You are an expert at summarizing long articles. Proceed to summarize the following text: 
most solid tumors eventually establish colonies in distant anatomical locations ; when these colonies become clinically detectable , they are called macrometastasis . while often there is a large burden from primary tumors , it is in fact metastatic disease that is responsible for most cancer fatalities @xcite . the creation of macrometastasis requires the successful completion of a sequence of difficult steps . first , cancer cells must gain access to the general circulation system via the process of intravasation . next , the cells must survive in the inhospitable environment of the circulatory system . following this , the tumor cells must exit the circulatory system ( extravasation ) at a distant site and initiate micrometastsis ( clinically undetectable population of tumor cells at a distant anatomical site ) . lastly , the micrometastsis must develop the ability to successfully proliferate in the distant site and grow into clinically identifiable macrometastasis . the completion of these steps is very difficult and only a small fraction of tumor cells are able to achieve this @xcite . however , due to the vast number of cells in most primary tumors , metastasis commonly occurs in later stage solid tumors .    there has been significant mathematical research in the design of optimal anti - cancer therapies . this has included studies on optimal chemotherapy , radiotherapy , and more recently targeted therapies and immunotherapy ( @xcite ) . since we are interested in radiotherapy we will focus on previous work in this field . the vast majority of modeling of radiotherapy response is based on the linear - quadratic model ( lq ) which says that tissue response is governed by the parameters @xmath1 and @xmath2 ( see e.g. , @xcite ) . specifically , following a single exposure to @xmath3 gray of radiation , the surviving fraction of viable cells is given by @xmath4 . an important question in this field is to decide on the optimal temporal distribution of a given amount of radiation , i.e. , how to kill the most tumor cells while inflicting the least amount of normal tissue damage . this is commonly referred to as the ` optimal fractionation problem . ' two possible solutions to this problem are hyper - fractionated and hypo - fractionated schedules . in hyper - fractionated schedules , small fraction sizes are delivered over a large number of treatment days , while in hypo - fractionated schedules , large fraction sizes are delivered over a small number of treatment days . if we minimize primary tumor cell population at the conclusion of treatment , it has been seen ( @xcite and @xcite ) that whether hyper or hypo - fractionation is preferable depends on the radiation sensitivity parameters of the normal and cancerous tissue . however we will observe in section 4 of this manuscript that when designing optimal treatments with the goal of minimizing metastatic production , hypo - fractionation is preferable for many parameter choices , and hyper - fractionation is only preferable sometimes when the @xmath0 value of the tumor is large . there have been a substantial number of works looking at optimal fractionation . the work @xcite considers dynamic design of fractionation schedules with incomplete repair , repopulation and reoxygenation . a more recent work @xcite considers the optimization problem associated with finding fractionation schedules under an lq model with incomplete repair and exponential repopulation . the authors theoretically establish the benefits of hypo - fractionation in the setting of a low @xmath0 value of the tumor . brenner and hall @xcite utilized the lq model in combination with the lea - catcheside function ( a generalization of the lq model that is useful at higher doses or prolonged doses ) to conclude that due to its slow response to radiation , prostate cancer can be treated equally effectively by either uniform radiation scheduling or hypo - fractionation ( which has fewer side effects ) . unkelbach et al . @xcite studied the interdependence between optimal spatial dose distribution and creation of fractionation schedules . another work @xcite utilized a dynamic programming approach to study the problem of optimal fractionation schedules in the presence of various repopulation curves . an important property common to all of these works is that they utilize an objective function that seeks to minimize final primary tumor population size in some sense . while this can be an important objective , in most cancers , it is ultimately metastatic disease that proves fatal . therefore , in this work , we study optimal fractionation schedules when using an objective function that seeks to minimize the total production of metastatic cells . the understanding of the metastatic process and how to respond to it has been greatly aided by the mathematical modeling community ( for an overview of this contribution see the recent review paper @xcite ) . in an interesting work @xcite , iwata et al . developed a set of differential equations governing the population dynamics of the metastatic population . a compelling work is the paper by thames et al . @xcite where they developed a mathematical model of the metastatic process to calculate risk from metastatic disease due to delay in surgery . hanin and korosteleva @xcite used a stochastic model to address questions such as : ( 1 ) how early do metastasis events occur , ( 2 ) how does extirpation of the primary affect evolution of the metastasis , and ( 3 ) how long are metastasis latent ? haeno and michor @xcite developed a multitype