Mathematical modelling of premature haematopoietic ageing in dyskeratosis congenita.

Hematopoiesis is a highly dynamic and tightly regulated biological process in which pluripotent HSCs and progenitor cells, located primarily in the bone marrow, continuously give rise to a wide spectrum of mature blood and immune cells 48 . These include erythrocytes, leukocytes, and platelets, which together sustain oxygen transport, immune defense, and hemostasis 49 . Over the course of an individual’s lifetime, the hematopoietic system is extraordinarily productive, generating blood and immune cells that are distributed throughout the human body. The demand for blood cell production is not fixed but rather scales with physiological requirements, most notably body weight and overall metabolic activity 50 . Larger individuals require proportionally greater hematopoietic output to maintain normal circulating cell counts and to meet the continuous turnover caused by programmed cell death and the clearance of senescent cells. A disruption in this finely balanced system can lead to hematological disorders. Clinical abnormalities associated with DC include bone marrow failure and pancytopenia 51 . In our framework, we have considered pancytopenia to be present when the population of hematopoietic cells declines to less than 10% of the expected level 52 , reflecting a profound loss of bone marrow function or hematopoietic reserve. This threshold serves as a critical marker of pathological suppression or failure of the hematopoietic system, with important implications for diagnosis, treatment, and prognosis. Figure 1 shows a schematic representation of normal hematopoietic cell production compared with the markedly reduced generation of erythrocytes, leukocytes, and platelets that characterizes pancytopenia. Fig. 2 Blood cell requirements for women (magenta) and men (blue). ( a ) The need for blood cells (BC) production is shown as a function of weight and age in a healthy lifestyle population who maintains their body weight. Of note, up to the age of 16 years there is no difference in weight. ( b ) The graph shows the number of hematopoietic cells required for women and men depending on age and body weight (solid lines). The graph also shows the number of cells in the case of pancytopenia, with a 10% of blood cell requirement (dotted lines). Blood cell requirements for women (magenta) and men (blue). ( a ) The need for blood cells (BC) production is shown as a function of weight and age in a healthy lifestyle population who maintains their body weight. Of note, up to the age of 16 years there is no difference in weight. ( b ) The graph shows the number of hematopoietic cells required for women and men depending on age and body weight (solid lines). The graph also shows the number of cells in the case of pancytopenia, with a 10% of blood cell requirement (dotted lines). Taking into account that the demand for blood cell production is proportional to body weight, we calculated the expected mean for women (magenta) and men (blue) depending on age (Figure 2 a). Weight data for children from 0 to 18 years of age are obtained from 53 . Up to the age of fifteen, the weight differences are relatively small between boys and girls. From 18 to 40 years women and men gain weight, up to 61 kilograms for women and up to 71 kilograms for men. There are great variations in weight interindividually depending on height and body mass index, but we have considered these data as average values to simplify the model. In 54 , the daily production of haematopoietic cells in a 70 kg male was estimated to be approximately \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4285\cdot 10^8$$\end{document} cells. This pool of cells would contain different cell types (monocytes, platelets, neutrophils, mature B cells, and mature T cells) as described in 54 . Based on this estimate, the annual number of haematopoietic cells produced per kilogram of body weight was calculated as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(4285/70)\cdot 365 \cdot 10^8$$\end{document} . Thus, the number of blood cells required up to the age of 40 years, increases for both women (magenta) and men (blue) up to the age of 16 years and then reaches a plateau (Figure 2 b). Men requirements for hematopoietic cells is higher compared to women after the age of 16 years (Figure 2 b). The graph also shows the number of hematopoietic cells produced in women and men with pancytopenia (dotted lines), which accounts for 10% of the total blood cell requirement. The primary objective of this study was to construct a mathematical framework that simulated the age at which individuals with DC develop pancytopenia, thereby enabling the prediction of its onset age. We employed the model ( 1 )-( 4 ), incorporating the assumption that, following mitotic replication, the rate at which cells exit the stem pool per unit time is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-p)m/ \mathcal {N}(t)$$\end{document} . Consequently, the total number of cells leaving the stem pool at time t is 11 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \displaystyle {\int _0^{x_H} \frac{(1-p)m}{\mathcal {N}(t)}N(x,t)\,dx}. \end{aligned}$$\end{document} Each cell possesses a proliferation potential that is determined by its generational age x , more precisely, this potential is given by the remaining replicative capacity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H-x$$\end{document} . Therefore, the maximum number of blood and immune cells that can be produced in a given year y , from cells leaving the haematopoietic stem cell compartment is expressed as 12 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \displaystyle {\int _{y-1}^y \left( \frac{(1-p)m}{\mathcal {N}(t)}\int _0^{x_H} N(x,t) 2^{x_H-x}\,dx\right) dt}. \end{aligned}$$\end{document} The central aim is to compare the demand for blood cells with the stem cell compartment’s production capacity, thereby identifying the age at which this balance falls below the threshold associated with pancytopenia.

