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[EQUATION] and let [MATH] . Then how [MATH] generates [MATH] individuals can be described by the finite-dimensional p.d.f.s of [MATH] |
[EQUATION] for every [MATH] . Overall, ( 34 ) describes the mapping [MATH] To better understand the construction, it is important to realize that for [MATH] |
both the input and the output are i.i.d. In other words, [MATH] generates i.i.d. population dynamics to simulate the real population dynamics produced by [MATH] , only that the transition equation in [MATH] is derived by mimicking how [MATH] generates each new individual on i.i.d. inputs and taking the population size ... |
In fact, under the i.i.d. assumption, deriving the transition equation for most operators is the easy part. The more difficult part is to prove the stacking property of [MATH] on [MATH] . To give an example, consider the transition equation ( ) constructed by Qi et al. in |
, which models the joint effects of proportionate selection and mutation. As our analysis in Section II shows, it does not hold under the assumption of exchangeability. However, if the modeling assumption is i.i.d., the transition equation can be immediately proved (see our analysis in Section II ). This also applies t... |
), where the transition equation is in fact constructed under the i.i.d. assumption. Therefore, in the following analyses, we do not refer to the explicit form of the transition equation, unless it is needed. We only assume that the transition equation is successfully constructed, and it has the form ( 33 ) which is de... |
The construction of the IPM also relates partly to condition in Theorem . Comparing with this condition, it can be seen that for a successfully constructed [MATH] , the following two facts are proved in the construction. |
1. [MATH] as [MATH] 2. [MATH] Of course, these two facts are not sufficient for this condition to hold. One still needs to prove [MATH] as [MATH] . In other words, one has to consider convergence of finite dimensional distributions. |
Finally, we sometimes use [MATH] for [MATH] if [MATH] is clear in the context. For example ( IV-B ) can be rewritten as [EQUATION] |
IV-C Analysis of the Mutation Operator Having derived sufficient conditions for the stacking property and constructed the IPM, we prove the convergence of the IPM of the mutation operator first. Mutation adds an i.i.d. random vector to each individual in the population. If the current population is [MATH] , then the po... |
Next, consider [MATH] . Recall that as an IPM, [MATH] simulates real population dynamics by taking i.i.d. inputs and producing i.i.d. outputs. If the marginal p.d.f.s of [MATH] and [MATH] are [MATH] and [MATH] , respectively, then [MATH] generates i.i.d. individuals whose p.d.f.s are [MATH] , where [MATH] stands for co... |
Theorem 10 (Mutation) Let [MATH] be the mutation operator, and [MATH] be the corresponding operator in the IPM constructed under the i.i.d. assumption, then [MATH] has the stacking property on [MATH] |
Proof. We use the notations and premises in Theorem . Refer to Fig. . In particular, the sequence [MATH] and the limit [MATH] are given and [MATH] as [MATH] |
Apparently, [EQUATION] Therefore, condition in Theorem is satisfied. Noting that condition in Theorem is equivalent to [MATH] , we prove this condition by proving that [MATH] for all [MATH] . Then by Theorem , condition in Theorem holds. Then, as both conditions in Theorem are satisfied, this theorem is proved. |
Now, we prove [MATH] for all [MATH] . First, note that [MATH] for all [MATH] [MATH] are i.i.d. and independent from [MATH] . In addition, for every [MATH] [MATH] |
Since [MATH] , it is apparent that [MATH] . Then by Theorem , we have [MATH] and [MATH] Consider the product space [MATH] . It is both separable and complete. Since [MATH] and [MATH] are independent, by Theorem , it follows that |
[EQUATION] Note that [EQUATION] where is the identity matrix of appropriate dimension and [MATH] is a function satisfying [MATH] . Apparently [MATH] is continuous. Then by ( 35 ), ( 36 ) and Theorem [MATH] for any [MATH] |
In the proof, we concatenate the input ( [MATH] ) and the randomness ( [MATH] ) of the mutation operator in a common product space, and represent [MATH] as a continuous function in that space. This technique is also used when analyzing other operators. |
IV-D Analysis of [MATH] -ary Recombination Consider the [MATH] -ary recombination operator and denote it by [MATH] . In [MATH] , the operator is denoted by [MATH] [MATH] works as follows. To generate a new individual, it first samples [MATH] individuals from the current [MATH] -sized population randomly with replacemen... |
[EQUATION] After the [MATH] parents are selected, [MATH] produces a new individual [MATH] following the formula [EQUATION] where [MATH] are random elements of [MATH] (recall that [MATH] and [MATH] are random elements of [MATH] modeling individuals in our framework). [MATH] are also independent of [MATH] , and the joint... |
