tag{33}$$

$$ RBCs_{n+1} = RBCs_{n}\times e^{-\alpha_{RBCs}(EPO(t),\, \mu_{RBCs}) \times \Delta t} \tag{34}$$

During the age range of 35 to 42 days, young red blood cells will undergo neocytolysis if the concentration of EPO in the blood drops below 9.8 mU/ml. The neocytolysis rate will increase as EPO levels drop until the level of EPO reach 3.3 mU/ml, when it reaches its maximum rate.

$$\alpha_{RBCs}(EPO(t), \mu_{RBCs}) = \alpha_{RBCs, \, r} + min\Big(\frac{c_E}{EPO(t)^{k_E}}, \, b_E \Big), \, for \, EPO(t) < \tau_E, \, 35 \, days \leq \mu_{RBCs} \leq 42 \, days \tag{35}$$

$$\alpha_{RBCs}(EPO(t), \mu_{RBCs}) = \alpha_{RBCs, \, r}, \, otherwise \tag{36}$$

Feedback via EPO

The natural release of EPO from the kidneys is dependent on the partial pressure of oxygen in the blood. If the partial pressure of oxygen in the blood decreases due to a lack of red blood cells, EPO release increases. This leads to an increase in red blood cell production, leading to a higher population of circulating red blood cells and a higher partial pressure of oxygen. The partial pressure of oxygen and the number of circulating red blood cell are therefore assumed to be directly proportional.

The amount \(E^{end}_{in}(t)\) of EPO released by the kidney can be estimated by the use of the total population of red blood cells \(RBCs(t)\), which consist of all circulating red blood cells:

$$ EPO^{end}_{in}(t) = \bigg(\frac{(a_3 - b_3)}{1+e^{k_3 \times \tilde{M}(t) - c_3}}+b_3\bigg)\ \times \frac{1}{TBV} \tag{37} $$

$$ \tilde{M}(t) = 10^{-8} \times \frac{RBCs(t)}{TBV} \tag{38} $$

The logistic equation that governs the natural release of EPO uses a scaled red blood cell count with \(TBV\) being the total blood volume.

Figure 19: The effect of red blood cell population on the release of EPO from the kidneys. The units of EPO released from kidneys is mU per ml per day.

The dynamics of the endogenous EPO concentration Eend(t) in plasma is modeled by the following ordinary differential equation:

$$\frac{dEPO_{end}(t)}{dt} = EPO^{end}_{in}(t) - k_{deg\,protein\,extracellular}EPO_{end}(t) \tag{39}$$

In the case of gene doping administration, the concentration of gene doping EPO at a given time is given by Equation 19. It is assumed that the degradation of natural EPO and gene doping EPO occurs at the same rate. The overall concentration of EPO in plasma consists of the naturally produced erythropoietin and the administered gene doping EPO:

$$ EPO(t) = EPO_{end}(t) + EPO_{doping}(t) \tag{40}$$

The parameters used in the model are based on the parameters derived in Fuertinger et al. 2012 except fora3and b3which are derived from WE and WL Owen Roberts 2011.

Table 4: Table of the parameters used in the red blood cell production model, their values, their units, and their meaning.

Parameters	Values	Units	Meaning
$$\beta_{BFU-E}$$	0.2	1/day	Proliferation rate for BFU-E cells
$$\beta_{CFU-E}$$	0.57	1/day	Proliferation rate for CFU-E cells
$$\beta_{Erythroblasts}$$	1.024	1/day	Proliferation rate for erythroblasts
$$\mu_{BFU-E, \, max}$$	7	Days	Maximal maturity for BFU-E cells
$$\mu_{CFU-E, \, min}$$	7	Days	Minimal maturity for CFU-E cells
$$\mu_{CFU-E, \, max}$$	13	Days	Maximal maturity for CFU-E cells
$$\mu_{Erythroblasts, \, min}$$	13	Days	Minimal maturity for erythroblasts
$$\mu_{Erythroblasts, \, max}$$	18	Days	Maximal maturity for erythroblasts
$$\mu_{Reticulocytes, \, max}(EPO(t))$$	18.75 to 21	Days	Maximal maturity for marrow reticulocytes
$$\alpha_{Reticulocytes}$$	0.089	1/day	Rate of ineffective erythropoiesis in the marrow reticulocytes stage
$$\alpha_{RBCs}$$	0.005	1/day	Intrinsic mortality rate for erythrocytes
$$a_1, \, b_1$$	0.35, 0.07	1/day	Constants for the sigmoid apoptosis rate for CFU-E cells
$$c_1, \, k_1$$	3, 0.14	Dimensionless, ml/mU	Constants for the sigmoid apoptosis rate for CFU-E cells
$$a_2, \, b_2$$	3.225, 2.475	Days	Constants for the sigmoid maturation velocity/marrow transit time for marrow reticulocytes
$$c_2, \, k_2$$	2.3, 0.2	Dimensionless, ml/mU	Constants for the sigmoid maturation velocity/marrow transit time for marrow reticulocytes
$$a_3, \, b_3$$	9.1, 0.2	Dimensionless, ml/mU	Constants for the sigmoid function governing the release of EPO from the kidneys
$$\mu_{RBCs, \, neocytolysis \, min}$$	35	Days	Lower bound of erythrocytes which are possibly exposed to neocytolysis
$$\mu_{RBCs, \, neocytolysis \, max}$$	42	Days	Upper bound of erythrocytes which are possibly exposed to neocytolysis
$$\mu_{RBCs, \, max}$$	141	Days	Maximal life span for red blood cells
$$b_E$$	0.1	1/day	Constant in the mortality rate for red blood cells
$$c_E$$	3.5	mU3/(ml3\(\times\)day)	Constant in the mortality rate for red blood cells
$$k_E$$	3	Dimensionless	Exponent in the mortality rate for red blood cells
$$\tau_E$$	9.8	mU/ml	EPO threshold for neocytolysis
$$k_{deg\,protein\,extracellular}$$	1.04	1/day	Degradation rate of EPO in the blood
$$S_0$$	108	1/day	Rate at which cells are committing to the erythroid lineage
$$TBV$$	5000	ml	Total blood volume