On the Probability of Random Genetic Mutations for Various Types of Tumor Growth

Tomasetti, Cristian

doi:10.1007/s11538-012-9717-1

Cited by 11 publications

(14 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Next we present some general results which are valid for a large class of growth functions. This extends the classic results found in [28,4,25,42] and recent work in [43,21] who considered the wild-type population growth rate to be time-dependent but coupled with the mutant growth rate. Secondly, rather than the total number of mutants, our primary interest is on the distribution of mutant number in the clones initiated by mutation events.…”

Section: Introductionsupporting

confidence: 86%

“…Some authors [21,43] have previously considered the case where all rates in the system are multiplied by a timedependent function, say z(τ). This is relevant in the scenario where both the wild-type and mutant populations have their growth restricted simultaneously by environmental factors, for example exposure to a chemotherapeutic agent.…”

Section: Time-dependent Rate Parametersmentioning

confidence: 99%

See 1 more Smart Citation

Universal Asymptotic Clone Size Distribution for General Population Growth

Nicholson

Antal

2016

Bull Math Biol

View full text Add to dashboard Cite

Deterministically growing (wild-type) populations which seed stochastically developing mutant clones have found an expanding number of applications from microbial populations to cancer. The special case of exponential wild-type population growth, usually termed the Luria–Delbrück or Lea–Coulson model, is often assumed but seldom realistic. In this article, we generalise this model to different types of wild-type population growth, with mutants evolving as a birth–death branching process. Our focus is on the size distribution of clones—that is the number of progeny of a founder mutant—which can be mapped to the total number of mutants. Exact expressions are derived for exponential, power-law and logistic population growth. Additionally, for a large class of population growth, we prove that the long-time limit of the clone size distribution has a general two-parameter form, whose tail decays as a power-law. Considering metastases in cancer as the mutant clones, upon analysing a data-set of their size distribution, we indeed find that a power-law tail is more likely than an exponential one.

show abstract

Section: Introductionsupporting

confidence: 86%

Section: Time-dependent Rate Parametersmentioning

confidence: 99%

Universal Asymptotic Clone Size Distribution for General Population Growth

Nicholson

Antal

2016

Bull Math Biol

View full text Add to dashboard Cite

show abstract

“…Several studies have been undertaken to explore the impact of non-exponential growth dynamics on the evolution of mutation-induced resistance [44, 56]. For example, Dewanji et al developed an extension of the Luria-Delbrück model that considered non-exponential growth dynamics [44], following the development of the tumorigenesis model by Lübeck and Moolgavkar [54].…”

Section: Previous Workmentioning

confidence: 99%

“…They showed that the inclusion of the possibility of cell death, as well as the assumption of Gompertzian growth of mutant cells, led to larger variations in the number of cells harboring resistance mutations as compared to earlier growth models. More recently, Tomasetti compared the impact of various growth laws on the dynamics of resistance [56]. He showed that the probability that a given random mutation will be present by the time a tumor reaches a certain size is independent of the type of curve assumed for the average growth of the tumor, for a general class of growth curves.…”

Section: Previous Workmentioning

confidence: 99%

Evolution of acquired resistance to anti-cancer therapy

Foo

Michor

2014

Journal of Theoretical Biology

230

178

View full text Add to dashboard Cite

Acquired drug resistance is a major limitation for the successful treatment of cancer. Resistance can emerge due to a variety of reasons including host environmental factors as well as genetic or epigenetic alterations in the cancer cells. Evolutionary theory has contributed to the understanding of the dynamics of resistance mutations in a cancer cell population, the risk of resistance pre-existing before the initiation of therapy, the composition of drug cocktails necessary to prevent the emergence of resistance, and optimum drug administration schedules for patient populations at risk of evolving acquired resistance. Here we review recent advances towards elucidating the evolutionary dynamics of acquired drug resistance and outline how evolutionary thinking can contribute to outstanding questions in the field.

show abstract

“…Theoretical advances includude the analysis of the probability distributions [8,9], asymptotic properties [10], numerical methods for fluctuation analysis [11], and the accuracy of estimates for the mutation rates [12]. Extensions of the theory include different cell cycle distributions and growth laws of wild type cells [13,14]. For a review and more recent advances see [15,16].…”

