propagation of acoustic waves in a heterogeneous medium plays an important role in many applications , for instance in seismic imaging in geophysics and in the exploration of hydrocarbons .this wave propagation is modeled by the linear wave equation :  [ eq : waveeq ]   where \times d ] with ;l^2(d)) ] , for some finite time , a finite interval with length , periodic boundary conditions and a given initial data .alternatively , we could consider and compactly supported initial data .we consider coefficients which are positive and bounded away from zero , that is as well as hlder continuous , , for some exponent . as we will see in the following ,it is more convenient to work with the variable instead of and the equation it satisfies ,  [ eq : forw ]   we assume that the initial data is hlder continuous for some :  we will see that some solution properties can also be obtained under the slightly weaker assumption in the following , we investigate to what extent the exponents , influence the regularity of the solution at a time and the convergence rate of the finite difference scheme as the mesh is refined .since the coefficient is not differentiable , it is possible that the solution is not differentiable in the classical sense either and only weak solutions to equation can be defined . by a weak solution to , we mean a function for some satisfying in the distributional sense , that is , for all smooth test functions , to deal with the possible non - differentiability of the solution , we will in a first step regularize the coefficient by convolving it with a smooth test function , given as where is an even function with the properties then we consider the solution of the equation  [ eq : forwdelta ]   the coefficient is smooth and therefore in particular lipschitz continuous and hence we can define classical solutions to in a standard way , using the method of characteristics : we let solve the ordinary differential equation  is constant along the charachteristics since thus the solution at time is given by therefore , if the initial data is in , the solution will be essentially bounded at any later time. using this , we can derive hlder continuity of the solution in time and space :  [ lem : hoeldercont ] assume that the coefficient in is bounded , i.e. satisfying , and that is hlder continuous with exponent , as in .then will be hlder continuous in space and time with exponent for any time , independently of .in particular , we have   we consider the characteristics equation once more and note that it is independent of the initial data of .hence any initial data will be propagated along the same characteristics , and we have for the difference corresponding to initial condition  with defined in . in particular , taking for some ( the case is analoguous ) , and omitting the index this implies the characteristics equation , , implies for any , for some . by the assumption on the boundedness of , , we can bound from above and below : and thus for any  } |{w^\delta}_0(x)-{w^\delta}_0(y))|\\ & \leq |{w^\delta}_0|_{c^{0,\gamma_\infty } }  { \overline{a}}^{\gamma_\infty}h^{\gamma_\infty}\\ & \leq |w_0|_{c^{0,\gamma_\infty } }  { \overline{a}}^{\gamma_\infty}h^{\gamma_\infty}\end{aligned}\ ] ] by the assumption on the initial data .taking the supremum over all , we obtain and thus the hlder continuity in time .to prove the hlder continuity in space , we note that , by the characteristics equation and the positivity of the coefficient , we have for some such that ( the characteristics starting at and passing through ) .this allows us to bound from below : and therefore this estimate is independent of and .hence which yields the hlder continuity in space .[ rem : moclp ] we note that the estimates in lemma [ lem : hoeldercont ] are independent of and the hlder coefficient of . moreover , if is bounded , or the initial data has compact support , this lemma implies that has a modulus of continuity in time , and a modulus of continuity of the same order in space : with for all .  subtracting a constant from equation and then multiplying by a regularized version of the sign function, we obtain the -norm conservation property of the equation : similarly , by multiplying with for even , we obtain conservation of -norms :   the moduli of continuity of lemma [ lem : hoeldercont ] and remark [ rem : moclp ] are independent of , which implies that the sequence of solutions is relatively compact in , ( by kolmogorov s compactness theorem ( * ? ? ?* theorem a.5 ) ) , and due to the compact embedding of the hlder spaces for and bounded domains , also relatively compact in , for any , and thus the limit function , is a weak solution of with the same moduli of continuity in space and time .moreover , the limit satisfies the entropy identities and .   in order to compute numerical approximations to ,we choose such that and discretize the spatial domain by a grid with gridpoints , and cell centers , .furthermore , we let and set , , where is such that . we define the averaged quantities and moreover , we denote , for a quantity , , defined on the grid , then we define approximations by letting , this is equivalent to which will yield an approximation to the solution of equation .[ lem : pdisc ] the approximations , , defined by the numerical scheme have the following properties :  1 .maximum principle : 2 .discrete entropy inequality in : discrete entropy inequality in : 3 .  bound on the discrete and -norms :   for all .writing as and taking the cfl - condition into account , we immediately see that is a convex combination of and and thus the approximations satisfy the maximum principle .  to obtain the discrete version of a krukov entropy inequality for the quantities ,we compute for a constant , moreover , we note that by and thanks to the cfl - condition , thus and similarly combining this with , we obtain .now we simply need to multiply the expression by , sum it over and set to obtain the bound on the discrete -norm of , . to prove the -entropy inequality, we note that the difference scheme is equivalent to for any constant , and then multiply both sides of equation by .subsequently , we use that once for the left hand side and once for the right hand side to get ( we write for convenience ) rearranging terms and using for the difference , this reads from which the claim follows using the cfl - condition . summing the discrete -inequality over and using induction over ,we furthermore obtain .thanks to the linearity of the transport equation , we obtain the following corollary of lemma [ lem : pdisc ] :  [ cor : contraction ] let denote the approximations computed by the scheme , for initial data and another approximation computed by for initial data .then we have for all .this follows from the fact that the differences satisfy due to the linearity of the scheme , together with lemma [ lem : pdisc ] .defining the piecewise constant approximations this corollary enables us to show that the piecewise constant function has a modulus of continuity in time :  [ lem : mocdisc ] the piecewise constant functions defined in have a modulus of continuity in time if the initial data satisfies for : or for : if the initial data is hlder continuous with exponent , the solution is approximately hlder continuous in time with the same exponent , i.e.   we observe that where we have denoted and is inductively defined as follows :  [ def : lambda ] we let and for all .then we define recursively by   an explicit expression for the coefficients for the third case is given by   [ cl : bla ] the coefficients satisfy and .equation follows by induction and using that for any .that the are nonnegative follows from the cfl - condition .thus we have , where we used claim [ cl : bla ] for the first inequality and the third equality , and the assumption on the initial data , , in the last inequality .similarly , we compute in the -setting , then applying corollary [ cor : contraction ] , we conclude .the approximate hlder continuity in time follows in a very similar way , instead of summing over , we take the maximum over all .  if has bounded variation , obtaining that the solution has bounded variation and so -moduli of continuity in time and space is much easier .similarly , if the initial data has a modulus of continuity of in , obtaining a rate is easier .the argument is similar to the one which will be outlined for the linear wave equation in section [ sec:1d ] but less technical .   in this section, we will prove a rate of convergence in of the numerical scheme to the limit of solutions of as .our approach is based on the doubling of variables technique developed by s. n. krukov  for nonlinear scalar conservation laws .the reason why we take this approach is the possible nondifferentiability of the solutions with which the doubling of variables technique can deal well . for this purpose ,we use the test function from , let and such that and and define the function by we note that the ( smooth ) solution to satisfies whereas the approximations satisfy , by the discrete entropy inequality where we have denoted   [ lem : ratetransport ] let satisfy . denote the solution of and the numerical approximation computed by scheme and defined in .assume that the initial data and is hlder continuous with exponent .then converges to the solution , , at ( at least ) the rate where is a constant depending on , , and but not on .inserting for and for in and , and integrating the respective equations over and respectively , and adding up , we have ( for convenience , we will omit writing the arguments of , , and in the following ) where .we have so that we can rewrite equation as we note that which means that we can rewrite as we begin by estimating the term . to do so , we note that and that because is constant with respect to and is compactly supported in the domain .therefore we can rewrite the term as and bound in the following way , using triangle inequality , } \int_{{d } } |{w^\delta}-{w^\delta}(t , y)|\ , dy \int_{0}^t \big|\partial_{ss}\omega_{{\epsilon_0}}(t - s+\xi)\big|\ , dsdtd\xi\\ & \leq \frac{c{{\delta t}}}{{\underline{a}}\ ,  { \epsilon_0}^2 } \int_{\nu}^{\tau}\sup_{s\in [ t-{\epsilon_0},t+2{\epsilon_0 } ] } \int_{{d } } |{w^\delta}-{w^\delta}(t , y)|\ , dy \ , dt\\ & \leq \frac{c{{\delta t}}t}{{\underline{a}}\,{\epsilon_0}^{2-\gamma_\infty } } \end{split}\end{aligned}\ ] ] where we have used lemma [ lem : hoeldercont ] for the last inequality .we proceed to estimating the term : similarly to the case of term , we use that thus subtracting this from , we can bound term using triangle inequality , }\int_{{d}}|{w^\delta}(s , y)-{w^\delta}(t , y)|\ , dy \int_0^t |\partial_s \omega_{{\epsilon_0}}|ds dt\\ & \leq c \frac{{{\delta x}}^{\alpha}+\delta^{\alpha}+{\epsilon_1}^{\alpha}}{{\underline{a}}^2 } { \epsilon_0}^{\gamma_\infty-1}\\ & \leq c \frac{\delta^\alpha+{\epsilon_1}^\alpha}{{\epsilon_0}^{1-\gamma_\infty } } , \end{split}\end{aligned}\ ] ] since we assumed .we continue to bound term .first , we observe that and that since is compactly supported .thus , we can estimate by it remains to relate the terms forming to the -norm of the differences .we have and can rewrite so that ^ 2}\mathbf{1}_{[\tau,\tau+{{\delta t}})}(t)\omega_{{\epsilon_0}}(t - s-{{\delta t}})\int_{{d}}|{w_{{{\delta x}}}}-{w_{{{\delta x}}}}(\tau,\cdot)|\ , dx\ , { d\underline{z}}\\ & \qquad + \frac{1}{{{\delta t}}{\underline{a}}}\int_{{d_t}^2}\omega_{{\epsilon_1}}\mathbf{1}_{[\tau,\tau+{{\delta t}})}(t)\omega_{{\epsilon_0}}(t - s-{{\delta t}})\left(|{w^\delta}-{w^\delta}(\tau , y)|+|{w^\delta}(\tau , y)-{w^\delta}(\tau , x)|\right)\ , { d\underline{z}}\\ & \quad\leq \frac{1}{{{\delta t}}{\underline{a}}}\int_{\tau}^{\tau+{{\delta t}}}\sup_{|h|\leq 2{\epsilon_0}}\int_{{d}}|{w_{{{\delta x}}}}-{w_{{{\delta x}}}}(t+h,\cdot)|\ , dx\,dt\\ & \qquad + \frac{1}{{{\delta t}}{\underline{a}}}\int_{[0,t]^2}\mathbf{1}_{[\tau,\tau+{{\delta t}})}(t)\omega_{{\epsilon_0}}(t - s-{{\delta t}})\int_{{d}}|{w^\delta}-{w^\delta}(\tau,\cdot)|\ , dx \,dsdt\\ & \qquad + \frac{1}{{{\delta t}}{\underline{a}}}\int_{{d}^2}\omega_{{\epsilon_1}}|{w^\delta}(\tau , y)-{w^\delta}(\tau , x)|\ , dxdy\\ & \quad \leq c { \epsilon_0}^{\gamma_\infty}+c { \epsilon_1}^{\gamma_\infty } \end{split}\end{aligned}\ ] ] in a similar way , defining we obtain thus , combining the estimates , , , , and , we have we let , we choose in this last expression , and to obtain the rate .note that the above lemma implies a rate of convergence in of at least for the variable .the main ideas for proving a rate of convergence in are similar to those in section [ ssec : l1rate ] , an additional tool involved is a type of grnwall inequality and the whole procedure is a bit more technical .we start by noting that the ( smooth ) solution to satisfies whereas the approximations satisfy , by the discrete entropy inequality , then we can prove  [ lem : ratetransportl2 ] let satisfy .denote the solution of obtained as the limit as in and the numerical approximation computed by scheme and defined in .assume that the initial data is hlder continuous with exponent .then converges to the solution , , at ( at least ) the rate where is a constant depending on , , and but not on .  as in the -case, we insert for and for in and , and integrate the respective equations over and respectively . then adding up, we have ( for convenience , we will again omit writing the arguments of , , and in the following ) where . by , we can rewrite equation as and by , this is equivalent to we start with the term . to do so, we use again and that , similarly to , it holds , hence we can rewrite the term as and bound in the following way , using triangle and cauchy - schwarz inequality , \big|\partial_{ss}\omega_{{\epsilon_0}}\big|\,dsdtd\xi\\ & \leq c\frac{2 { { \delta t}}}{{\epsilon_0}^2}\int_{\nu}^\tau\!\ ! \sup_{s\in[t-2{\epsilon_0},t+2{\epsilon_0}]}\left(\int_{{d}^2 } \frac{|{w^\delta}-{w^\delta}(t , y)|^2}{a_{{{\delta x}}}}\omega_{{\epsilon_1}}\ , dxdy\right)^{\frac{1}{2}}\!\ ! \sup_{s\in[t-2{\epsilon_0},t+2{\epsilon_0}]}\left(\int_{{d}^2 }\frac{|{w_{{{\delta x}}}}-{w^\delta}|^2}{a_{{{\delta x}}}}\omega_{{\epsilon_1}}\ , dxdy\right)^{\frac{1}{2}}\!\ !dt\\ & \leq c\frac{{{\delta t}}}{\sqrt{{\underline{a}}}{\epsilon_0}^{2-\gamma_\infty}}\int_{\nu}^\tau  \sup_{s\in[t-2{\epsilon_0},t+2{\epsilon_0}]}\left(\int_{{d}^2 } \frac{|{w_{{{\delta x}}}}-{w^\delta}|^2}{a_{{{\delta x}}}}\omega_{{\epsilon_1}}\ , dxdy\right)^{\frac{1}{2 } }  dt \end{split}\end{aligned}\ ] ] we denote and observe that }\left(\int_{{d}^2 } \frac{|{w_{{{\delta x}}}}(t , x)-{w^\delta}(s , y)|^2}{a_{{{\delta x}}}(x)}\omega_{{\epsilon_1}}\ , dxdy\right)^{1/2}dt\\ & \quad \leq  \int_{\nu}^\tau\biggl\{\sup_{s\in[t-2{\epsilon_0},t+2{\epsilon_0}]}\left(\int_{{d}^2 } \frac{|{w^\delta}(t , y)-{w^\delta}(s , y)|^2}{a_{{{\delta x}}}}\omega_{{\epsilon_1}}\ , dxdy\right)^{1/2}\\ & \quad \hphantom{\leq  \int_{\nu}^\tau\biggl\ { }  + \left(\int_{{d}^2 } \frac{|{w_{{{\delta x}}}}-{w^\delta}(t , y)|^2}{a_{{{\delta x}}}}\omega_{{\epsilon_1}}\ , dxdy\right)^{1/2}\biggr\}dt\\ & \quad\leq c t{\epsilon_0}^{\gamma_\infty } + \int_{\nu}^\tau \left(\int_{{d}^2 } \frac{|{w_{{{\delta x}}}}(t , x)-{w^\delta}(t , y)|^2}{a_{{{\delta x}}}}\omega_{{\epsilon_1}}\ , dxdy\right)^{1/2}dt\\ & \quad\leq c t{\epsilon_0}^{\gamma_\infty } + \int_{\nu}^\tau \biggl\ { \left(\int_0^t\int_{{d}^2 } \frac{|{w_{{{\delta x}}}}(t , x)-{w^\delta}(s-{{\delta t}},y)|^2}{a_{{{\delta x}}}}\omega_{{\epsilon_1}}\omega_{{\epsilon_0}}\ , dxdyds\right)^{1/2}\\ & \quad \hphantom{\leq c t{\epsilon_0}^{\gamma_\infty } + \int_{\nu}^\tau\biggl\{}+ \left(\int_0^t\int_{{d}^2 } \frac{|{w^\delta}(t , y)-{w^\delta}(s-{{\delta t}},y)|^2}{a_{{{\delta x}}}}\omega_{{\epsilon_1}}\omega_{{\epsilon_0}}\ , dxdyds\right)^{1/2}\biggr\}dt\\ & \quad \leq c t{\epsilon_0}^{\gamma_\infty } + \int_{\nu}^\tau \sqrt{\kappa(t)}\,dt . \end{split}\end{aligned}\ ] ] therefore , the term can be bounded as latexmath:[\[\label{eq : al2 }  we proceed to estimating the term : we again use that which admits us to rewrite the term and estimate as follows : }\left(\int_{{d}^2}|{w^\delta}-{w^\delta}(t , y)|^2\omega_{{\epsilon_1}}\,dxdy\right)^{1/2}\\ & \hphantom{\leq c\frac{{{\deltax}}^{\alpha}+\delta^{\alpha}+{\epsilon_1}^{\alpha}}{{\underline{a}}^{3/2}}\int_{\nu}^\tau } \times\sup_{s\in[t-{\epsilon_0},t+{\epsilon_0}]}\left ( \int_{{d}^2 } \frac{|{w_{{{\delta x}}}}-{w^\delta}|^2}{a_{{{\delta x } } } } \omega_{{\epsilon_1}}dxdy\right)^{1/2}\int_0^t\!|\partial_s \omega_{{\epsilon_0}}|\ , dsdt\\ & \leq c \frac{{{\delta x}}^{\alpha}+\delta^{\alpha}+{\epsilon_1}^{\alpha}}{{\epsilon_0}^{1-\gamma_\infty}}\int_{\nu}^{\tau}\sup_{s\in[t-{\epsilon_0},t+{\epsilon_0}]}\left ( \int_{{d}^2 } \frac{|{w_{{{\delta x}}}}-{w^\delta}|^2}{a_{{{\delta x } } } } \omega_{{\epsilon_1}}dxdy\right)^{1/2 } dt\\ & \leq c \frac{{{\delta x}}^{\alpha}+\delta^{\alpha}+{\epsilon_1}^{\alpha}}{{\epsilon_0}^{1-\gamma_\infty}}\left ( { \epsilon_0}^{\gamma_\infty } + \int_{\nu}^{\tau}\sqrt{\kappa(t ) } dt\right ) \end{split}\end{aligned}\ ] ] where we have used for the last inequality .we continue to estimate the term .we have , using hence we can estimate the term by we have and consequently , where we choose and such that , i.e. and for some and notice that hence we can rewrite equation as now we use the following simple adaption of ( * ? ? ?* [ theorem 5 , page 4 ) :  [ lem : discretegronwall ] let , , for some satisfy for all , for some .then   using this lemma with and , we obtain the estimate   next , we relate the -norm of the difference to . indeed , in a similar way, we can show and therefore , with , letting and inserting the definitions of and , this is    now we can choose , and to balance the errors and finally obtain .define . then by , .moreover , subtracting the expression for from the expression for , we have since , we can divide both sides of the above equation by to obtain using induction over , we obtain hence by the definition of . using that , this proves the claim .in this section , we run a few numerical experiments to compare the theoretically established rates to experimentally observed ones . as a model coefficient , we choose a sample ( single realization ) of a log - normally distributed random field , which was generated using a spectral fft method from a given covariance operator which we assume to be log - normal , so that the covariance operator completely determines the law of .it is easy to check that this coefficient is uniformly positive , bounded from above and hlder continuous with exponent .see figure [ fig : coef1d ] for an illustration of the coefficient .used for the numerical experiments for the transport equation .,scaledwidth=80.0% ]  for the function , we choose the product of the lipschitz continuous hat function with a modification of the weierstrass function for different parameters : that is , we add a constant such that becomes strictly positive , and truncate after terms , and define .it can be shown that is nowhere differentiable but hlder continuous with exponent . as a computational domain , we take  ] .the current paper is therefore an attempt to design robust numerical approximations for the one - dimensional transport and the wave equation with rough , i.e. , only hlder continuous coefficients . for low enough hlder exponent ,this regularity requirement is less than the one in  , and also our assumptions on the regularity of the solution are weaker .however , our results ( so far ) restrict to the one - dimensional case.  we propose upwind finite difference approximations and show that these approximations converge as the mesh is refined .furthermore , we establish rigorous convergence rates of these approximations .the obtained rates explicitly depend on the hlder exponent of the material coefficient as well as the modulus of continuity in or of the initial data .the rates of convergence are obtained by a novel adaptation of the krukov doubling of variables technique from scalar conservation laws to our linear system setting . in particular , we prove that for coefficients which are hlder continuous with exponent and initial data that is hlder continuous with exponent , the solution of the transport equation and its approximation have the same hlder continuity and the approximations converge with rate at least in and ( c.f . theorem [ lem : ratetransport ] and [ lem : ratetransportl2 ] ) . for the wave equation, we could show that if the initial data has a modulus of continuity of in , then the solution will inherit it . in this case , the finite difference approximations converge at rate at least .the numerical experiments demonstrate the near sharpness of this rate .we also show rates of convergence under the assumption that the numerical approximations have lower moduli of continuity , however , in this case , we can not prove that the numerical approximations actually inherit those .we conclude with a brief discussion on possible limitations and future extensions of our methods :  * we consider finite difference discretizations in the current paper .the formal order of accuracy of our three - point finite difference schemes is one .one can argue that analogous to linear hyperbolic systems with smooth coefficients , one can obtain higher rates of convergence by designing schemes with a larger stencil ( a higher formal order of accuracy ) .we find that prospect unlikely to hold in practice on account of the lack of smoothness of the coefficient .furthermore , the irregularities of the coefficient are not localized .hence , one can not expect any localization of singularities in the solution and its derivatives .this is in marked contrast to nonlinear systems of conservation laws where discontinuities such as shocks and contact discontinuities separate smooth parts of the flow .thus , high - resolution finite difference schemes perform better than low order schemes for conservation laws .such a situation does not hold for wave propagation in a rough medium .we expect that the low - order schemes presented here are not only simple but also optimal in this case .