branching process model to study metastasis and in particular the probability of metastasis being present at diagnosis . in a follow up work @xcite , they used a mathematical model to study metastasis data in recently deceased pancreatic cancer patients . in a recent work @xcite , diego et al . used an ode model to study the relationship between primary and metastatic cancer sites , and in particular , makes predictions about the clinical course of the disease based on the parameter space of their ode model . the remainder of the paper is organized as follows . in section [ sec : metsprod ] , we discuss a model for metastasis production and how it can be used to develop a function that reflects metastatic risk . next , in section [ sec : opt ] , we describe the optimization model and solution approach . finally , in section [ sec : num ] , we present numerical results in the setting of breast cancer . we start by assuming that the total population of primary tumor cells at time @xmath5 is given by the function @xmath6 . note we will assume throughout this work that the population of cells is large enough that we can treat the population as a deterministic function . we then assume that each tumor cell initiates a successful macrometastasis at rate @xmath7 per unit of time . this is similar to the modeling approach taken in @xcite where they were able to fit their model to metastasis data from patients . if we are interested in the time horizon @xmath8 $ ] then our total rate of production of successful macrometastasis on that interval is @xmath9 in particular @xmath10 , the number of successful metastasis established in the time interval in @xmath8 $ ] , is a poisson random variable with mean @xmath11 and thus the probability of metastasis occurring is @xmath12 . therefore , in order to minimize @xmath13 , it suffices to minimize @xmath11 .    in the rate @xmath11 , we assume that every cell is capable of metastasis . in the geometry of the actual tumor , it might be the case that only those cells on the surface of the tumor are capable of metastasis , or only those cells in close proximity to a blood vessel are capable . therefore , we consider the generalization @xmath14 for @xmath15 . this is similar to the model of metastasis creation in @xcite ; there they refer to the parameter @xmath16 as the fractal dimension of the blood vessels infiltrating the tumor , if we assume that the tumor is three dimensional with a two dimensional surface and that all cells on the surface are equally capable of metastasis then we can take @xmath17 . however if we assume that only a small fraction of cells on the surface are capable of metastasis we could take e.g. , @xmath18 . notice that in order to minimize @xmath19 , we do not need to know the parameter @xmath20 , which is difficult to measure . note that we are using a rather simplistic model for the metastasis production in that we assume at most only two rates of metastasis for the primary tumor cells . in reality , it is likely that the rate of metastasis for a given cell will be a complex function of its position , migratory potential , and its oxidative state . however , given the lack of data available , we found it preferable to work with this relatively simplistic model that does not require the knowledge of any intricate parameters . in addition , since the primary goal of this work is to introduce a novel objective function , we feel that adding further biological details can be saved for further exploration . lastly and importantly , variants of this relatively simplistic model have been matched to clinical metastasis data @xcite . our goal is to determine an optimal radiotherapy fractionation scheme that minimizes the probability that the primary tumor volume metastasizes . given the discussion from the previous section , this equates to minimizing @xmath19 ; however , for simplicity , we will use an approximate objective that uses a summation rather than an integral during therapy . let the time horizon @xmath21 consists of treatment days and a potentially long period of no treatment at the end of which metastatic risk is evaluated . suppose a radiation dose @xmath22 is delivered at time instant @xmath5 , for @xmath23 , then we choose to minimize @xmath24 where @xmath25 is the number of cells immediately before the delivery of dose @xmath26 . during the course of the treatment , we assume exponential tumor growth with a time lag . beyond that , we use the gomp - ex model for tumor growth because we are evaluating long - term metastatic risk and the exponential model will give unreasonably large values over such a long time period . the gomp - ex law of growth assumes that the cellular population expands through the exponential law @xcite initially , when there is no competition for resources . however , there is a critical size threshold @xmath27 such that for @xmath28 the growth follows the gompertz law . thus , we have @xmath29 for @xmath30 and @xmath31 for @xmath32 $ ] where @xmath33 and @xmath34 are tumor tissue sensitivity parameters , @xmath35 is the tumor doubling time in units of days , @xmath36 is the tumor kick - off time , @xmath37 is the tumor size at the starting of treatment , @xmath38 is the carrying capacity and @xmath39 is a constant related to the proliferative ability of the cells . the expression @xmath40 is defined as @xmath41 . @xmath42 is the time when the tumor population reaches size @xmath27 and can be computed using following equation @xmath43 we also use the concept of biological effective dose ( bed ) to constrain the side - effects in the oar around the tumor . we assume that a dose @xmath3 results