A Method for Modeling Growth of Organs and Transplants Based on the General Growth Law: Application to the Liver in Dogs and Humans
AA Method for Modeling Growth of Organs andTransplants Based on the General Growth Law:Application to the Liver in Dogs and Humans
Yuri K. Shestopaloff * , Ivo F. Sbalzarini Research and Development Lab, Segmentsoft Inc., Toronto, Ontario, Canada, MOSAIC Group, Center of Systems Biology Dresden (CSBD), Max Planck Institute ofMolecular Cell Biology and Genetics, Dresden, Germany
Abstract
Understanding biological phenomena requires a systemic approach that incorporates different mechanisms acting ondifferent spatial and temporal scales, since in organisms the workings of all components, such as organelles, cells, andorgans interrelate. This inherent interdependency between diverse biological mechanisms, both on the same and ondifferent scales, provides the functioning of an organism capable of maintaining homeostasis and physiological stabilitythrough numerous feedback loops. Thus, developing models of organisms and their constituents should be done within theoverall systemic context of the studied phenomena. We introduce such a method for modeling growth and regeneration oflivers at the organ scale, considering it a part of the overall multi-scale biochemical and biophysical processes of anorganism. Our method is based on the earlier discovered general growth law, postulating that any biological growthprocess comprises a uniquely defined distribution of nutritional resources between maintenance needs and biomassproduction. Based on this law, we introduce a liver growth model that allows to accurately predicting the growth of livertransplants in dogs and liver grafts in humans. Using this model, we find quantitative growth characteristics, such as thetime point when the transition period after surgery is over and the liver resumes normal growth, rates at which hepatocytesare involved in proliferation, etc. We then use the model to determine and quantify otherwise unobservable metabolicproperties of livers.
Citation:
Shestopaloff YK, Sbalzarini IF (2014) A Method for Modeling Growth of Organs and Transplants Based on the General Growth Law: Application to theLiver in Dogs and Humans. PLoS ONE 9(6): e99275. doi:10.1371/journal.pone.0099275
Editor:
Manlio Vinciguerra, University College London, United Kingdom
Received
March 10, 2014;
Accepted
May 12, 2014;
Published
June 9, 2014
Copyright: (cid:2)
Data Availability:
The authors confirm that all data underlying the findings are fully available without restriction. All used data were published and references topublications are in the article.
Funding:
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests:
There are no competing interests. Although one of authors, Yuri Shestopaloff, works in a commercial company Segmentsoft Inc. as aDirector of Research & Development Lab, as you correctly noted, the study has nothing to do with the business activity of the company. This study is an entirelypersonal undertaking conducted in a spare time. The situation allows me taking unpaid breaks in order to do these studies and visiting other organizations. Inparticular, part of this study was done at Max Planck Institute of Molecular Cell Biology and Genetics, where I stayed as an invited Visiting Professor (without beingpaid as well). The topic of the study was not included into any official Institute’s research plans, and no funds were allocated for it. So that formally it can beconsidered as a self-initiated research with no funding. This does not alter our adherence to PLOS ONE policies on sharing data and materials.* E-mail: [email protected]
Introduction
First we introduce the earlier discovered general growth law andits mathematical representation, the growth equation, and apply ittowards modeling growth of livers and liver transplants in dogsand humans (the first article) and finding liver metabolism (thesecond article). Then, we present a review of presently availablemodels from the perspective of developing a general framework formodeling biological phenomena, and how the general growth lawcan benefit it. Such a framework, if created correctly, would uniteand mutually reinforce available methods and provide directionsand guidance for the development of multi-scale models of livingorganisms and their constituents, such as organs and cells, as wellas allow model verification and subsequent refinement. Such aframework is especially important given the many practicalproblems whose solution requires a transition to systemic under-standing of living organisms, so that on this well founded basis thefollowing practical applications and methods could be introduced in diverse areas, such as medicine, pharmacology, biology,biotechnology, etc.Developing such a framework, indeed, became a necessity giventhe launch of projects aiming at the creation of models oforganisms and organs to be used in medicine, pharmacology,biology, evolutionary and developmental studies, etc., such as, e.g.,the Virtual Liver Network (VLN) [1], the Recon-2 project onhuman metabolism [2], the virtual liver project [3], the whole-body model [4], the Physiome Project on cardiac electrophysiol-ogy [5], the BlueBrain project on modeling the brain cortex, andothers. Such models have different levels of generality addressingcertain phenomenological, structural, and organizational aspects.However, since the different mechanisms and systems in organismsclosely interrelate, the adequacy and usefulness of models wouldbe improved by including additional mechanisms and compo-nents, through interlacing different factors, and unification ofmethodological approaches based on a general framework. ethods
1. The general growth law
Growth regulation and modeling growth of cells, organs, andwhole organisms is an area of intensive study. Approaches rangefrom studies of biomolecular growth mechanisms and growthfactors, to developmental and systems biology methods. Forinstance, in [6], authors argue that changes during growth, such asprogressive decline in proliferation, ‘‘results from a geneticprogram that occurs in multiple organs and involves the down-regulation of a large set of growth-promoting genes.’’ The authorsfurther note that ‘‘This program does not appear to be drivensimply by time, but rather depends on growth itself, suggesting thatthe limit on adult body size is imposed by a negative feedbackloop.’’ They consider different cellular events that could beinvolved in cooperatively providing commensurate growth oforgans and whole organisms. An important inference is therecognition of the existence of feedback mechanisms between thecurrent integral state of a growing organ or an organism (which theauthors call ‘‘growth itself’’) and triggering particular growthmechanisms into cooperative action.Reference [7] considers growth hypotheses based on morpho-gen gradients. They conclude that the growth phenomenon isdriven by a combination of different factors. A similar view isexpressed in [8], which considers growth from a systems-biologyperspective. The author suggests that ‘‘developing systems devotea considerable amount of cellular machinery to the explicitpurpose of control’’, although he does not specify what this‘‘controlling machine’’ consists of, or what are the coordinatingand managing mechanisms.All cited articles converge to the conclusion that growth isdriven by the cooperative working of many different factors, whoseaction, besides other possible mechanisms, is regulated byfeedback loops. In [6], a guiding mechanism is placed into a‘‘genetic program that occurs in multiple organs’’, which ‘‘dependson growth itself’’. In other words, the authors assume that thegeneral governance and coordination of biomolecular growthmechanisms resides at the molecular level. Articles [7,8] supportsimilar ideas, that biomolecular mechanisms govern and coordi-nate the multitude of interacting mechanisms constantly synthe-sizing and degrading molecules within cells, managing cooperative growth of multiple cells, and growth of different organs andsystems in the whole organism. These governing molecularmechanisms are assumed in [6] to be a ‘‘genetic program’’ thathas to have a complexity on the order of that of the biochemicalmachinery itself. But still we are unable to explain the coordinatedgrowth of organs and systems within an organism. Such inter-organ genetic regulation would amount to unmanageable com-plexity and consequently to extreme vulnerability and instability,which we do not observe in nature.We hence take the view that biochemical mechanisms execute operations in such a manner that one operation faithfully followsanother, so that there is no need for a run-time scheduler.However, such a sequence of operations had to be evolutionarilydeveloped and organized over a long time. Some researchersassume that such sequences of events are somehow stored in DNA.However, the existence of genes does not explain neither how theaforementioned sequence of events has evolved, nor does it givesatisfactory answers as to how it unfolds in a particular growth andreplication scenario on the cell, organ, and organism levels. So,there should be other than purely genetic mechanisms responsiblefor growth control. Examples of such views can be found in aseminal work by D’Arcy Thompson [9], and the book [10]. Recent studies [11–17] (the most important and comprehensivework is [17]), discovered that, indeed, such a regulatorymechanism exists at higher-than-molecular scales, which is calledthe general growth law. This law universally operates at scalesranging from cells and cellular components to organs and wholeorganisms. It is responsible for the evolutionary development ofsequentially executed biochemical mechanisms in developingorganisms, as well as for unfolding these sequences of events inparticular growth and replication scenarios of cells, organs, andorganisms. During growth, the general growth law imposes certainconstraints on the amount of produced biomass, which accord-ingly causes changes in composition of biochemical reactions insuch a way that the growing entity proceeds through the growthcycle. The same mechanism is also the major player securing balanced growth of different organs and systems in an organism [17].Mathematically, the general growth law is represented by growthequations, which come in different forms depending on thereplication and growth scenario.Previously, the general growth law and the growth equationhave been successfully used for studying and modeling the growthand replication mechanisms in unicellular organisms, such asfission yeast and its mutants, amoeba, and
S. cerevisiae [13,17].Here, we propose and demonstrate a method for modeling growthof multi-cellular organs. We present mathematical forms of thegrowth equations for modeling the growth of transplanted livers,liver grafts, and liver remnants in dogs and humans. The purposeof this study is twofold: First we develop a general method, whichcan be thought of as a methodological framework, that allows todescribe, predict, and understand different aspects of the growth oforgans, such as finding the rate of growth and its dynamics, theprogression in changes of size and geometry, the size (meaningmass and volume) of an organ, identifying certain qualitativephases of growth, etc. Although in this work the proposed methodis exemplified by studying the growth of transplanted livers in dogsand humans, the approach itself is of a general nature and can beused in similar applications, including growth of artificial organs,such as kidneys or hearts [18]. The second purpose of this work isto continue the study of the general growth law, includingverification aspects. It is also the first time that the general growthlaw is applied on the organ scale.Growth and replication of living species are governed bybiophysical mechanisms on molecular and higher levels. Thegeneral growth law and its mathematical representation, thegrowth equation, formulate how nutrients are distributed athigher-than-molecular levels and uniquely relate it to metabolicand geometric properties of the growing organism and itsconstituents, such as organelles, cells, and organs. The generalgrowth law is based on conservation of mass with regard tonutrients, since nutrients are digested in biochemical reactions, forwhich the law of conservation is valid.Any living organism is an open system that consumes nutritionalresources, which are balanced between two main activities vitallyimportant for any organism: supporting existing biomass, the so-called maintenance resources, and the resources that are used forsynthesis of new biomass. This distribution of resources is notarbitrary, but represents a tradeoff that is uniquely defined in everyphase of growth and replication and on each spatial scale. Theparameter that mathematically defines this resource division iscalled the growth ratio [17]. It naturally depends on the geometry(shape) of the growing object and, indirectly, on the properties ofits biochemical machinery. An optimum distribution of nutritionalresources has likely emerged from evolutionary pressures.As organs grow, more and more resources are required formaintenance, leaving less resources for biomass production, since he nutrient-supplying ability of the environment and themetabolic abilities of the cells are limited. Nutrients, regardlessof how they are supplied, are received through the surface (of theorgan or its blood vessels), while they have to support thefunctioning of mass in the volume . Since volume increases fasterthan surface area when organisms grow, the nutrient supply perunit volume is fundamentally limited. At some point, the amount ofnutrients per unit volume decreases to a level that is just sufficientto support maintenance needs, and no nutrients are left forbiomass production. This effectively imposes limits on themaximum size of growing organisms and their constituents(besides the specific properties of the biochemical machinery,which, in this regard, plays the role of an execution mechanism).Note that the