* we present the analysis only in one space dimension and for uniform grids .the extension of the finite difference scheme to the two and three dimensional wave equation is straightforward , however , it is not easy to show that the solution computed in this way has a modulus of continuity , which is fundamental to obtaining convergence rates using our technique .in fact , we do not know currently if the approximations do have a modulus of continuity . obtaining more insight intothe regularity of the approximations is one of the objectives of our current research efforts .we would furthermore like to extend the method and convergence analysis to unstructured grids . *the numerical experiments for the transport equation ( section [ ssec : exprates ] ) suggest that the rate from theorem [ lem : ratetransport ] may not be sharp .consequently , we plan to experiment more in order to find out if the rate is sharp or not , and otherwise try to improve the estimate .* we restrict ourselves to acoustic wave propagation in rough media in this paper. however , elastic wave propagation also involve media with material properties that lead to rough , hlder continuous coefficients .the extension of these methods to such problems will be considered in a forthcoming paper .another possible direction of research would be to prove a rate of convergence for numerical methods that approximate electromagnetic wave propagation in heterogeneous media .possible extensions to nonlinear wave equations , and discontinuous and time - dependent coefficients will also be considered .this project was supported in part by erc stg .n 306279 , sparccle .the author thanks jonas ukys ( eth ) for providing the sample coefficient and helpful discussions on the topic .many thanks also to nils henrik risebro ( uio ) and siddhartha mishra ( ethz ) for discussions and advice in the course of preparing this manuscript .b. s. jovanovi .convergence of finite - difference schemes for hyperbolic equations with variable coefficients ., 72(6):t493t496 , 1992 .bericht ber die wissenschaftliche jahrestagung der gamm ( krakw , 1991 ) .

Abstract: we prove convergence rates of explicit finite difference schemes for the linear advection and wave equation in one space dimension with hlder continuous coefficient . the obtained convergence rates explicitely depend on the hlder regularity of the coefficient and the modulus of continuity of the initial data . we compare the theoretically established rates with the experimental rates of a couple of numerical examples .


complex systems in diverse fields are often represented as weighted networks .inferring information transfer between network elements from such representations can provide important insights into system structure and perturbations .this paper proposes a general methodology to quantify such transfer based on information theory and probability on graphs .+ previous attempts to infer dynamics from weighted networks include an interaction model based on electrical circuitry , to discover active pathways contributing to the pathogenesis of the brain cancer glioblastoma multiforme .reference uses a model of infection transmission , proportional to interaction frequency , to identify the spread of disease through social networks .such case by case approaches have proved informative , however they are often tailor made to their applications and the _ general _ quantification of information transfer in weighted networks currently lacks a theoretical foundation .+ the purpose of this paper is to construct an information theoretic measure , _ network transfer entropy _ , quantifying the directed amount of information transferred between any two vertices in a weighted network , with minimal assumptions and general applicability . following construction ( section ii ), we demonstrate the measure on simple synthetic networks and a biological signalling network ( section iii ) . + in the construction of general measures ,one aims at an insight into the theoretical concepts governing the process being studied .we demonstrate that the network transfer entropy framework can be interpreted in the context of metric spaces ( section iv ) . in this construction onedefines a family of mappings from the space of weighted networks ( represented by matrices ) to a family of metric spaces , whose elements describe possible signal dynamics on networks .+ we prove a convergence principle , demonstrating that a form of convergence of weighted networks in the metric space implies convergence in the constructed metric space of signal dynamics .this result shows that in our general framework , deformation of network structure influences network dynamics in an intuitive way .this result has real world implications in , for example , network drug design , where one wishes to modify the chemical affinities of interacting proteins in a pathological signalling network ( _ i.e. _ modify the edge weights ) to restore a healthy signalling regime ( _ i.e. _ modify the dynamics ) .finally , we motivate how certain further theoretical problems in network evolution and network perturbation can be approached within this framework .+   in this appendix we provide certain mathematical derivations and technical methodologies to accompany the main text . in appendixb we derive a closed form expression for the probability distribution  ] .+ given a model of interactions , we wish to quantify paths of high or low traffic through the network . to proceed in this aim it suffices to identify the directed amount of information transferred between any two pairs of network vertices during a period of system evolution .to achieve this we again turn to schreiber s methodology . in the derivation of transfer entropy the directed amount of information transferred from a process to a process is formulated as the incorrectness of the assumption that is not conditional on .this quantity can be expressed as the _ kullback - leibler divergence _  where for .thus to quantify the amount of information vertex in our network transfers to vertex over paths of length we must derive a distribution for in which vertex sends no information to vertex .we must then compute the informational distance between this distribution and the above distribution  ] . herewe diverge from , however , as the kullback - leibler divergence \log \frac { p[x^{i}_{n}|\vec{{x}}_{0 } ] } { p[x^{i}_{n}|\vec{{x}}_{0},j]}\ ] ] is only well defined provided =0\ } \subset \ { x : p[x^{i}_{n}=x|\vec{{x}}_{0}]=0\},\ ] ] which is an assumption that does not hold in general .consider , for example , a directed graph on two vertices and , with a single directed edge oriented from to ; if we assign the isd as , then it is trivial that = \delta_{x}^{1} ] .+ thus to quantify the directed amount of information transferred from vertex to we must employ a different measure of statistical distance .there are several possible choices available , among the most promising are the _ jensen - shannon divergence _ , which is a linear combination of kullback - leibler divergences and the statistical distance introduced by wootters .both measures are theoretically rich ; wootter s measure was designed as a distinguishably distance between pure quantum states after a finite number of observations , and applies equally well to distinguishing two probability distributions .the measure also has a geometric interpretation in the context of hilbert space .the jensen shannon divergence of two distributions , quantifies the total kullback - leibler divergence from each distribution to the average of the two , and thus is a measure of distributional similarity .the jensen shannon divergence is also the square of a metric over the space of probability distributions on a measurable set .these two measures have been shown to agree to second order in a quantum mechanical framework .+ we will use the jensen shannon divergence defined by where  ] is inherently asymmetric , and thus quantifies information transfer through a network in a directed sense , permitting the inference of causality .in order to demonstrate the use of nte we consider three examples , two synthetic networks and one application to biological signal transduction . to evaluate nte in these examples , we estimated the probability distributions  ] for all , using a simulation .we also devised a method to estimate the statistical error in the probability distributions ( see appendix c ) .+ the first and most simple example we consider is a directed path of length 5 with equal edge weights ( figure 1 ) .this induces the stochastic matrix     the structure of this network provides a completely predictable path for signal transfer and thus is ideal for proving the capability of our measure to detect information transfer .we consider two isds on this network , firstly , where the first vertex in the path is given an initial signal and all other vertices have no signal to transfer .if we number the vertices 1 to 5 from the start of the path to the end , then it is clear that for this isd , over path length , vertex 1 sends information to vertex 2 , and no other vertices communicate , for vertex 1 sends information to vertex 3 and vertex 2 sends information to vertex 3 and no other vertices communicate , and similarly we can compute all pairwise information transfer events up to beyond which all signal is absorbed at vertex 5 and can not be transmitted through the network .this pattern is precisely what is seen if we calculate the nte between all vertex pairs over different path lengths ( figure 1 ) .+ we next consider the isd , on the same network , in order to demonstrate the ability of the nte measure to discern between situations where networks with identical edge weights have different starting signal states .one would expect that with this isd , for , rather than just vertex 1 forwarding information to vertex 2 , we have vertex forwarding information to vertex for , and similar extensions for longer path lengths .the nte measure can detect these differences due to initial signal distribution ( figure 1 ) .