[Paleopsych] Euroresidentes.com: Spanish scientists use maths to cure terminal liver cancer

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Subject: Euroresidentes.com: Spanish scientists use maths to cure terminal liver

News from Spain: Spanish scientists use maths to cure terminal liver cancer
Tuesday, May 31, 2005
[The report itself appended after the comments. It may be best just to get the 
PDF. Go to the end for the URL.]

    By using a mathematical formula formula designed to strengthen the
    immune system, a team of scientists in Spain have succeeded in curing
    a patient who was in the last stage of terminal liver cancer.
    The team of researchers from the [4]Complutense University in Madrid
    believe that this discovery could open new doors for the treatment of
    solid cancerous tumours.
    The new treatment was developed in 1998 by a team led by Antonio Bru,
    a physicist who bases his theory on the idea that the evolution of
    solid tumors depends on a mathematical equation which defines their
    biological growth. An equation is then obtained in the laboratory and
    used to design a therapy to destroy the tumor.
    The scientists, who have carried out successful tests on mice over the
    past few years, announced yesterday that the only human experiment
    they have carried out so far has been a complete success.
    Apparantly the patient was suffering from liver cancer which had been
    diagnosed by his doctors as terminal and in its final stage. The
    scientists used a mathematical formula to create a treatment based on
    neutrofiles that strengthened the patient's immune system. The patient
    responded well to the treatment immediately and has since made a total
    recovery and has returned to work.
    The treatment produces no side effects.The Spanish scientists believe
    that their theory could be applied to treat all kinds of solid tumors
    although they will need to carry out many more tests on human patients
    before they can be sure.
    Update 1/06/2005:
    Today, the Complutense University (whose switchboards have apparantly
    been innundated with phone calls from people wanting to find out more
    about this news item) has published a communication on its website
    with a brief communication from Prof. Antonio Bru. The full article is
    [5]here. Below is a translation of Professor Bru's brief note which
    appears at the end of the article:
    Given the expectation generated by the news of the publication of the
    article Regulation of neutrophilia by granulocyte colony-stimulating
    factor: a new cancer therapy that reversed a case of terminal
    hepatocarcinoma in the Journal of Clinical Research, I would like to
    make the following points:
    1) The proposed treatment is still at an experimental stage and needs
    much wider experimentation before it can be validated.
    2) For this reason, at this moment there is no treatment protocol
    which enables it to be applied as a general treatment.
    3) Given that it is impossible for the Complutense University of
    Madrid to answer all the phone calls received, and bearing in mind how
    they can disrupt normal teaching and research activity, please send
    any enquiries to the following email address: bru at mat.ucm.es
    Dr. Antonio Brú, Departamento de Matemática Aplicada
    Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid
    [6]The Universal Dynamics of Tumor Growth by Antonio Bru and his team
    of researchers.

    posted by Euroresidentes, Spain. @ [7]11:30 AM


      * At [8]9:00 AM, Anonymous said...
        Additionally, they proved that with a sample of one, the variance
        of their data is zero, therefore irrefutable.

      * At [9]9:48 AM, cnycompguy said...
        well anonymous, i'd like to see what you've been doing in the time
        that these people have been trying, quite successfully, to save
        lives. I bet if you developed a terminal cancer you'd fucking be

      * At [10]9:51 AM, Anonymous said...
        Oh, how this comment is clever. Have you ever had a relative dying
        of cancer?

      * At [11]9:59 AM, Anonymous said...
        8 relatives died of cancer (including parents). Forget about zero
        variance and let them work (another anon just passing by)

      * At [12]10:08 AM, Anonymous said...

      * At [13]10:21 AM, Anonymous said...
        Anything that produces these kind of results, even if it is by
        chance, is worth delving into. Just give them money and time.

      * At [14]11:17 AM, Anonymous said...
        Good! Keep up the good work!

      * At [15]11:21 AM, Jill said...
        Fantastic news. Wow. I agree with the comment from
        last-anonymous-but-one. Give them funds and encouragement and see
        if this proves to be the breakthrough we've been waiting for ever
        since cancer research began. Well done Spain!!

      * At [16]11:24 AM, Anonymous said...
        If my liver cancer ever gets to the stage of being terminal I'll
        be glad of any work they'd done.

      * At [17]12:27 PM, [18]Pablo said...
        Although being optimistic we should also be careful and a bit
        I think more trials are needed and that will take time and money.

      * At [19]12:49 PM, Laura said...
        Congratulations Spain!!

      * At [20]12:58 PM, Anonymous said...
        These are really good news!!!
        I hope they can try it on more people to see the outcome.
        To first anonymous: ANY research done to save lives and improve
        quality of life souldn't be undervalued.
        Keep up the good work Spain!

      * At [21]1:41 PM, Anonymous said...
        Let's not get over excited here .... anonymous #1 is voicing a
        concern and it might all be too good to be true. The success rate
        would more likely not ever be 100% exactly, so lets try to keep it
        in perspective.
        Yes, it sounds promising and should be investigated some more and
        given it's effects on the current clinical thinking on and
        therapies against tumour growth, this study, even with whatever
        flaws may come to light, could kick start a whole new direction of
        attack against cancer.
        So well done Bru et al, keep up the good work.

      * At [22]1:49 PM, Anonymous said...
        what on earth did the equation actually "do" ? Does anybody know

      * At [23]3:06 PM, Greg said...
        No, sorry, no idea at all. The mind boggles. But I bet it isn't
        your good old straightforward x+y=z

      * At [24]3:39 PM, Anonymous said...
        Anonymous #1 is only partially correct. Yes, there is only one
        human sample, but they have done extensive lab studies on rats.
        The mechanics of the study are not species specific. Treatments
        based on the use of a specific drug are much harder to translate
        between species. This treatment is based on the fundamental
        development of tumors. It does not suffer similar limitations. My
        guess as to why there is only one human guiney pig is that earlier
        research has probably inspired others to study the effect on
        As such, time is the essence. The sad reality of learned studies
        these days is centered on the "publish or perish" mentality.
        Still, this kind of info SHOULD be out quickly. Even if it was not
        well grounded, a patient who has already been given up for dead
        might like to try it.

      * At [25]3:54 PM, Anonymous said...
        Irrefutable and anecdotal. Patients sometime get better, sometime
        diagnoses are
        incorrect. Without a controlled study this is just dangerous hope
        inflating hype.

      * At [26]4:18 PM, Anonymous said...
        The equation being talked about is actually a generalized growth
        model that appears to properly describe and predict tumor growth
        within a host body. This model isn't strictly theoretical or
        anecdotal, as it is based on a large amount of experimental data
        on cancer growth.
        This model predicts the known and documented problems with current
        cancer therapies (such as chemo), as well as the problem with
        biospies being unreliable for prediction, another known fact.
        Furthermore, this model suggests more effective methods for curing
        cancer based on affecting the growth factors of the tumor. These
        methods appear to have worked in at least one human subject who
        was beyond hope of cure by any conventional means.
        However, these methods aren't really the point to the research.
        The point is just understanding how cancer works, with the
        assumption that such an understanding will make curing cancer
        easier and more reliable. They certainly have enough data to make
        a good case for why their model of cancer growth is a good one.

      * At [27]4:37 PM, Anonymous said...
        One problem is this model is based on solid tumor growth. Not all
        cancers are solid. Malignant cancers tend to have no well defined
        boundary because the cancerous cells invade the surrounding tissue
        rather that pushing it out of the way when they go through

      * At [28]4:44 PM, Lucien Chardon said...
        I would tend to think that all problems can be solved by math.
        Algorythms are everywhere and they shape nature and the human
        body. Like someone renwoned said : if you have aa problem with
        your interface throw in a algorythm and it keeps the doctor

      * At [29]4:54 PM, Anonymous said...
        I don't really know that much about malignant cancer growth, but I
        don't see why oddly-shaped tumors would make them any less solid.
        Mathematically, they are still volumes, and they still have a
        surface and an interior. The basic growth model should still
        apply, it just gets more complicated.

      * At [30]4:55 PM, Anonymous said...
        The problem with this kind of news is that cancer sufferers, like
        myself, know that it probably won't be available for some years,
        until it's too late. This makes it much harder to stay positive.
        This kind of news should be repressed until the cure can be
        available to all. And what about the cost factor? Will it be out
        of reach to the ordinary person, like so many other treatments?