nutrient concentration in the surrounding environ-ment (for instance, in the blood flow) cannot increase endlessly too,as well as the capacity of an organism or its constituent, such as acell or a liver, to process nutrients. So, in one way or another, atsome point of growth, the amount of nutrients per unit volume willbe capped.The growth ratio , which defines the fraction of nutrients that goesto biomass production, depends on the geometric shape of theorgan. It is defined as follows: Let us assume that nutrientavailability and the biochemical specifics of an organ that receivesnutrients through its surface, allow the organ to grow to amaximum volume of V MAX with a maximum surface of S MAX ~ S ( V MAX ) . We define the dimensionless relative surface R S and the relative volume R V as: R S ~ S ( V ) S ( V MAX ) ð Þ R V ~ VV MAX ð Þ Then, the growth ratio G R , which is also dimensionless, isdefined as: G R ~ R S R V { ð Þ Although this parameter is described in terms of geometriccharacteristics, it is closely related to the biochemistry of the organ,since it defines how much nutritional resources are used forgrowth, in other words, for biomass production, while the rest isused to support the organ’s maintenance needs. The particularform of the growth equation depends on the growth scenario. Forinstance, when nutrients are supplied through the surface, thegrowth equation can be written as: p c ( X ) dV ( X , t ) ~ ð S ( X ) k ( X , t ) | dS ( X ) | R S R V { (cid:2) (cid:3) dt ð Þ . Here, X is the spatial coordinate, p c is the density of the tissuemeasured in kg = m , t is time, k is the specific influx, which is thenutrient influx per unit surface per unit time measured in kg = ( m | sec ) , p c ( X ) dV ( X , t ) is the change in mass, and dS ( X ) is the elementary surface area. In case when the specific influxdoes not depend on the location of an elementary surface area,equation 4 simplifies to p c ( X ) dV ( X , t ) ~ k ( t ) | S | R S R V { (cid:2) (cid:3) dt ð Þ where S is the total surface through which nutrients are supplied.Equation 4 has a simple interpretation: The left-hand siderepresents the mass increment. The right-hand side represents thetotal influx through the surface, that is the term Ð S ( X ) k ( X , t ) | dS ( X ) , multiplied by the growth ratio R S = R V { ð Þ , so that this product defines the amount of nutrientsthat is available for biomass production.Note that the maximum size of a growing organism or an organcan vary, since the size and shape can change during growthdepending on many factors, such as nutrient availability,temperature, etc. This fundamental property of every growthphenomenon is exactly what the growth equation incorporatesthrough the introduction of a maximum size that can depend onother parameters. This property can be illustrated as follows: Itwas experimentally found in [19] that cells placed from anutritionally poor into a nutritionally rich environment grownoticeably bigger. Similarly, suppose an organ started to grow in anutritionally poor environment so that it is destined to have asmaller final size [17]. If, during growth, the nutritionalenvironment becomes richer, the organ’s final size can be larger.So, unless conditions for the whole growth period are known atthe onset of growth, the final size is generally unknown. However,in many instances, the final size of a growing organ is known fromprior information, for example when the organ’s mass is a well-defined fraction of the mass of the whole organism.Another approach to finding the maximum size is the following:In an extensive review [20] on tissue growth, the authors note: ‘‘Asurprising result of this type of modeling (allometric) is that themass of an organism during its growth process can be predictedbased on metabolic processes in its cells.’’, referring to resultsobtained in [21]. If we take a look at the growth equation,equation 4, then the ‘‘surprising’’ result finds a rational explana-tion. According to the growth equation, the rate of biomasssynthesis, and consequently the final size, depends on the nutrientinflux consumed by the growing organ, which is defined by the metabolic abilities of the cells to process nutrients for biomasssynthesis and maintenance, which explains the aforementionedresult in [21]. In fact, the dependence of an organ’s final size onthe metabolic properties of its cells and on nutrient availability was first inferred from the growth equation, and then the search in theliterature confirmed this fact.Mathematically, this property can be expressed as a power law[21], that is ‘‘If y is the length scale of the organ, and x is the lengthscale of the body, they can often be related by a power law of theform y ~ x b , for constant a and b ’’. Reference [22] furtheradvances this result allowing finding the maximum size of a grownorganism based on metabolic properties of its cells. The authorsproved that ‘‘the mass of a wide variety of animal species grewaccording to the equation dmdt ~ am = { bm , where a , b areconstants (different for each species), which are dependent on themetabolic characteristics of the cells. The key assumption here isthat the metabolic rate B depends on the total body mass m through the power law relation B ! m = which is true for a widerange of biological organisms [23].’’ sage of the fact that the maximum size of a grownmulticellular organism depends on the metabolic activity of itscells is facilitated by the growth equation as follows: According toequation 5, the increase of biomass at any given moment isproportional to the nutrient influx k , while the functionaldependence of the change of nutrient influx for the same organismis similar in a wide range of growth scenarios [13,17]. So, once weknow the minimum k min and maximum k max nutrient influxes,corresponding accordingly to the minimum and maximummetabolic rates of the cells and the minimum m min and maximum m max masses of the organism, we can find the maximum massresulting from influx k as m ~ m min z ( m max { m min )( k { k min ) = ( k max { k min ) . Here, we assume that nutrient influxes k , k min , k max relate to the same phase of growth, let us say to the beginning, and k min ƒ k ƒ k max .When one does not know how the nutrient influx varies duringgrowth, and consequently how the maximum size changes, thediscussed approaches produce approximate values of maximumsize. Finding the maximum size is by no means restricted to thedescribed methods. Other considerations and approaches can beused too.So, the variable maximum size of an organism in the growthequation is just a reflection of the fact that, generally, themaximum final size is a value that is fundamentally unknown at thebeginning of growth, since the change of growth conditionschanges the maximum final size (unless we know how allparameters, which influence the growth, dynamically changeduring the whole growth period). However, in many instances,when conditions of growth are stable, the maximum size can bepredicted with reasonable accuracy for practical purposes.