+ the next network we consider is a slight extension to the deterministic path which constitutes a directed feedback from vertex 2 to vertex 4 weighted ( figure 2 ) .this induces the stochastic matrix     the network introduces some indeterminism in that if vertex 4 holds signal , it can either forward it to vertex 5 with probability or feedback the signal to vertex 2 with probability .this essentially sets up a feedback loop which dampens the signal received at node 5 over a given path length , by a factor dependant on .we calculated the nte for all vertex pairs in this altered path for and found that as is increased the nte to vertex 5 from all other vertices falls , as expected ( figure 2 ) .thus in the context of these very simple synthetic networks , the use of nte as a tool for detecting information transfer is clear .+ we next demonstrate nte in a real world biological network .to do this we consider the human primary naive cd4 + t cell intracellular signalling network analysed by sachs _et al_. , consisting of 11 vertices ( figure 3 ) . in this network vertices are proteins which can be phosphorylated and directed edges connect kinases ( capable of phosphorylating proteins ) with their targets .the kinases must be in an active state before they can phosphorylate a target ; activity can be achieved by either phosphorylation by an upstream kinase or activation by a reagent .et al_. generated data accompanying this network consisting of quantification ( by flow cytometry ) of the amount of phosphorylated protein for each vertex in the network following ten independent perturbations .the network perturbations consisted of the administration of reagents which could either activate or inhibit the kinase activity of particular vertices . +    to apply nte we considered two of these perturbations , firstly treatment with anti - cd3 and anti - cd28 to activate the t - cells and induce flux through the network , secondly treatment with anti - cd3 , anti - cd28 and psitectorigenin , a reagent which specifically inactivates pip2 .we computed the nte between all vertex pairs in the network over paths of length for the two perturbations as described in appendix d ( see figure 5 in appendix d , for matrices of ntes for each perturbation ) .+ we found that in the psitectorigenin treated network information transfer from pip2 to the rest of the network was reduced over all path lengths ( figure 3 ) .specifically , information transfer from pip2 to pip3 was greatly reduced and information transfer from pip2 to plcg was reduced over paths of maximal length greater than one ( implying pip3 received less information from plcg via pip2 under psitectorigenin treatment ) . at longer path lengths we also see a reduced information transfer from pip2 to akt and p38 in the psitectorigenin treated network .this indicates that specific inhibition of pip2 can lead to decreased akt and p38 activation downstream of pip2 signalling .+ interestingly , we also notice that in the pip2 inhibited network , there is increased information transfer from pka to akt and from pkc to p38 .this points at a compensatory mechanism , in which inhibition of pip2 leading to reduced akt and p38 activation is compensated for by pkc dependant p38 activation and pka dependant akt activation . +thus our nte measure is capable of providing novel insights into signalling mechanisms in biological networks .in defining our nte measure we have additionally constructed a family of mappings from the space of weighted networks to a family of metric spaces , in which elements of the metric spaces correspond to signal dynamics on the networks .the mappings and structure of the metric spaces are parameterised by the path length parameter , the isd and the topology ( _ i.e. , _ the zero pattern , but not the edge weights ) of the network and their construction is explained in detail in appendix b. + this formalism allows a more theoretical treatment of dynamics on networks from the perspective of metric spaces , and permits a coupling between weighted network structure and dynamics . in certain fields , understanding the reaction of network dynamics to perturbations of edge weights is of great importance .this is particularly true of network drug design , in which one is interested in sequentially deforming the quantitative strengths of interactions in a pathological signalling network ( via drugs ) into those of a healthy network , with the aim of establishing a healthy gene expression dynamic and mitigating the pathology .if this notion of treatment is logical within our framework , then one would postulate that convergence in weight distribution of a sequence of networks to a limit distribution ( in a matrix metric space ) would imply convergence of the corresponding sequence of dynamics to the dynamics of the limit network ( in the network dynamic metric space ) .we state and prove a theorem in appendix e which establishes this postulate as true .this result demonstrates an intuitive coupling between network structure and dynamics within our framework . + further theoretical questions may consider which isds are maintained under different networks , these represent persistent ( attractor ) states of the network information distribution . to identify such states we note that every graph ( if we permit self edges at every vertex to represent a non - zero probability of signal maintenance ) admits a disjoint vertex cycle decomposition .thus there is always a way of sending signal around the network , without combining signal from two vertices at any one vertex .this implies that for every weighted network , with self edges , there must exist a permutation matrix , which admits at least one vector satisfying ( _ e.g. , _ the vector ) , such that >0 ] for a given weighted network , with corresponding stochastic matrix ( see main text ) we first note that = \sum_{\vec { x } } p[\vec { x}_{n}=\vec { x}|\vec { x_{0}}]\delta_{x_{i}}^{y},\end{aligned}\ ] ] where denotes the kronecker delta of and .+ in addition by the markovian nature of our dynamic model  = \sum_{\vec { x}_{1},\ldots , \vec { x}_{n-1}}p[\vec { x}_{n}=\vec{{x}}|\vec { x}_{n-1}]\ldots p[\vec { x}_1|\vec { x}_{0}],\end{aligned}\ ] ]  reducing our problem to the calculation of the transition probabilities  ] , it suffices to compute ,\ ] ] which is simply the probability of the signal dynamic being selected from . by model constructionthis can be expressed as whence combining this with and we derive the closed form expression   = \sum_{a_{1},\ldots , a_{n } \in \mathcal{a } } \delta _ { ( { a_{1}^{t}\ldots a_{n}^{t}\vec { x}_0})_{i}}^{y } \prod_{r=1}^{n } \prod_{k=1}^{n } \sum_{j=1}^{n } p_{kj}{(a_{r}})_{kj}.\end{aligned}\ ] ]   we demonstrated above how , for a specific network , and isd we can calculate a set of matrices describing possible signal dynamics over a single time - step of our model , as well as a probability distribution describing the signal on the entire network after time steps .we will denote these constructs for the weighted network by and  ] is a measure over the finite set which we will denote .if we denote the space of probability measures over by , then it is clear that for any two weighted networks and with the same topology , , the probability distributions  ] are elements of .+ it has been shown that for any measurable space , the square root of the jensen shannon divergence induces a metric on the space  , thus the quantity ,p_{w_{2}}[\vec { x}_{n}|\vec{{x}}_{0}])}\ ] ] computes a metric distance between the probability distributions describing the dynamics on and . + thus our nte framework results in a mapping from the space of weighted networks to a family of metric spaces in which elements of the metric space represent possible signal dynamics .network transfer entropy is formulated as the jensen shannon divergence between two probability distributions .as explored above we can derive closed form expressions for these probability distributions , however , their evaluation can be computationally expensive , if there are multiple vertices of large degree .this is because a main step in the evaluation of the expressions is constructing the set of possible signal dynamics over a single time step , which for a network on vertices is of dimension .moreover , the time complexity of evaluation scales exponentially with the path length parameter .+ for most networks , however , estimation of the probability distributions involved in the nte expression can be done efficiently . as the model underlying these distributions is a discrete time markov chain , with a discrete state space, we can employ monte carlo simulation for any isd to provide realisations of the signal distribution on the entire network , for any path length parameter . from these realisations the probability of a specified signal level at vertex , given an isd and path length parameter , can be estimated as the proportion of simulations in which the specified level is achieved .+ two major considerations need to be addressed to ensure accurate estimation from this procedure .firstly , it is clear that the more simulations of the model performed , the more accurate the estimate of the probability distribution , moreover the estimate computed from simulations will converge to the true distribution as . thus it is essential to select sufficiently large to ensure that the estimated distribution is sufficiently near the true distribution with high probability .secondly , given a specified it is important to establish how the error in estimating the probability distributions translates to error in estimating the nte .+ to address the first issue , we consider only the full network _i.e. , _ without any vertices set to absorbing state , as the stochastic matrix for the full network will have the fewest deterministic vertices , and thus will be the hardest to estimate probability distributions for .for each probability to be estimated we construct a trace plot describing the change of the estimate with the number of simulations .this plot allows us to assess convergence of the estimate as is increased .we select the number of simulations for each network as the maximal such that , the shape of the trace plots indicates convergence and the estimates ( for every vertex signal probability ) at and differ by no more than .