      * At [31]5:54 PM, Anonymous said...
        "An algorithm a day keeps the doctor away." How many cancer
        patients are lined up outside their University today? Having
        watched my father-in-law deteriorate over the past two years, even
        if the cure rate only matches that of chemo- and radiation
        therapies, if it saves some of the side-effects it's worth more
        funding and research.

      * At [32]6:08 PM, Anonymous said...
        I'm sorry you are suffering from cancer. I feel bad for you. But
        suppressing news of progress just to avoid giving you hope isn't
        really the direction to go.
        If anything, it would slow down funding which would slow down
        progress. If it bothers you that much to read about it, there IS a
        simpler solution.

      * At [33]9:37 PM, [34]Warren Feltmate said...
        This post has been removed by the author.

      * At [35]9:41 PM, [36]Warren Feltmate said...
        Have many of you read the paper they posted? It's a long read, as
        to be expected, but it's quite informative and looks to be very
        promising. It's posted at the end of the post, but here's another
        link if you missed it.
        Treatment Paper

      * At [37]10:11 PM, Anonymous said...
        I'm both hopeful and skeptical.
        Mathematics is an idealization, an abstraction of reality. A
        formula can predict what might happen in a double blind study of a
        larger population but it's not always 100% accurate.
        I'm sure that the zero side effects may be true in the model, but
        in group of real patients, that's not any more likely than a 100%
        cure rate.

      * At [38]1:07 AM, Anonymous said...
        Any step forward in correcting the biological data corruption know
        as Cancer is definitely a step in the right direction.
        Wish all those people working on this, not only in Spain, but
        around the world all the best to find a cure for this scurge on

      * At [39]3:27 AM, Anonymous said...
        There's no information here, at all. I don't get why this is even
        news. Of course new therapies for serious illness are exciting,
        but seriously, what a lot of nothing from an information

      * At [40]8:17 AM, Anonymous said...
        Having accurate (even fractal!) models to predict the weather
        doesn't make it any easier for us to change tomorrow's weather.
        Nor does accurate knowledge about how a child's brain grows allow
        us to necessarily change that growth.
        The referenced paper proposes that knowing the growth behaviour of
        the tumour can be used to better time chemotherapy, but I could
        not see any mention in the paper about the fortunate person whose
        liver cancer went into ?remission.

      * At [41]4:30 PM, Anonymous said...
        Some people said it was impossible to cross the horizon because
        boats will fall, other people said earth was flat, and some others
        said "it will be only 4 computers in the whole world". All of them
        was scientists too.
        Give Dr. Bru time, be skeptical, but don´t close your mind to
        Keep going Bru!!!!.

      * At [42]2:10 AM, Anonymous said...
        and doesn't the medical profession just talk about your chance of
        surviving cancer over a 5 year horizon. They don't say cured for
        all time that I know of.

      * At [43]8:53 AM, [44]Giu1i0 Pri5c0 said...
        Well, the first part of your statement is very true. Even if this
        develops into an operational therapy, it will take years and
        probably won't be effective in all cases.
        But I don't agree on repressing news. Science advances by
        spreading information and making it available to all. If the
        information is available, another researcher may be able to make
        further advances. A rich donor may fund the work of Prof. Bru. A
        patient with early stage cancer may refuse giving up and fight
        harder to stay alive until an effective therapy is developed...
        --- In reply to: "The problem with this kind of news is that
        cancer sufferers, like myself, know that it probably won't be
        available for some years, until it's too late. This makes it much
        harder to stay positive. This kind of news should be repressed
        until the cure can be available to all. And what about the cost
        factor? Will it be out of reach to the ordinary person, like so
        many other treatments?"

      * At [45]5:34 PM, Anonymous said...
        "dangerous", "hype"... well, some never get it.
        read: it's not about the clinical results. it's about the math
        that led to it all.
        endavant, bru et al!


    1. http://www.euroresidentes.com/Blogs/Spain_News/atom.xml
    2. http://www.euroresidentes.com/Blogs/Spain_News.htm
    3. http://www.euroresidentes.com/euroresiuk/indexuk.html
    4. http://www.ucm.es/
    6. http://www.biophysj.org/cgi/content/abstract/85/5/2948


The Universal Dynamics of Tumor Growth -- Brú et al. 85 (5): 2948
Biophysical Journal

    Antonio Brú^ *, Sonia Albertos^  {dagger} , José Luis Subiza^
    {ddagger} , José López García-Asenjo^  § and Isabel Brú^ ¶

    ^* CCMA, Consejo Superior de Investigaciones Científicas, 28006
    Madrid, Spain; ^{dagger} Servicio de Aparato Digestivo, Hospital
    Clínico San Carlos, 28003 Madrid, Spain; ^{ddagger} Servicio de
    Inmunología, Hospital Clínico San Carlos, 28003 Madrid, Spain; ^§
    Servicio de Anatomía Patológica, Hospital Clínico San Carlos, 28003
    Madrid, Spain; and ^¶ Centro de Salud La Estación, 45600 Talavera de
    La Reina, Toledo, Spain

    Correspondence: Address reprint requests to Antonio Brú, Serrano 115,
    28015 Madrid, Spain. Tel.: 34-91-7452500; E-mail:


    Scaling techniques were used to analyze the fractal nature of^
    colonies of 15 cell lines growing in vitro as well as of 16^ types of
    tumor developing in vivo. All cell colonies were found^ to exhibit
    exactly the same growth dynamics--which correspond^ to the molecular
    beam epitaxy (MBE) universality class. MBE^ dynamics are characterized
    by 1), a linear growth rate, 2),^ the constraint of cell proliferation
    to the colony/tumor border,^ and 3), surface diffusion of cells at the
    growing edge. These^ characteristics were experimentally verified in
    the studied^ colonies. That these should show MBE dynamics is in
    strong contrast^ with the currently established concept of tumor
    growth: the^ kinetics of this type of proliferation rules out
    exponential^ or Gompertzian growth. Rather, a clear linear growth
    regime^ is followed. The importance of new cell movements--cell^
    diffusion at the tumor border--lies in the fact that tumor^ growth
    must be conceived as a competition for space between^ the tumor and
    the host, and not for nutrients or other factors.^ Strong experimental
    evidence is presented for 16 types of tumor,^ the growth of which cell
    surface diffusion may be the main mechanism^ responsible in vivo.
    These results explain most of the clinical^ and biological features of
    colonies and tumors, offer new theoretical^ frameworks, and challenge
    the wisdom of some current clinical^ strategies.^


    Tumor growth is a complex process ultimately dependent on tumor^ cells
    proliferating and spreading in host tissues. The search^ for the
    underlying mechanisms of tumor development and progression^ has been
    largely focused on the molecular changes accounting^ for the malignant
    phenotype at the cell level, while our knowledge^ on tumor growth
    dynamics has remained scarce. In part, this^ has been due to
    difficulties in developing tools able to describe^ growth processes
    associated with disordered phenomena. As with^ many natural objects,
    cell colonies are fractal (Losa et al.,^ 1992[39] [fig-down.gif] ;
    Cross et al., 1995[40] [fig-down.gif] ; Losa, 1995[41] [fig-down.gif]
    ), and a description of^ their very complex contours using classical
    Euclidean geometry^ is very difficult to provide. However, the
    contours of objects^ can give valuable indications about their dynamic
    behavior,^ and the fractal nature of the contours of tumors/cell
    colonies--with^ their scale invariance (self-affine character)--allow
    scaling^ analysis to be used to determine this.^

    A very important implication of the spatial and temporal symmetries^
    of tumors is that certain universal quantities (termed critical^
    exponents) can be defined which allow the characterization of^ tumor
    growth dynamics. In turn, this allows the main physical^ mechanisms
    responsible for their growth processes to be determined.^

    The current view of tumor growth kinetics is based on the general^
    assumption that tumor cells grow exponentially (Shackney, 1993[42]
    [fig-down.gif] ).^ Such kinetics agrees with the unlimited
    proliferative activity^ of tumor cells recorded in early, mainly in
    vitro, studies.^ However, a number of poorly explained issues remain
    in disagreement^ with an exponential regime of cell proliferation. For
    example,^ there is an evident discrepancy between the exponential
    tumor^ growth theory and experimental data obtained from tumor cells^
    growing in vivo: tumor doubling times have been found to greatly^
    exceed cell cycle times. Lower-than-expected activity of tumor^ cells
    and greater-than-expected aneuploidy have also been consistently^
    found. These issues are of great importance since both radiotherapy^
    and chemotherapy are entirely based on cytokinetics.^