In this work we sought to mathematically model the dynamics of HSC population in cases of DC, as one of the main causes of mortality in DC patients is bone marrow failure 7 . DC manifests with different symptoms and at different ages, according to the severity of the disease and is mainly caused by mutations in genes that affect telomerase and shelterins 4 , 10 , 48 . Thus, telomere maintenance is altered in these patients 6 , 11 , 77 , causing excessive telomere shortening which negatively affects organ function 8 . One consequence of the telomere dysfunction in the bone marrow includes pancytopenia, observed in 50 to 90% of DC cases 10 . DC severity varies significantly based on initial telomere length and level of telomerase activity, with some patients facing critical blood cell deficiencies in infancy while others remain stable until adulthood, a phenomenon which is well documented clinically and contributes to generational anticipation of telomere biology disorders 78 . To model the evolution of HSC we used a non-local diffusion–advection model with zero-flux boundary conditions 45 , which incorporates telomerase activity into the generational–temporal dynamics, a feature that is essential for accurately representing the aging processes within the hematopoietic compartment. Our model represents a simplified view of the complex mechanisms underlying HSC evolution, which includes various cell types, such as quiescent and self-renewing cells 79 . Although HSCs are represented as a homogeneous pool in the model, individual cells are allowed to differ in telomere length, which introduces a certain degree of heterogeneity (variable x ). Telomere length can be assessed at three levels of resolution: the individual telomere, the single cell, and the tissue level. In this study, we focus on the cellular level by considering the average telomere length per cell. While this approach may underestimate the number of senescent cells, the model remains sensitive to the accumulation of cells in a state of cell cycle arrest. Furthermore, it captures temporal and generational changes in the distribution of telomere lengths. On the other hand, even though our numerical experiments used a fixed initial HSC pool size, the model could change the initial condition if one wished to represent the variability among DC patients in this way; in our study, we modulated disease severity using the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} , which represents the maximum proliferation potential of HSCs. In DC population, this potential is markedly reduced compared to healthy populations. Using this approach, we estimated the maximum annual production of blood and immune cells derived from cells exiting the HSC compartment, thereby enabling a quantitative assessment of the relationship between physiological demand and the system’s productive capacity. Several gene mutations are known to cause DC. Each mutation may lead to different rates of telomere shortening, potentially affecting the proliferative capacity. However, these differences were not explicitly incorporated into the model in order to keep the framework as simple and tractable as possible. When modeling severe cases of DC, such as Hoyeraal-Hreidarsson syndrome 56 , Revesz syndrome 58 and young adults affected by DC, we took into consideration that the telomere length in blood cells from these patients is short 5 , below the first percentile, even at very early ages 73 . In addition, if the normal telomere loss in healthy people is of 0.05 to 0.1 kb per year, DC patients lose from 0.02 to 0.34 kb per year 38 , 73 . These very short telomeres limit cell division 80 . Our model assumed that the generational age of the stem cell population was relatively young in newborns, meaning that cells still would have potential to divide and give rise to the different haematopoiteic cell types 81 . However, as age increased, the population distribution shifted towards values corresponding to older generational age. In this regard, at the age of 10 years, the generational age of the population would be old, and at the age of 15 years, the possibility of cell division would be almost null. These stages of older generational age in the stem cell pool could limit tissue homeostasis. One of the reasons for a limitation in cell division is telomere shortening 82 , as discussed above, as very few short telomeres could limit cell division 80 . Indeed, when stem cells are forced to maintain the cell demands of the body, but their potential to divide is low (short telomere length), stem cells have the dilemma of self-renewal, to maintain stem cell population, or differentiation, to generate the various blood-cell types 52 . Ultimately, it will cause pancytopenia and bone marrow failure. In fact, the potential for cell production was low, according to our model. One important reason is that telomerase activity is very low (s=0.1 in our model) or null in these patients, because the telomerase genes (TERT and TERC) as well as other genes involved in telomerase function are mutated in these individuals 4 , 6 , 7 . Our results showed that while the need for hematopoietic cells increased from the time of birth to the age of 15 years, the cell production of these patients was dramatically decreasing. At an early age, the potential cell production reached the pancytopenia stage. The best-case scenario is when it happens at the age of 15 years, in individuals with better cell division potential, mostly influenced by the generational age of their cell population. The analysis of DC patients that develop the disease after the age of 15 years showed that lower levels of telomerase activity ( s=0.1 ) and older generational age of the stem cell population, would cause pancytopenia earlier in life. In contrast, for a similar telomerase activity ( s=0.1 ) in combination with good potential for cell production, pancytopenia would appear later in life. These cases happen usually due to heterozygous mutations, such autosomal dominant in TERC and DKC1 gene 4 and in several genes 8 . In adults, the main symptoms are BMF, pulmonary fibrosis and liver cirrhosis 6 although the spectrum of the symptoms is diverse 6 , 8 , 10 . Our model also revealed that higher rates of telomerase activity in patients with a better division potential would develop the disease after the age of 60 years. Indeed, in these patients the levels of telomerase activity are different. For instance, some heterozygous mutations in TERT gene seem to have higher telomerase activity than mutations in TERC gene, although at lower levels compared to healthy people 66 . These different levels of telomerase activity may reveal as diverse manifestations of the disease 66 . Apart from the health difficulties that DC imposes on patients, one extra issue for them is fertility 29 – 31 , 33 . Reports show that some of these patients had pregnancy complications that ranged among those who could not become pregnant to those who suffered miscarriage, preeclampsia, placenta previa, cytopenia during pregnancy, fetal distress, etc 30 , 31 . In the case of DC patients, possible strategies to improve fertility would not include gene therapy with telomerase gen, as these patients are cancer prone 7 and a high expression of telomerase 75 , 83 may be detrimental for them. Instead, we model danazol treatment which would produce a moderate level of telomerase reactivation, since it is produced from the endogenous promoter, and has been used in these patients 5 , 38 . Regarding danazol treatment, it is important to note that approximately 80% of the patients responded to danazol 38 , thus, in about 20% of patients with DC telomerase activity may not increase, or produce any benefit in the hematopoietic compartment after danazol treatment 38 . Indeed, we modelled low doses of danazol, as the maximum dose (800 mg/day) may alter liver function 38 and had other side effects 4 . Our model showed that moderate telomerase reactivation by danazol in the cases with aged cell population, the time to pancytopenia onset would be delayed. However, the level of cell production required by the body could not be attained using the proposed values for s , as these remain purely theoretical due to the scarcity of empirical data needed to support them. Only in cases where the cell population was younger, with a better division potential, would blood cell production be close to what is required, thus the risk of pancytopenia would be lower. Although the expected effect of danazol seems to be promising according to the model, and data from other reports 5 , 10 , 38 , more studies would be needed to test the effects of danazol treatment on fertility in DC patients.

