antenna ( a.k.a .mimo : multiple - input multiple - output ) techniques have been extensively studied to improve the spectral efficiency of mobile communication systems , and are envisioned to be ubiquitously supported to accommodate the exponential growth of future wireless service demands .mimo communications , however , requires the knowledge of the channel state information ( csi ) at the transmitter for precoding and at the receiver for signal detection .a common approach is to allocate a certain amount of time and power resources to construct and transmit training signals for acquiring csi .the impact of csi acquisition on the overall performance of a mimo system has been investigated under various performance measures , such as channel minimum mean - square error ( mmse ) , bit error rate ( ber ) , and channel input output mutual information ( mi ) . among these measures ,mi is advantageous in that it characterizes the fundamental tradeoff between achieving high - quality channel estimate and the information throughput .one one hand , to achieve a high - quality channel estimate , more time and power resources should be allocated for training , leaving the data transmission with little time and power . on the other hand , if too little resources are allocated to training , information throughput deteriorates due to channel mismatch .the exact throughtput characterization of the training - based mimo system is difficult , but a tight mi lower bound was derived in .the authors in also discussed the tradeoff from the perspective of degrees of freedom ( dof ) .later , coldrey et al . established the optimal tradeoff by assuming independent and identically distributed ( i.i.d . )rayleigh fading and exploiting the rotational invariance property of an i.i.d .gaussian channel matrix .recently , the incorporation of mimo into multiuser cellular systems has attracted much research interest , especially in a massive mimo setup where users communicate with a base station ( bs ) equipped with a large - scale antenna array .a multiuser mimo system can be treated as a virtual mimo system without cooperation at the transmitter side .it exhibits some new features compared with conventional mimo .first , geographically separated users in general experience significantly different large - scale fading caused by path loss and shadowing . that is , the near - far effect is indispensible in modelling a multiuser mimo system .second , the total transmission power of a mimo system is usually constrained , while that of a multiuser mimo system scales with the number of active users .third , large antenna arrays may be deployed at bss , focusing energy into ever - smaller spatial regions to bring huge improvements in system throughput and energy efficiency ; but at the same time , large antenna arrays creates a lot more channel links than ever before , which imposes a heavy burden on the acquisition of csi .the above features imply that the pilot design presented in ( under the rayleigh fading assumption and a total transmission power constraint ) is not necessarily good for a multiuser mimo system .new insights and guidelines must be developed to better understand the tradeoff between the channel estimation quality and the information throughput in a multiuser mimo system .  in this paper , we investigate the fundamental throughput limit of a training - based mimo multiple - access system , where each transmission frame consists of a training phase for acquiring csi and a data - transmission phase for information delivery .we assume that the system consists of an -antenna bs and single - antenna user terminals , forming an -by- virtual mimo channel with coherence time ( during which the channel state is assumed to be constant ) .each user suffers both large - scale fading caused by the near - far effect and small - scale fading caused by the multipath effect .as large - scale fading varies much slowly in magnitude of time than small - scale fading , we assume that the large - scale fading coefficients of users are known _ a priori _ at the bs , while the small - scale fading coefficients of users are to be estimated using training sequences . our target is to characterize and optimize the throughput of a training - based multiuser mimo system over the parameters including the pilot symbols , the time allocation coefficient ( which specifies the fraction of the training phase in a transmission frame ) , the power allocation coefficient of each user , and the user number , etc .  due to the near - far effect , the distribution of the mimo channel matrix is rotationally variant .as a result , the technique developed in is not applicable to a mimo multiuser system .instead , we use the majorization theory as a basic tool to derive upper and lower bounds of the system throughput .we show that the derived throughput lower bound is asymptotically optimal for throughput maximization in the high snr regime .we also show that the upper and lower bounds are reasonably tight in various system settings .we note that the results in this paper is applicable to an arbitrary antenna and user configuration . further , to establish a close link with massive mimo , we use the random matrix theory to derive a closed - form expression of the system throughput as , , and scale at the same speed towards infinity , under the assumption of uniform large - scale fading ( i.e. , users are co - located ) .an interesting finding of this work is that , in a training - based multiuser mimo system , the optimal training length for throughput maximization is usually less than the number of active users .this is in contrast with the case of conventional mimo in which the optimality always occurs at ( implying that each user has one separate time slot for channel estimation ) . to understand this fact, we first note that a mimo multiuser system does not reduce to conventional mimo even if uniform large - scale fading is assumed .the key difference is that the total transmission power of a mimo multiuser system scales linearly with the number of active users , while that of a conventional mimo system is usually limited by a fixed total power budget .consequently , for a multiuser mimo system , the scalability of the total transmission power shifts the optimality point from to .particularly , when the optimality occurs at , there is not enough degrees of freedom to design orthogonal training sequences ( with length ) for all the users .this is again different from the case of conventional mimo in which the optimal training sequences are always orthogonal .  