+ to address the second issue of error in the nte after selecting , we computed multiple ( ) estimations of the signal probability distributions for the full network from simulations .we then computed , for every estimate pair , the jensen shannon divergence between the two estimates of the signal distributions at each vertex .this jensen shannon divergence , computed for vertex , can be interpreted as the nte from a vertex to , when sends no information to , if the estimation is perfect , this quantity should be zero .as the estimation is imperfect we obtain estimates of the error in the nte , deriving from error in the estimation of the probability distributions from simulation , for each receiving vertex . from these estimateswe can estimate the first two moments of the error distribution , a nte for a given receiving vertex is considered not attributable to error , provided it lies at least two standard deviations above the maximal error observed for the vertex .to compute the ntes over the two perturbations of the biological signalling network considered in the main text we must first define the isd and the edge weights from the data . as the kinases in the networkmust be phosphorylated to phosphorylate their direct targets , two connected proteins with highly positively correlated phosphorylation levels across single cell observations under a given perturbation , are likely interacting .thus a suitable edge weight which captures the strength of a phosphorylation interaction represented by an edge under a given perturbation is , where is the pearson correlation of phosphorylated protein levels across single cell measurements under perturbation .+ defining the isd is less trivial and requires consideration of the question we wish to answer and some technicalities . to determine the differences in information transfer between the two perturbations, it makes sense to consider an isd which quantifies the difference in phosphorylated protein levels between the two treatments .a technical issue to consider is how different signal distributions on the same network lead to different nte values between vertices .it is clear that weighting a vertex with a non - zero signal leads to a higher nte value between that vertex and its downstream interaction partners ( depending on the value of ) than weighting the vertex with zero signal .thus vertices with an information deficit in one perturbation versus another should be weighted with zero signal in that perturbation and a non - zero signal in the other .+ a more subtle issue concerns the number of unique signal values attainable at each vertex for a given isd and how this relates to the nte .it is somewhat intuitive that if the inputs to a given vertex each have a unique initial signal value , then the range of values attainable by the receiving vertex will be more diverse than if all the inputs had the same initial signal value .thus one may hypothesise that the nte from one vertex to another vertex , with multiple inputs , will be larger if the input vertices have unique initial signal values than if they have identical signal values .+ to explain this concept , consider the network shown in figure 4 , consisting of one central vertex with 3 possible inputs , which are each as likely to forward signal to central vertex as they are to forward signal to a separate independent neighbour .we consider the effect of the isd on the nte from the circled vertex to the central vertex for path length parameter via two cases . in the first scenariothe initial signal at every input vertex is identical ( in this example this signal value is one ) , while the signal at every other vertex starts at zero . in the second caseevery input vertex is given a unique value ( in our example these values are simply 1 , 2 and 3 ) , while the other vertices are again given initial signals of 0 .consider the probability distribution of signal at the central vertex after a single signal transfer event .for the first isd , this distribution can take 4 unique values ( namely 0 , 1 , 2 and 3 ) , however for the second isd , with unique signal values at the inputs , the distribution can take 7 values ( integers 0 through 6 ) .now consider the signal distribution at the central vertex given that we prevent the circled vertex from sending information .for the first isd , this distribution now admits only 3 possible values ( 0 , 1 and 2 ) , while for the second isd only 4 possible values are now attainable ( 0 , 2 , 3 and 5 ) .thus the size of the coding alphabet " at the central vertex after removal of input from the circled vertex has shrunk from 4 to 3 under the first isd but from 7 to 4 under the second isd ( a much greater fall ) .consequentially , we notice that the nte from the circled vertex to the central vertex is lower for the first isd than the second isd .+ if we follow this concept that inputs with unique initial signal send more information to their outputs , it is logical that vertices with significantly higher signal in one perturbation versus another should be given a unique initial signal value in the first perturbation , reflecting their capacity to send more information about the network .+ all that remains now is to consider how to assign initial signal to vertices which do not display a great difference in signal distribution across the two perturbations .one solution to this problem is to assign all these vertices an identical initial signal , in this way they can transfer more information than vertices with a signal deficit in one perturbation versus another but less information than vertices with a signal surplus . + guided by these concepts we constructed the isd for each perturbation as follows .we utilised the limma package in r to compute t - values testing , for each vertex in the network , the hypothesis that the phosphorylated protein level of the vertex was significantly different in the two treatments .if for a given vertex the phosphorylated protein levels were significantly lower ( ) in one perturbation versus another it was assigned an initial value of zero in that perturbation and a unique initial value ( here chosen as the absolute t - statistic of the test ) in the other perturbation .all vertices which did not display significant changes between the two perturbations were assigned the same non - unique initial value of 1 in both perturbations .+ the isds and edge weights for the two perturbations alongside nte matrices computed for each perturbation are provided in figure 5 .in this appendix we prove the following theorem  let be a sequence of weighted networks on vertices of fixed topology , , and let ^{n \times n} ] be a stochastic matrix of topology . if in , , as , then for fixed isd and path length parameter , the signal distributions \rightarrow p_{p}[\vec { x}_{n}|\vec{{x}}_{0}]\ ] ] as in the metric space    firstly we define the norm of a matrix  a well - known and easy to derive bound on spaces , which holds for any is + fix . as in , we have that : there exists such that for all , + let us define the matrix via it is clear that .+ we now consider for a fixed isd and path length parameter the distributions  ] , which we will hereafter refer to as and .it was shown above that    the set of possible signal dynamics for a given weighted network was also explicitly constructed above and was shown to depend only on the network topology and not on the edge weights of the network .consequentially as the sequence and the network have the same topology , the set of possible signal dynamics is the same for every element of the sequence and the network .+ consider expanding the product  \notag\\ & \ldots &  \bigg[\big(\sum_{j=1}^{n } ( p^{m}_{1j}(a_{n})_{1j } -\sum_{j=1}^{n }\delta^{p}_{1j})(a_{n})_{1j}\big ) \ldots  \big(\sum_{j=1}^{n } ( p^{m}_{nj}(a_{n})_{nj } -\sum_{j=1}^{n } \delta^{p}_{nj})(a_{n})_{nj}\big)\bigg].\end{aligned}\ ] ]  grouping together terms we can express the product as   + o(\epsilon).\end{aligned}\ ] ]  we will denote the second term in the above expression by  \end{aligned}\ ] ]  substitution of into yields  .\end{aligned}\ ] ]  clearly from the first term can be expressed    for the second term , notice that    where the second inequality follows from and by construction of the set . the final inequality follows from the facts that and .given these bounds and eq .it follows that an identical argument can be used exchanging and , in which case the sign of in changes to positive , however the bound established in bounds the modulus of and thus will always be greatest than the largest negative or largest positive value of . thus we obtain the symmetric bound let us define it follows from and that and thus it follows that    by algebra of limits , it is clear that as  and whence it follows that as and the theorem is true .+ we note that the theorem also holds if the topologies of the sequence of weighted networks and limit network are different from one another , the proof of this statement follows precisely as above , the only difference being the set utilised is that induced by the complete graph topology .as mentioned in the main text the evolution of self - assembling networks can also be considered in our framework via an application of dynamic programming . to see this we consider the space ( explicitly constructed above ) containing matrix representations of all possible single path length , signal forwarding choices , induced by the complete graph on vertices .we note that for every weighted network , , on vertices , where , the corresponding stochastic matrix can be expressed as a convex combination of elements in , where , , for all .if one interprets the space as a state space of possible choices of signal dynamics through the network and considers as a policy , giving a probability distribution of selecting a given global signal dynamic from the state space , that has been obtained by some optimality criterion , then one has a dynamic programming framework for network dynamic evolution .we note that one can calculate the policy explicitly , as $ ] .thus we have information to guide to construction of an optimality criterion describing network evolution .forms of such a criterion can be posited and parameterised for different systems and suitable parameter regimes can be reverse engineered from the policy solution .

Abstract: a measure is derived to quantify directed information transfer between pairs of vertices in a weighted network , over paths of a specified maximal length . our approach employs a general , probabilistic model of network traffic , from which the informational distance between dynamics on two weighted networks can be naturally expressed as a jensen shannon divergence . our _ network transfer entropy _ measure is shown to be able to distinguish and quantify causal relationships between network elements , in applications to simple synthetic networks and a biological signalling network . we conclude with a theoretical extension of our framework , in which the square root of the jensen shannon divergence induces a metric on the space of dynamics on weighted networks . we prove a convergence criterion , demonstrating that a form of convergence in the structure of weighted networks in a family of matrix metric spaces implies convergence of their dynamics with respect to the square root jensen shannon divergence metric .