Dyskeratosis congenita (DC) is a rare inherited disease that presents a variety of clinical features, from mucocutaneous defects such as nail dystrophy, reticular pigmentation and oral leukoplakia 1 , 2 , to bone marrow failure (BMF), immunodeficiency, a high risk of cancer and liver fibrosis among others 3 – 5 . Indeed, in tissues with high turnover (such as hematopoietic system) the disease manifests as stem cell failure and appears early in life, with severe symptoms. In contrast, in low turnover tissues, the disease manifests later in life 6 . The major causes of mortality in this disease are BMF, which affects to 85% of DC patients 7 , and immune disease 4 . The origin are several pathogenic germline variants in at least 16 genes related with telomere maintenance 4 , 5 , 8 . Thus, DC is a syndrome that affects telomerase activity leading to short telomeres in most cases 5 , 8 , which in mice 9 and humans 1 , 4 , 5 , impair tissue regeneration and organ function. The prevalence of DC is estimated in one in a million 5 , 10 , 11 . Telomeres, which protect chromosomes from degradation and fusions 12 – 14 , consist of long repeats of the sequence TTAGGG, coated by a complex of proteins called Shelterin which have a pivotal function in telomere capping 15 . In humans, telomeres are about 10 to 15 kb long 16 , 17 and shorten as cells divide 18 along their lifespan, at a rate of 31 to 72 bp/year 19 – 21 . Telomere attrition is caused during DNA replication, as polymerases cannot copy the very ends of chromosomes 22 , 23 . In fact, telomeres can predict the time of replicative senescence 24 as these in vitro cultured cells stop dividing after a defined number of divisions 25 . Telomerase, a reverse transcriptase, counteracts telomere shortening 26 . However, variants in genes that affect telomerase activity or stability such as TERT, TERC, DKC1, TERC, TERT, NOP10, NHP2 and TCAB, as well in genes related to telomere protection (TINF2) and maintenance such as MDM4, RPA1, NAF1, CTC1, RTEL1,PARN, etc 2 , 4 , 5 , 8 , 10 cause telomerase activity to be low or null. Telomere erosion and reduced fertility are reported in mice models of DC 27 , 28 and human DC patients 29 – 31 , who, in addition, have complications for fertility preservation 29 . Interestingly, the age of menarche and menopause happen at normal ages 30 , 31 . Both women 32 and men 33 with DC have lower levels of anti-Mullerian hormone, and pregnant women with DC have more complications including miscarriages 30 , and cytopenias with the need of transfusions and fetal illnesses 30 , 31 . In vitro fertilization technologies facilitate births of a healthy baby in the case of monogenic disease 34 . Androgens have been used to treat bone marrow failure syndromes 35 , 36 and aplastic anemia 37 . However, danazol, also used in patients with mutations in the TERT gene 38 produces less androgenic side effects in women, and has been widely used with endometriosis patients 39 , 40 and women with diminished ovarian reserve 41 . It acts through the estrogen response elements present in the promoter of TERT, increasing telomerase activity in vitro 42 . Previous modeling studies have shown that replicative senescence in HSCs significantly influences long-term hematopoietic output and population aging 43 . The mechanisms governing hematopoietic cells are complex, so any mathematical model must necessarily involve simplifications. In their work, Marciniak-Czochra and colleagues 43 developed a compartmental model of hematopoietic stem and progenitor cells, where each compartment represents a stage of differentiation and is subject to a finite replicative potential. The model in 44 examines the progression of chronic myeloid leukemia across five hematopoietic cell compartments (cycling stem cells, quiescent stem cells, progenitor cells, differentiated cells, and terminally differentiated cells) through a system of 10 ordinary differential equations. These models account for the heterogeneity of hematopoietic cells through different compartments, but each compartment is considered homogeneous. However, we consider a single heterogeneous HSC compartment in which the division potential of each cell depends on its generational age. The diffusion–-advection model developed by 45 , incorporating zero-flux boundary conditions, is used here to simulate the generational and temporal dynamics of a hematopoietic stem cell (HSC) population in order to investigate the progression of DC. In contrast to 46 , 47 , our formulation explicitly includes telomerase activity within the generational–-temporal dynamics, as this component plays a critical role in shaping the evolution of proliferative potential and, consequently, the aging of the population. We utilized the model described in 45 to estimate the maximum annual output of blood and immune cells derived from cells exiting the HSC compartment. This estimation facilitates a quantitative comparison between the physiological demand for blood cells and the productive capacity of the HSC compartment, thereby enabling the identification of the age at which this balance declines below the threshold associated with pancytopenia. We further simulated the impact of therapeutic interventions on telomere length, specifically modeling treatment with the synthetic sex hormone danazol, which has been shown to reactivate expression of the endogenous telomerase gene 5 , 10 , 38 . Fig. 1 Scheme of normal and aberrant function of the bone marrow. On the left, the figure depicts normal production of blood cells. On the right part of the figure, abnormal function of bone marrow, showing decreased production of blood cells, characteristic of pancytopenia. Figure was created with Biorender. Scheme of normal and aberrant function of the bone marrow. On the left, the figure depicts normal production of blood cells. On the right part of the figure, abnormal function of bone marrow, showing decreased production of blood cells, characteristic of pancytopenia. Figure was created with Biorender.