2. Modeling growth of whole livers transplanted fromsmall dogs into large dogs
In [24], the authors measured the growth of whole livers thatwere transplanted from small dogs, whose weight was (13 : + : kg , to large dogs with weights (23 : + : kg . The control groupconsisted of dogs with similar weights. In that group, livers fromdonor dogs with weights (19 : + : kg were transplanted torecipient dogs with weights (18 : + : kg . The goal of theexperiments was to find out which factors define the final size oftransplanted livers. It turned out that the liver volumes (andaccordingly their masses, since the density of a liver is relativelyconstant) grows to a final size defined by a certain, stable fractionof the overall body mass. In other words, there is no ‘‘memory’’ ina small liver of a small dog that it is small or that it belonged to asmall dog. This confirms that the molecular pathways are diligent reactive executers of instructions at the cellular level, but not morethan that, while the organ’s size and geometrical characteristics aredefined by other mechanisms. From this study, we consider inmore detail two data sets of liver volume over time, from two dogs,for a total observation period of 30 days. Dog livers have ashape that is largely defined by the anatomical location and byadjoining organs. We model a dog liver as a partial torus, cutthrough its plane of symmetry. The parameters defining the torusare: initial distance d b (index ‘b’ stands for ‘‘beginning’’, i.e., at theonset of growth) between the torus center and the center of thecircle that creates the torus, the initial ( r b ) and final ( r e ) radii of thetorus at the beginning and at the end of growth, and the number P that defines which fraction of the torus is left (like 2/3 of the totalcircle). The ends of the torus are capped by two hemispheres, andthen the whole shape is cut through its plane of symmetry (seefront and side views in Fig. 1). This shape imitates the growth of a liver that increasesproportionally in all dimensions, so that once we know how muchthe radius of the torus increases (which is defined by the final livervolume), we can find how the size changes of all other dimensions.For example, the distance d is defined through a scaling coefficient C ~ r e = r b as d e ~ Cd b . The four parameters ( r b , r e , d b , P ) uniquelydefine the shape of the dog liver before and after the growthperiod. We assume that a liver grows proportionally in alldimensions. This is a reasonable, albeit not confirmed assumption,since no indications were made in [24] with regard to the shape ofthe liver at intermediate phases of growth. Using the notationintroduced in Fig. 1, the volume V and the surface S of the livermodel are: V ( r , d ) ~ P p r ( p d z = r ) S ( r , d ) ~ P p r (2 p d z d z r ) ð Þ Accordingly, the relative surface R S and the relative volume R V , which we need for the growth equation, are: R S ~ S ( r , d ) = S ( r e , d e ) ð Þ R V ~ V ( r , d ) = V ( r e , d e ) ð Þ For known relative surface and relative volume, the growth ratio G can be found using equation 3.In order to formulate the growth equation, we have to definethe nutrient influx. In a liver, nutrients are supplied through theblood, which flows through the liver as driven by blood pressure.In the portal veins, the blood pressure drops from 130 to 60 mmwater. After passing the sinusoids the pressure further drops to20 mm water. We assume that the amount of nutrients supplied toevery position in the liver is the same, and that each unit of volumeconsumes the same amount of nutrients. Under these assumptions,the growth equation becomes: pdV ( r , d ) ~ K | ( V ( r , d ) = V b ) | R S R V { (cid:2) (cid:3) dt ð Þ where p is the liver density, which we assume to be constant, K isthe total influx of nutrients supplied to the liver by the blood per Figure 1. Front and side views of a partial, sliced torus. nit volume per unit time, t is time, V b is the initial liver volume,and ( R S = R V { is the growth ratio. We normalize K = 1, since itsvalue defines the unknown time-scaling coefficient.The assumption of constant density of the liver is well justified,given its anatomical and cytological uniformity [25,26]. Althoughthe composition of the nutrients received by the hepatocytesdepends on the location along the sinusoid, the amount ofnutrients available per unit volume is assumed to be constant [25].This is reflected in equation 9 by the multiplier ( V = V b ) .We numerically solve equation 9 using the rectangular rule fornumerical integration, i.e., by dividing the ranges of the radius r and the distance d into equal intervals and computing theappropriate function values at the centers of the intervals (recallthat r/d = const ). All variables except t in equation 9 depend onvolume, so that we collect them on the left-hand side, andintegrate over the range of r (and correspondingly d ) in order toobtain time. Thegeometric parameters of the growing livers from two dogs, as takenfrom [24], are summarized in Table 1. This is a complete set ofparameters required to compute the growth curve using equation9. Then, we scale the obtained growth curve along the time axisonly, in order to adjust the time scale to experimental data. Notethat this is not a data fitting procedure, because we first computed thegrowth curve, and only after that compared it to experimental data.The scaling along the time axis does not change the shape of thegrowth curve, but rather amounts to identifying time scale K of theobserved dynamics.Due to transition processes occurring in a transplanted liverafter resection and surgery, the transplanted liver initially does notgrow the same way it would normally grow, and less hepatocytesare involved in replication compared to a normally growing liver.When the liver grows normally, its size increase is described by thegrowth equation, which represents the evolutionarily optimizedgrowth scenario, securing the shortest growth time. According to[25], which considers hepatectomy with significant resections,gradually all hepatocytes become involved in replication. Theauthors say: ‘‘After tissue loss, residual hepatocytes are activated toproliferate within few hours; hepatocytes proliferation begins atthe portal ends of plates …, and successive waves of hepatocytesproliferation ultimately involve virtually all residual hepatocytes.Hepatocytes proliferation is followed sequentially by proliferationof sinusoidal endothelial cells and macrophages, and the other cellsof parenchymal matrix’’. However, towards the end of growth,more and more hepatocytes switch to a quiescent state, since at theend the liver growth decelerates. Another possibly contributingfactor could be a slowing of the hepatocyte cell cycle toward the end of growth, but according to [27] switching to a quiescent stateis the main cause of growth deceleration. So, although accordingto [25] there is a relatively long phase of growth when all hepatocytes become involved in liver regeneration (in donors andrecipients), when the liver is only reduced little by resection, andalso towards the end of growth, a noticeable fraction ofhepatocytes are in a quiescent state.In order to identify the time point after which the entire livergrows normally (according to the general growth law), we firstassume that the entire liver grows normally from the beginningand compare the so-obtained growth curve to the experimentaldata. The point after which the curve agrees with the data is thetime when the entire liver grows normally. When the resected liverpart is significant, then, according to [24], this is also the pointafter which all hepatocytes are involved in replication. Then, oncewe know when the entire liver begins to regenerate normally, wecan model the preceding phase of partial growth with gradualinvolvement of hepatocytes.Using equation 9, we compute growth curves for dog livers andcompare the results with experimental data from [24], as shown inFig. 