    In a previous article (Brú et al., 1998[43] [fig-down.gif] ) we
    mathematically^ described the growth dynamics of colonies derived from
    a tumor^ cell line (rat astrocyte glioma C6), which raised reasonable^
    doubts about the exponential cell proliferation theory. The^ novel
    approach used in this study was based on fractality (Mandelbrot,^
    1982[44] [fig-down.gif] ) and scale invariance of the colony contour.
    These cells^ form colonies that are fractal objects which can be
    characterized^ by a fractal dimension (a measure of their degree of
    complexity).^ This allows the use of scaling analysis (Mandelbrot,
    1982[45] [fig-down.gif] ; Barabási^ and Stanley, 1995[46]
    [fig-down.gif] ; Brú et al., 1998[47] [fig-down.gif] ) for
    determining^their dynamic behavior, which was found compatible with
    the^ molecular beam epitaxy (MBE) universality class (Brú^ et al.,
    1998[48] [fig-down.gif] ).^

    In the present study, different cells lines and different types^ of
    solid tumor were studied to determine whether such growth^ dynamics
    also apply to them. In the case of cells cultivated^ in vitro, all
    cells growing as colonies have dynamics compatible^ with the MBE
    universality. These dynamics are characterized^ by: 1), a linear
    growth rate, 2), the constraint of growth activity^ to the outer
    border of the cell colony or tumor, and 3), diffusion^ at the colony
    surface. In this work, the term "linear" means that^ the colony radius
    grows linearly with time. With respect to^ tumors growing in vivo,
    common characteristics were seen in^ all cases, several of which were
    common to those of tumors growing^ in vitro. In all cases, growth in
    these in vivo tumors was limited^ to the tumor border ([49]Figs.
    6-[50]8), which could indicate^ that the mechanisms at work in vitro
    are also those at work^ in vivo. The Discussion will provide clinical
    and biological^ evidence that this is the case.^

      FIGURE 6  Spatial distribution of cell proliferation in colonies.
    (a) A clone of HT-29 cells formed after 260 h of culture and labeled
    with bromodeoxyuridine (BrdU). (b) Cells scored as BrdU positive. (c)
    Three different regions can be distinguished: an inner region of
    radius r[1] = R/2 practically without activity, an intermediate region
    from r[1] to r[2] = 0.8R with a linear increase in activity, and a
    third region from r[2] to r[3] = R which has half of the whole colony
    activity. The outer region has 20% of the whole colony surface and 47%
    of total activity. (d) Various contours have been traced according to
    a division of the colony (from the center of its mass) into 10 inner
    contours of radii R/10, 2R/10, 3R/10, .... 9R/10 and R (where R is the
    whole colony radius). Taking into account the number of BrdU stained
    cells, the spatial distribution of active cells is determined as a
    function of the radius. In the inset, the cumulative activity rate is
    plotted as a function of the colony surface. To a large extent, cell
    proliferation is seen to be located at the colony border. It is very
    important to note that (as seen in [53]Fig. 4) for HT-29 line cells, a
    growing time of 260 h still corresponds to an exponential regime, in
    which contact inhibition is still very scarce. Only after 400 h does
    growth of the colony reach a linear regime with respect to the radius.
    From this moment, colony activity is constrained more and more to the
    border, as expected from its dynamics.

      FIGURE 8  Spatial distribution of cell proliferation in tumors. (a)
    A human colon adenocarcinoma. (b) Cells scored as Ki-67 positive. (c)
    Tumor mass has been divided into three regions having 50%, 30%, and
    20% of the whole tumor surface (inside to outside). The distribution
    of cell proliferation in these three regions (having a mean radius of
    R/2, 8R/10, and R, respectively, R being the mean radius of the tumor)
    is 6%, 14%, and 80%, respectively. (d) Various contours have been
    traced according to a division of the tumor (from the center of its
    mass) into 10 inner contours of radii r/10, 2r/10, 3r/10, .... 9r/10,
    and r (where r is the whole tumor radius). Taking into account the
    number of Ki-67 stained cells, the spatial distribution of active
    cells is determined as a function of the radius. In the inset, the
    cumulative activity rate is plotted as a function of the colony
    surface. Spatial activity distribution corresponding to the three
    regions determined by r[1] = R/2, r[2] = 8R/10, and r[3] = R is shown.
    Cell proliferation is mainly located at the tumor border, rather than
    randomly and homogeneously throughout the tumor as might be expected.

    As shown in this article, any type of tumor developing in vivo^ has
    most of its cell proliferation constrained to the border.^ This may
    indicate that cell surface diffusion is the main mechanism^
    responsible for growth in any type of tumor.^


    Cell lines

    Cell lines were obtained from the Servicio de Inmunología,^ Hospital
    Clínico San Carlos (Madrid, Spain) and from^ ATCC (American Type Cell
    Culture, Rockville, MD).^

    Cell colonies

    Cell colonies were formed in 5-cm-diameter petri dishes by shedding^
    disaggregated cells at low density (1000 to 5000 cell/ml) in^ a
    culture medium that completely covered them. The medium employed^ for
    HT-29, HeLa, 3T3, 3T3 K-ras, and 3T3 V-src cell lines was^ RPMI 1640,
    2mM L-glutamine, 80 µg/ml gentamicin, and 10%^ fetal bovine serum
    (FBS). For the C6 cell line, a mixture of^ Dulbecco modified Eagle
    medium (DMEM) and F12 Ham's mixture^ (F12) in a 1:1 ratio supplemented
    with 10% FBS was used; for^ MCA3D, AT5 and Car B was supplemented with
    Ham's mixture and^ 10% FBS. For HT-29 M6, C-33 A, Saos-2, VERO C, and
    Mv1Lu, DMEM^ supplemented with 10% FBS was employed. After 48 h of
    culture,^ various individual clones containing 4-8 cells were chosen^
    for study. Cultures were maintained in a 5% CO[2] atmosphere at^ 37°C,
    carefully changing half of the culture medium every^ three days.^

    Tumor sections

    All human tumors were spontaneous tumors surgically removed^ from
    human patients at the Hospital Clínico San Carlos^ (Madrid, Spain).
    Tissue sections (4 µm thick) were obtained^ from paraffin-embedded
    material on poly-L-lysine-coated glass^ slides. After
    deparaffinization and rehydration, the sections^ underwent microwave
    treatment three times for 5 min. Endogenous^ peroxidase activity was
    blocked with hydrogen peroxide for 15^ min. Sections were incubated
    with Ki-67 (MIB-1, diluted 1:50;^ Immunotech, Marseille, France) for 1
    h at room temperature;^ they were then incubated with biotinylated
    secondary antibody^ for 20 min, followed by treatment with
    streptavidin-biotin-peroxidase^ complex (LSAB kit, Dako, Milan, Italy)
    for 20 min at room temperature.^ The sections were rinsed with several
    changes of phosphate-buffered^ saline (PBS) between steps. Color was
    developed with diaminobenzidine^ tetrahydrochloride. Light
    counterstaining was performed with^ hematoxylin.^

    Bromodeoxyuridine labeling

    Cell cultures were pulsed with bromodeoxyuridine (BrdU) (10^ µM) for 1
    h at 37°C. After washing with prewarmed^ Hanks' balanced salt
    solution, cells were fixed (8 min) with^ methanol:acetone (2:1) at
    -20°C, washed with PBS, and submitted^ to further incubation (1 h)
    with 1M HCl. BrdU was immunodetected^ by means of anti-BrdU specific
    antibodies, using a secondary^ antibody coupled with peroxidase and
    the diaminobenzidine tetrahydrochloride-substrate^ chromogen system.^

    Counting procedure

    Ki-67 positive cells were defined as having brown nuclear staining.^
    Ki-67 score was expressed as the percentage of positive cells^
    relative to the total number of tumor cells. For each slide,^ the
    number of positive cells was counted. These evaluations^ were
    performed without knowledge of clinicopathological data.^