in a homogeneous dose @xmath44 in the @xmath45th oar , where @xmath46 is the sparing factor ; for the heterogeneous case , it is possible to use the approach in @xcite . the bed in the @xmath45th oar is defined by @xmath47_i } } \right),\ ] ] where @xmath48_i}$ ] is an oar tissue sensitivity parameter . the bed can be derived from the lq model and is used to quantify fractionation effects in a clinical setting . note that we can also have multiple bed constraints on a single oar , e.g. , to model early and late effects . the optimization problem of interest is @xmath49 s.t . @xmath50_i}}\right ) \leq c_i,\ \ i=1,\dots , m\ ] ] where @xmath51 is a constant that specifies an upper bound to the @xmath45th bed in the oar and @xmath52 is the total number of oars in the vicinity of tumor . note that we do not work directly with the quantity @xmath53 during therapy but instead with its approximation @xmath54 . here , @xmath55 is an upper bound for @xmath53 and is a good approximation for @xmath53 if the impact of the exponential growth term in ( [ eq : linearquadraticexp ] ) is relatively small compared to the dose fraction terms , which is typically the case for most disease sites . formulation is a nonconvex quadratically constrained problem . such problems are computationally difficult to solve in general . however , we can use a dynamic programming ( dp ) approach with only two states to solve this deterministic problem , similar to the work in @xcite . the states of the system are @xmath56 , the cumulative dose , and @xmath57 , the cumulative dose squared , delivered to the tumor immediately after time @xmath5 . we have @xmath58 @xmath59 now we can write the dp algorithm ( forward recursion ) as @xmath60 , & t\le n_0 - 1\\ \min_{d_{t}\ge0}[x_0^\xi e^{-\xi\left ( \alpha_t ( u_{t-1}+d_{t } ) + \beta_t ( v_{t-1}+d_{t}^2)-\frac{\ln 2}{\tau_d}(t - t_k)^+\right)}+j_{t-1}(u_{t-1},v_{t-1})]+\int_{n_0}^{t}(x_t)^\xi , & t = n_0\end{cases}\ ] ] with @xmath61 , @xmath62 . we set the function @xmath63 to be @xmath64 if @xmath65_i } } v_t > c_i\ ] ] for any @xmath66 . since there are only two state variables , we can solve our optimization problem by discretizing the states and using this dp algorithm . we solve the optimization problem based on the radiobiological parameters for breast cancer . we consider two different normal tissues @xcite , heart and lung tissue . for each normal tissue , we define the maximal toxicity @xmath67_i}}\frac{d_i^2}{n_i}\ ] ] where @xmath68 and @xmath69 are tissue specific parameters and define the maximum total dose @xmath68 delivered in @xmath69 fractions for each oar . a standard fractionated treatment is to deliver @xmath70 gy to the tumor with @xmath71 gy fractions @xcite . the tolerance bed values ( @xmath51 ) for various normal tissues were computed based on the standard scheme . hence all bed in oar associated with optimal schedules obtained in this section are less than or equal to their corresponding bed in standard schedule , i.e. @xmath72 gy and @xmath73 for @xmath74 . all radiobiological parameter values used are listed in table [ tabledata ] along with their sources . .breast tumor and normal tissues parameters [ cols="<,<,<,<",options="header " , ] in this work , we have considered the classic problem of optimal fractionation schedules in the delivery of radiation . we have however done this with the non - traditional goal of minimizing the production of metastasis . this is motivated by the fact that the majority of cancer fatalities are driven by metastasis @xcite , and that this disseminated disease can be very difficult to treat . we addressed this goal by considering the optimal fractionation problem with a novel objective function based on minimizing the total rate of metastasis production , which we argue is equivalent to minimizing the time integrated tumor cell population . we were able to numerically solve this optimization problem with a dynamic programming approach . we computed radiotherapy fractionation schedules that minimized metastatic risk for a variety of parameter settings . the resulting optimal schedules had an interesting structure that was quite different from what is observed from the traditional optimal fractionation problem where one is interested in minimizing local tumor population at the end of treatment . in the traditional optimal fractionation problem it was observed in @xcite that if @xmath48_t}\le\min_i\{{[\alpha/\beta]_i}/\gamma_i\}$ ] then a hypo - fractionated schedule is optimal and if @xmath48_t}>\min_i\{{[\alpha/\beta]_i}/\gamma_i\}$ ] then a hyper - fractionated schedule is optimal . in contrast , in the current work we observed that if we are evaluating metastatic risk at the conclusion of therapy then independent of the relationship between @xmath48_t}$ ] and @xmath75_i}/\gamma_i\}$ ] , the resulting fractionation schedules for minimizing metastatic risk is a hypo - fractionated structure with large initial doses that taper off quickly . this is due to the structure of the objective function . in order to minimize the time integrated tumor cell population immediately after treatment , it is necessary to quickly reduce the tumor cell population since this is the high point of the tumor cell population over the course of the treatment . if we think of the tumor cell population as quite dangerous due to its metastasis potential , then it is natural to want to reduce their population as quickly as possible . we observed that the structure of the optimal schedule depends on the length of time for which we evaluated metastatic risk . in particular if we evaluate metastasis risk in a long time frame ( several