Each cell possesses a characteristic range of telomeric lengths, with a maximal and minimal threshold. The minimal telomeric length, commonly referred to as the Hayflick limit, corresponds to the critical threshold beyond which further replication is no longer possible. We denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} the maximal proliferative potential of a cell, namely, the maximum number of divisions a cell can undergo prior to entering the senescent state. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in [0,x_H]$$\end{document} and let N ( x ,  t ) denote the population density at generational age x and time t . We introduce the following parameters: m , the mitotic replication rate per cell per unit time; p , the probability of symmetric division yielding two daughter stem cells, both progressing to the subsequent generational state; and 1-p , the probability of asymmetric division, where one daughter cell progresses to the next generational state while the other differentiates and leaves the stem pool. Furthermore, d denotes the mortality rate per cell per unit time, and r represents the telomerase activity rate per cell per unit time, acting to rejuvenate the cell and thereby extend its replicative potential. We consider the following diffusion-advection initial and boundary value problem (IBVP) introduced in 45 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & N_t (x,t)=\frac{1}{\mathcal {N}(t)}(D N_{xx}(x,t)-v N_x(x,t)+ \rho N(x,t)), 0< x0, \end{aligned}$$\end{document} 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & DN_{x}(0,t)-v N(0,t)=0,\, t>0, \end{aligned}$$\end{document} 3 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & DN_{x}(h,t)-v N(h,t)=0,\, t>0, \end{aligned}$$\end{document} 4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & N(x,0)=f(x), \quad 0< x< x_H, \end{aligned}$$\end{document} where 5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {N}(t) = \int _0^{x_H} N(x,t)dx. \end{aligned}$$\end{document} is the total population, D=(m(1+p)+r)/2 is the diffusion constant, v=m(1+p)-r the advection coefficient, and ρ =mp-d can be interpreted as an effective proliferation rate. The equation ( 1 ) must be supplemented with suitable zero-flux boundary conditions ( 2 )-( 3 ) to guarantee that no cell population density N ( x ,  t ) either leaves or enters the interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in [0,x_H]$$\end{document} . Finally, a regular enough initial condition ( 4 ). The IVBP ( 1 )-( 4 ) was solved numerically by the method of lines. We used a uniform mesh on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,x_H]$$\end{document} for discretising the variable x . Let M be a positive integer and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h=x_H/M$$\end{document} the space step. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_j=j h$$\end{document} for j=0,1,… ,M be the nodes. We denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_j(t)=N(x_j,t)$$\end{document} . We employed second-order centred finite differences to discretise \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{xx}(x_j,t)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{x}(x_j,t)$$\end{document} for j=1,… ,M-1 . Finally, we implemented the following finite differences of second order to approximate the boundary conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \displaystyle \frac{D}{h}\left( -\displaystyle \frac{3}{2}N_0+2N_1-\displaystyle \frac{1}{2}N_2\right) =v N_0,\\ & \displaystyle \frac{D}{h}\left( \displaystyle \frac{1}{2}N_{M-2}-2N_{M-1}+\displaystyle \frac{3}{2}N_M\right) =v N_M, \end{aligned}$$\end{document} then 6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & N_0=\displaystyle \frac{1}{\frac{3}{2}+\frac{vh}{D}}\left( 2N_1-\frac{1}{2}N_2\right) , \end{aligned}$$\end{document} 7 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & N_M=\displaystyle \frac{1}{-\frac{3}{2}+\frac{vh}{D}}\left( \frac{1}{2}N_{M-2}-2N_{M-1}\right) . \end{aligned}$$\end{document} After the discretization in the variable x , we arrived at a system of ordinary differential equations in the variable t for the vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {N}(t)=[N_1(t), \cdots ,N_{M-1}(t)]^T,$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \vec {N}'(t)= & \frac{1}{I_T^C(N(x,t),h)}(D A_{2,h}\vec {N}(t)-v A_{1,h}\vec {N}(t)+\rho \vec {N}(t)\\ & +\displaystyle \frac{D}{h^{2}} \left( \begin{array}{c} \displaystyle \frac{(2N_1-\frac{1}{2}N_2)}{\frac{3}{2}+\frac{vh}{D}} \\ 0\\ \vdots \\ 0\\ \displaystyle \frac{(\frac{1}{2}N_{M-2}-2N_{M-1})}{-\frac{3}{2}+\frac{vh}{D}}\end{array}\right) -\displaystyle \frac{v}{2h} \left( \begin{array}{c} -\displaystyle \frac{(2N_1-\frac{1}{2}N_2)}{\frac{3}{2}+\frac{vh}{D}} \\ 0\\ \vdots \\ 0\\ \displaystyle \frac{(\frac{1}{2}N_{M-2}-2N_{M-1})}{-\frac{3}{2}+\frac{vh}{D}}\end{array}\right) ), \end{aligned}$$\end{document} where 8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I_T^C(N(x,t),h) = h\left( (N_0(t)+N_M(t))/2 +\sum _{j=1}^{M-1}N_j\right) , \end{aligned}$$\end{document} is the trapezoidal composed rule for approximating the integral ( 5 ), 9 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{2,h}=\displaystyle \frac{1}{h^2}\left( \begin{array}{rrrrr} -2& 1& 0& \cdots & 0\\ 1& -2& 1& \cdots & 0\\ \vdots & \vdots & \ddots & & \vdots \\ 0& \cdots & 1& -2& 1\\ 0& \cdots & 0& 1& -2 \end{array} \right) , \end{aligned}$$\end{document} 10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{1,h}=\displaystyle \frac{1}{2h}\left( \begin{array}{lcrcc} 0& 1& 0& \cdots & 0\\ -1& 0& 1& \cdots & 0\\ \vdots & \ddots & \ddots & \cdots & \vdots \\ 0& \cdots & -1& 0& 1\\ 0& \cdots & 0& -1& 0 \end{array} \right) . \end{aligned}$$\end{document} Spatial discretization ensures second-order accuracy, while the IBVP is transformed into a system of ordinary differential equations. The Matlab stiff solver ode15s was chosen for its efficiency in handling such system.

