Estimating an initial telomere length distribution from the Laplace transform of its senescence times distribution

Abstract

This work follows from a previous study on the estimation of an initial distribution of telomere length from a senescence times distribution done in [10.1051/m2an/2026022, J. Olay{é}]. In this previous study, we have presented an estimation method based on the fact that our telomere shortening model can be approximated by a transport equation. This method has encouraging results, but fails to provide a good estimation when the variability of the initial telomere length distribution is too small. We improve here this method by approximating our model with an advection-diffusion equation, which allows us to better take into account the randomness of the shortening values. We show that under this approximation, there exists a simple link between the Laplace transform of the initial telomere length distribution and that of the senescence times distribution. Then, by using a numerical method for inverting Laplace transforms called Gaver-Stehfest algorithm, we exploit this link to construct a new estimator.

Figures (11)

• Figure 1: Representation of the set $Q_N(\alpha,\beta) =\left\{p\in\mathbb{C}\,|\,Re(p) > \alpha,\,Re(q_N(p)) > \beta\right\}$ (in red) for different values of $(\alpha, \beta)\in\left(\mathbb{R}_{-}\right)^2$, and comparison with vertical lines (in blue), when $b = 1$, $g = 1_{[0,1]}$ and $N = 40$. We see that as the sets $\left(Q_N(\alpha,\beta)\right)_{(\alpha,\beta)\in\left(\mathbb{R}_{-}\right)^2}$ have a conical shape, they do not contain vertical lines. Since the set $\mathcal{P}_N$ defined in \ref{['eq:definition_P']} has the same form as the sets $\left(Q_N(\alpha,\beta)\right)_{(\alpha,\beta)\in\left(\mathbb{R}_{-}\right)^2}$, this figure illustrates that $\mathcal{P}_N$ does not contain an interval of the form $[\gamma - i\infty,\gamma + i\infty]$, where $\gamma\in\mathbb{R}$.
• Figure 2: Estimation results with the estimator $\widehat{n}_0^{(N,K)}$ defined in \ref{['eq:estimator_gaver_stehfest_noise_free']} when $b = 1$, $g = 1_{[0,1]}$, $N = 40$, $K=250$ and $n_0\in\left\{h_{9,12},h_{16,16},h_{25,30},h_{49,50}\right\}$. Comparison with the results obtained with the estimator $\widehat{n}_0^{(\text{old},N)}(x) =\frac{1}{bm_1}n_{\partial}^{(N)}\left(\frac{x}{bm_1}\right)$ for all $x\geq0$, constructed in olaye_transport_2025. For plotting the curves, the numerical precision of the computations was set to $200$ digits.
• Figure 3: Evolutions of the estimation errors in $L^1$-norm of $\widehat{n}_0^{(N,K)}$ and $\widehat{n}_0^{(\text{old},N)}$ as a function of $N\in \left\{1 + 4\ell\,|\,\ell\in\llbracket0,25\rrbracket\right\}$, when $b = 1$, $g = 1_{[0,1]}$, $N = 40$, $K=250$ and $n_0 = h_{9,12}$. Comparison with the curves of the functions $N\in\mathbb{R}_+^{*} \mapsto \frac{C_1}{N}$ and $N\in\mathbb{R}_+^{*} \mapsto \frac{C_2}{N^2}$, where $C_1 > 0$ and $C_2 > 0$ are estimated thanks to a constrained least-squares regression on the last error points. For plotting the curves, the numerical precision of the computations was set to $200$ digits.
• Figure 4: Estimation results with the estimator $\widehat{n}_0^{(N,K)}$ defined in \ref{['eq:estimator_gaver_stehfest_noise_free']} when $b = 1$, $g = 1_{[0,1]}$, $N = 40$, $K=250$ and $n_0\in\left\{h_{100,120},h_{225,200},h_{324,330},h_{400,500}\right\}$. Comparison with the results obtained with the estimator $\widehat{n}_0^{(\text{old},N)}(x) =\frac{1}{bm_1}n_{\partial}^{(N)}\left(\frac{x}{bm_1}\right)$ for all $x\geq0$, constructed in olaye_transport_2025. For plotting the curves, the numerical precision of the computations was set to $200$ digits.
• Figure 5: Comparison of the absolute errors between the Laplace transform of a Log-normal distribution with parameters $\left(\log(2),\alpha\right)$ and the Laplace transform of Gamma distributions with parameters $\left(\ell^{*}(\alpha),\beta^{*}(\log(2),\alpha)\right)$, $\left(\ell^{**}(\alpha),\beta^{**}(\log(2),\alpha)\right)$ and $\left(\ell^{***}(\alpha),\beta^{***}(\log(2),\alpha)\right)$ (see \ref{['eq:parameters_gamma_same_mean']} and \ref{['eq:other_possible_parameters_gamma_distribution']}), when $\alpha\in\{0.2,0.4\}$.

Theorems & Definitions (18)

• Proposition 2.1: Well-posedness of \ref{['eq:scaled_model']}
• Remark 2.2
• Remark 2.3
• Theorem 2.4: Bounds on approximation errors
• Proposition 2.5: Link between the Laplace transforms
• Lemma 3.1: Alternative representation of a solution to \ref{['eq:approximation_transport_diffusion']}
• Lemma 3.2: Maximum principle
• proof
• Proposition 3.3: Existence and uniqueness of \ref{['eq:PDE_second_derivative_advection_diffusion']}
• Proposition 3.4: Explicit solution to \ref{['eq:PDE_second_derivative_advection_diffusion']}