Our formulation seems strange at first sight, but it covers many real world recombination operators. For example, consider [MATH] and [MATH] . This operator is the crossover operator taking the mean of its two parents. On the other hand, if [MATH] and the distributions of [MATH] and [MATH] satisfy |
[EQUATION] where [MATH] constructs a diagonal matrix from its inputs, [MATH] are i.i.d. random variables taking values in [MATH] satisfying [MATH] , then this operator is the uniform crossover operator which sets value at each position from the two parents with probability [MATH] |
Consider the IPM [MATH] . As stated in Section IV-B , we do not give the explicit form of the transition equation in [MATH] . We assume that the IPM is successfully constructed, and the transition equation is derived by taking [MATH] in ( 32 ). The reason for this approach is not only because deriving the transition eq... |
The following theorem is the primary result of our analysis for the [MATH] -ary recombination operator. Theorem 11 [MATH] -ary recombination) |
Let [MATH] be the [MATH] -ary recombination operator, and [MATH] be the corresponding operator in the IPM constructed under the i.i.d. assumption, then [MATH] has the stacking property on [MATH] |
We prove that [EQUATION] as [MATH] for any [MATH] . Then by Theorem , the conclusion follows. The overall idea to prove ( 39 ) is that we first prove the convergence in distribution for the [MATH] selected parents, then because the recombination operator is continuous, ( 39 ) follows. |
First, we decompose the operator [MATH] [MATH] generates the [MATH] c.i.i.d. outputs one by one. This generation process can also be viewed as first selecting the [MATH] groups of [MATH] parents at once from the first [MATH] elements of the input (in total the intermediate output is [MATH] parents not necessarily disti... |
Consider [MATH] . Let [MATH] and [MATH] . Let [MATH] be described by the probability [EQUATION] In essence, [MATH] describes how to select the [MATH] parents from [MATH] |
Consider [MATH] . Let [EQUATION] Let [MATH] . Let [MATH] be described by [EQUATION] in which [MATH] , where [MATH] are decided by the recombination operator [MATH] as in ( 38 ), and [MATH] are independent for different [MATH] . In essence, [MATH] describes how to generate the [MATH] individuals from the [MATH] parents. |
Now it is obvious that [MATH] . Therefore, [EQUATION] for all [MATH] and [MATH] Next, consider [MATH] . Let [MATH] , we prove that |
[EQUATION] 43 ) is almost obvious because both operators generate i.i.d. outputs, and both marginal p.d.f.s of the outputs follow the same distribution decided by [MATH] on [MATH] i.i.d. parents from [MATH] In other words, [MATH] is a model of [MATH] on i.i.d. inputs. The outputs they generate on the same i.i.d. input ... |
Since [MATH] , by ( 43 ), [EQUATION] Then ( 39 ) is equivalent to [EQUATION] as [MATH] for any [MATH] To prove ( 45 ), we prove the following two conditions. |
1. [MATH] , such that for all [MATH] [MATH] uniformly as [MATH] , i.e. [MATH] as [MATH] 2. [MATH] as [MATH] and [MATH] is i.i.d. |
These two conditions correspond to the conditions in Theorem . Since [MATH] is from [MATH] to [MATH] , we cannot directly apply Theorem . However, it is easy to extend the proof of Theorem to prove that these two conditions lead to [MATH] as [MATH] . Then, by ( 41 ) it is apparent that [MATH] is a continuous function o... |
In the remainder of the proof, we prove conditions and . These conditions can be understood by replacing the top line with [MATH] in Fig. |
Proof of Condition Since [MATH] (recall that [MATH] can be viewed both as a mapping from [MATH] to [MATH] and from [MATH] to [MATH] ), [MATH] is continuous (see Example 1.2 in |
). Since [MATH] , by Theorem [MATH] . Apparently, [MATH] is i.i.d. Therefore condition is proved. It is worth noting that this simple proof comes partly from our extension of [MATH] to inputs [MATH] . In fact, the only requirement for [MATH] is ( 43 ), i.e. [MATH] should model [MATH] on i.i.d. inputs. By defining [MATH... |
Proof of Condition To prove condition , we first give another representation of [MATH] , where [MATH] and [MATH] . This representation is based on the following mutually exclusive cases. |
1. The [MATH] parents chosen from [MATH] by [MATH] are distinct. 2. There are duplicates in the [MATH] parents which are chosen from [MATH] by [MATH] |
Let [MATH] be random variables taking values in [MATH] , with probability [EQUATION] Let [MATH] follow the conditional distribution of the [MATH] parents when [MATH] , and [MATH] follow the conditional distribution of the [MATH] parents when [MATH] , then [MATH] can be further represented as |
[EQUATION] For our purpose, it is not necessary to explicitly describe the distribution of [MATH] and [MATH] . The only useful fact is that by exchangeability of [MATH] |