the disparity between the optimal and is further enlarged by the near - far effect .in fact , for throughput maximization in the considered multiuser mimo system , a portion of users with relatively poor channel quality should be kept silent in transmission .this is because , with the near - far effect , the channel qualities of users vary significantly from each other .the channels of far - off users are so weak that it will be a waste of resource for throughput enhancement if any time or power is allocated to these users . as such , a good strategy is to inactivate these far - off users in transmission , which translates to an enlarged gap between the optimal and .existing work related to the throughput analysis of multiuser mimo systems includes .specifically , marzetta examined the training - throughput tradeoff in a multiuser mimo broadcast channel by assuming orthogonal training sequences .kobayashi et .al studied the throughput optimisation of a multiuser mimo system by taking into account the effect of channel estimation error and finite channel state feedback .chi et al . considered the pilot sequence design for a multiuser mimo ofdm system , and derived the optimal pilot design in the sense of minimizing the mean - square error ( mse ) of the channel estimation .given the above work , the characterization of the optimal training - throughput tradeoff is still missing in the literature , which motivates the work presented in this paper .the remainder of this paper is organized as follows .first , in , we describe our system model and formulate the throughput optimization problem .then , in section [ section iii ] and section [ section iv ] , we establish upper and lower bounds of the system throughput under the assumption of arbitrary large - scale fading . later in section [ section v ] , we derive the optimal design of the system parameters under the assumption of uniform large - scale fading .we conclude the paper in section [ section vi ] .bold upper - case letters denote a matrix and bold - lower case letters denote a column vector .for a matrix , the element of row and column is denoted as . denotes an indentity matrix , where is an integer .the superscripts , stand for the transpose and hermitian transpose , respectively .the operators , , represent the inverse , the determinant , and the trace of a matrix , respectively .we use to denote the norm of a vector , and to denote the kronecker product .the vector inequality means that majorizes ; log denotes logarithm with base 2 . ; represents the diagonal matrix with the element being ; represents the diagonal matrix obtained by setting the off - diagonal elements of to zeros ; represents the vector specified by the diagonal of , with the entry of being . for a square matrix , represents the vector of the eigenvalues of ( counting multiplicity ) arranged in a descending order .consider a multiuser mimo system , where single - antenna users deliver information to an -antenna base station ( bs ) .assume that both and are very large but finite .the channel is block - fading , i.e. , the channel keeps invariant within coherence time . the corresponding channel model for a frame of symbols is given by  [ channel ] or equivalently   where represents the received signal matrix at the bs , and ^{\rm t}\!\in\!\mathbb{c}^{k\times t} ] , the optimal for is given by where is the root of    for a general snr , we have no explicit expression of the optimal in terms of .but we have a closed - form expression of the optimal in the high snr regime , as presented below .[ corollary 2 ] for any given  ] .   in this subsection ,we present an approximate expression of the throughput by using the random matrix theory .let be a non - zero eigenvalue of .then  [ propo : eig_dis ] as with fixed ratios of and , the asymptotic distribution of is given by  [ eq : f_beta ]  alignat=2 & , &  [ eq : f_beta_a ] + & , & > [ eq : f_beta_b ] ,  where , , and .recall that the elements of are independently drawn from .then , the proof is immediate by noting theorem 2.35 in .  with proposition [ propo : eig_dis ], the throughput in can be approximated by .it is known that the large - system approximation is accurate even when the system parameters are relatively small .compared with , an advantage of is that no monte carlo simulation is required in evaluating the throughput .also , provides a simpler analytical throughput characterization when applying the analytical methods in this work to a massive mimo setup .alignat=2 & ( 1- ) n _a^b(1 + ) d , & kt [ eq : r_asyform_a ] + & ( 1- ) n _ a^b(1+)d , & k > t .[ eq : r_asyform_b ]  [ eq : r_asyform ]     we now present numerical results to verify the analysis .[ fig : approximation - error ] illustrates the throughputs given by and against with various values of .the settings are , , db , , and . in [ fig : approximation - error ] , we see that the two curves always coincide with each other .this demonstrates that is indeed a good approximation of the throughput in .[ fig : optimal_k ] illustrates the optimal user number versus under various snr values .the simulation settings are as follows : , , .we see that the optimal is always no less than , and it converges to for an arbitrary value of when goes to infinity .this is in well agreement with theorem [ thm3 ] and corrolary [ corollary 2 ] .   vs the time allocation factor under various values of snr . ][ fig : r_n_t1000 ] illustrates the throughput against the snr with and . the optimal pilot design and the random pilot designare considered in simulation .the optimal pilot design is given by theorem [ thm3 ] .for random pilot design , we set and ; each element of the pilot matrix is independently drawn from a complex gaussian distribution , and then each row of is scaled to meet the power constraint for each user .in both designs , the optimal is obtained through an exhaustive one - dimensional search .we see that the optimal pilot design significantly outperforms the random pilot design especially in the medium to high snr regime .we see that the throughput increases with .this is because a larger implies a higher power gain of the receiving antenna array .moreover , we also include the upper and lower bounds developed in sections [ section iii ] and [ section iv ] .we see that these bounds are very tight in the medium to high snr regime .   under different values of .in this paper , we analyzed the tradeoff between the channel estimation quality and the information throughput of a training - based multiuser mimo system .we first studied the pilot design , and established upper and lower bounds for throughput maximization .we then considered the optimization of power coefficients for training and data transmission .we showed that the established lower bound is asymptotically optimal in the high snr regime .our analysis revealed that the optimal training design strategies for a multiuser mimo system are very different from those for conventional mimo .for example , we showed that the optimal training sequences for throughput maximization in a multiuser mimo system are not necessarily orthogonal to each other .also , due to the near - far effect , the optimal training design strategy for throughput maximization is to inactivate a portion of users with the weakest channels in transmission .these findings will provide insights and guidance for the practical design of a training - based multiuser system .  future research may arise in a number of directions .for example , the work in this paper was focused on the case of a single bs with multiple antennas , without considering the interference from other base stations .bs cooperation can significantly mitigate such interference ; see , e.g. , and the references therein . how to characterize the tradeoff between the channel estimation overhead and the information throughput in a cellular network with bs cooperationwill be a challenging research topic deserving future endeavour .as problems ( [ eq : optprob_1 ] ) and ( [ eq : optprob_5 ] ) have the same objective function , it suffices to show that the feasible region of ( [ eq : optprob_5 ] ) contains that of ( [ eq : optprob_1 ] ) , or in other words , to show that ( [ eq : optprob_5_b ] ) is implied by ( [ eq : optprob_1_b ] ) . to proceed ,let the singular value decomposition of be where is a diagonal matrix with non - negative diagonal elements , and and are unitary matrices .then , ( [ eq : optprob_1_b ] ) can be equivalently expressed as where is the element of ; is the diagonal element of for and otherwise .note that , and that is a concave function of .then , we obtain where the first step is from the jensen s inequality , and the second step from ( [ eq1 ] ) . noting that ( [ eqx3 ] ) is equivalent to ( [ eq : optprob_5_b ] ) , we conclude the proof of theorem [ thm1 ] .   