Section: Introductionmentioning

confidence: 99%

Cellular replication limits in the Luria–Delbrück mutation model

Rodriguez-Brenes

Wodarz

Komarova

2016

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

In this appendix reference to equations that appear in the main text are labeled with arabic numerals, e.g. (1). Equations that appear in this appendix are labelled with the letter "A" followed by an arabic numeral, e.g. (A1). Cellular replication limitsProposition S1. "If the x j (t) are defined by system [1], x k (0) = 1, and x j (0) = 0 for j < k, then ."Proof: From [1] we haveẋ j = 2q x j+1 − x j . Hence, using the initial conditions and integration by parts we find: x j (t) = e −t t 0 2q x j+1 (s) e s ds for 0 < j < k, and x 0 (t) = e (q−1)t t 0 2q x 1 (s)e (1−q)s ds. Given that x k (t) = e −t we can prove by induction that x j (t) = e −t (2qt) k−j (k−j)! for j > 0 and x 0 (t) = 2 k e −t ∞ n=k (qt) n n! . The total cell population X tot (t; q) = k ρ=0 x ρ (t) is then given by:Luria-Delbrück formulation where ψ(z; t, s) = e ize γ(t−s) . Let f (s, z) and g(s) be two functions with continuous second order partial derivates, then:If we make f (s, z) = e ize γ(t−s) −1, we have f (s, 0) = 0 and ∂f (s,0) ∂z = ie γ(t−s) . If we also make g(s) = νX(s), then from the integral expression for Ψ and [A2] we find:Proposition S3. "In the Luria-Delbruck formulation, when the wild type population is modeled with Eq. 3, we have:Proof: Substituting Eq. 3 in the article for X(s) in [A3] we have:For the special case where the growth rate of both populations are equal (γ = 2q − 1), integrating [A6] and simplifying we find [A4]. Expressions of the form e ax Γ(k, bx)dx will show often in our calculations. If a = b, a closed expression for this antiderivative can be found using integration by parts and the fact thatWe can use [A7] to find E[Y (t)] wl when γ = 2q − 1. In this case E(Y (t)) wl is given by Eq. A5.Proposition 4. "In the Luria-Delbruck formulation, when the wild type population is modeled with Eq. 3, we have:(2γ = 2q − 1)Proof: From [5] the characteristic function of the number of mutants Ψ(z; t) = expwhere ψ(z; t, s) = e ize γ(t−s) . Let f (s, z) and g(s) be two functions with continuous second order partial derivates, then:If we make again g(s) = νX(s) and f (s, z) = e ize γ(t−s) − 1, then= −e 2γ(t−s) , and substituting this expression into (A9) we find: Proof: I. From the introduction in the article of the Lea-Coulson formulation, the probability generation function of the number of mutants Y (t) equals:where φ(z; t, s) is given by [19]. ) + E[Y (t)] − E[Y (t)]2 . Again making g(s) = νX(s) and f (s, z) = φ(z; t, s) − 1 and using equations [A2] and [A9], we find:Substituting γ by α−β in Eqs. A3 and A10. It follows that the integrand in [A15] equals E[Y (t)]−V[Y (t)](LD), a n (t, s)z n is the power series representation of φ(z; t, s)−1 centered around z = 0, and X(s) is given by [3], then t 0 νX(s) a n (t, s)z n ds = t 0 νX(s)a n (t, s)z n ds."Proof: The interchange is licit when the growth of the wild type population is exponential, i.e. when X(t) = e δt for any δ > 0 (reference [?]). Hence, since X(t;q) ≤ e (2q−1)t , it follows directly from the dominated convergence theorem that the interchange is also licit when t...

show abstract

On the Probability of Random Genetic Mutations for Various Types of Tumor Growth

Cited by 11 publications

References 33 publications

Universal Asymptotic Clone Size Distribution for General Population Growth

Universal Asymptotic Clone Size Distribution for General Population Growth

Evolution of acquired resistance to anti-cancer therapy

Cellular replication limits in the Luria–Delbrück mutation model

Contact Info

Product

Resources

About