2. For dog 1 (Fig. 2A) the first experimental point is not takeninto account, since it corresponds to the not-yet transplanted liver,when it was weighted right after hepatectomy, while the rest of theexperimental points correspond to results obtained by CTscanning. The second point marks the beginning of growth whereless hepatocytes than in normal growth are involved in regener-ation. As one can see from the graph, this point is off the growthcurve computed under the assumption of normal growth.For the rest of data, the correspondence between theexperimental results and the computed growth curve is very good,which is an indication that the proposed approach produces a realdependence. So, the growth equation can serve as an adequatetool for modeling the growth of dog livers.The computed growth curve for the second dog (Fig. 2B) is alsoin good agreement with the experimental data after some initialdivergence, which indicates the time it takes for the liver to engagein normal growth.Our model has hence allowed us to identify the time point afterwhich the maximum number of hepatocytes are involved in theregeneration process. This is the point where the growth curvecomputed under the assumption of normal proliferation starts toagree with the experimental data. For large resections, accordingto [25], at this stage ‘‘virtually all residual hepatocytes’’ areinvolved in proliferation. In case of usual hepatectomy, whenroughly 30% is removed from the donor liver, according to [25],this will be the point where normal growth resumes (points V and V in Fig. 2, which we refer to as ‘‘joining’’ points). Table 1.
Dimensions of geometric models used for computing dog liver growth, taken from (24).
Parameter Dog 1 Dog 2
Initial torus radius r b r e d b d e .3. Modeling partial growth of livers in dogs. The phasebefore the joining points is characterized by partial growth whereless hepatocytes than in case of normal growth are contributing toorgan increase. As we discussed before, liver transplants do notstart growing entirely at once after transplantation, since the liverstructure is built sequentially starting from proliferation ofhepatocytes. Also, not all hepatocytes are activated for prolifera-tion simultaneously, but are gradually engaged in the proliferationprocess from the portal ends of plates. However, it is presentlyunknown what fraction of hepatocytes, relative to normal growthor total volume, are involved in proliferation at the beginning, andwhen all hepatocytes become involved in regeneration, althoughthese are important characteristics which are directly related totransplantation outcome and recovery process. Our model allowsanswering these questions.In order to extend our model to partial growth, given theobserved slower growth at the beginning, we assume that a smallerfraction of hepatocytes is initially involved in proliferation. Theindication that ‘‘ hepatocytes proliferation begins at the portal ends of plates ’’[25] means that a fairly large part of the total liver volume isinvolved in proliferation from the very beginning. For simplicity,we assume that the proliferating hepatocytes are uniformly distributed across the entire organ. If shells of proliferation wouldexist, as they do for example in other organs in which activelygrowing areas are located at the periphery, then we wouldconsider such a growing shell and compute the value of the growthratio for this shell only, and accordingly would apply volumetriccharacteristics to the shell as well.During partial growth, we distinguish the growing part of theliver (we call it the ‘‘active’’ part below) from the part of the liverthat does not participate in regeneration (the ‘‘passive’’ part). Wetake into account that the passive part still requires nutrients formaintenance, but do not contribute to biomass production. Moreand more cells from the passive part get activated until the wholeorgan contributes to growth. Computationally, this means that ateach integration time step we transfer an elementary volume fromthe passive part to the active part.We model the reduction of the passive part V P during growthas: V P ~ V b (1 { A ) | V J V b { V C V J V b { V b (cid:2) (cid:3) p ð Þ where A is the fraction of the initial active part, V b is the initialliver volume; V J is the relative volume at the joining point, V C isthe total volume of the growing liver, p is a power that allowsvarying the functional dependence V P ( V C ) , choosing differentconcave and convex shapes. Equation 10 reflects the monotonicincrease of the active liver volume. The exponent p accounts fordeviations from purely linear increase (when p ~ ). Bothparameters p and A are found by fitting to experimental data,including the one before the ‘‘joining point’’. The volumes at thejoining points for dog 1 and dog 2, according to our previouscomputations, are V J ~ : and V J ~ : . Note thatequation 10 is constructed in such a way that the passive volumebecomes zero when V C ~ V J V b . The active growing part is thecomplement of the passive part, hence V A ~ V C { V P ð Þ The growth equation for a partially growing liver is: pdV ( r , d ) ~ K | ( V A = V b ) | R S R V { (cid:2) (cid:3) dt ð Þ Equations 10–12 define a complete system of equationsrequired for computing growth curves for partially growing livers.Computed growth curves for the whole growth cycle, including thephase of partial growth, are shown in Fig. 3 for both dogsconsidered.The parameters p and A reveal that initially about half of theorgan is engaged in proliferation, and that proliferation increasesuntil normal growth is reached. The computed increase of theactively growing part of the liver, defined as a ~ ( V b { V P ) = V b , isshown in Fig. 3 for both dogs (dashed curves). If the resection issmall, and the liver size is close to original, then there is no growthphase in which all hepatocytes are involved in proliferation. In thiscase, our considerations are valid with regard to the fraction ofhepatocytes contributing to normal, evolutionarily developedgrowth as described by the growth equation.Is the result of identifying p and A from the data unique, orcould there be several combinations of p and A that would lead tosimilar results? We claim that the found values are unique and Figure 2. Growth of small livers in big dogs. obust. The reason is that parameter A defines the shape of the whole growth curve, while parameter p affects only the shape of thegrowth curve before the joining point. Besides, the shape of thegrowth curve is very sensitive to the value of A , indicating that thisparameter has a good identifyability; changing A by few percentincreases the difference between the computed growth curve andthe experimental data by tens of percent. In fact, A influences theshape and location of the growth curve substantially more than p, even during partial growth. We hence first only find the value of A such that we obtain the least diversion of the computed growthcurve from the experimental data. Then we identify p to bestrepresent the partial growth phase. This alternating optimizationscheme is then iterated until convergence is reached. While theresults are very sensitive to the value of A , the value of p causesorders of magnitude less change in the diversion from experimen-tal data.Overall, the correspondence between the computed growthcurves and experimental data can be considered good within theentire growth period. Results and discussion for Models of Dog Livers