for ease of disscussion, we first ignore the effect of in throughput maximization .the minimization of ( so as to maximize the throughput ) is discussed at the end of the proof .  from theorem 4.3.26 in , we obtain the definition of the majorization `` '' can be found in appendix [ appendix a ] . from the matrix theory , and share the same set of nonzero eigenvalues .that is , and have the same set of nonzero elements .thus , problem ( [ eq : optprob_5 ] ) can be recast as to maximize the rate in ( [ eqxqq1 ] ) subject to ( [ eq : optprob_52_b ] ) and ( [ eq4 ] ) . as the rate in ( [ eqxqq1 ] ) is monotonically increasing in each , we see that the maximum of the problem occurs when ( [ eq : optprob_52_b ] ) takes the equality . therefore , problem ( [ eq : optprob_5 ] ) can be rewritten as  [ eq : optprob_10 ]   where for , and for .we now solve problem ( [ eq : optprob_10 ] ) for two different cases , namely , and .for the case of , we have .note that in ( [ eqxqq1 ] ) is symmetric and concave with respect to . from proposition c.2 in , in ( [ eqxqq1 ] ) is schur concave .thus , the optimal to problem ( [ eq : optprob_10 ] ) is given by .this implies that the optimal to ( [ eq : optprob_5 ] ) satisfies ( [ eq5 ] ) , which proves the first half of the theorem .  for the case of , as the rank of is at most , we have .thus , unlike the case of , we can not set to . then , from the schur concavity of the throughput function , the optimal should be the smallest vector that majorizes .noting the equivalence between problems ( [ eq : optprob_5 ] ) and ( [ eq : optprob_10 ] ) , we see that the optimal to ( [ eq : optprob_5 ] ) satisfies the two conditions in the second half of the theorem .  to complete the proof of theorem [ thm : optimal_pilots_content ] , we still need to show that the above pilot design of also minimizes the equivalent noise power in ( [ sigma_v ] ) ( and hence indeed maximizes the throughput ) . from ( [ sigma_v ] ) , we see that minimizing is equivalent to maximizing where is defined in ( [ eq : x_tilde ] ) . clearly , is maximized when the equality in ( [ eq : optprob_52_b ] ) holds .this agrees with the optimality condition ( [ eq5 ] ) for the case of ; it also agrees with condition 1 in theorem [ thm : optimal_pilots_content ] for the case of .satisfying conditions 1 and 2 is presented in remarks [ rmk1 ] and [ rmk2 ] . ]this concludes the proof .let be the submatrix of that consists of the first columns of .then , . as , be approximated by .then , the throughput can be written as then , the optimization problem reduces to  [ eq : optprob_3011 ]   the solution to problem ( [ eq : optprob_3011 ] ) is described as follows . notethat is concave in , and that is convex in .thus , in ( [ eqqqx1 ] ) is concave in , and so problem ( [ eq : optprob_3011 ] ) is a convex problem . by solving the kkt conditions ,we obtain and , which concludes the proof .   from corollary [ corollary 1 ], we see that algorithm [ alg2 ] outputs the optimal in the sense of throughput maximization when .the corresponding optimal and at high snr are given by theorem [ thm7 ] , which is in agreement with the statement in theorem [ thm6 ] .then , it suffices to focus on the case of , i.e. , is a tall matrix .  without loss of generality, we assume that there are active users with . , we have . ]let be the pilot matrix for those active users , and be the corresponding diagonal fading - coefficient matrix .we consider the limiting process of as , where is an arbitrary constant invariant to .let the compact svd of be , where satisfies , is a diagonal matrix with the diagonal elements being the singular values , satisfies , and is the rank of .then , as , we obtain    thus , as , the equivalent noise power in ( [ sigma_v ] ) satisfies where is the diagonal matrix obtained by deleting the rows and columns of corresponding to inactive users .note that for , i.e. , in ( [ eqqq2 ] ) is unbounded when .this implies that the optimal number of active users for throughput maximization can not exceed .moreover , from theorem [ thm7 ] , to maximize throughput in the case of , all the users should be active .therefore , the optimal number of active users for throughput maximization is given by .we now discuss how to select the active users . without loss of generality, we assume that users are the selected active users .this implies that .then , as , the throughput in ( [ eqxqq1 ] ) can be approximated by , where is given by ( [ eqq2 ] ) , is given by ( [ eqi1 ] ) for , and consists of the first columns of .further note that , to maximize at high snr , for .then , the throughput is given by ( [ eqqqx1 ] ) with replaced by . from ( [ eqqqx1 ] ) , we see that the optimal choice to maximize the throughput is to select users with the maximum values . finally , the optimal for the active users are given by theorem [ thm7 ] , which concludes the proof .following the proof of theorem [ thm : optimal_pilots_content ] , we first ignore the effect of in throughput maximization. we will discuss the minimization of ( so as to maximize the throughput ) at the end of the proof .with and , we can rewrite ( [ eq : r_x_tilde ] ) as \ ] ] where , and where the last step follows from the singular value decomposition : . then , as has the same distribution as does , we can further rewrite ( [ eq0x ] ) as \ ] ] where is given by also , the eigenvalue vector of can be expressed as , where is the diagonal element of . from theorem 4.3.26 in , we have , where is an all - one vector of an appropriate size . then , problem ( [ eq : optprob_1 ] ) can be recast as  [ eq : optprobb_1 ]   note that in ( [ eqxqq1 ] ) is symmetric and concave with respect to . from proposition c.2 in , in ( [ eqxqq1 ] )is schur concave .thus , the optimal should be the smallest vector that majorizes . more specifically , for the case of , zero padding in is not necessary .thus , the optimal is simply taken as .this implies that . for the case of , as the rank of is at most , we see that is padded with at least zeros .then , the optimal is given by and .this implies that ( [ eq : xpx ] ) holds .  to complete the proof, we still need to show that the above pilot design of minimizes the equivalent noise power in ( [ sigma_v ] ) .the detailed argument is very similar to the corresponding part of the proof of theorem [ thm : optimal_pilots_content ] in appendix [ a2 ] . we omit the details for brevitythis concludes the proof of theorem [ thm3 ] .we first show that for any given , the optimal always satisfies . to see this , we need the following two facts : 1 ) is monotonically increasing in when ; 2 )  ] and }\leq\cdots\leq y_{[n]} ] are in an increasing order , and the second inequality follows from . together with ( [ eqq10_b ] ) , we have }\leq\frac{j - m}{k - m}\sum_{i = m+1}^{k}y_{[i ] } = \!\!\sum_{i = m+1}^{j}x_{[i]}^*,\ j = m+1,\cdots , k.  \end{aligned}\ ] ] the proof concludes by combining ( [ eqq11 ] ) and ( [ eqq12 ] ) .the authors would like to thank ...   l. zheng and d. n. c. tse , `` communication on the grassmann manifold : a geometric approach to the noncoherent multiple - antenna channel , '' _ ieee trans .inf . theory _ ,2 , pp . 359383 , feb . 2002 .b. hassibi and b. m. hochwald , how much training is needed in multiple - antenna wireless links ? 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Abstract: in this paper , we endeavour to seek a fundamental understanding of the potentials and limitations of training - based multiuser multiple - input multiple - output ( mimo ) systems . in a multiuser mimo system , users are geographically separated . so , the near - far effect plays an indispensable role in channel fading . the existing optimal training design for convenitional mimo does not take the near - far effect into account , and thus is not applicable to a multiuser mimo system . in this work , we use the majorization theory as a basic tool to study the tradeoff between the channel estimation quality and the information throughput . we establish tight upper and lower bounds of the throughput , and prove that the derived lower bound is asymptotically optimal for throughput maximization at high signal - to - noise ratio . our analysis shows that the optimal training sequences for throughput maximization in a multiuser mimo system are in general not orthogonal to each other . futhermore , due to the near - far effect , the optimal training design for throughput maximization is to deactivate a portion of users with the weakest channels in transmission . these observations shed light on the practical design of training - based multiuser mimo systems .   training - based multiuser mimo , throughput maximization , massive mimo