cancer is an evolutionary process taking place within a genetically and functionally heterogeneous population of cells that traffic from one anatomical site to another via hematogenous and lymphatic routes .the population of cells associated with the primary and metastatic tumors evolve , adapt , proliferate , and disseminate in an environment in which a fitness landscape controls survival and replication .tumorigenesis occurs as the result of inherited and acquired genetic , epigenetic and other abnormalities accumulated over a long period of time in otherwise normal cells .before we can typically detect the presence of a tumor , the cells are already competing for resources in a darwinian struggle for existence in tissues that progressively age and evolve .it is well established that the regenerative capacity of individual cells within a tumor , and their ability to traffic multi - directionally from the primary tumor to metastatic tumors all represent significant challenges associated with the efficacy of different cancer treatments and our resulting ability to control systemic spread of many soft - tissue cancers .details of the metastatic and evolutionary process are poorly understood , particularly in the subclinical stages when tumors are actively developing but not yet clinically visible . it could be argued that in order to truly understand cancer progression at the level in which quantitative predictions become feasible , it is necessary to understand how genetically and epigenetically heterogeneous populations of cells compete and evolve within the tumor environment well before the tumor is clinically detectable .additionally , a better understanding of how these populations develop resistance to specific therapies might help in developing optimal strategies to attack the tumor and slow disease progression .evolutionary game theory is perhaps the best quantitative framework for modeling evolution and natural selection .it is a dynamic version of classical game theory in which a game between two ( or more ) competitors is played repeatedly , giving each participant the ability to adjust their strategy based on the outcome of the previous string of games . while this may seem like a minor variant of classical ( static ) game theory , as developed by the mathematicians von neumann and morgenstern in the 1940 s , it is not . developed mostly by the mathematical biologists john maynard smith and george price in the 1970s and martin nowak and karl sigmund more recently , this dynamic generalization of classical game theory has proven to be one of the main quantitative tools available to evolutionary biologists ( with a mathematical bent ) whose goal is to understand natural selection in evolving populations . in this biological context , a strategy is not necessarily a deliberate course of action , but an inheritable trait . instead of identifying nash equilibria , as in the static setting ,one looks for evolutionary stable strategies ( ess ) and fixation probabilities of a subpopulation .this subpopulation might be traced to a specific cell with enhanced replicative capacity , for example , that has undergone a sequence of mutations and is in the process of clonally expanding .a relevant question in that case is what is the probability of fixation of that subpopulation ?more explicitly , how does one subpopulation invade another in a developing colony of cells ?one game in particular , the prisoner s dilemma game , has played a central role in cancer modeling ( as well as other contexts such as political science and economics ) .it was originally developed by flood , dresher and tucker in the 1950s as an example of a game which shows how rational players might not cooperate , even if it seems to be in their best interest to do so .the evolutionary version of the prisoner s dilemma game has thus become a paradigm for the evolution of cooperation among a group of selfish individuals and thus plays a key role in understanding and modeling the evolution of altruistic behavior .perhaps the best introductory discussion of these ideas is found in dawkins celebrated book , the selfish gene .the framework of evolutionary game theory allows the modeler to track the relative frequencies of competing subpopulations with different traits within a bigger population by defining mutual payoffs among pairs within the group .one can then define a fitness landscape over which the subpopulations evolve .the fitness of different phenotypes is frequency dependent and is associated with reproductive prowess , while the ` players ' in the evolutionary game compete selfishly for the largest share of descendants .our goal in this article is provide a brief introduction to how the prisoner s dilemma game can be used to model the interaction of competing subpopulations of cells , say healthy , and cancerous , in a developing tumor and beyond .an evolutionary game between two players is defined by a 2 x 2 payoff matrix which assigns a reward to each player ( monetary reward , vacation time , reduced time in jail , etc . ) on a given interaction .let us call the two players a and b. in the case of a prisoner s dilemma game between cell types in an evolving population of cells , let there be two subpopulations of cell types which we will call ` healthy ' , and ` cancerous ' .we can think of the healthy cells as the subpopulation that is cooperating , and the cancer cells as formerly cooperating cells that have defected via a sequence of somatic driver mutations .imagine a sequence of ` games ' played between two cells ( a and b ) selected at random from the population , but chosen in proportion to their prevalence in the population pool .think of a cancer - free organ or tissue as one in which a population of healthy cells are all cooperating , and the normal organ functions are able to proceed , with birth and death rates that statistically balance , so an equilibrium healthy population is maintained ( on average ) .now imagine a mutated cell introduced into the population with enhanced proliferative capability as encoded by its genome as represented as a binary sequence of 0 s and 1 s carrying forward its genetic information ( which is passed on to daughter cells ) . a schematic diagram associated with this process is shown in figure 1 .we can think of this cancer cell as a formerly cooperating cell that has defected and begins to compete against the surrounding population of healthy cells for resources and reproductive prowess .from that point forward , one can imagine tumor development to be a competition between two distinct competing subpopulations of cells , healthy ( cooperators ) and cancerous ( defectors ) .we are interested in the growth rates of a ` tumor ' made up of a collection of cancer cells within the entire population , or equivalently , we are interested in tracking the proportion of cancer cells , , vs. the proportion of healthy cells , , in a population of cells comprising the simulated tissue region .  to quantify how the interactions proceed , and how birth / death rates are ultimately assigned , we introduce the 2 x 2 prisoner s dilemma payoff matrix :    the essence of the prisoner s dilemma game is the two players compete against each other , and each has to decide what best strategy to adopt in order to maximize their payoff .this 2 x 2 matrix assigns the payoff ( e.g. reward ) to each player on each interaction .my options , as a strategy or , equivalently , as a cell type , are listed along the rows , with row 1 associated with my possible choice to cooperate , or equivalently my cell type being healthy , and row 2 associated with my possible choice to defect , or equivalently my cell type being cancerous .your options are listed down the columns , with column 1 associated with your choice to cooperate ( or you being a healthy cell ) , and column 2 associated with your choice to defect ( or you being a cancer cell ) .the analysis of a rational player in a prisoner s dilemma game runs as follows .i do not know what strategy you will choose , but suppose you choose to cooperate ( column 1 ) . in that case , i am better off defecting ( row 2 ) since i receive a payoff of 5 instead of 3 ( if i also cooperate ) .suppose instead you choose to defect ( column 2 ) .in that case , i am also better off defecting ( row 2 ) since i receive a payoff of 1 instead of 0 ( if i were to have cooperated ) .therefore , _ no matter what you choose , i am better off ( from a pure payoff point of view ) if i defect_. what makes this game such a useful paradigm for strategic interactions ranging from economics , political science , biology , and even psychology is the following additional observation . _ you will analyze the game in exactly the same way i did ( just switch the roles of me and you in the previous rational analysis ) , so you will also decide to defect no matter what i do_. the upshot if we both defect is that we will each receive a payoff of 1 , instead of each receiving a payoff of 3 if we had both chosen to cooperate .the defect - defect combination is a nash equilibrium , and yet it is sub - optimal for both players and for the system as a whole .rational thought rules out the cooperate - cooperate combination which would be better for each player ( 3 points each ) and for both players combined ( 6 points ) .in fact , the nash equilibrium strategy of defect - defect is the worst possible system wide choice , yielding a total payoff of 2 points , compared to the cooperate - defect or defect - cooperate combination , which yields a total payoff of 5 points , or the best system - wide strategy of cooperate - cooperate yielding a total payoff of 6 points .the game becomes even more interesting if it is played repeatedly , stochastically , and with spatial structure with each player allowed to decide what strategy to use on each interaction so as to accumulate a higher payoff than the competition over a sequence of games . in order to analyze this kind ofan evolving set - up , a fitness function must be introduced based on the payoff matrix a. let us now switch our terminology so that the relevance to tumor cell kinetics becomes clear .when modeling cell competition , one has to be careful about the meaning of the term ` choosing a strategy ' .cells do not choose a strategy , but they do behave in different ways depending on whether they are normal healthy cells cooperating as a cohesive group , with birth and death rates that statistically balance , or whether they are cancer cells with an overactive cell division mechanism ( as triggered by the presence of oncogenes ) and an underactive ` break ' mechanism ( as triggered by the absence of tumor suppressor genes ) . in our context , it is not the strategies that evolve , as cells can not change type based on strategy ( only based on mutations ) , but the prevalence of each cell type in the population is evolving , with the winner identified as the sub - type that first saturates in the population .consider a population of cells driven by a stochastic birth - death process as depicted in figure [ fig1:fig1 ] , with red cells depicting cancer cells ( higher fitness ) and blue cells depicting healthy cells ( lower fitness , but cooperative ) .we model the cell population as a stochastic moran process of cells , ` ' of which are cancerous , ` ' of which are healthy .if each cell had equal fitness , the birth - death rates would all be equal and a statistical balance would ensue . at each step ,a cell is chosen ( randomly but based on the prevalence in the population pool ) and eliminated ( death ) , while another is chosen to divide ( birth ) .if all cells had equal fitness , the birth / death rates of the cancer cells would be , while those of the healthy cells would be . with no mechanism for introducing a cancer cells in the population , the birth / death rates of the healthy cells would be 1 , and no tumor would form ..9   ( ) , a mutation may occur .( b ) the number of cancer cells , , is defined on the state space where is the total number of cells .the cancer population can change at most by one each time step , so a transition exists only between state and , , and . ][ fig1:a ]  0.9   ( ) , a mutation may occur .