Based on the computed growth curves, we can make severalobservations: First, using the proposed method in case ofsubstantial resections, we are able to accurately estimate whenpartial growth is complete and the entire liver begins to grow (or,in case of small resections, when normal growth begins). This isimportant information, which previously could not be obtainedfrom observations. In the case of the first dog, whose transplantedliver was larger relative to the final size, this happened after 3 days,while the liver of the second dog began regenerating normally (andin this case apparently entirely because of the small original size)after 6 days.Although the second dog had a noticeably smaller initial liversize compared to the final size, the fractions of volumescorresponding to the joining points (relative to the final liver size)for both dogs are remarkably close and within overlapping errormargins: v J ~ V J = V e ~ : = : ~ : + : v J ~ V J = V e ~ : = : ~ : + : Here, we take into account that the error of volumemeasurement by CT scan is about 5% [16], which translates toerrors of + : and + : for dogs 1 and 2, respectively.From this, one may conclude that regardless of the initial size ofa liver transplant, the liver begins to normally regenerate when itsrelative size (relative to the final size) is about 54%. Beforereaching the joining point, the liver can grow only partially. Onepossible explanation for this is that a liver, which is smaller thanthe size corresponding to the joining point, is under stress, and itsfirst priority is to support the functioning of the organism, whileless nutritional resources and less hepatocytes can be allocated forgrowth.We were also able to quantify another important parameter: thefraction of liver that is involved in regeneration from the verybeginning. In the case of the first dog, which had a larger initialliver transplant, this was about + : of the entire liver, whilein the smaller liver transplant of the second dog about + : of the initial liver volume contributed to regeneration. (The errorwas estimated for a 5% change of the average deviation of thecomputed growth curve from the experimental data). Overall, wesee that in both cases a significant portion of the liver is involved inregeneration from the beginning.The rates of liver growth in both dogs were almost identical.Although we do not have data for smaller fractions for the firstdog, because the original size of the liver transplant was larger, wecan compare rates of liver growth in the last two days before thejoining points. For the first dog the relative increase in volumeduring these two days was D v & : , while for the second dogit was D v ~ V J = V & : , where V is the liver’s relativevolume on day 4. Given the measurement error of about 2%, thisis a remarkable similarity in the rates of liver regeneration. Ofcourse, having results from only two dogs does not allow definitiveconclusions.Lastly, we were able to identify the rate at which the passive partof the liver joins active regeneration. Fig. 3 (dashed lines) showstwo convex curves with exponents 0.85 and 0.83 for the first dogand the second dog, respectively. So, the rate at which ‘‘passive’’liver parts become ‘‘active’’ is similar for both dogs, and itaccelerates toward the ‘‘joining’’ point. Both features are Figure 3. Partial growth of the liver in two dogs. hysiologically justified, since fundamental mechanisms of liverregeneration should not significantly differ across differentspecimens, while acceleration of the rate is facilitated by livergrowth, which allows devoting progressively more resources toregeneration, while continuing to support the physiologicalrequirements of the organism.
Modeling Liver Growth in Humans
We further validate our model using experimental data on thegrowth of transplanted livers in humans from [28,29]. We modelhuman liver geometry as a prism with one edge cut as shown inFig. 4, based on liver description from [26]. In clinical practice,either the right or left liver lobe is transplanted, leaving the donorwith the remaining lobe. In Fig. 4, the right lobe is on the left, andvice versa, since this is how livers are presented in the anatomyliterature. In the previous case of dogs, whole livers weretransplanted and then grew in size at constant shape. Here, thegrowing liver changes shape as a single lobe regenerates to a fullliver. The geometric form of the liver hence changes duringgrowth. The geometric characteristics of whole livers, as takenfrom [26,28,29], are given in Table 2.Using the same consideration as for dog livers, we can alsoassume that the nutrient influx per unit volume in human livers isconstant. This assumption and the parameters from Table 2constitute a complete set of parameters required to compute thegrowth curve for the model of a human liver. Other parameters,such as graft lengths, can be computed using the formulas below.For the simulations, we also need the angle a at the prism’s base(see Fig. 4), which is defined as: tg a ~ L = ( B X { B ) ð Þ Let us denote t ~ tg a for brevity. Then, in the notation of Fig. 4,we can find the surface and volume of the cut prism as: V ( B , L , t ) ~ W ( BL z L = (2 t )) ð Þ S ( B , L , t ) ~ W (2 B z L = t ) z WL (1 z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z = t q ) z L (2 B z L = t ) ð Þ The equivalent expressions in terms of the larger base B X aremore convenient for computations of the right lobe: V ( B X , L , t ) ~ W ( B X L { L = (2 t )) ð Þ S ( B X , L , t ) ~ W (2 B X { L = t ) z WL (1 z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z = t q ) z L (2 B X { L = t ) ð Þ equation 17 is the sum of the areas of all prism faces.The mass or volume of the graft taken from the donor fortransplantation is usually recorded as a fraction of the total size ofthe donor’s liver. We denote this fraction F (fraction). Then, incase of a left-lobe transplant, we find initial length L b as follows:We rewrite equation 16 as: FV ( B , L , t ) ~ W ( BL z L = (2 t )) ð Þ Solving this equation for L bL , we find: L bL ( B , t , W , F ) ~{ Bt z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B t z tFV = W q ð Þ Substituting the volume V from equation 14 into equation 19,we can rewrite it as: L bL ( B , t , F ) ~{ Bt z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B t z FBLt z FL p Similarly, we can find the initial length L bR for a transplantedright lobe: Figure 4. Geometric model of a human liver.