    Image processing

    Colonies were photographed at 24-h intervals during the study^ (for
    over 1400 h in some cases) using an inverted phase-contrast^
    microscope (Diaphot, Nikon, IZASA S.A., Madrid, Spain). Cell^ colonies
    showing adhering growth were considered as two-dimensional^ systems.
    Photographs were scanned with a final resolution of^ 1.3 µm/pixel.
    Cell colony profiles were hand-traced. Scaling^ analysis and other
    measurements were then performed on these^ profiles with in-house
    computer software.^

    Determination of the fractal dimension of interfaces
    To determine the fractal dimension value, data were treated^ using
    three different methods: the box counting method, the^ yardstick
    method, and the y-/x-variance relationship. As expected,^ for any
    given interface, total coincidence between the three^ methods was

    Fractals and scaling analysis

    In a previous article (Brú et al., 1998[64] [fig-down.gif] ), we
    established^ the fractal nature of the contours of rat astrocyte
    glioma C6^ colonies and used scaling analysis to show that the colony
    growth^ dynamics belonged to the MBE class. The analysis of
    tumor/colony^ contours was based on the fractal geometry established
    by Mandelbrot^ (1982)[65] [fig-down.gif] , and on the scale invariance
    of fractal interfaces. The^ same techniques are used in the present
    paper to show that all^ tumors/colonies have these same dynamics.^

    Fractal interfaces--for example, those shown in [66]Fig. 1^ which
    correspond to different culture times of a HeLa cell colony--show^
    temporal and spatial invariances during the process of roughening.^
    The increase in irregularity of a front--or roughening^ (roughness is
    a useful quantitative measurement of the irregularity^ of an interface
    such as that of a tumor or cell colony contour)--is^ generally
    analyzed in terms of a time- and position-dependent^ function called
    the "local width function" or "interface width,"^ w(l,t). This is
    defined as the root mean square of the deviations^ of an interface
    about its mean value and is defined by the relationship ^

    [fd1_1.gif] (1)

    as a function of the arc length l and the time^ t, where L is the
    length of the whole contour, r[i] the distance^ from the center of the
    tumor mass to the point i of the interface,^ and < r[i] > the average
    radius of the arc length ([67]Fig. 2). The term^ < . > [l] is the
    local average of subsets of arc length l, and {.}[L]^ is the overall
    average of the system (Brú et al., 1998[68] [fig-down.gif] ).^

      FIGURE 1  Cell colony contours. Contours of a C6 cell line at
    different culture times. Morphology of tumor contours determines the
    dynamic behavior of growth by means of the scale invariances of their
    complex structures.

      FIGURE 2  Interface width of a tumor or cell colony border. The
    interface width is calculated for sectors with an arc length l. For
    each arc length, the mean value of the interface and the fluctuations
    around it are calculated to obtain the corresponding interface width.
    For a given value of the length of a sector, all values of the
    interface width are averaged to obtain the final value of the arc
    length l, w(l,t). The power law behavior of w(l,t) versus l provides
    the local roughness critical exponent.

    As a result of the fractal nature of the interface--or^ the cell
    colony contour--the interface width possesses^ a series of both
    spatial and temporal invariances which provide^ the basis for scaling
    analysis (see [73]Appendix A). All these invariances^ exhibit power
    law behavior, and for each type of invariance^ a critical exponent can
    be defined as the power law exponent.^ The power law behavior arises
    from the dependence of the interface^ width on the observation length
    and timescales. Usually, these^ interfaces become more and more rough
    as time goes by until,^ in some cases, the interfaces are always of
    the same roughness--they^ reach saturation.^

    How this roughening process develops both in time and in space^ is
    described by five critical exponents. Two of these exponents^ are
    related to the geometry of the system, quantifying its roughness^ on
    two scales: at the small scale of the system, i.e., the local^
    roughness critical exponent {alpha} [loc], and at system size scale,
    i.e.,^ the global roughness critical exponent {alpha} [glob]. The
    third exponent^ is related to the development of the interface width
    with time:^ ß, the growth exponent. A further exponent is the^ dynamic
    exponent, z, which is related to the correlation time^ of the
    interface. The physical meaning of this exponent is related^ to the
    celerity by which the information about points growing^ on the
    interface is transmitted across the interface. Finally,^ when the
    development of the interface is anomalous from a dynamic^ point of
    view, it is defined by another growth exponent, ß^*,^ which describes
    this anomaly in time. By determining this set^ of five exponents,
    which are not independent (see [74]Appendix A),^ the dynamics of the
    interface can be known, as well as the main^ mechanisms responsible
    for growth. The dynamics of a process^ is written in its interface,
    and this information can be extracted^ by determining these critical

    All known dynamic processes have been classified into just a^ few
    universality classes, each comprising all those physical^ processes
    with the same type of dynamics, and each characterized^ by having a
    different set of values for this series of critical^ exponents.
    Further, each universality class reflects the main^ conditioning
    factor responsible for growth, which can be described^ by a continuum
    stochastic equation.^


    Fractality of contours

    The first condition that must be fulfilled to apply scaling^ analysis
    techniques is that the growth behavior of a process^ lie in the
    fractal nature of the interface.^

    The contours of a series of colonies of different cell lines^ and
    tumors were morphometrically analyzed to calculate their^ geometric
    dimensions ([83]Tables 1 and [84]2) (see Materials and Methods).^ In
    all cases, a noninteger value, i.e., a fractal dimension^ (d[f]), was
    found. The values lay in the range 1.05-1.30,^ characteristic of any
    fractal object. These values are in good^ agreement with previous
    determinations of the fractal dimensions^ in melanomas and skin
    lesions (Claridge et al., 1992[85] [fig-down.gif] ; Cross^ et al.,
    1995[86] [fig-down.gif] ). The complexity of the contours of both
    colonies^ and tumors is due to the complexity of the growth process
    rather^ than the individual characteristics of the cells that compose^
    them. In all the tumors/colonies studied, the dynamics were^ the same,
    and, therefore, fractal dimensions do not seem to^ depend on the
    morphological characteristics of the cells but^ rather on the growth
    process. The differences found in the fractal^ dimensions might be
    related to the growth medium (in vivo versus^ in vitro) or to specific
    conditions in the culture or of the^ growth process.^

      TABLE 1  In vitro cell lines

      TABLE 2  Human and animal tumors

    During colony growth, the value of d[f] remained constant in all^ the
    cell lines studied. It is remarkable that the cell line^ HT-29 (colon
    adenocarcinoma) had an in vitro fractal dimension^ of 1.12 and a
    corresponding in vivo value of 1.30 ([91]Tables 1^ and [92]2). This
    difference indicates the greater complexity of^ the host tissue
    wherein tumors grow compared to the culture^ medium in which cell
    colonies are cultivated (much more homogeneous).^ From these results,
    the fractality of all the tested cell lines^ and tumors was
    established, and contour fractality allowed scaling^ techniques to be
    used to obtain growth dynamics.^

    Growth dynamics

    The dynamics and basic mechanisms of growth processes can be^ fully
    described by the set of five critical exponents determined^ by scaling
    analysis. Two of these exponents are directly related^ to the border's
    shape: local roughness, {alpha} [loc], and global roughness,^ {alpha}
    [glob]. The other three are related to the development over time^ of
    the contours: ß and ß^* (the growth exponents),^ and z (the dynamic

    The 15 cell lines were grown in culture to determine the critical^
    exponents of their colony contours by analyzing them at intervals^ of
    24 h. In all cases, the following characteristic values were^ obtained
    ([93]Figs. 3 and [94]4): {alpha} [loc] = 0.9 ± 0.1, {alpha} [glob] =
    1.5^ ± 0.15, ß = 0.38 ± 0.07, ß^*^ = 0.15 ± 0.05, and z = 4.0 ± 0.5
    ([95]Table 1). These^ values indicate that the growth dynamics of cell
    colonies correspond^ to the MBE universality class (Das Sarma et al.,
    1994[96] [fig-down.gif] ) ( {alpha} [loc]^ = 1.0, {alpha} [glob] =
    1.5, ß = 3/8, ß^* = 1/8, and^ z = 4.0). This is described by the
    following linear continuum^ equation (Das Sarma et al., 1994[97]
    [fig-down.gif] ; Brú et al., 1998[98] [fig-down.gif] ; Kessler^ et
    al., 1992[99] [fig-down.gif] ): ^

    [fd2_2.gif] (2)

    where h(x,t) is the^ position on the tumor or colony border, K is the
    surface diffusion^ coefficient (which is independent of critical
    exponents), F^ is the growth rate, and {eta} (x,t) is random noise
    where < {eta} (x,t) > =^ 0 and the correlation < {eta} (x,t) {eta}
    (x',t') > = 2D {delta} (x-x') {delta} (t-t') is seen.^ The first term
    on the right side of the equation implies that^ the growth process is
    characterized by the surface diffusion^ of cells (Brú et al.,
    1998[100] [fig-down.gif] ).^

      FIGURE 3  Scaling analysis of the colony interface width. The
    interface width is shown against window size for the HT-29 (colon
    adenocarcinoma) cell line at different times (t = 288 h (black), 624 h
    (green), 986 h (red), 1203 h (blue), and 1348 h (cyan)). From the
    shape, the value of the local roughness exponent {alpha} [loc] = 0.91
    ± 0.10 is obtained. In the inset of this figure, the transformation of
    w(l,t) into w(l,t)/l^ {alpha} and l into l/t^(1/z)) shows that these
    curves collapse into a single, universal curve with z = 4.0 and
    {alpha} [glob] = 1.5.