melanocytic nevi are benign proliferations of melanocytes , the skin cells that produce the pigment melanin .they are by definition benign , but 50% of malignant melanomas arise from pre - existing nevi .current diagnostic methods of melanocytic lesions are based on hystopathology and dermoscopy which reveal wide morphological diversity and evolution patterns whose origin is still unknown . leaving aside the controversial cases of spitz and blue nevi , neviare categorized from dermoscopic analysis into globular , reticular , structureless brown and mixed patterns ( the latter are subdivided into mixed pattern with globular structureless areas in the center or mixed pattern with globuls at the periphery ) .the classical theory of nevi evolution formulated in 1893 by unna claims that nevi originate from melanocytes proliferating at the dermo - epidermal junction ( lentigo simplex and junctional nevus ) that form nests ( compound nevus ) and eventually migrate completely into the dermis ( dermal nevus ) .more recently , cramer described an opposite model known as the theory of upward migration .cramer suggested that precursor cells of melanocytes deriving from pluripotent stem cells of the neural crest wander during embryogenesis along nerves into the dermis , mature here and finally migrate as functional melanocytes into the epidermis .cramer s last migration step , however , was never broadly accepted .more recently kittler et al . suggest that nevi can migrate not only vertically but also horizontally , explaining how nevi can expand in the course of time .the underlying limitation of these theories is that they are based on hystopathological observation only , and do not reflect the dynamics of an individual nevus . from a diagnostic point of viewit would be extremely useful to correlate the morphological features of nevi to the degree of malignancy or pre - malignancy .this issue is intricate because nevi proliferate in a complex microenvironment that can mediate cell behavior through the composition , structure , and dimensionality of the extracellular matrix ( ecm ) , the polymeric scaffold that surrounds cells within tissues .for example , a malignant phenotype can be reverted into a nonmalignant one by specifically blocking aberrant adhesion of the cancer cell to its extracellular scaffold .recent research shows that mechanical properties of the tumor microenvironment and of the normal tissue can influence tumor growth and dynamics , in way that is still poorly understood .mechanical stresses such as those experienced by cancer cells during the expansion of the tumor against the stromal tissue have been shown to release and activate growth factors involved in the progression of cancer . moreover, the stiffness of the matrix surrounding a tumor determines how cancer cells polarize , adhere , contract , and migrate , and thus regulates their invasiveness .forces exerted by cancer cells as they migrate through the ecm has been quantified accurately using traction force microscopy another possibility is that mechanical stresses directly regulate the growth and death rates of cancer cells as shown by montel et al . who induce osmotic stress by adding dextran , a biocompatible polymer that is not metabolized by cells .several studies in the literature indicate important changes in cellular functioning due to osmotic pressure , but the stresses involved ( in the mpa range ) were orders of magnitude larger than those ( in the kpa range ) studied in refs .it is interesting to notice that compressive stresses of slightly less than 1 kpa applied through a piston were recently found to induce a metastatic phenotype in cancer cells . while osmotic pressure may have a different origin than compressive mechanical pressure the effect on cells is exactly the samethis was demonstrated experimentally in refs . by applying osmotic pressure directly on the cells , by adding dextran to the growth medium , or indirectly , by placing the cells inside a dialysis bag which was then placed into a dextran solution . in both cases ,the effect of pressure on the cells was found to be exactly the same .this is not surprising , since osmotic pressure corresponds indeed to a _real _ mechanical pressure on the membrane due to the collisions with solute molecules .  in a recent paper , simonsen andcoauthors showed that interstitial fluid pressure ( ifp ) was associated with high geometric resistance to blood flow caused by tumor - line specific vascular abnormalities in xenografted tumors from two human melanoma lines with different angiogenic profiles . in another recent paper ,wu and colleagues investigated how nonlinear interactions among the vascular and lymphatic networks and proliferating tumor cells might influence ifp transport of oxygen , and tumor progression .they also investigated the possible consequences of tumor - associated pathologies such as elevated vascular hydraulic conductivities and decreased osmotic pressure differences .all these parameters might affect microenvironmental transport barriers , and the tumor invasive and metastatic potential , opening interesting new therapeutic approaches .in general , understanding the influence of mechanical stress on cancer growth could shed new light on tumor development and progression .  to investigate the role of the environment in the growth of melanocytic nevi, we introduce a computational model of elastic cells proliferating in a layered non - linear elastic medium representing the skin .several computational models have been introduced in the past to simulate cancer growth in vitro and in vivo .they range from individual cell models , to lattice models and continuum models .the aim of our investigation is to correlate the morphological features of the growing nevus with the location of the originating melanocyte and the mechanical properties of the environment . to this end , a natural strategyis provided by individual cell models where each cell can be mechanically deformed and moves according to mechanical forces due to other cells and to the ecm .models of this kind have been used in the past to simulate the growth of the epidermis and the growth of cancer cell colonies inside a confining medium .we first report experimental results on melanoma cell lines showing that a low osmotic pressure reduces proliferation , in agreement with other reports for other tumors .we have performed experiments on two cell lines , the first obtained from a primary tumor and the second from a metastatic one , both from the same patient .we find that melanoma cells stemming from a primary tumor are less sensitive to pressure than the metastatic ones .we find a similar results by growing the same cells in collagen coated plates , simulating the effects of the ecm .melanoma is widely