( b ) the number of cancer cells , , is defined on the state space where is the total number of cells .the cancer population can change at most by one each time step , so a transition exists only between state and , , and . ][ fig1:b ]  now , introduce one cancer cell into the population of healthy cells , as shown in figure [ fig1:fig1 ] . at each step, there would be a certain probability of this cell dividing ( ) , being eliminated ( ) , or simply not being chosen for either division or death ( ) .based on this random process , it might be possible for the cancer cells to saturate the population , as shown by one simulation in figure [ fig2 ] depicting cells , with initially cancer cell , and healthy cells .however , the growth curve would not show any distinct shape ( figure [ fig2 ] ( black ) ) , and might well become extinct after any number of cell divisions , as opposed to reaching saturation .but we emphasize that without mutational dynamics , heritability , and natural selection operating on the cell population , the shape of the growth curve would look random , and we know this is not how tumors tend to grow .by contrast , figure [ fig2 ] ( red ) shows a gompertzian growth curve starting with exponential growth of the cancer cell subpopulation , followed by linear growth , ending with saturation .the growth rate is not constant throughout the full history of tumor development , but after an initial period of exponential growth , the rate decelerates until the region saturates with cancer cells .the basic ingredients necessary to sustain gompertzian growth seem to be : an underlying stochastic engine of developing cells , mutational dynamics , heritability , and a fitness landscape that governs birth and death rates giving rise to some sort of natural selection .cells for generations shows no particular shape .a single simulation of the moran process ( red ) with selection ( ) and mutations ( ) gives rise to the characteristic s - shaped curve associated with gompertzian growth.,scaledwidth=70.0% ]   each of the cells in our simulated population carries with it a discrete packet of information that represents some form of molecular differences among the cells . in our model , we code this information in the form of a 4-digit binary string from 0000 up to 1111 , giving rise to a population made up of 16 distinct cell types . at each discrete step in the birth - death process ,one of the digits in the binary string is able to undergo a point mutation , where a digit spontaneously flips from 0 to 1 , or 1 to 0 , with probability .the mutation process is shown in figure [ fig1:fig1 ] , while a mutation diagram is shown in figure [ fig3 ] in the form of a directed graph .this figure shows the possible mutational transitions that can occur in each cell , from step to step in a simulation .a typical simulation begins with a population of healthy cells , all with identical binary strings 0000 .the edges on the directed graph represent possible mutations that could occur on a given step .the first 11 binary string values ( 0 - 10 ) represent healthy cells in our model that are at different stages in their evolutionary progression towards becoming a cancer cell ( the exact details of this genotype to phenotype map do not matter much ) .mutations strictly within this subpopulation are called passenger mutations as the cells all have the same fitness characteristics .the first driver mutation occurs when a binary string reaches value 11 - 15 .the first cell that transitions from the healthy state to the cancerous state is the renegade cell in the population that then has the potential to clonally expand and take over the population .how does this process occur ?     at the heart of how the prisoner s dilemma evolutionary game dictates birth and death rates which in turn control tumor growth , is the definition of cell fitness .let us start by laying out the various probabilities of pairs of cells interacting and clearly defining payoffs when there are i cancer cells , and healthy cells in the population .the probability that a healthy cell interacts with another healthy cell is given by , whereas the probability that a healthy cell interacts with a cancer cell is .the probability that a cancer cell interacts with a healthy cell is , whereas the probability that a cancer cell interacts with another cancer cell is .  in a fixed population of cells , with i cancer cells ,the number of healthy cells is given by .the average payoff of a single cell ( ) , is dependent on the payoff matrix value weighted by the relative frequency of types in the current population :      here , , , , are the entries in the prisoner s dilemma payoff matrix ( [ eqn1 ] ) . for the prisoner s dilemma game ,the average payoff of a single cancer cell is always greater than the average payoff for a healthy cell ( figure [ fig4:c ] ) . with the invasion of the first cancer cell, the higher payoff gives a higher probability of survival when in competition with a single healthy cell .selection acts on the entire population of cells as it depends not on the payoff , but on the effective fitness of the subtype population .the effective fitness of each cell type ( , ) is given by the relative contribution of the payoff of that cell type , weighted by the selection pressure :    and the fitness of the cancer cells as :    the probability of birthing a new cancer cell depends on the relative frequency ( random drift ) weighted by the effective fitness , and the death rate is proportional to the relative frequency .the transition probabilities can be written :        in the event of the first driver mutation , the first cancer cell is birthed . at the beginning of the simulation ,the effective fitness of the healthy population is much greater than the fitness of the cancer population ( figure [ fig4:b ] ) .this is because although the single cancer has a higher _ payoff _ than any of the healthy cells , the number of healthy cells far outnumber the single cancer cells .that single cancer cell initiates a regime of explosive high growth and the fitness of the cancer population steadily increases .cancer cells are continually competing with healthy cells and receiving a higher payoff in this regime ( compare the payoff entries of a cancer cell receiving vs a healthy cell receiving ) . at later times, growth slows because cancer cells are competing in a population consisting mostly of other cancer cells .the payoff for a cancer cell is dramatically lower when interacting with a cancer cell ( observe the payoff entry of both cancer cells receiving when interacting ) . as the cancer population grows , the payoff attainable decreases and growth slows .in addition , the average fitness of the total population steadily declines because each interaction derives less total payoff , from to ..9   cells , , ) is plotted for 20,000 cell divisions to show the cancer cell population ( defectors ) saturating .the pink lines delineate the regions of tumor growth ( defined by the maximum and minimum points of the second - derivative of ) .( b ) fitness of the healthy population , cancer population , and total population plotted for the range cancer cell proportion .( c ) average payoff of a single healthy cell , cancer cell , and all cells plotted for the range cancer cell proportion . ]  [ fig4:a ]  .9   cells , , ) is plotted for 20,000 cell divisions to show the cancer cell population ( defectors ) saturating .the pink lines delineate the regions of tumor growth ( defined by the maximum and minimum points of the second - derivative of ) .( b ) fitness of the healthy population , cancer population , and total population plotted for the range cancer cell proportion .( c ) average payoff of a single healthy cell , cancer cell , and all cells plotted for the range cancer cell proportion . ]  [ fig4:b ]  .9   cells , , ) is plotted for 20,000 cell divisions to show the cancer cell population ( defectors ) saturating .the pink lines delineate the regions of tumor growth ( defined by the maximum and minimum points of the second - derivative of ) .( b ) fitness of the healthy population , cancer population , and total population plotted for the range cancer cell proportion .( c ) average payoff of a single healthy cell , cancer cell , and all cells plotted for the range cancer cell proportion . ]  [ fig4:c ]  this complex process of competition among cell types and survival of subpopulations , where defection is selected over cooperation , produces a gompertzian growth curve shown in figure [ fig5 ] , and compared with a compilation based on a wide range of data first shown in .it is now well established that tumor cell populations ( and other competing populations , such as bacteria and viral populations ) generally follow this growth pattern , although the literature is complicated by the fact that different parts of the growth curve have vastly different growth rates , and it is nearly impossible to follow the growth of a population of cancer cells _ in vivo _ from the first cancer cell through to an entire tumor made up of cells .growth rates are typically measured for a short clinical time period , and then extrapolated back to the first renegade cell , and forward to the fully developed tumor population .cells , , ) is overlaid on data from a normalized " gompertzian .values for m and w were chosen by implementing a least - squares fit to the data over a range of ( ) , and ( ) .pink lines delineate regions of growth ( defined by the maximum and minimum points of the second - derivative of ).,scaledwidth=70.0% ]   insights into the process by which growth rates vary and conspire to produce a gompertzian shape can be achieved by positing that growth is related to molecular and cellular heterogeneity of the developing population .indeed , an outcome of the model is that molecular heterogeneity ( i.e. the dynamical distribution of the 4-digit binary string 00001111 making up the population of cells ) drives growth .consider entropy of the cell population as a measure of heterogeneity :    ( here , log is defined as base 2 ) .the probability measures the proportion of cells of type , with representing the distribution of binary strings ranging from 0000 to 1111 .we typically course - grain this distribution further so that cells having strings ranging from 0000 up to 1010 are called ` healthy ' , while those ranging from 1011 to 1111 are ` cancerous ' .then growth is determined by :    it follows from ( [ eqn10 ] ) that the cancer cell proportion can be written in terms of entropy as :    this relationship between growth of the cancer cell population and entropy is pinned down and detailed in .we consider it to be one of the key emergent features of our simple model .a typical example of the emergence of genetic heterogeneity in our model system is shown in the form of a phylogenetic tree in figure [ fig6 ] .this particular tree is obtained via a simulation of only 30 healthy phenotypic cells ( 0000 ) , which during the course of a simulation expand out ( radially in time ) to form a much more heterogeneous population of cells at the end of the simulation . in our model , the genetic time - history of each cellis tracked and the population can be statistically analyzed after the simulation finishes .  , ) .pathways are color coded to indicate genetic cell type.,scaledwidth=70.0% ]figure [ fig7 ] shows the clear advantage of early stage therapy in our model system .we compare the effect of therapy given at an early stage , mid - stage , and late stages of the gompertzian growth of the tumor .the black gompertzian curve is the freely growing cancer cell population .the blue curve shows the cancer cell population diminishing to zero in time of continual therapy .the red curves shows the same therapy administered at a later stage in the growth phase ( three stochastic simulations ) , diminishing to zero in time .the yellow curves show the therapy ( five stochastic simulations ) administered even later , diminishing in time . in one of the simulations ,the growing cancer cell population overcomes the killing effect of the therapy .the purple curves shown in the figure all are late stage therapies which do not kill the full population of cancer cells . in these cases , the growth of the cancer cell population outstrips the ability of the simulated therapy to kill them off . clearly , the earlier in the growth phase the therapy is administered , the less time is required to kill off the full population of cancer cells .but since the cancer cell population grows at very different rates throughout the full gompertzian history of the developing tumor , this balance of kill cycles based on dose density and accelerated growth due to fitness advantage needs to be quantitatively determined .cells , , ) with no therapy is plotted ( black ) . the stochastic response to therapyis shown for 4 different time points in tumor progression .therapy is designed to kill off " cancer cells with potency equal to their proportion . the first therapy ( blue , beginning at cell divisions )has the shortest time to total elimination of cancer cells .the second ( red at cell ) has a longer time to total elimination .the kill effect is diminished further for the third and fourth therapy ( yellow , and purple , cell divisions ) and some stochastic simulations are not able to eliminate all cancer cells but only delay saturation.,scaledwidth=90.0% ]  an established empirical law which relates drug dose density to its effectiveness in killing off cancer cells is known as the ` log - kill ' law .the log kill law states that a given dose of chemotherapy kills the same fraction of tumor cells ( as opposed to the same number of tumor cells ) , regardless of the size of the tumor at the time the therapy is administered , a consequence of exponential growth with a constant growth rate .this effect is best illustrated on a dose - response curve , plotting the dose density , , with respect to the probability of tumor cell survival , .the dose density is simply the product of the drug concentration , , and the time over which the therapy is administered , :    thus , the log - kill law states the following :    as an example , if there are 1000 cancer cells in a population , and the first therapy dose kills off 90% of them , then after the first round of therapy there will be 100 cancer cells remaining . if a second round of therapy is administered , exactly as the first round , starting soon enough so that no new cancer cells have formed , then this next round will also kill off 90% of the cells , leaving 10 cells , and so on for each future round of therapy . in a sense , since the first round killed 900 cells , while the second identical round killed only 90 cells , the population gets increasingly more difficult to kill off using the same treatment on each cycle .the log - kill law , a fundamentally static law ( does nt say anything about the relationship of the fraction of cells killed vs. the growth rate of the tumor ) , is verified in our model system , as shown in figure [ fig8:a ] . on the x - axis, we increase the dose density d , and we plot the number of surviving cancer cells .the slope of this straight line ( verifying the log - kill law ) can be thought of as the rate of regression of the tumor , ..45  cells , , ) where therapy is administered with varied dose densities ( , ) .the rate of tumor regression , , the slope of the red curve fit , is constant for a short period of simulated time where the tumor growth rate is approximately constant .this is the log - kill law .