      FIGURE 4  Scaling analysis of the colony power spectrum. This figure
    shows the structure factors of an HT-29 (colon adenocarcinoma) cell
    line at different times (t = 288 h (black), 624 h (green), 986 h
    (red), 1203 h (blue), and 1348 h (cyan)). The global roughness
    exponent is obtained from the shape of these curves (2 {alpha}
    [glob]+1 = -4.0), which gives a value for this critical exponent of
    1.5. Transforming S(k,t) into S(l,t).k^2^ {alpha} +1 and k into
    k.t^(1/z), these curves collapse into a single universal curve as seen
    in the inset of this figure, with z = 4.0 and {alpha} [glob] = 1.5.

    Host-tumor interfaces were used to calculate the values of local^ (
    {alpha} [loc]) and global ( {alpha} [glob]) roughness for the 16 tumor
    types investigated.^ These values were characteristically 0.9 ± 0.1
    and 1.5^ ± 0.15, respectively. The time-related critical exponents^
    (ß, ß^*, and z) could not be calculated for^ obvious reasons.^

    These values of the critical exponents of tumor roughness do^ not
    agree with the theoretical values of the MBE universality^ class in
    2+1 dimensions for linear systems and in which the^ system size does
    not change. However, it must be taken into^ account that, in this
    case, there are two very important qualitative^ differences that could
    cause the values of the exponents to^ vary: the symmetry of the system
    is circular, not linear, and^ the system size varies with time.
    Presently, the values that^ would be obtained for the critical
    exponents in the latter case^ are unknown and require investigation in
    future work. In any^ event, as the following section shows, both
    experimental and^ clinical evidence indicate that, for tumors in vivo,
    the dynamics^ behaves as though the main mechanism responsible for
    growth^ were cell diffusion at the interface. Therefore, in both
    cases,^ a growth pattern characterized by the following can be
     1. Cell^ diffusion at the colony or tumor borders;^
     2. Cell proliferation^ mainly restricted to the colony or tumor^
        border, i.e., growth^ is greatly inhibited inside the colony^ or
     3. A linear^ growth rate for both colonies and tumors.^

    Experimental assessment of the features imposed by MBE class dynamics

    Cell surface diffusion

    MBE dynamics implies surface diffusion of cells, i.e., their^ movement
    along the tumor/colony border, not their free movement^ away from it.
    This should not be entirely surprising since cell^ movement is a
    well-known phenomenon, as is the increase in motility^ of tumor cells.
    However, the diffusion associated with MBE dynamics^ is not random,
    but more frequent toward places where there is^ a large coordination
    number (in this case derived from the number^ of cells that surround a
    given cell). Therefore, diffusion to^ zones with greater local
    curvatures, i.e., with larger coordination^ numbers, should be
    expected. Preliminary studies recording cell^ colonies by time lapse
    video suggest that, at least for HT-29^ cells, this is the case
    ([105]Fig. 5). However, more work is needed^ to address the
    biomolecular features of this, especially since^ it constitutes the
    main mechanism of colony and tumor proliferation.^

      FIGURE 5  Cell surface diffusion. The dynamic behavior of cells
    growing in a colony is compatible with the molecular beam epitaxy
    universality class, of which surface diffusion is characteristic,
    i.e., cells located at the growing interface tend to migrate along the
    colony border. To show this movement, a clone of HT-29 cells formed
    after 300 h of culture was recorded by time lapse video. The figure
    shows different steps of this movement: (a) the arrow indicates a cell
    just after division; (b-e) the arrows follow this cell to show how it
    moves along the colony border; (e) the arrow shows the resting site of
    this cell at the interface. The local curvature radius is positive at
    the initial (a) and negative at the final (e) sites, consistent with
    predictions derived from molecular beam epitaxy dynamics and reflected
    experimentally for the first time here.

    Cell proliferation is restricted to the colony or tumor border
    To experimentally support this second characteristic of MBE^ dynamics,
    actively proliferating cells within colonies and tumors^ were labeled
    with bromodeoxyuridine (BrdU) and Ki-67, respectively.^ [108]Fig. 6
    shows a representative colony labeled after 260 h of^ culture. The
    proliferative activity was located mainly within^ the external
    portion. It must be borne in mind that at this^ time the greater part
    of the colony is not inhibited. This is^ confirmed by the velocity
    curve. However, a clear tendency toward^ the restriction of cell
    proliferation to the edge of the colony^ was seen. Thus, the outer
    region occupied 20% of the colony's^ surface but included 47% of all
    proliferating cells. [109]Fig. 7^ shows that cell proliferation is
    further restricted to the colony^ border in HeLa cells grown over a
    longer period (380 h). Similar^ results were obtained when analyzing
    tumor specimens ([110]Figs. 8^ and [111]9). Active Ki-67 cells were
    clearly concentrated in the^ external portion of tumors. [112]Fig. 8
    represents a colon adenocarcinoma.^ In this case, 80% of the active
    cells were found in the outer^ 20% of tumors; only 6% were found in
    the innermost 50% of the^ tumor. This constraint of cell proliferation
    to the border was^ also obtained even for polypous carcinomas.
    [113]Figs. 6 c; [114]7, bottom,^ [115]8 c; and [116]9 show that, in
    every case, the number of proliferating^ cells increases as a function
    of the colony or tumor radius.^ This indicates a relationship between
    the ability to proliferate^ and spatial distribution within the colony
    or tumor. As a consequence,^ these data also indicate that
    proliferation is inhibited in^ the innermost areas.^

      FIGURE 7  Spatial distribution of cell proliferation in colonies. As
    in [119]Fig. 6, a spatial study of cell proliferation was made. This
    case corresponds to a HeLa cell line after 360 h of culture time.
    Cells in mitosis are stained brown. The three different regions in the
    figure contain 50%, 30%, and 20% of total tumor surface, respectively
    (inside to outside); cell proliferation is therefore mainly restricted
    to the border. As time progresses, cell proliferation will be more and
    more restricted to the colony border.

      FIGURE 9  Colony growth. Development over time of the mean radius of
    a colony of HT-29 (colon adenocarcinoma) cells. The mean radius shows
    a linear regime with time (linear fit shown in red), which gives a
    constant interface speed of 0.29 µm/h. This result is incompatible
    with the general assumption that tumors grow exponentially. An
    exponential regime is observed only at very early times, during which
    all cells are active. Later, in the linear regime, a very important
    cell fraction is partially contact-inhibited, and the majority of
    colony activity is constrained to a very fine band at the border, as
    suggested by the universal dynamic behavior determined for the growth
    process of any type of colony. This result supports the impossibility
    of the division of all cells in the colony, which would give an
    exponential regime for radius or size at any time.