years ) after therapy , and it holds that @xmath48_t}>\min_i\{{[\alpha/\beta]_i}/\gamma_i\}$ ] , then the optimal schedule will have either medium initial doses tapering off slowly ( that is , a schedule where @xmath76 ) or equal - dosage routine where we have @xmath77 . this is in contrast to when we evaluate metastatic risk over the course of the treatment period , in which case a hypo - fractionated schedule is nearly always optimal . note that if our evaluation time - frame of metastatic risk is several years after the conclusion of therapy , then we want to minimize the tumor cell population as soon as possible to reduce initial metastatic risk , but we also need to avoid a large tumor population in the long time period after the conclusion of treatment . in many of our results , we observed the optimality of hypo - fractionated schedules . interestingly , previous clinical trials , e.g. , the fast trial for breast cancer @xcite and the chhip trial for prostate cancer @xcite , show the benefit of hypo - fractionation . it should be noted that the motivation for these trials was the low @xmath48_t}$ ] value in prostate and breast tumors . our results indicate even when the @xmath48_t}$ ] ratio of the tumor is very large , it may still better to deliver a hypo - fractionated schedule when taking metastatic risk into account . the current work provides a possible new motivation for considering hypo - fractionated schedules , i.e. , metastasis risk reduction .    in this paper , we have assumed the delivery of single daily fractions and have not considered alternate fractionation schemes such as chart @xcite that deliver multiple fractions a day . some of our results indicate optimal hypo - fractionation schedules ( as obtained for the t=0.07 case ) . one way to deliver a large amount of dose in a very short period of time is to use such schemes like chart . however , additional modeling is needed to incorporate incomplete sublethal damage repair due to short inter - fraction time periods . another possible interpretation of this work is to view the output @xmath53 as the risk of the tumor developing resistance to a chemotherapeutic treatment . this can be achieved by simply viewing the parameter @xmath20 as the rate at which tumor cells develop drug resistance . due to the severe consequences of drug resistance , this is also an interesting direction for further exploration . we feel that this type of work minimizing metastatic production opens the potential for a new line of research in the radiation optimization community as well as cancer biology . in particular , there are several important biological phenomena that we have not included . this includes oxygenation status ( and history ) of cells as well as the vascular structure of the tumor of interest . lastly , a potentially interesting extension of this work could be to validate our predictions in animal models of metastatic cancer . t.  bortfeld , j.  ramakrishnan , j.  n tsitsiklis , and j.  unkelbach . optimization of radiation therapy fractionation schedules in the presence of tumor repopulation . forthcoming in informs journal on computing . , 2013 . m.  mizuta , s.  takao , h.  date , n.  kishimoto , k.  sutherland , r.  onimaru , and h.  shirato . a mathematical study to select fractionation regimen based on physical dose distribution and the linear - quadratic model . , 84:829833 , 2012 . g.  tortorelli , et al . standard or hypofractionated radiotherapy in the postoperative treatment of breast cancer : a retrospective analysis of acute skin toxicity and dose inhomogeneities . , 13(1):230 , 2013 . j.  owen , a.  ashton , j.  bliss , j.  homewood and et al . . effect of radiotherapy fraction size on tumour control in patients with early - stage breast cancer after local tumour excision : long - term results of a randomised trial . , 7:467471 , 2006 . d. dearnaley , i. syndikus , g. sumo , m. bidmead , d. bloomfield , c. clark , ... & e. hall . conventional versus hypofractionated high - dose intensity - modulated radiotherapy for prostate cancer : preliminary safety results from the chhip randomised controlled trial . , 13:4354 , 2012 .
metastasis is the process by which cells from a primary tumor disperse and form new tumors at distant anatomical locations . 
 the treatment and prevention of metastatic cancer remains an extremely challenging problem . 
 this work introduces a novel biologically motivated objective function to the radiation optimization community that takes into account metastatic risk instead of the status of the primary tumor . in this work 
 , we consider the problem of developing fractionated irradiation schedules that minimize production of metastatic cancer cells while keeping normal tissue damage below an acceptable level . a dynamic programming framework is utilized to determine the optimal fractionation scheme . 
 we evaluated our approach on a breast cancer case using the heart and the lung as organs - at - risk ( oar ) . for small tumor @xmath0 values , 
 hypo - fractionated schedules were optimal , which is consistent with standard models . 
 however , for relatively larger @xmath0 values , we found the type of schedule depended on various parameters such as the time when metastatic risk was evaluated , the @xmath0 values of the oars , and the normal tissue sparing factors . 
 interestingly , in contrast to standard models , hypo - fractionated and semi - hypo - fractionated schedules ( large initial doses with doses tapering off with time ) were suggested even with large tumor @xmath1/@xmath2 values . 
 numerical results indicate potential for significant reduction in metastatic risk .