The boundary planedefines the initial volume of the transplanted lobe. It can be shiftedalong the direction of arrow A bR ( B X , t , F ) ~ B X t { ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B X t { FB X Lt z FL q ð Þ Equations 15–20 uniquely define the shapes of the transplantedand remaining liver grafts.
Reference [28] focused on the growth of the remaining parts oflivers in donors, whose safety was a primary goal of that study.Fig. 5A presents results for 27 male donors who had their rightlobes removed, so that their left lobes had to regrow to full livers.The data points show average and standard deviation valuesacross all 27 donors. Comparing with female donors (Fig. 5B), thegrowth phase in male donors lasted longer. There are severalplausible explanations for this difference, but in lack of experi-mental data no sound conclusion can be made. The authors alsonoted that ‘‘Female donors had significantly slower liver regrowthwhen compared to males at 12 months ( : + : versus : + : )’’. This result is almost surely due to higher metaboliccapacity of female livers required to support pregnancy (thisfeature has been discovered and discussed in the second article onliver metabolism), so that neither female liver transplants nor liverremnants in females need to grow as big as in males, since theirhigher metabolic capacity allows supporting metabolic require-ments by having smaller size.Given the inter-patient fluctuations, the computed growth curvefrom our model corresponds well with the experimental data.Overall, the present model also accurately reproduces the growthdynamics of organs whose geometric shape changes duringgrowth. In [29], the authors studied the growth of livers in both donorsand recipients for both left-lobe donors and left-lateral-sectiondonors. It was discovered that ‘‘Livers of the right lobe donorgroup regenerated fastest in the donors group…’’. Our model,which is based on the general growth law, readily explains thiseffect. Looking at Fig. 4, the thickest part of the liver has to regrowduring the regeneration of a left lobe. When the right lobe grows,then the thinnest part of the liver (on the right in Fig. 4) has toregrow. Such different growth geometries lead to different valuesof the growth ratio and consequently to different rates of growthfor left and right lobes. Fig. 6 shows the change in value of thegrowth ratio during growth of right and left lobes when the relativeinitial volumes are the same (females). We see that the value of the growth ratio for the left lobe is higher than for the right lobe,although all other parameters are the same, except geometry.Since the amount of nutrients available for biomass production isdefined by the growth ratio, this means that for the same nutrientsupply the amount of biomass produced per unit time is higher ingrowing left lobes than it is in right lobes, which, indeed, wasexperimentally observed [29].The aforementioned difference in growth ratios between rightand left lobes creates a difference in the rate of biomass productionof several percent. For instance, after 108 days the rates of biomassproduction between the right and left lobes differ by about 7%.Note that the discussed difference in values of growth ratioswhen the relative initial volumes of right and left lobes are thesame is not the only factor contributing to different rates of livergrowth. Differences in the initial relative volume also influence thevalue of the growth ratio and consequently the rate of biomassproduction, as seen for the male cohort. In Fig. 6, the right loberemnant in males began to grow at a size of about 69% of the finalliver volume, while the size of left lobe remnants was about 47% ofthe final volume. Both remnants grew to about the same final size.As we can see, the value of the growth ratio for the left loberemnant is significantly higher than for the right lobe, which in thiscase is due to different geometries of right and left lobes, and alsodue to different initial volumes. (Strictly speaking, different initialvolumes also influence geometrical characteristics, but the result ofthis influence is of lesser value.) The simultaneous action of the twodiscussed effects explains the observed differences - the fastergrowth of left lobes in donors that donated right lobes.
Developing Integral Models
The liver is an important part of any organism. Its workinginterrelates to other organism systems and organs. On the otherhand, it is a separate organ with specialized functions, whoserelationship to other organs and systems can be described in termsof inputs and outputs. Indeed, given the multitude of different,often interrelated, factors that affect liver function, its mathemat-ical modeling presents a challenge. In such a situation, it isespecially important to provide a robust and adequate modelingstructure (including hierarchical relationships) that would incor-porate different scales, from molecular mechanisms to the wholeorgan. In this regard, the proposed model presents a valuable and,in fact, unique development, since the general growth law isapplicable at dimensional scales from cellular components toentire organisms, thus providing a universal conceptual approachand the same uniform mathematical apparatus for different scalelevels. This way, all meaningful parameters, such as, for instance,nutrient influxes, can be related from the lower scale level to theupper one, up to the integral parameters characterizing the whole
Table 2.
Geometric parameters of initial liver grafts and whole livers (from [26,28,29]).