[EQUATION] To put it another way, [MATH] and [MATH] both follow the same distribution of [MATH] distinct individuals from the current exchangeable population [MATH] . Also note that [MATH] are i.i.d. random variables. They are independent of [MATH] and [MATH] |
Now consider [MATH] for any [MATH] . By conditioning on whether the [MATH] parents are distinct, we have [EQUATION] Then by ( 48 ), |
[EQUATION] Since [MATH] [MATH] and [MATH] are all less than or equal to [MATH] [EQUATION] i.e. [MATH] for all [MATH] . Taking supremum over all [MATH] , we have |
[EQUATION] The left hand side of ( 50 ) is the total variation distance between [MATH] and [MATH] . It is an upper bound of the Prokhorov distance (see |
for its definition and properties). Since the bound [MATH] is uniform with respect to [MATH] and [MATH] as [MATH] , we have [EQUATION] |
This is exactly condition . Therefore this theorem is proved. Or, if we do not want to use the total variance distance, we have the following result for any [MATH] -continuity set [MATH] |
[EQUATION] Since we already proved [MATH] , by ) in Theorem [MATH] . Then apparently ( 52 ) converges to [MATH] . Noting that [MATH] is arbitrary, by applying ) in Theorem again, [MATH] is proved. |
We give a brief discussion of the proof. In our opinion, the most critical step of our proof is decomposing the [MATH] -ary recombination operator to two sub-operators, one is responsible for selecting parents ( [MATH] ), the other is responsible for combining them ( [MATH] ). In addition, for parent selection, the sub... |
Another point worth mentioning is the choice of Theorem in our proof. Though Theorem and Theorem are symmetric, the difficulties of proving them are quite different. In fact, it is very difficult to prove the uniform convergence condition in Theorem |
Finally, our proof can be easily extended to cover [MATH] -ary recombination operators using uniform sampling without replacement to select parents for each offspring. The overall proof framework roughly stays the same. |
IV-E Summary In this section, we analyzed the simple EA within the proposed framework. As the analysis shows, although the convergence of IPM is rigorously defined, actually proving the convergence for operators usually takes a lot of effort. We derived sufficient conditions under which the convergence of IPM is guaran... |
To appreciate the significance of our work, it is worth noting that in the convergence of the IPMs of the mutation operator, the uniform crossover operator and the proportionate selection operator was not properly proved, and the issue of stacking of operators and iterating the algorithm was not addressed at all. In th... |
as special cases, the convergence of the IPMs of mutation and uniform crossover are actually proved in this paper. Besides, our proof does not depend on the explicit form of the transition equation of the IPM. As long as the IPM is constructed under the i.i.d. assumption, our proof is valid. |
As a consequence of our result, consider the explicit form of the transition equation for the uniform crossover operator derived in Section II in |
. As the authors’ proof was problematic and incomplete, the derivation of the transition equation was not well founded. However, it can be seen that the authors’ derivation is in fact equivalent to constructing the IPM under the i.i.d. assumption. Since we have already proved the convergence of IPM of the [MATH] -ary c... |
regarding the explicit form of the transition equation can be retained. Conclusion and Future Research In this paper, we revisited the existing literature on the theoretical foundations of IPMs, and proposed an analytical framework for IPMs based on convergence in distribution for random elements taking values in the m... |
Perhaps the most immediate topic is to analyze the proportionate selection operator in our framework. The reason that the mutation operator and the [MATH] -ary recombination operator can be readily analyzed is partly because they do not use the information of the fitness value. Also to generate a new individual, these ... |
Theorem 12 (Analysis under the Prokhorov metric) Let [MATH] be the combined operator of mutation and proportionate selection in the simple EA, and [MATH] be the IPM constructed under the i.i.d. assumption with the transition equation ( ). Assume the objective function [MATH] and the conditional p.d.f. for mutation [MAT... |
1. [MATH] as [MATH] 2. [MATH] as [MATH] Comparing with Theorem , it can be seen that condition in Theorem is proved. The only difference is that condition in the theorem requiring the uniform convergence of [MATH] as [MATH] has not been proved yet. |
Let [MATH] stand for total variation convergence. Our analysis of proportionate selection under the total variation distance yields the following results. |
Theorem 13 For c.i.i.d. operators, if [MATH] , then [MATH] uniformly with respect to [MATH] , i.e. [MATH] as [MATH] Theorem 14 For the proportionate selection operator, [MATH] as [MATH] for all [MATH] |