Patients with DC show alterations in telomere maintenance, which lead to very short telomeres. In women, this condition affects fertility, complicating egg preservation 29 and pregnancy, along with fertility outcomes. Women with DC experience higher rates of recurrent miscarriages in the mid-trimester compared to the general population and these losses occurred at younger ages than expected 68 , 69 . Other complications are placenta previa and higher rates of primary cesarean deliveries 68 . Both mother and the fetus may be affected by the development of aplastic anemia during pregnancy, and this brings many complications such as intrauterine growth restriction, premature birth, neonatal death, preeclampsia, and other adverse maternal outcomes 70 , 71 . In addition, cytopenias in women with DC, may worsen during pregnancy, requiring transfusions; these cytopenias could lead to adverse maternal and fetal outcomes 68 , 72 , 73 . Fig. 7 Simulation of danazol treatment to reactivate endogenous telomerase activity in DC patients. The settings for the simulation were \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} 31, 32 and 33 (left, central and right panels respectively) and very low telomerase activity with s=0.1 (black line) along with danazol treatment for 2 years. In the three graphs treatment started at age 23 and telomerase activity improved according to s values of 0.2, 0.3 and 0.4, as indicated inside the figure. The magenta line represents the HCF need and the red line represents the 10% of the HCF need. It is observed that normal levels could not be reached. Treatment with danazol would only partially mitigate the accelerated telomere attrition, as reported in 38 . Because patients with DC do not recover substantial telomerase activity, the decline in blood cell production is only modestly slowed and does not return to pretreatment levels. Simulation of danazol treatment to reactivate endogenous telomerase activity in DC patients. The settings for the simulation were \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} 31, 32 and 33 (left, central and right panels respectively) and very low telomerase activity with s=0.1 (black line) along with danazol treatment for 2 years. In the three graphs treatment started at age 23 and telomerase activity improved according to s values of 0.2, 0.3 and 0.4, as indicated inside the figure. The magenta line represents the HCF need and the red line represents the 10% of the HCF need. It is observed that normal levels could not be reached. Treatment with danazol would only partially mitigate the accelerated telomere attrition, as reported in 38 . Because patients with DC do not recover substantial telomerase activity, the decline in blood cell production is only modestly slowed and does not return to pretreatment levels. One possible strategy to improve TL could be the reactivation of telomerase with sexual steroids 42 , 74 . Indeed, danazol has been used in women with diminished ovarian reserve resulting in improved rate of mature oocyte production 41 and in patients with telomere disease, improving the symptoms of cytopenia 38 . The advantage of danazol is the reactivation of the endogenous gene, meaning that the production of telomerase would have normal limits compared to other treatments. For instance, gene therapy produced higher levels of telomerase as the gene is under very strong promoters 75 . Because patients with DC have a higher risk of cancer 4 , 10 , in the case of women with DC the dose used should probably be moderated. The mathematical model was designed for the use danazol two years before pregnancy occurred, with the aim of sustaining the production of hematopoietic cells during pregnancy. In our model, danazol treatment starts in adulthood, at the age of 23 years, because mice work has shown that telomerase reactivation at adult age has the same cancer incidence as controls 75 . Figure 7 displays the simulation of danazol treatment for two years starting at age 23 for values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} 31, 32 and 33. Telomerase reactivation in blood cells from DC patients treated with danazol in vitro is rather low 76 and the values for telomerase activity fold increase after danazol treatment in patients are not available. Thus, we propose theoretical values for the parameter ranking from 0.1 to a maximum of 0.4, to analyse to what extent different levels of telomerase activity would stabilize the hematopoietic pool. The black line shows the gradual decrease of the blood cell production that happens with age in cases of DC, which are below the normal need (Figure 7 , left graph). When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} is higher the blood cell production decreased below the normal need at older ages (central and right graphs). The administration of danazol may partially attenuate the accelerated telomere shortening described in 38 . However, because telomerase activity is not substantially restored in patients with DC, blood cell production continues to decline, albeit at a slower rate. By the age of 23 years, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=31$$\end{document} is the worst-case scenario and danazol treatment slows the loss of blood cells, although the decline is already very advanced. Case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=32$$\end{document} represents an intermediate situation, while in case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=33$$\end{document} the condition is not yet as severe, and the slowing of blood cell loss with danazol treatment may still produce some benefit. These results highlight the critical dependence of treatment outcome on the initial proliferative capacity of the hematopoietic stem cell population, emphasizing that earlier intervention or higher baseline proliferation potential may be necessary to achieve clinically meaningful improvements.