    Linear growth rate
    In all studied cases, tumor radius grows linearly with time.^ The
    growth rate of colonies was obtained by plotting the variation^ of the
    mean radius as a function of time.^

    Common to all cell lines, colony growth was dominated by a linear^
    growth regime throughout the culture period (up to 1400 h).^ This
    regime is, in most cases, preceded by an exponential transitional^
    phase lasting 200 h on average. The slope of the linear regime^
    indicates the average growth velocity of the colony, which was^
    different depending on the cell line ([122]Table 1) and growth
    substrate^ (not shown). It is important to note that the mean growth
    velocity^ of the colonies is characteristic of the process, and not
    of^the cell line. It probably depends on a variety of external^
    factors such as the experimental conditions, the available nutrients,^
    type of medium, etc.^

    Changing the substrate did not modify the dynamic behavior,^ except
    for the average growth velocity. [123]Fig. 9 shows the corresponding^
    growth rate analysis of the HT-29 cell line. In this case, the^
    exponential regime was one of the longest, lasting a little^ less than
    400 h. Though exponential phases were shorter, similar^ results were
    found for all the cell lines studied. The inset^ of [124]Fig. 9 shows
    the growth rate in semilogarithmic representation.^ It is not an
    exponential process; if it were, a straight line^ would be obtained.^

    In vivo tumor growth rate could not be measured directly as^ explained
    above. However, the restriction of cell proliferation^ to the tumor
    contour mathematically implies a linear growth^ rate.^


    Elucidating the basic mechanisms of tumor growth is one of the^ most
    intricate problems in the field of tumor biology, and one^ of its
    major challenges. Many attempts have been made in recent^ decades to
    obtain a mathematical model that would allow us to^ discern these
    basic features of cell and tumor growth (Shackney,^ 1970[133]
    [fig-down.gif] ; Durand, 1990[134] [fig-down.gif] ; Gatenby and
    Gawlinsky, 1996[135] [fig-down.gif] ; Byrne, 1997[136] [fig-down.gif]
    ;^ Hart et al., 1998[137] [fig-down.gif] ; Scalerandi et al.,
    1999[138] [fig-down.gif] ; Drasdo, 2000[139] [fig-down.gif] ; Kansal^
    et al., 2000[140] [fig-down.gif] ; Sherrat and Chaplain, 2001[141]
    [fig-down.gif] ; Ferreira et al., 2002[142] [fig-down.gif] ).^ Several
    different hypotheses have been postulated to describe^ the main
    conditioning factor of tumor growth, and nutrient competition^ between
    tumor cells or tumor and host cells is currently the^ most accepted.
    This concept of tumor growth is a legacy of an^ older problem, that
    concerning the growth of bacterial colonies.^ In the latter, it has
    been fully shown that the main mechanism^ is nutrient competition.
    However, this cannot be extrapolated^ to tumor growth. First, the
    majority of these models reproduce^ patterns with a roughness exponent
    of {alpha} [loc] = 0.5, which is in^ good agreement with the Eden
    model (Eden, 1961[143] [fig-down.gif] ). Second, the^ kinetic behavior
    that reproduces these models is Gompertzian.^ However, it should be
    mentioned that there is often no qualitative^ or quantitative
    comparison made of these models. The roughness^ of simulated patterns
    obtained from mathematical models that^ consider nutrient competition
    is largely in good agreement with^ that corresponding to the Eden
    model, i. e., {alpha} [loc] = 0.5. Other^ types of mathematical models
    also partially reproduce some features^ of tumor growth. Nevertheless,
    the majority are very restrictive^ in their hypothesis or use a series
    of conditions that are insufficient^ to reproduce the main features of
    tumor growth.^

    This work provides a very extensive and detailed study of pattern^
    morphology both of tumors and cell colonies. From the behavior^ of the
    corresponding contours, both in time and with length^ scales, and by
    applying scaling techniques, the dynamics and^ the main mechanism
    responsible for tumor growth can be extracted^ without the need of any
    hypothesis. This is one of the major^ advantages of scaling analysis.
    Scaling techniques used to analyze^ the fractal nature of cell
    colonies growing in vitro, and of^ tumors developing in vivo, showed
    them to exhibit exactly the^ same growth dynamics independent of cell
    type. These dynamics^ are compatible not with the currently and widely
    accepted idea^ of Gompertzian growth, but with the MBE universality
    class,^ which involves a linear growth regime. It should be
    remembered^that the concept of a Gompertzian growth regime is based
    on^the exponential growth of cells, and an exponentially decaying^
    growth rate is assumed. The Gompertz law is considered a robust^
    feature of the nutrient-limited model of cancer growth. However,^
    according to the present analysis, the main mechanism responsible^ for
    tumor progression would be cell diffusion at the tumor border.^

    Strikingly, the dynamics obtained for the studied tumors and^ cell
    colonies are the same for any cell proliferation process,^ independent
    of cell line or the in vivo or in vitro nature of^ growth. These
    dynamics, which are also obeyed in other phenomena^ such as crystal
    growth, possess the property of super-roughness.^ This means that the
    traditional Eden growth model (Eden, 1961[144] [fig-down.gif] ),^
    conceived to satisfy cell proliferation processes, does not^ explain
    tumor growth. The Eden model is the simplest growth^ model that can be
    defined based on random particle deposition^ and aggregation. The
    surface of an Eden cluster obeys scaling^ dynamics with {alpha} [loc]
    = {alpha} [glob] = 0.5. Far from this behavior, however,^ the present
    results show that cell proliferation dynamics exhibit^ super-roughness
    ( {alpha} [glob] >= 1.0) as an effect of surface^ diffusion, a process
    which tends to smooth the tumor or colony^ borders. The stochastic
    nature of duplication induces cell colony^ or tumor roughness, but
    this is generally counterbalanced by^ a smoothing or ordering process
    due to the mobility of generated^ cells. Cell diffusion on the tumor
    or colony border tends to^ counterbalance the effect of random
    duplication. There is, therefore,^ a mean doubling time (the duration
    of the cell cycle) with some^ dispersion around this value.
    Physically, this effect is described^ in [145]Eq. 2 by the noise

    In addition, as in the case of crystal growth where evaporation^
    effects do not alter dynamic behavior in MBE universal dynamics,^ the
    movement of cells away from the colony/tumor does not influence^ the
    growth process.^

    The widely accepted concept of tumor growth kinetics is based^ on the
    assumption that tumor cells grow exponentially. However,^ it is
    generally recognized that this assumption is applicable^ to virtually
    no solid tumor growing in vivo (Shackney et al.,^ 1978[146]
    [fig-down.gif] ). Given an exponential-like growth regime, doubling
    times^ would be similar to the total duration of the cell cycle.
    Nevertheless,^ tumor doubling times are strikingly longer than cell
    cycle times,^ e.g., more than 100-fold in breast carcinomas (Shackney,
    1993[147] [fig-down.gif] ).^ These differences are even more
    remarkable in large tumors.^ This is currently explained as a
    consequence of tumor cell loss^ and/or a low rate of cell production
    because of nutrient deprivation^ and/or waste product accumulation
    (Shackney, 1993[148] [fig-down.gif] ).^

    Based on the results, it can be stated that tumor growth would^ be
    well described by a linear regime. It is then needless to^ account for
    the different incidental processes that might explain^ disagreements
    between a theoretical basis--the supposed^ exponential regime--and
    experimental observations which^ appear to show growth to be linear.
    It is important to again^ point out that, in this paper, a linear
    process means one in^ which rate changes with time in a completely
    linear way.^

    As already described, this linear regime implies that there^ are less
    actively proliferating cells, and that these are not^ randomly
    distributed throughout the whole volume of the tumor,^ but
    homogeneously constrained to the border. Only when the colony^ is
    small enough to assume that most cells are located at the^ growing
    border is the growth regime depicted by an exponential--but^ still

    Other than the growth dynamics of any type of tumor/colony being^ the
    same, the most important result of this study is perhaps^ that cell
    movement occurs at their surface. This type of movement^ ([149]Fig. 5)
    invalidates the hypothesis that the main mechanism^ responsible for
    tumor growth is nutrient competition between^ cells. As seen in
    [150]Fig. 10, newly generated cells move to sites^ with a higher
    coordination number, i.e., with a higher number^ of neighboring cells.
    This movement is that predicted by MBE^ dynamics and, from a
    mathematical point of view, is the movement^ originated by the
    fourth-order derivative in [151]Eq. 2.^

      FIGURE 10  Cell surface diffusion. A schematic diagram of surface
    diffusion at the tumor border. A new cell born in 1 migrates until a
    neighboring position, 2, in which the local curvature of the interface
    is higher and the coordination number is greater than at its original