believed to be originating from melanocytes , which are not easily cultivated in vitro .therefore melanoma cell lines is the most sensible in vitro model for nevi .we expect that the effect of pressure on proliferation seen in melanoma cells will be even stronger for melanocytes since melanoma cells are much more resistant .numerical simulations of the model yield different growth patterns for nevi originating in the epidermis , either close to the basal membrane or to the stratum corneum , or in the dermis .different initial locations give also rise to different mechanical properties as we show by measuring compressive stresses on the proliferating melanocytes .compressive stresses are fluctuating strongly inside the nevus , with an average stress that typically increases until the ecm breaks leading to stress drop . a crucial role in the growth pattern is played by the basal membrane , separating the epidermis from the dermis . depending on its mechanical properties ,the membrane can resist the pressure induced by the nevus or fail leading to the invasion of the dermis .in this case , invasion is not correlated with an increased malignancy of the tumor but just to mechanics .this observation of course does not exclude that malignant phenotypes may be induced by mechanical stresses as previously reported in the literature .it is in fact known that malignant cells break the basal membrane by matrix metalloproteinases ( mmps ) .we simulate this process explicitly showing that it accelerates the invasion of the dermis , but the general morphology remains the same .we simulate an individual cell based model for the growth of melanocytic nevi in the skin .we consider a two dimensional vertical section of the skin ( see figure [ fig:1]a ) consisting of two main layers , the _ epidermis _ and the _ dermis _ , separated by the _ basal membrane_. the deepest layer of the skin , the hypodermis , is not considered in the present study . in our model , the dermis constitutes the bottom part of the skin , whereas the upper layer is represented by the epidermis and by its most superficial part , the _stratum corneum_. the model describes the skin as a highly organized elastic organ , whose mechanical properties are specific of any of its constituent layers .periodic boundary conditions are applied to any of the skin layers along the horizontal direction , while open boundary conditions are imposed in the vertical directions .[ [ section ] ]  the dermis is arbitrarily divided into two anatomical regions : the papillary and reticular dermis .we consider only the papillary layer , whose thickness we set to .we model the dermis as a network of cells with an average radius , arranged in a disordered lattice and connected by non - linear springs .the disordered lattice is obtained starting from a triangular lattice , displacing the nodes randomly by a small amount .the resulting configuration is triangulated by a voronoi algorithm .nearest neighbors cells are connected by non - linear springs with a rest length equal to the resulting distance between the displaced cells .cells here represents fibrolasts , macrophages , and adynocytes present inside the papillary layer , while springs model the effect of the ecm , composed by collagen fibers and elastin .we set the average length of these springs at .the dermis also contains several irregularities such as blood vessels , lymph vessels , nerve endings and skin appendages such as air follicules , small hair muscles and sebaceous glands , but we ignore these details in the present model .the mechanical behaviour of the dermis has been studied experimentally using in - vivo suction and tension tests as reported in  . in these studies ,the mechanical response of the dermis layer to an external stress was found to obey to the following non - linear _ james - green - simpson _ pressure - displacement relationship  where and represent respectively the stress and the stretch of the tissue , the strain being just .the expression ( [ stress_nh ] ) is derived from the strain energy density function usually adopted to mimic the non - linear behaviour of the skin  .here we use the same law to model the deformation of the ecm springs , setting their elastic constant to and , and introduce a failure stress at , at which the spring breaks .the non - linear elastic behavior of the springs is illustrated in fig [ fig:1]b , showing that when the stress overcomes the spring breaks and the stress goes to zero .[ [ section-1 ] ]  the epidermis is a multilayered tissue composed of the stratum granulosum , stratum spinosum and stratum corneum. since the elastic properties of the stratum corneum are cosiderably different from the other two , we will consider it separately .the thickness of the epidermis ( stratum granulosum and stratum spinosum ) is and it is composed mainly by keratinocytes , whose average radius is . in the model ,keratinocytes are placed at the nodes of a disordered lattice , obtained as discussed above for the dermis .the turnover rate of melanocytes is assumed to be much faster than the natural turnover rate of keratinocytes , modelled explicitly in ref . but ignored here .this is justified since the typical turnover for melanoma cells is 1 - 2 days which is much smaller than the turnover of keratinocytes ( 50 - 70 days ) . to simulate the packing of keratinocytes in the epidermis , we chose the radius of each cell after the voronoi triangulation so that no neighboring cell overlaps .cells are then coupled by non - linear springs accounting for the neo - hookean behavior observed in in - vivo experiments  .experiments suggest that the non - linear pressure - displacement relation in eq .( [ stress_nh ] ) captures the mechanical properties of the epidermal layer with a good accuracy using the same parameters values obtained for the dermis  . in the model , we link the cells by non - linear springs with , and . the average length of the links between cells coincides with the average cell radius , hence two keratinocites may often be in contact and will thus interact as we discuss below .[ [ section-2 ] ]  the stratum corneum is composed of dead corneocytes of an average radius of .although it may consists of up to 15 - 20 layers of corneocytes , in our simplified numerical model only one single layer is taken into consideration .experiments  have provided evidence that the stratum corneum mechanical properties are consistent with a linear hookean law . here , we consider elastic cells linked by linear springs with young s modulus and failure stress set to .the stratum corneum cells are linked to the epidermis by linear springs whose young s modulus is .[ [ section-3 ] ]  the basal membrane constitutes and elastic sheet separating the dermis from the epidermis .it has a complex structure composed of the lamina reticula and lamina basale connected by collagen fibrils ; usually the thickness is around . in our modelthe basal membrane is represented by an assembly of fictitious cells of radius , connected by linear springs with young s modulus which we set either or .the failure stress is of the springs is set to .the fictitious cells composing the basal membrane have hertzian repulsive interactions with the other cells thus providing an effective barrier to their motion . to represent the highly corrugated structure of the membrane, we use a simple periodic function with amplitude equal to and period .dermis and epidermis cells are connected to the basal membrane from below and above respectively , these springs obey to a linear hookean force with young s modulus equal to . in the model , the basal membrane can be broken mechanically , but we also consider the production of mmps . to model this effect , we stipulate that at each time step each cell composing the nevus can dissolve one link with a probability that decreases with the distance from the cell : with and .