Theorem 15 [MATH] if and only [MATH] uniformly with respect to [MATH] , i.e. [MATH] as [MATH] Comparing with Theorem , Theorem 13 proves condition requiring the column-wise uniform convergence in Fig. . Theorem 14 proves convergence of finite-dimensional distributions of the last row in Fig. . However, Theorem 15 state... |
In summary, our results show that proving the convergence of [MATH] is more difficult under the total variation metric than under the Prokhorov metric, while in proving the uniform convergence of [MATH] , it is the other way around. |
We think further analysis on proportionate selection can be conducted in the following two directions. 1. In the analyses we tried to prove the stacking property on [MATH] for the IPM of proportionate selection. Apart from more efforts trying to prove/disprove this property, it is worth considering modifying the space ... |
2. Another strategy is to bypass the sufficient conditions and return to Definition to prove [MATH] for every [MATH] . This is the original method. In essence, it requires studying the convergence of nesting integrals. |
Apart from proportionate selection, it is also worth studying whether other operators, such as ranking selection, can be analyzed in our framework. As many of these operators do not generate c.i.i.d. offspring, it makes deriving the IPM and proving its convergence difficult, if not impossible. In this regard, we believ... |
Finally, it is possible to extend the concept of “incidence vectors” proposed by Vose to the continuous search space. After all, as noted by Vose himself, incidence vectors can also be viewed as marginal p.d.f.s of individuals. As a consequence, the cases of EAs on discrete and continuous solution spaces indeed do bear... |
# Source: arxiv 1511.05046 # Title: Mathematical Analysis of a Clonal Evolution Model of Tumour Cell Proliferation # Sections: all # Downloaded: 2026-03-03T01:59:32.577576+00:00 |
Mathematical Analysis of a Clonal Evolution Model of Tumour Cell Proliferation Abstract. We investigate a partial differential equation model of a cancer cell population, which is structured with respect to age and telomere length of cells. We assume a continuous telomere length structure, which is applicable to the cl... |
Key words and phrases: Cancer modelling, structured populations, semigroups of operators, asymptotic behaviour, telomere self-renewal |
1991 Mathematics Subject Classification: 35Q92, 35B35, 92C37 This work was completed with the support of the Royal Society. 1. Introduction |
Mathematical models of tumour growth provide insight into the dynamic characteristics of tumour cell populations. Important issues are cell proliferation, cell heterogeneity, and cell differentiation. These issues are currently examined in two hypothesized models of tumour evolution: the cancer stem cell (CSC) model an... |
. Both models have scientific support, as well as therapeutic implications, and in fact, both models may be involved in the development of a tumour. The essential distinction of the two models is the role of self-renewal in specific cells, and the fraction of the total cell populations that these cells comprise. Here s... |
The CSC model hypothesizes that a very small sub-population of tumour stem cells generate the entire tumour cell population . In mathematical treatments, these stem cells have the ability to self-renew indefinitely, and through sequential mutations generate all the heterogeneous and differentiated cell types comprising... |
. This stem cell population lies at the apex of a hierarchical structure of cell types, and tumour evolution is dependent on their unconstrained self-renewing ability |
The CE model hypothesizes that a tumour population is composed of multiple genetically identical clones, which have the possibility of mutation, selection, and expansion |
. In the CE model all undifferentiated cells have a self-renewing capacity for contributing to the tumour evolution. These cells, however, are not at the apex of a hierarchal tree, but rather dispersed widely throughout the tumour cell population as a large fraction of the total tumour cell count. |
A central element of cell self-renewal is the Hayflick limit, which constrains differentiated cell lines to a finite number of divisions |
. As differentiated cells divide, telomeres (nucleotide sequences at the ends of chromosomes) shorten until a critical limit is reached, and further divisions are prohibited |
. The existence of a mechanism which reverses telomere shortening was predicted several decades ago. The CSC model hypothesizes that cancer stem cells circumvent telomere shortening by using the enzyme telomerase to replace their telomeres, and thus obtain the ability to divide indefinitely |
. Recently, it has been shown that around 90% of all types of human cancer exhibit a form of telomerase activation . In the CSC model this property resides in an extremely small sub-population, from which all the differentiated tumour cells derive. In contrast, the CE model hypothesizes that a large number of undiffere... |