    Tumors are surrounded by a very thin acidic environment as a^ result
    of cell metabolism (these cells mainly consume glucose^ and secrete
    lactic acid, increasing the acidity of the environment).^ Following
    the rules of cell surface diffusion as in MBE dynamics,^ the final
    position of a diffusing cell will be in a region in^ which the
    quantity of nutrients or oxygen is lower since it^ becomes surrounded
    by a greater number of cells. Moreover, as^ a consequence of cell
    metabolism, the pH of this region will^ be lower than at the cell's
    initial position. The lack of oxygen^ in the concave regions where new
    cells deposit does not constitute^ an obstacle to tumor growth since
    cell proliferation is supported^ by anaerobic respiration (Eskey et
    al., 1993[154] [fig-down.gif] ). Oxygen is a limiting^ factor only for
    functions such as differentiation, respiration,^ and mechanical work.
    This movement determines the mechanism^ responsible for the growth
    dynamics: as this work has determined^ both theoretically and
    experimentally, it is not possible to^ conceive tumor growth merely as
    a process of nutrient competition.^ On the contrary, this movement can
    be understood as the search^ for space by tumoral cells. In its
    initial position in [155]Fig. 10,^ the mechanical pressure the new
    cell undergoes is greater^ than in its final position. This obliges
    that tumor growth be^ considered a process in which a mass grows and
    looks for space^ to avoid the mechanical response of both the host
    tissue and^ the immune response.^

    This has a number of consequences with respect to the treatment^ of
    solid tumors. First, the effectiveness of chemotherapy becomes^
    dependent on the specific surface of tumors. Given that the^
    proliferating cells sensitive to antiproliferative agents are^ mainly
    associated with the surface of tumors, then the effectiveness^ of
    chemotherapy must decrease as tumor size increases. For this^ reason,
    the current log-kill concept of chemotherapy assumes^ a constant
    effect at random (Skipper et al., 1970[156] [fig-down.gif] ), but it
    fails^ experimentally in large tumors (Shackney, 1970[157]
    [fig-down.gif] ; Skipper et al.,^ 1970[158] [fig-down.gif] ). The
    concept of log kill rests on the fact that each chemotherapeutic^
    cycle kills 90% of all cells in proliferation. But if proliferative^
    cells are restricted to the border of the tumor and are not^ randomly
    distributed, as this work argues, the relative fraction^ of cells in
    proliferation compared to the total number of cells^ in the whole
    tumor is clearly much smaller. Chemotherapy would^ certainly kill all
    the cells on the border--but the inner^ cells, prevented from
    proliferating by the pressure exerted^ on them through the lack of
    space, would escape the effect of^ the therapeutic agent. They would
    therefore survive to become^ the new peripheral, proliferative, layer.
    However, their number^ would be again small in comparison to the total
    number of cells^ of the tumor--and so the process repeats itself. The
    efficacy^ of chemotherapy would be less than expected if all the
    cells^in the tumor were randomly proliferating. It is important to^
    note that both primary tumors and metastases show the same growth^
    dynamics ([159]Table 2). It is also well known that hypoxia is
    associated^ with resistance to radiation therapy and chemotherapy
    (Harris,^ 2002[160] [fig-down.gif] ). This is also an important point
    to consider in developing^ therapy strategies if, following MBE
    dynamics, cells migrate^ to positions where they are more likely to
    suffer hypoxia ([161]Figs. 5^ and [162]10).^

    Second, aneuploidy (Caratero et al., 1990[163] [fig-down.gif] ;
    Tomita, 1995[164] [fig-down.gif] ), along^ with other genetic
    abnormalities (Sun et al., 1998[165] [fig-down.gif] ; Ried et^ al.,
    1999[166] [fig-down.gif] ), is more frequent than expected in advanced
    solid^ tumors, and less frequent in early stage than in advanced
    cancer.^ The genetic mutation rates in tumor cells are thought to be^
    linked to the number of mitotic cell divisions (Nicholson, 1987[167]
    [fig-down.gif] ).^ If an exponential growth regime is assumed, each
    cell must undergo^ 32 divisions to form a 2 cm^3 tumor ( ~ 4 x 10^9
    cells). However,^ in a linear growth regime, the number of divisions
    by cells^ on the surface would be ~ 30 times greater than at the
    center.^ Naturally, this leads to a higher frequency of genetic
    abnormalities^ in cells at the growing tumor border. In this way, if
    we consider^ that metastases are generated from cells from the border
    of^ the primary tumor (Fukakawa, 1997[168] [fig-down.gif] ), it is
    completely coherent^ that metastatic cells would be always more
    aneuploid than those^ of primary tumors. A linear growth regime
    provides a much better^ explanation of this than does exponential

    Another implication of a linear growth regime is that the most^
    malignant cells should be located at the tumor border. This^ is
    because cells become more malignant as the number of chromosomal^
    aberrations increases (Rasnick and Duesberg, 1999[169] [fig-down.gif]
    ) (i.e., as^ the number of cell divisions increases). Given enough
    time,^ the accumulation of aberrations would probably lead to cell^
    death, but tumors become mortal for the patient before this^ point is
    reached. Accordingly, the malignancy of cells should^ increase along
    the tumor radius: the further from the center,^ the more malignant the
    cell should be. One of the important^ clinical consequences of this is
    that it explains the discrepancy^ between anatomopathological analysis
    of biopsies and the diagnosis^ of many cancers. The doctor who
    performs the biopsy usually^ takes a sample from the center of the
    tumor to be sure that^ what is taken corresponds to the lesion. But if
    growth is linear,^ and the malignancy of cells increases along the
    tumor radius,^ such a biopsy would always take the least malignant
    cells and^ might lead to diagnostic error (Liberman et al., 2000[170]
    [fig-down.gif] ).^

    A major phenotypic hallmark of tumor cells is thought to be^ the lack
    of inhibition of the cell proliferation process. However,^ a
    downregulation of cell proliferation is shown by the present^ BrdU
    (for in vitro cell colonies) and Ki-67 (for in vivo tumors)^ labeling
    data ([171]Figs. 7 and [172]8). Some type of inhibition of cell^
    proliferation must therefore be operating on cells inside tumors.^
    This has been observed experimentally on numerous occasions.^
    Traditionally, it has been ascribed to necrosis, probably as^ a result
    of poor vascularization. In the present experiments,^ the same type of
    behavior is seen. However, at no time could^ the inhibition of
    proliferation have been due to central necrosis^ since none of the
    tumors became necrotic. Further, in the in^ vivo studies, no
    correlation was found between the presence^ of blood vessels inside
    tumors and any increase in proliferation.^ However, the proposed model
    offers a new interpretation for^ the inhibition of cell proliferation
    inside tumors. The tumor^ contour is super-rough, indicating that
    tumors adopt the best^ shape for bearing the "pressure" exerted by the
    host organ and^ the inflammatory response, and it is these "pressure
    effects"^ that may be inhibiting proliferation of cells. Cells inside^
    the tumor can proliferate if they have room to do so, but at^ the
    moment cell density becomes so high that there is no longer^ any
    space, inhibition begins. Durand (1990)[173] [fig-down.gif] showed
    that quiescent^ cells, when extracted from tumors and cultured,
    recover their^ proliferative capacity and resume their preinhibition
    cell cycle.^ This inhibition does not exist at the tumor border. The
    spatial^ distribution of mitotic cells in [174]Figs. 6-[175]8 fit a
    barometric^ distribution, i.e., an exponential distribution in good
    agreement^ with the argument derived from surface cell movement and
    the^ concept of tumor growth as a search for space. A growing tumor^
    has therefore to release enough space at the host-tumor interface.^
    This requirement is in line with the critical roles assigned^ in
    cancer invasion to the development of an acidic environment^
    destroying parenchymal cells at the host-tumor interface (Gatenby^ and
    Gawlinsky, 1996[176] [fig-down.gif] ) and/or the presence of tumor
    metalloproteinases^ cleaving the extracellular matrix (Sato et al.,
    1994[177] [fig-down.gif] ; Egeblad^ and Werb, 2002[178] [fig-down.gif]
    ), and also with the notion that tumor cells require^ enough motility
    to invade--growing cells cannot simply^ be pushed along a solid
    substratum to which they are adhered^ (Abercrombie, 1979[179]
    [fig-down.gif] ). Thus, the dynamics, which has been verified^ in all
    studied cases, predict that tumor growth might be constrained^ by host
    tissue resistance if no space is released, as suggested^ by the
    suppression of tumorigenesis when tumor cells lack proper^ matrix
    metalloproteinases (Wilson et al., 1997[180] [fig-down.gif] ) or when
    there^ is an increase in matrix proteins at the stroma-stroma border^
    (Bleuel et al., 1999[181] [fig-down.gif] ). In addition, a loss of
    tumorigenicity^ might occur if such a space is refilled with host
    cells more^ resistant to an acidic microenvironment. The fact that
    tumor^ cells transduced with certain cytokines lose their
    tumorigenicity^ by a mechanism involving a strong recruitment of
    neutrophils^ (Hirose et al., 1995[182] [fig-down.gif] ; Musiani et
    al., 1996[183] [fig-down.gif] ; Milella et al.,^ 1999[184]
    [fig-down.gif] ) supports this possibility. These cells are resistant
    to^ extracellular acidosis (Gukovskaya et al., 1992[185]
    [fig-down.gif] ; Serrano et^ al., 1996[186] [fig-down.gif] ) and may
    compete for space at the acidic host-tumor^ interface.^