[ [ section-4 ] ]  cells are modelled as non - linear elastic spheres and , for the sake of simplicity , we employ the same mechanical constants for all the cells composing the epidermis the dermis and the nevus . when two cells are in contact they repel by a finite - thickness hertzian law  where is the cell s young s modulus , set to  , is the poisson s ratio , with and radii of the contact cells , is the indentation depth and ( see fig .[ fig:1]c ) .we notice that cells are often modelled as incompressible ( ) , but the value of poisson ratio is difficult to estimate experimentally .individual cell models use compressible cells as we do and employ  . herewe use but its precise value is not really important in the framework of our simulations : changing would just imply a modification of the pre - factor of eq .[ hertz ] , leaving qualitatively unaffected the numerical results hereby displayed .we notice that , although eq.([hertz ] ) refers to the experimental situation of a microsphere indenting a finite - sized soft gel , it captures the non - linear corrections to the usual hertzian law , needed when large deformations occur . in most simulations ,nevi cells are interacting by hertzian repulsion only , while the other cells are typically linked also by non - linear spring .we have also tested for the effect of adhesion between nevi cells introducing a radial force , with .adhesive forces only act when the indentation depth is larger than .[ [ section-5 ] ]  in the simpler implementation of the model , nevi cells have no direct interaction with the ecm . an indirect interactionexist since nevi cell interact with other cells which are held together by the ecm .this indirect interaction is , however , weak inside the dermis . to overcome this limitation, we also consider a direct interaction between nevi cells and the ecm . in practice , we follow the same strategy used to model the basal membrane and place three fictitious cells with radius on each ecm link . in this way , nevi cells interact directly with the ecm .[ [ section-6 ] ]  the quasi - static growth of the nevus is simulated by randomly selecting a single cell among those belonging to the dermis or epidermis .this cell represents the first proliferating cell and therefore does not present any linkage with the surrounding skin network .the growth process is obtained by duplication of the initial cell , so that the new cell is still in contact with the first one , but its growth direction is picked at random , i.e. lattice - free : such a choice is motivated by the fact a proliferative population will always eventually fill the lattice , suppressing cell - to - cell crowding effects .the insertion of a new cell encompasses an overall rearrangement of the entire network , which is achieved by minimization of the stress within the system .since growing cells are completely disconnected from the surrounding skin tissue , they only interact through hertzian - like forces ( eq . [ hertz ] ) and possibly through adhesion .two different choices of protocol are applied to the duplication mechanism : a cell can either be picked randomly among those composing the nevus , or the cell can be selected with a rate that depends on the compressive stress . in particular, the probability is assigned to each cell , where represents compressive stress of the -th cell and is the average stress .finally , the cell that duplicates is selected with probability .[ [ section-7 ] ]   we test the effect of low osmotic pressure ( from 0.1kpa to 1kpa ) on cellular proliferation of primary and metastatic human melanoma cell lines , igr39 and igr37 , respectively at short ( 3 days ) and long time ( 6d ays ) .as shown in fig .[ fig:2 ] , the treatment with 1kpa for 6 days decreases the cellular proliferation of igr39 while igr37 are only slightly affected . according to table1 , the osmotic pressure increases cell death to necrosis while the percentage of cells in early apoptosis slightly decreases at low pressures and then returns to the original value at higher pressure ( table1 ) .we are tempted to attribute this last result to a statistical fluctuation .we also perform a colony formation experiment using the crystal violet assay on primary and metastatic melanoma cells , following the prescriptions of ref. . in fig[ fig:2bis]a we report the cumulative colony size distribution , where is the number of cells in each colony , for igr39 cells with or without osmotic pressure .each distribution is obtained by combining the result of colonies obtained in six different wells .typical images of the colonies are reported in fig [ fig:2bis]c .we fit the distributions with the solution of a continuum time branching process in which each cell divides with rate which yields a cumulative colony size distribution after days given by  from the fit , we estimate the average division rate as divisions / day in normal conditions and divisions / day under an osmotic pressure of 1kpa . in fig .[ fig:2bis]b we report the average colony size with its standard error for igr39 and igr37 cells with and without osmotic pressure .while osmotic pressure hinders colony growth in both cases , the effect is much stronger for primary tumor cells than in metastatic ones .   to confirm that the general effect of stress we observe on growing melanoma cells is not restricted to osmotic pressure , we perform proliferation experiments using collagen coated plates simulating the effect of to the extracellular matrix on the cells . as shown in fig .[ fig:2tris ] , the presence of collagen significantly reduces proliferation in primary melanoma cells ( igr39 ) while the effect is much weaker in metastatic cells ( igr37 ) .we next induce an osmotic pressure of 1kpa in the collagen coated plates and find no additional significant result with respect to the case with collagen alone .we can interpret this result by assuming that collagen already induces a strong stress on the cells so that additional osmotic stress has no significant effect of the growth .we can thus suggest that osmotic pressure and collagen both act in similar ways on melanoma cells : they reduce proliferation but the effect is much stronger in primary cells than in metastatic ones .[ [ section-8 ] ]   we perform numerical simulations of the growth of melanocytic nevi for different conditions . in particular , we study the morphology resulting from the growth using different locations of the initial seed .as illustrated in fig [ fig:3 ] , we consider four type of initial locations : deep inside the epidermis ( fig .[ fig:3]a and movie s1 ) , in the epidermis but close to the basal membrane , either in the minima ( fig .[ fig:3]b , fig .[ fig:3]e and and movies s2-s3 ) or on the maxima of the membrane ( fig .[ fig:3]c , fig . [ fig:3]f and and movie s4-s5 ) , or inside the dermis ( fig .[ fig:3]d and movie s6 ) .for initiating cells close to the basal membrane , we consider two different scenarios with a strong ( fig .[ fig:3]b and [ fig:3]c ) or weak membrane ( fig .[ fig:3]e and [ fig:3]f ) .[ fig:3 ] indicates that the morphology of the nevus depends substantially on the location of the initiating cell .when the nevus starts up in the epidermis the growth is mostly horizontal and far from the dermis ( fig .[ fig:3]a ) . for initiating cells residing close to a strong basal membranethe growth follows the profile of the membrane ( fig .[ fig:3]b ) or expands vertically in the epidermis ( fig . [ fig:3]c ) depending on whether the initiating cells lies in the minima or maxima defined by the membrane profile .when the membrane is weak , however , it is easily broken by the growing cells that then invade the dermis ( fig . [ fig:3]e and fig .