In order to model the dynamics of self-renewing cells lineages, which correspond physiologically to chromosomal telomere lengths, is it necessary to track all cells through successive generations. Many mathematical treatments of telomere structure in cell population dynamics have been developed |
. The CSC model has been treated for example in , where telomere shortening is investigated in continuum differential equation models. The key focus of these treatments is that mother stem cells produce two daughter cells, one of which is a stem cell, and the other, a differentiated cell, with a limited number of futur... |
The objective of this paper is to develop a mathematical analysis for the alternative CE model and to quantify its dynamic properties. Our model here incorporates continuum, rather than discrete, telomere lengths in cells. This idealised continuum telomere length is assumed for convenience, to avoid unwieldy compartmen... |
. The distribution of daughter cell telomere lengths is governed by a mathematically formulated rule that assigns the telomere restoration property to cells which may be viewed as those capable of indefinite divisions in each of the diverse clonal sub-populations. |
Our CE model belongs to the class of continuum structured population models, with age and time as dynamic variables, and telomere length as a population structure variable. In the past three decades physiologically structured population models have been increasingly utilised to shed light on some important phenomena of... |
. The power of structuring a population with respect to physiological variables is of great value in understanding the evolution of biological populations. There is an increasing literature of physiologically structured population models with more than one (physiological) structuring variable. The development of a unif... |
We first consider the following linear model for an age and telomere length structured proliferating cancer cell population. [EQUATION] |
Above [MATH] stands for the density of cells of age [MATH] , having telomere length [MATH] at time [MATH] . We assume a maximum cell age denoted by [MATH] and a maximum telomere length denoted by [MATH] . The population count at time [MATH] of cells with age between [MATH] and [MATH] and telomere length between [MATH] ... |
[EQUATION] and the total population of all cells at time [MATH] is [EQUATION] [MATH] quantifies the natural mortality of cells of age [MATH] and telomere length [MATH] . A mother cell of age [MATH] and telomere length [MATH] divides into two daughter cells of age [MATH] having (possibly) different telomere lengths, at ... |
[EQUATION] From a probabilistic interpretation of [MATH] it would be natural to normalise the maximal telomere length [MATH] to [MATH] . Throughout we retain [MATH] as a general parameter. We will impose further regularity assumptions on the model ingredients later on. Note that our model ( 1.1 )-( 1.3 ) can be conside... |
2. Existence of the governing linear semigroup Our starting model ( 1.1 )-( 1.3 ) is a linear one, moreover the telomere length [MATH] only plays an important role in the somewhat unusual boundary condition ( 1.2 ). Hence to establish the existence of the governing linear semigroup (and therefore the existence of mild ... |
; see also and for similar general results. It is very natural to apply the boundary perturbation result of Greiner, since the unperturbed generator (arising from equation ( 1.1 ) with zero flux boundary condition) is readily shown to generate a translation semigroup. Moreover, it has the added advantage that the spect... |
. For basic definitions and results not introduced in the section we refer the reader to To apply the perturbation result of Greiner we set the framework as follows. Assume that [MATH] and [MATH] , and that all of the model parameters are non-negative. In particular, it is natural to assume that [MATH] does not vanish ... |
For the linear problem ( 1.1 )-( 1.3 ), since we are dealing with density functions, the natural choice of state space is the following Lebesgue space. |
[EQUATION] Elements of [MATH] above can be understood as equivalence classes of measurable functions [MATH] on the square [MATH] such that [MATH] We further set [MATH] . We define the operators [MATH] and [MATH] as follows |
[EQUATION] with [EQUATION] We further introduce the norm [EQUATION] With the [MATH] norm [MATH] is complete, and the maximal operator [MATH] is continuous and linear. Furthermore, we define |
[EQUATION] Then [MATH] is also continuous and linear, and we have Im [MATH] . We denote by [MATH] the restriction of [MATH] to Ker [MATH] . It is then clear that [MATH] generates the strongly continuous and nilpotent shift semigroup [MATH] , explicitly given as |
[EQUATION] In particular note that for [MATH] we have [MATH] for any initial condition [MATH] We now define the bounded linear perturbing operator [MATH] as follows |
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