Fig. 3 Time evolution of hematopoietic stem cell population for patients with DC. Graph depicts the population density evolution with the generational age \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H = 30$$\end{document} and s = 0.1 . The top left graph represents the starting point (newborns). The top right graph represents the generational age of HSC at the age of five years. The left bottom graph represents HSC evolution at the age of 10 years and the bottom right graph, the evolution of HSC at the age of 15 years. Note that the population increasingly concentrates in the higher-right end of the generational age interval, corresponding to the more aged cells. Time evolution of hematopoietic stem cell population for patients with DC. Graph depicts the population density evolution with the generational age \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H = 30$$\end{document} and s = 0.1 . The top left graph represents the starting point (newborns). The top right graph represents the generational age of HSC at the age of five years. The left bottom graph represents HSC evolution at the age of 10 years and the bottom right graph, the evolution of HSC at the age of 15 years. Note that the population increasingly concentrates in the higher-right end of the generational age interval, corresponding to the more aged cells. Cosgrove and coworkers estimate that the number of haematopoietic stem cells that are active at any time ranges from 25, 000 to 1, 300, 000 54 . Then, in the numerical experiments, we assumed that initially the total population of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_0=10^6$$\end{document} stem cells is at generational age 0 and we considered an initial density function of Gaussian (normal) type 13 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} N(x,0) =\displaystyle \frac{n_0}{\sqrt{2\pi \sigma ^2}}e^{-(x-\nu )^2/2\sigma ^2}, \end{aligned}$$\end{document} for ν = 1 and σ = 0.001 . In this way the population distribution was concentrated at the beginning of the interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,x_H]$$\end{document} . Some parameters of our model were computed according to data from 47 : the probability of symmetrically division of stem cells p=0.35 , the effective proliferation rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m/n_0=0.95 \,(\text{ year}^{-1})$$\end{document} and the effective mortality rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d/n_0=0.34 \,(\text{ year}^{-1})$$\end{document} . Regarding the effective telomerase rate, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r/n_0=s \,(m/n_0)\,(1+p).$$\end{document} The parameter s was entered as the proportionality factor between r and m(1+p) , i.e. r=m(1+p)  s . Small s values represent low telomerase activity, in particular s=0 means no telomerase activity. Conversely, values of s close to 1 denote high telomerase activity. There is evidence for severe defects in telomerase function and telomere maintenance in stem cells from DC patients 1 . There are several causes and degrees of the disease, depending on mutations in different genes, affecting telomeric length and telomerase activity in different proportions and resulting in different life expectancies. The maximum proliferation potential of HSCs at 50% percentile was taken as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=100$$\end{document} in the former article 45 . In the current, we have considered values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} varying between 23 and 35 depending on the severity of the disease. The IVBP ( 1 )-( 4 ), with the specified parameters, was solved numerically using the method of lines with a spatial discretisation step size of h=0.2 . The resulting system of ordinary differential equations was subsequently integrated with the MATLAB solver ode15s, employing a relative tolerance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.25 \times 10^{-14}$$\end{document} and an absolute tolerance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-16}$$\end{document} . As an illustrative example, the time evolution of the population distribution from 0 to 15 years for hematopoietic stem cell population with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=30$$\end{document} and a telomerase activity s=0.1 is presented in Figure 3 . At the age of 5 years, the population shifted almost to the middle of the generational age range. After 10 years the population shifted further to the right, to the range between 10 and 30. While at 15 years the population shifted even further to the right, accumulating in the 20–30 generational age range, which represents a rather aged population. This evolution indicates a substantial reduction in the proliferative potential of the hematopoietic stem cell compartment, reflecting the progressive aging and diminished regenerative capacity of the cell population. Fig. 4 Simulation of the most severe cases of DC. ( a ) The graph shows the production potential of the cells leaving HSCs with respect to time measured in years. The blue solid line represents the blood cell requirements depending on age. The red solid line corresponds to 10% of the blood cell requirement. The black lines correspond to the estimated maximum potential for haematopoietic cell production of DC patients with very low telomerase activity s=0.1 and the smallest telomere length values corresponding to values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} from 23 to 28. ( b ) Quadratic polynomial curve of pancytopenia onset. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{onset}1(x_H)\thickapprox 0.1341 x_H^2 - 4.393 x_H + 31.65$$\end{document} . Note that the onset of pancytopenia shows a range between 2 years ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=23$$\end{document} ) and 14 years ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=28$$\end{document} ). Simulation of the most severe cases of DC. ( a ) The graph shows the production potential of the cells leaving HSCs with respect to time measured in years. The blue solid line represents the blood cell requirements depending on age. The red solid line corresponds to 10% of the blood cell requirement. The black lines correspond to the estimated maximum potential for haematopoietic cell production of DC patients with very low telomerase activity s=0.1 and the smallest telomere length values corresponding to values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} from 23 to 28. ( b ) Quadratic polynomial curve of pancytopenia onset. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{onset}1(x_H)\thickapprox 0.1341 x_H^2 - 4.393 x_H + 31.65$$\end{document} . Note that the onset of pancytopenia shows a range between 2 years ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=23$$\end{document} ) and 14 years ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=28$$\end{document} ). The most severe cases of DC appear at early ages, such as the Hoyeral-Hreidarson syndrome 55 , 56 , which appears during infancy and whose frequent symptoms are cerebellar hypoplasia, intrauterine growth retardation and pancytopenia. This disease is mainly caused by mutations in DKC1 10 , 57 , and other genes, TERT and TERC, TINF2, WRAP53, RTEL, PARN and ACD (reviewed in 4 , 6 , 10 . Another severe case of DC is the Ravesz syndrome, discovered in a six-month old patient who also developed BMF 58 . In most cases, this condition happens before the age of 2 years 59 . Mutations causing this syndrome have also been found in TINF2 59 , which encodes TIN2 protein, a component of the shelterin complex 60 . Mutations in this gene cause telomere unprotection, leading to very short telomeres 59 . Finaly, Coats plus syndrome, which also appears at an early age, affects the vasculature of retina, bones, brain and gastrointestinal system 61 and is caused by mutations in CTC1 62 whose protein is part of the CST complex and plays an essential role in telomere maintenance (reviewed in 10 ). A mutation in POT1, that encodes for a protein of the shelterin complex 63 , has also been described 64 . In this context, we sought to model cases in which pancytopenia developed during childhood (Figure 4 ). To this end, the estimated maximum hematopoietic cell production potential (PHCP) for DC patients with very low telomerase activity ( s=0.1 ) and low telomere lengths, corresponding to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} values from 23 to 28, was studied. The model showed very low levels of PHCP for DC patients, from birth to the age of 15 years (panel a). These maximum levels crossed with the red line which represents the onset of pancytopenia. Lower \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} values result in PHCP curves crossing the red line at earlier ages, indicating earlier pancytopenia onset. None of the cases maintained the normal PHCP (blue line) requirements after the age of 5 years in enfants with more division potential. A quadratic fit for the age of pancytopenia onset as a function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{onset}1(x_H)\thickapprox p_1 x_H^2 + p_2 x_H + p_3$$\end{document} , shows that the onset can happen at a range of ages from 1.7 to almost 14 years in these patients. In individuals with severe DC, pancytopenia typically manifests during childhood. However, there are documented cases where BMF presents after the age of 15. Indeed, 80 % of patients with DC develops BMF at the age of 30 years 10 . A study by Fogarty and coworkers 65 assessed two families with adult-onset pancytopenia and identified novel mutations in the TERC gene. These cases did not exhibit the typical physical signs of DC but had significantly shortened telomeres, reduced hematopoietic function, and elevated serum erythropoietin and thrombopoietin levels. Thus, BMF due to DC can occur in phenotypically normal adults and may mimic acquired aplastic anemia 4 , 5 , 10 . Taking into consideration that the median age of DC diagnosis is 15 years, some patients, particularly those with autosomal dominant inheritance, present the disease at a later age 65 . This delayed onset can complicate the diagnosis, as the disease may not be immediately recognized in adults. These findings emphasize the importance of studying telomere length, pancytopenia and BMF in adults, even in the absence of classic clinical features. Patients exhibited short telomeres and a severe reduction of telomerase activity, ranging from 5– 15% . The simulation of DC due to mutations in genes that give rise to later onset of pancytopenia in which patients exhibited a severe reduction of telomerase activity, is shown in Figure 5 . The parameters range from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=29$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=35$$\end{document} and s=0.1 . The black lines correspond to the estimated maximum production of hematopoietic cells in DC. Depending on the number of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} the onset of pancytopenia (the cross between black lines and red line) can vary from the age of 17 to the age of 50 years (Figure 5 a). The model shows how the normal requirement of cells is maintained until the age of 30 years (cross between the black lines and magenta (women) and blue (men). Afterwords, patients with DC do not show normal levels of hematpoietic cells (Figure 5 a). The onset of pancytopenia happens at earlier age in men (solid line) compared to women (dotted line) particularly when the proliferation potential is low (Figure 5 b). In all cases, the onset of pancytopenia occurred approximately one year earlier in men than in women. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=29$$\end{document} it occurred at age 17 for males and 18 for females, while for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=35$$\end{document} it occurred at age 47 for men and at age 48 for women. In summary, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} increased, the onset of pancytopenia occurred progressively later, indicating that higher initial proliferation potential postponed the development of the condition. Fig. 5 Simulation of DC cases whith pancytopenia onset after the age of 15. (a) The graph shows the estimated maximum production of hematopoietic cells in DC patients from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=29$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=35$$\end{document} and s=0.1 . The blue solid line represents the blood cell requirements for men depending on age (HCM need). The magenta solid line represents the blood cell requirements for women depending on age (HCF need). The red solid line and the red dotted line correspond to 10% of the blood cell requirement for men and women respectively. (b) The graph shows the onset of pancytopenia versus maximum proliferation potential for HSCs. Note that the onset of pancytopenia is increasing as the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} increases. Simulation of DC cases whith pancytopenia onset after the age of 15. (a) The graph shows the estimated maximum production of hematopoietic cells in DC patients from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=29$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H=35$$\end{document} and s=0.1 . The blue solid line represents the blood cell requirements for men depending on age (HCM need). The magenta solid line represents the blood cell requirements for women depending on age (HCF need). The red solid line and the red dotted line correspond to 10% of the blood cell requirement for men and women respectively. (b) The graph shows the onset of pancytopenia versus maximum proliferation potential for HSCs. Note that the onset of pancytopenia is increasing as the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} increases. Mutations in several genes associated with telomere maintenance have been identified in DC patients 2 , 5 , 6 , 10 . Most mutations alter telomerase activity, decreasing or eliminating its enzymatic activity. For instance, mutations in the TERC gene, encoding the RNA component of telomerase, have been linked to autosomal dominant forms of DC, which are less severe 5 , 6 . These mutations result in decreased levels of TERC 66 , leading to impaired telomerase function and shortened telomeres. Similarly, mutations in the TERT gene, encoding the telomerase reverse transcriptase, have been associated with DC and result in reduced telomerase activity 67 . Additionally, mutations in genes such as NOP10, NHP2, and WRAP53, which are involved in telomerase RNA processing and assembly, have been implicated in DC 2 ,. These mutations disrupt the proper formation of the telomerase complex, further contributing to decreased telomerase activity and telomere shortening. Collectively, these genetic alterations underscore the critical role of telomerase in maintaining telomere integrity and highlight its dysfunction as a central mechanism in the pathogenesis of DC. Fig. 6 DC severity in cases with pancytopenia onset after the age of 20 years. The graph shows the quadratic polynomial curve of pancytopenia onset in function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} and s (TA factor) variables. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{onset}2(x_H,s)\thickapprox -64.81 + 1.596 x_H -214.4 s + 0.04286 x_H^2 + 7.389 x_H s + 6.667 s^2$$\end{document} . Note that the lower the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} and s values were, the earlier the onset of pancytopenia started. DC severity in cases with pancytopenia onset after the age of 20 years. The graph shows the quadratic polynomial curve of pancytopenia onset in function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} and s (TA factor) variables. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{onset}2(x_H,s)\thickapprox -64.81 + 1.596 x_H -214.4 s + 0.04286 x_H^2 + 7.389 x_H s + 6.667 s^2$$\end{document} . Note that the lower the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} and s values were, the earlier the onset of pancytopenia started. The model ( 1 )-( 4 ) has been solved for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} values from 30 to 35 and s values from 0.1 to 0.4, the telomerase activity parameter, and the onset of pancytopenia has been calculated in each case. Fitting the data of pancytopenia onset to a quadratic polynomial curve was obtained when: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{onset}2(x_H,s)\thickapprox p_{00} + p_{10}x_H + p_{01}s + p_{20}x_H^2 + p_{11}x_H s + p_{02}s^2$$\end{document} , where coefficients (with 95% confidence bounds) were \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{00}= & -64.81\, (-92.64, -36.99),\\ p_{10}= & 1.596 \, (-0.1128, 3.306),\\ p_{01}= & -214.4 \, (-226, -202.7),\\ p_{20}= & 0.04286 \, (0.0166, 0.06911),\\ p_{11}= & 7.389\, (7.046, 7.732),\\ p_{02}= & 6.667 \, (0.1173, 13.22). \end{aligned}$$\end{document} The fitting surface is depicted in Figure 6 . The analysis indicated that lower values of both variables ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_H$$\end{document} and TA factor) were associated with an earlier onset of pancytopenia. A spectrum of disease severity was observed, which appeared to be closely related to both the initial telomere length and the level of telomerase activity. Patients with shorter telomeres and lower telomerase activity generally exhibited more severe manifestations. In contrast, patients with relatively longer telomeres or higher telomerase activity tended to present milder clinical phenotypes. These findings highlight the combined influence of telomere reserve and enzymatic activity on the progression and expressivity of the disease.