( b ) the process to find the rate of tumor regression , , is repeated for a range of instantaneous tumor growth rates ( ) .the norton - simon hypothesis predicts that is proportional to , indicated by a linear fit ( red).,title="fig : " ]  [ fig8:a ]  .45  cells , , ) where therapy is administered with varied dose densities ( , ) .the rate of tumor regression , , the slope of the red curve fit , is constant for a short period of simulated time where the tumor growth rate is approximately constant .this is the log - kill law .( b ) the process to find the rate of tumor regression , , is repeated for a range of instantaneous tumor growth rates ( ) .the norton - simon hypothesis predicts that is proportional to , indicated by a linear fit ( red).,title="fig : " ]  [ fig8:b ]  so how is the rate of regression , , related to the growth rate of the tumor , ? this is relevant , since we know from the shape of the gompertzian curve , the growth rate is highest ( exponential ) at the beginning stage of tumor development and lowest at the late saturation stage .the norton - simon hypothesis states that the rate of regression is proportional to the instantaneous growth rate for an untreated tumor of that size at the time therapy is first administered .faster growing tumors ( early stage ) should show higher rates of regression than more slowly growing tumors ( late stage ) .this hypothesis is also verified in our model system , and shown clearly in figure [ fig8:b ] .the reality of this growth - dependent tumor regression rate effect ( where early stage faster growing tumors are more vulnerable to therapy than later stage more slowly growing tumors ) combined with the fact that the first round of therapy is more effective than future rounds of identical therapies , dramatically reinforces the need to administer drug treatment early in tumor progression when growth rates are high and there are fewer cancer cells to kill off . as drug concentration , ,is kept constant , the effect of treatment is reduced at later times in tumor development ( figure [ fig7 ] ) until the same drug concentration is simply unable to overcome the tumor growth rate .so how do these molecular and cellular growth details manifest themselves on the larger scales associated with metastatic progression patterns in patients ? despite the fact that disease progression patterns can vary from patient to patient , if a sufficiently large cohort of patients with similar characteristics is tracked over the course of the disease , statistical patterns emerge and can be exploited to build dynamical models of large scale progression .this lies at the heart of the models described in for lung cancer progression , and for breast cancer progression .  as an example of the kinds of whole - body scale models that can be built ,consider first the tree - ring diagram shown in figure [ fig9]a .the diagram encapsulates the entire progression history of a cohort of 289 primary breast cancer patients tracked at the memorial sloan kettering cancer center for a 20 year period .all of the patients entered the cohort with a primary breast tumor , but no metastatic tumors .the inner ring , shown in pink , represents this cohort when they entered the study .as time progresses , the rings grow out , surrounding the inner breast ring . the first ring out shows the metastatic tumor distribution associated with first recurrence .the sector sizes represent the percentage of patients in this group . likewise , the second ring out represents the distribution of tumors on second recurrence , and so forth for the further rings out .hence , subsequent rings outward represent the tumor distributions as time progresses , with each patient history depicted on a ray going out from the center of the ring diagram .we caution that despite our usage of the term ` tree - ring ' diagrams for these representations , the thickness of the rings are all equal , hence do not reflect the time between subsequent recurrences ( timescales of progression are documented and modeled in ) .the power of the diagrams is that in one quick glance , one gains an appreciation for the statistical complexity of the disease . from them, one can also calculate the probability of the disease ` transitioning ' from one site to another as the disease progresses ( called transition probabilities ) .these can then be used to create a single markov transition matrix for each cancer type , which quantitatively encodes much of the information associated with the disease .figure 9b shows the markov transition graph from the last metastatic site to the deceased state for the cohort from figure [ fig9]a .the sites are ordered clockwise from the most probably last metastatic site , to the least probable .figures [ fig9]c and [ fig9]d show reduced markov diagrams for two specific important sub - groups of breast cancer patients , her2 + patients , and er-/her2- patients .generally speaking , her2 + patients have the poorest prognosis .the red sites in these reduced diagrams ( bone , lung / pleura , chest wall , ln ( mam ) ) in figure [ fig9]c , and bone , ln ( mam ) , chest wall in figure 9d are the spreaders associated with these groups . the blue sites ( liver , ln ( dist ) , brain for fig .[ fig9]c ; ln ( dist ) , lung / pleura , liver , brain for fig .[ fig9]d ) are the sponges .it is interesting to note that lung / pleura switches from a spreader in the her2 + sub - group to a sponge in the er-/her2- sub - group , suggesting a possible biological difference of the site in the different groups that correlates with different survival probabilities .it is important to keep in mind that no mathematical model captures all aspects of reality , so choices must be made which involve prioritizing the features that are most essential in capturing the essence of a complex process and which are not .most experts now agree that the evolutionary processes in a tumor played out among subpopulations of competing cells are key to understanding aspects of growth and resistance to chemotherapy , which will ultimately lead the way toward a quantitative understanding of tumor growth and cancer progression .the paradigm of the cancer cell subpopulation and the healthy cell subpopulation competing as the defectors and cooperators in a prisoner s dilemma evolutionary game has been useful in obtaining a quantitative handle on many of these processes and frames the problem in an intuitive yet predictive way .nonetheless , the mathematical ` taste ' of the modeler plays a role in what techniques are selected and ultimately where the spotlight shines .this fact makes clinicians uncomfortable and can lead to deep suspicion of the mathematical modeling enterprise as a whole .are nt the outcomes and predictions of mathematical models a straightforward consequence of the model assumptions ?once those choices are made , is nt the cake already baked ?so why should we be surprised if you tell us it tastes good ?why not simply use tried and true statistical tools like regression methods to curve - fit the data directly , with no built in assumptions , and be satisfied with uncovering correlations and trends ? clinicians ( and experimentalists , in general ) feel that they are dealing directly with reality , so why mess around with ` toy ' systems based on possibly ` ad hoc ' or incorrect assumptions that create artificial realities that may or may not be relevant ? to a theoretician , calling their assumptions ad hoc , as opposed to natural , is as insulting as calling a clinician sloppy and uncaring ( try this for yourself at the next conference you go to ! but please use the term ` somewhat ad hoc ' to lessen the blow . ) and if you want to deliver an even harsher insult , you could comment that the model seems like an exercise in curve fitting .but the usefulness of mathematical models built on simplified assumptions is well established in the history of the physical sciences , as detailed beautifully in peter dear s book , _ the intelligibility of nature : how science makes sense of the world _ .bohr s simple model of the structure of the atom was crucial in moving the community forward towards a deeper understanding of cause and effect , and ultimately pushing others to develop more realistic atomic models .the same could be said for many other important , but ultimately discarded models of reality ( e.g. the notion of aether used as a vehicle to understand the mysterious notion of action - at - a - distance ) now relegated to footnotes in the history of the physical sciences .lessons from this history highlight the importance of using the principle of occam s razor ( law of parsimony ) as a heuristic guide in developing models : ( 1 ) keep things simple , but not too simple ; ( 2 ) see what can be explained by using a given set of assumptions , and try to identify what is either wrong or can not be explained ; ( 3 ) add complexity to the model , but do this carefully .since ultimately , the model will always be wrong ( with respect to some well chosen and specific new question being posed about a system ) , it is important that it be _ useful as a vehicle of intelligibility _ associated with the set of questions surrounding the phenomena it was built to explain .answers to some new questions will be found using the model as a temporary crutch , and new questions will emerge in the process that had not yet been asked , as their relevance had never previously been realized .a new quantitative language will emerge in which aspects of the model will be associated with the underlying reality it is attempting to describe , predictions will be easier to frame and test , and shortcomings will be exposed . in his famous article , eugene wigner writes compellingly that ` the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. we should be grateful for it and hope that it will remain valid in future research and that it will extend , for better or for worse , to our pleasure , even though perhaps also to our bafflement , to wide branches of learning . '  in general , the more complex the model ( as measured , for example , by the number of independent parameters associated with it ) , the less useful it will be , and the less likely it is to be adopted by the community at large .after all , if the model is as complex as the phenomena it was built to understand , why not stick with reality ?effective models can be thought of as _ low - dimensional approximations of reality _ , surrogates that help us bootstrap our way forward .they arise as the outcome of a complex balancing act between simplicity of the ingredients , and complexity of the reality the model is meant to describe .they generally do not arise in a vacuum , but are built in the context of informed and sustained discussions among people with different expertise . in the context of medical oncology ,this means physical scientists developing ongoing interactions with clinical oncologists , radiologists , pathologists , molecular and cell biologists and other relevant medical specialists .appropriate data is a necessary ingredient in developing and testing any successful model , and treasure troves of medical data sit unexamined in patient files and government databases across the country waiting to be put to good use .there is no doubt that they are telling an interesting and important story that we have yet to fully understand .it is not currently possible for the computer to simulate all of the complex , relevant , and systemic ingredients at play to faithfully recreate all aspects of cancer progression and treatment response in patients .it is hard to imagine that a deep and actionable understanding can ever be obtained without the combined use of data , models , and computer simulations to help guide us and highlight some of the underlying causal mechanisms of this complex and deadly disease .

Abstract: tumor development is an evolutionary process in which a heterogeneous population of cells with differential growth capabilities compete for resources in order to gain a proliferative advantage . what are the minimal ingredients needed to recreate some of the emergent features of such a developing complex ecosystem ? what is a tumor doing before we can detect it ? we outline a mathematical model , driven by a stochastic moran process , in which cancer cells and healthy cells compete for dominance in the population . each are assigned payoffs according to a prisoner s dilemma evolutionary game where the healthy cells are the cooperators and the cancer cells are the defectors . with point mutational dynamics , heredity , and a fitness landscape controlling birth and death rates , natural selection acts on the cell population and simulated ` cancer - like ' features emerge , such as gompertzian tumor growth driven by heterogeneity , the log - kill law which ( linearly ) relates therapeutic dose density to the ( log ) probability of cancer cell survival , and the norton - simon hypothesis which ( linearly ) relates tumor regression rates to tumor growth rates . we highlight the utility , clarity , and power that such models provide , despite ( and because of ) their simplicity and built - in assumptions .   _ keywords _ : cancer model ; evolutionary game theory ; moran process ; gompertzian tumor growth ; tumor heterogeneity ; birth - death process