[ fig:3]f ) .the growth is more radial when the initiating cells is in the minimum of the basal membrane ( fig .[ fig:3]e ) and more vertical when it lies close to the maximum ( fig .[ fig:3]e ) .we record the compressive stresses sustained by the melanocytes during the growth of the simulated nevi . a pictorial view of the stress in each cell is shown in fig .[ fig:3 ] , where the color scale depicts compressed cells with a scale going from yellow ( low stress ) to red ( high stress ) .the images show that stresses are in general heterogeneously distributed and , as illustrated by the supplementary movies , fluctuate considerably after each division step .we notice that typically the most stressed cells are located in the inner part of the nevus , reproducing a general feature observed in tumor spheroids .furthermore , we see the formation of stress chains , typical of granular assemblies . to better quantify the evolution of the compressive stress we measure the average compressive stress experienced by the cells as a function of the number of cells in the nevus .the results reported in fig .[ fig:4 ] correspond to the different cases explored in fig .[ fig:3 ] . in all cases , pressure builds up as cells are duplicating until at some point the connective tissue breaks drastically reducing the average compressive stress experienced by the cells .this effect is particularly clear in the case of a nevus growing in the epidermis ( see fig . [ fig:4]a ) .the large error bars reported in fig . [ fig:4 ] indicate that stresses vary widely between individual cells but also among different realizations of the process .to better quantify the stress fluctuations experienced by individual cells , we report in fig .[ fig:5 ] an estimate of the distribution ( probability density function ) of the compressive stresses for different initial conditions .the distribution is sampled over different realizations and over different cells . since the distribution is expected to change with the number of cells present in the nevus ( as shown from the error bars in fig .[ fig:4 ] ) , we consider only the case for which the average pressure is highest ( the peak stress in each curve in fig .[ fig:4 ] ) . in all casesthe distribution displays an approximately exponential tail ( i.e. ) .compressive stresses are much higher for nevi growing in the epidermis than in the dermis .this is due to the general assumption made in the model that cells are more packed in the epidermis than in the dermis .when we include a direct interaction between the nevi cells and the ecm , as discussed in the model section , compressive stresses in the dermis increases but the general features of the results are unchanged ( see fig .[ fig : ecm ] and movie s7 ) .we have also tested for the effect of adhesion between nevi cells and found very little differences with respect to the case in which adhesion is absent ( see fig . [fig : ecm ] ) .   the results reported in fig .[ fig:3]e and [ fig:3]f refer to the case in which the basal membrane breaks due to the action of mechanical forces .simulations of the rupture of the basal membrane induced by mmps are reported in fig .[ fig : mpp ] and in movie s8 . in particular ,the movie s8 illustrates how the effect of mmps leads to a much rapid invasion of the dermis .this is because mmps typically break the basal membrane in different locations , while when the membrane ruptures mechanically in one location the stress in the rest of the membrane is released preventing further rupture .finally , we notice that the average compressive stress in the nevus is reduced when mmps are present ( see fig .[ fig : mpp]b ) .   inspired by the experimental results on primary melanoma that indicating that compressive stress inhibits growth , we perform the simulations with a stress dependent division rate , so that compressed cells divide less .the morphology of the resulting nevi changes only slightly with respect of the previous case ( compare fig .[ fig:6 ] with fig.[fig:3 ] ) , but we observe the development of incipient finger - like structures ( see in particular fig.[fig:6]c and [ fig:6]f ) .we also measure the average compressive stress and find that it is typically smaller than in the previous case ( see fig .[ fig:7 ] ) .this is due to the fact that in this case growth occurs in regions that are less compressed so that there is more space to accommodate new cells .   in fig .[ fig:8 ] , we report two examples of common melanocytic nevi as revealed by histological sections . in fig .[ fig:8]a , we suggest that an interaepithelial juction nevus that is all confined in the epidermis ( compare with fig . [ fig:3]b ) .views at different magnifications show that nevi cells are closely packed and press against the basal membrane as indicated by rounded pocket shown in the left panel . in fig .[ fig:8]b , we report an example of dermal nevus ( compare with fig . [ fig:3]d ) . in this case, cells are less packed and are scattered all through the dermis without pressing on the basal membrane .these images are in good qualitative with the results of our model that show a tendency for nevi cells to press from the epidermis into the dermis rather than the reverse .this effect is likely due to the different mechanical properties of the two skin layers , as assumed in the model .melanoma is a very aggressive tumor , being chemio- and radio - resistant once it becomes metastatic .its incidence is increasing both in europe and in us , making it a current challenging problem for research .it is commonly believed that nevi may progress into melanoma , or alternatively regress by differentiating .since diagnosis is mostly based on dermoscopic and histological analysis , it is extremely important to correlate the morphological features of nevi to the degree of malignancy or pre - malignancy of the lesions . since nevi proliferate inside a complex microenvironment , we investigated the role of mechanics and geometry on the morphology and internal stresses of melanocytic nevi . using human melanoma cell lines ,we confirm also for melanoma a negative effect of the mechanical stress on cellular proliferation as already reported for other tumors .we show that osmotic pressure is more effective in primary melanoma cells than in metastatic ones , suggesting that metastatic cells come from a subpopulation more aggressive and insensitive to the mechanical properties of the environment .we corroborate this result by experiments of cell growth in collagen coated plates where again primary cells are more affected than metastatic ones .this suggests that cells composing melanocytic nevi may be sensitive to stress as well , an issue that we investigated computationally .we have devised a computational model for the growth of melanocytic nevi in a layered tissue representing the skin .numerical simulations of the model show that the morphology of the resulting nevus depends considerably on the environment in which the growth takes place , in the epidermis close or far from the basal membrane or in the dermis .this is interesting since environmental properties , in this case the mechanical behavior of the tissue , is rarely considered in the progression of nevi .simulations indicate that nevi are subject to fluctuating compressive stresses due to the tissue elasticity .if we introduce a stress dependence proliferation rate , we observe a decrease in the overall stress .our computational model is appropriate to describe a benign nevus , since we do not consider active motion of the cells but only their quasi - equilibrium displacement in response to elastic forces . to model the growth of melanoma cells, one should implement their mobility and their ability to degrade the surrounding extracellular matrix , by expressing mmps .we have incorporated this last aspect in the simulations showing that the invasion of the dermis occurs more rapidly .it is remarkable that many intriguing features observed in melanoma , such as the formation of rough tumor boundaries and the invasion of the dermis from the epidermis , are observed from simple rules combining mechanics and geometry , although we simplified many of the irregularities present in the dermis . as a matter of factwe expect that , while the overall shape of the nevus can be influenced by the degree of accuracy in the dermis description , the qualitative picture remains unchanged .a limitation of our model lies in its two - dimensional nature that , while it greatly simplifies the numerical calculations , could affect the values of the quantities we measure .we expect , however , that the general phenomenology we observe should be unchanged in a more realistic three - dimensional situation .