    These pressure effects, as an inhibiting factor of tumor cell^
    proliferation, are in good agreement with previous reports on^ solid
    state stress in tumor spheroids (Haji-Karim and Carlsson,^ 1978[187]
    [fig-down.gif] ; Mueller-Klieser, 1997[188] [fig-down.gif] ; Acker,
    1998[189] [fig-down.gif] ; Hamilton, 1998[190] [fig-down.gif] ;
    Kunz-Schugart^ et al., 1998[191] [fig-down.gif] ; Santini and
    Rainaldi, 1999[192] [fig-down.gif] ). These spheroids are^ clusters of
    cancer cells that have been widely used in the laboratory^ to study
    the early stages of avascular tumor growth, the response^ to external
    factors such as supplied nutrients or growth inhibitory^ factors,
    cellular differentiation, and cell-cell interactions,^ and have even
    been used in therapeutically oriented studies.^ Helmlinger (1997)[193]
    [fig-down.gif] showed that solid stress inhibits their growth.^ The
    pressure exerted by the host over the tumor could explain^ the
    deviation of the tumor growth rate from a pure linear regime.^ This
    effect is not present in the two-dimensional in vitro cell^ colonies
    of our study, and this might be responsible for the^ pure linear
    regime of the growth rate even after very long periods^ ([194]Fig.
    10). Cell colonies in vitro should undergo a pure linear^ regime until
    they reach whole confluence.^

    In summary, this article shows that tumor cells of widely different^
    genetic backgrounds share a common behavior. When tumors grow^ in
    vitro, this behavior is completely compatible with MBE universality^
    dynamics. Further, there is sufficiently abundant and clear^
    biological and clinical evidence to suggest that this is also^ the
    case in vivo, although further work is needed to confirm^ this. In any
    case, a universal tumor growth dynamics is observed^ for any type of
    tumor in vivo, independently of any other characteristic^ of tumoral
    cell lines. This dynamics is always governed by processes^ of cell
    surface diffusion. However, more work is needed to fully^ determine
    the whole dynamical behavior of tumor growth. The^ fractality of the
    contour of all the studied cell colonies and^ tumors has been
    demonstrated. Scaling techniques show that in^ vitro and in vivo cell
    proliferation would obey the same dynamics,^ independent of cell line
    or any other characteristic. These^ universal dynamics are compatible
    with a linear growth regime,^ a result in contrast with the currently
    accepted exponential^ or Gompertzian models of tumor growth. The main
    mechanism responsible^ for tumor progression, as for any cell
    proliferation process,^ is cell diffusion on the tumor border. These
    results incorporate^ the new concept that the major conditioner of
    tumor growth is^ space competition between tumor and the host, which
    is more^ important than nutrient competition or angiogenesis, etc.
    The^latter must be considered, in some cases, as necessary or as^ a
    coadjuvant condition of tumor growth, but their effects mainly^
    consist of modifying the growth rate--perhaps simply allowing^ it or
    not. These results invalidate the current concept of cell^
    proliferation and offer a unified view of tumor development.^ The
    dynamics involved provide coherent explanations where the^ traditional
    model cannot. Despite the importance of characteristics^ common to the
    dynamics of the in vivo growth of different tumors,^ more work is
    needed to completely characterize them. It should^ not be forgotten
    that, independent of interpretations, this^ article shows for the
    first time that different tumors have^ common characteristics such as
    the distribution of cell proliferation^ and their characteristic forms
    (that would imply common basic^ growth processes), determined via the
    critical exponents of^ local and global roughness.^

    As a result, some important features of cancer can be better^
    explained. Moreover, some clinical strategies may need to be^


    In this procedure, the critical exponents are the so-called^ local
    roughness of the interface, {alpha} [loc], the interface global^
    roughness, {alpha} [glob], the dynamic exponent, z, the growth
    exponent^ ß, and the critical exponent ß^*. These critical^ exponents
    originate as a result of the power law behavior of^ the geometry, and
    the development in time of the interface (tumor-host^ surface) (Brú et
    al., 1998[203] [fig-down.gif] ). This power law behavior^ is
    associated with two quantities used in the description of^ tumor cell
    colonies. The first is the mean radius of the colony^ border: ^

    [fd3_3.gif] (3)

    Its development over time^ gives the growth velocity of the tumor. The
    second is the rough^ aspect which can be quantified in terms of the
    standard deviation^ of the mean radius, denominated the width of the
    interface: ^

    [fd4_4.gif] (4)

    where < . > [l] represents the local average of subsets^ of the arc of
    length l and {.}[L] the average of the whole system.^ These
    fluctuations around the average position of the external^ cells of
    colonies grow in time in a power law fashion, w(l,t)^ ~ t^ß, with a
    characteristic critical exponent ß,^ the growth exponent. In the same
    manner, if we select small^ windows over the whole tumor, the larger
    the size of the window,^ the greater the width of the interface. These
    spatially growing^ fluctuations also follow a power law, w(l,t) ~ l
    {alpha} [loc], with another^ characteristic exponent, {alpha} [loc,],
    the local roughness exponent,^ which can also be obtained from the
    scaling behavior of the^ correlation functions. The behavior described
    above cannot be^ used at all scales in a finite-size system such as a
    tumor because^ the fluctuations cannot grow indefinitely. Therefore,
    there^ must exist a point at which these temporal fluctuations
    saturate,^ a situation that is not common in systems with circular
    symmetry.^ This critical time is called the saturation time (t[s]) and
    its^ dependence with the system size provides a new critical
    exponent:^the dynamic exponent z. These results for the interface
    width^ can be summarized as follows (Barabási and Stanley, 1995[204]
    [fig-down.gif] ): ^

    [fd5_5.gif] (5)

    The last magnitude used in this analysis was the^ spectrum of the
    tumor profiles. This quantity measures the characteristic^ length of
    interface structures formed by solid cell colonies^ in their growth
    process. Computing the power spectra as the^ Fourier transformation of
    the interface, h(x,t), a power law^ behavior is established with an
    exponent referred to as global^ roughness, {alpha} [glob] (Barabási
    and Stanley, 1995[205] [fig-down.gif] ; Brú^ et al., 1998[206]
    [fig-down.gif] ; López et al., 1997[207] [fig-down.gif] ): ^

    [fd6_6.gif] (6)

    where^ k is the momentum and s the structure factor.^

    None of these critical exponents are independent, but are related^ by
    the following: ^

    [fd7_7.gif] (7)

    and ^

    [fd8_8.gif] (8)

    where ß^* is another critical exponent.^ Therefore, the whole set of
    critical exponents that determine^ the dynamics of a growth process is
    {alpha} [loc], {alpha} [glob], ß,^ ß^*, and z.^


    We thank Eliezer Shochat, José Antonio Cuesta, and Rodolfo^ Cuerno for
    fruitful discussions, Jesús Martín^ Tejedor for help, David Casero and
    Susana García for^ technical assistance, and Dirk Drasdo for reading
    the final^ manuscript. Our special thanks to Ysmael Alvarez and
    Granada^ Alvarez for time lapse video filming, and Luis Ortega for
    analyzing^ animal tissue sections.^

    Submitted on June 27, 2002; accepted for publication July 23, 2003.


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