[ [ section-9 ] ]  a master solution of dextran at 10% ( w / v ) was formed ( dextran from leuconostoc spp , fluka ) and the diluted to the desired concentration with complete medium .transformation from dextran concentration to osmotic pressure was performed according to the calibration curve measured in ref . .[ [ section-10 ] ]  human igr39 and igr37 cells were obtained from deutsche sammlung von mikroorganismen und zellkulturen gmbh and cultured as previously described .igr39 was derived from a primary amelanotic cutaneous tumor and igr37 from an inguinal lymph node metastasis in the same patient .[ [ section-11 ] ]  the cells were plated on pre - coated collagen type i dishes ( sigma ) according to the manufacturer s instructions .  [ [ section-12 ] ]  cells are plated on 6 multiwells , fixed with 3.7% paraformaldeide ( pfa ) for 5 minutes and then stained for 30min with 0.05% crystal violet solution .after two washing with tap water , the plates are drained by inversion for a couple of minutes . in order to control the merging of different colonies ,the experiments are performed with different initial cell concentrations as described recently by our group .data analysis of the resulting colonies has been performed according to ref . .[ [ section-13 ] ]  the sulforhodamine b ( srb ) assay is used for cell density determination based on the cellular protein content according to ref .  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[ cols="^,^,^,^,^ " , ]c ) when in contact , the cells interact by a finite - thickness hertz law .the corresponding zero - thickness law is reported for comparison . ]  ) are denoted with . ]that is reduced to under osmotic pressure .b ) the average value of the colony size distribution with the associated standard error for igr39 and igr37 cells .statistically significant results according to the ks test ( ) are denoted with .c ) the images show two representative examples of the colonies for 0 kpa and 1kpa conditions . ]  .the graph shows that the presence of collagen slows cell growth both in primary ( igr39 ) and metastatic ( igr37 ) cells but the effect is much stronger for primary cells . additional osmotic pressure does not significantly alter the results .statistically significant results , according to the ks test ( ) , are denoted with . ]   according to the color bar .the maximum and minimum for each configuration are equal to ( in kpa ) : a) , b) , c) , d) , e) , f) .a ) nevi grown from melanocytes residing inside the epidermis tend to grow horizontally and do not spread towards the basal membrane .b ) nevi grown on the minima of a strong basal membrane tend to grow roughly parallel to the membrane itself .c ) nevi growing from the maxima of a strong basal membrane tend to grow vertically in the epidermis .d ) dermal nevi tend to have radial shape .when the basal membrane is weak , e ) nevi growing from the minima of the basal membrane invade the dermis in a radial fashion , while f ) when they start from maxima of the basal membrane the invasion of the dermis occurs more vertically . ]  , for the same conditions as in fig .[ fig:3 ] .the error bars represent the standard error of the mean .the stress is averaged over all the cells in a nevus and over at least ten statistically independent realizations of the growth process .the different panels represent different initial locations : a ) in the middle of the epidermis , b ) in the minima of a strong basal membrane , c ) in the maxima of a strong basal membrane , d ) in the dermis , e ) in the minima of a weak basal membrane , f ) in the maxima of a weak basal membrane . ]   of the compressive stresses ( in kpa ) experienced by melanocytes in a nevus .the results are sampled over different realizations of the process . since the distribution also changes with time, we only consider the time at which the average compressive stress is highest .the different panels represent different initial conditions corresponding to fig .[ fig:3 ] : a ) in the middle of the epidermis , b ) in the minima of the basal membrane ( for strong and weak membranes ) c ) in the dermis , d ) in the maxima of the basal membrane ( for strong and weak membranes ) . ]b ) b ) dermal nevi are confined in the dermis ( compare with fig .[ fig:3]d ) .cells are loosely packed and do not touch the basal membrane .the basal membrane is located at the separation between the upper ( epidermis ) and lower ( dermis ) skin layers and indicated by an arrow in panel a ) .images courtesy of dr .claudio clemente ][ [ section-18 ] ]  , but here the growth depends on the compressive stress acting on each cell .growing melanocytes are shown with a varying color that reflects their compressive stresses according to the color bar .the maximum and minimum for each configuration are equal to ( in kpa ) : a)7.46 , b) , c ) d) e ) f ) . ]  , for the same conditions as in fig .[ fig:6 ] .the error bars represent the standard error of the mean . the different panels represent different initial locations : a ) in the middle of the epidermis , b ) in the minima of a strong basal membrane , c ) in the maxima of a strong basal membrane , d ) in the dermis , e ) in the minima of a weak basal membrane , f ) in the maxima of a weak basal membrane . ]

Abstract: melanocytic nevi are benign proliferations that sometimes turn into malignant melanoma in a way that is still unclear from the biochemical and genetic point of view . diagnostic and prognostic tools are then mostly based on dermoscopic examination and morphological analysis of histological tissues . to investigate the role of mechanics and geometry in the morpholgical dynamics of melanocytic nevi , we study a computation model for cell proliferation in a layered non - linear elastic tissue . numerical simulations suggest that the morphology of the nevus is correlated to the initial location of the proliferating cell starting the growth process and to the mechanical properties of the tissue . our results also support that melanocytes are subject to compressive stresses that fluctuate widely in the nevus and depend on the growth stage . numerical simulations of cells in the epidermis releasing matrix metalloproteinases display an accelerated invasion of the dermis by destroying the basal membrane . moreover , we suggest experimentally that osmotic stress and collagen inhibit growth in primary melanoma cells while the effect is much weaker in metastatic cells . knowing that morphological features of nevi might also reflect geometry and mechanics rather than malignancy could be relevant for diagnostic purposes .