Parameter Male Female
Width of a whole liver (relative units) 1, 2 1, 2Small base B of a whole liver (in units of width) 1 1Large base of a whole liver B X (in units of width) 3.5 3.5Length of a whole liver (in units of width) 2.9 2.9Initial volume (percentage of the original donor liver) 48.5 59.6Final volume (percentage of the original donor liver) 85.42 79.58Relative final volume (relative to minimum) 1.7612 1.5605doi:10.1371/journal.pone.0099275.t002 rgan. Note that exactly the same concept and mathematicalframework is entirely applicable to other systems, organs and theirsubcomponents, which is a consequence of the universal nature ofthe general growth law [17]. Such a universal approach significantly simplifies mathematical modeling of organisms and their constit-uents.However, would the above be sufficient to develop an adequate biophysical and biochemical model of a complex organ? In principle,the answer is yes. The proposed approach resolves severalfundamental issues defining success of any modeling, such asmodel uniqueness, stability, scalability and integrity. In practicalapplications, much attention is given to biochemical mechanismsbecause of their importance for medical, pharmaceutical andbiotechnological purposes. In this regard, presently the relation-ship between the integral characteristics, such as nutrientconsumption or geometrical form of an organ or cells, andbiochemical reactions are very weakly explored, if at all. This iswhy the ‘‘biochemical part of the story’’ is usually self-enclosed,although it is far from being self-sufficient. In such a situation, theuse of the general growth law and models developed on its basis,like the one which we introduced in this work, become of criticalimportance, since they enable to directly relate integral charac-teristics, such as nutrient influx and amount of produced biomass,to composition of biochemical reactions and to geometric size andshape. In this arrangement, the amount of produced biomass (which in turn is defined by the growth ratio) is a leading indicatorthat defines composition of biochemical reactions. This fact is wellstudied at a cellular level [13,17] with the aid of methods ofmetabolic flux analysis. However, according to the general growthlaw, the same is true at the organ level. Thus, we acquire a veryimportant universal link between the composition of biochemicalreactions, integral nutrient influxes and biomass production at the organ level . Of course, further studies are required to realize thispotential. Model Applications
The introduced model and the obtained results can be appliedin different areas of biology and medicine. Real phenomena, bytheir nature, are multifactorial. One of the advantages of theproposed model is that it provides a general understanding of anorgan’s growth dynamics in relation to many other factors. Inother words, it allows seeing the overall, often dynamicallychanging, picture. For instance, in liver transplantation thepatient’s safety and fast recovery are priorities. Although therewere successful transplantations when donors were left with onlyabout 30% of their original liver volume [28,29], in otherinstances, donors with a substantially bigger part of livers died, soit is a combination of different factors that secures positiveoutcome. In [28], the authors list diverse reasons for rejectingdonors, which confirms this fact. So, any additional informationcan potentially be useful if it is correctly interpreted.We found that in the case of dogs there is apparently a stablerelative size of a regenerating liver, equal to roughly 54% of thegrown organ, when the normal growth begins. In case of humans,a similar effect most likely exists, so that finding such a value forpeople would allow having a reliable quantitative parameter relatedto successful recovery. We were also able to evaluate thepercentage of liver mass actively involved in proliferation belowthis threshold depending on the phase of growth. This is also avaluable parameter which serves as a good indicator of themetabolic stress the liver transplants (or the liver remnants indonors) are subjected to, since at this critical stage of growth the
Figure 6. Change of the growth ratio for growing right and leftliver lobes.
The effect is due to changing liver geometry duringgrowth. Scenarios are presented when initial lobe volumes are differentand when they are the same.doi:10.1371/journal.pone.0099275.g006
Figure 5. Growth of remaining left liver lobes in human donors. iver has to support both the metabolic needs of an organism and,at the same time, its own growth.Another discovered result useful for clinical and other applica-tions relates to the close relationship between the size of a growingliver and its biochemical properties. What is even more important,we were able to introduce a quantitative parameter, the growthratio, which quantifies such a relationship through the amount ofproduced biomass. In fact, the found relationship unambiguouslyworks in both directions, that is, once we know the current size of agrowing liver, we can make predictions about the composition ofbiochemical reactions. Inversely, once we know certain specificbiochemical characteristics, we can evaluate the relative size of agrowing liver compared to its final size, which would be a nicenoninvasive inexpensive method for controlling the recoveryprocess. Such a possibility is confirmed by observations from [24]with regard to ornithine decarbohylase, whose concentrationdepends on the phase of growth. Since biochemical reactions donot proceed in isolation, but are tightly interrelated to each otherwithin the same biochemical machinery, this approach lookspromising, since knowledge of the content of several substancesfundamentally allows restoring the overall composition of biochem-ical reactions.Close values of biomass increase rates, which we obtained fordogs, present another observation worthy of attention, since if it isvalid for people, it allows introducing a quantitative referencevalue, to which the recovery process may refer to.Abilities of livers to regenerate depend on their metaboliccapacity, which is indirectly evidenced by results obtained inreferenced works. We already briefly discussed that the metaboliccapacity of female livers is noticeably higher than that of malelivers, of which the smaller final size of livers in females [28] is oneof the effects. Such a sexual distinction is an important factor to betaken seriously in clinical practice. It means that a female donorcan be safely left with a smaller part of liver than a male donor.For male donors, the size of liver remnants is more critical forsuccessful recovery, all other factors being equal.The mere fact that the liver size and its metabolic capacityinterrelate also provides interesting possibilities. Of course, lifestyleinfluences metabolic requirements, and accordingly affects liversize. However, when all other factors are equal, a smaller liverwould mean higher metabolic capacity. So that maybe a smallliver is not a so restrictive factor for transplantation purposes,although in the study [28] ‘‘inadequate liver volume’’ contributedto 19.5% of donor rejections.Of course, the considered examples by no means exhaustpossible clinical and other applications of the presented model andobtained results. This is a general model, which is based on afundamental law of nature, so that it can be used for a very widerange of purposes. In this section, we just scratched the surfacediscussing examples of possible applications.
Conclusions
Based on the earlier discovered general growth law, presented in[17], we proposed a macroscopic model for volumetric growth oforgans that accounts for quantitative characteristics of growth andfor the geometric shape of the organ. We exemplified the use ofthe resulting model by applying it to modeling growth oftransplanted livers and to identifying characteristics of growinglivers in dogs and humans. We validated the model by comparisonwith available experimental data from the literature on growth ofliver transplants in dogs and liver grafts and remnants in humans.In the case of dogs, we modeled growth of whole livers, so that wehave had a proportional increase of the whole organ, whose shape thus did not change during growth. In the case of humans, wemodeled growth of liver grafts obtained from donors as a result ofhepatectomy, and liver remnants in donors, so that the liver hasbeen changing its geometric form during growth.We made the following observations:1. A dood agreement between experimental data and thetheoretically predicted growth curves for growing livers, livergrafts, and liver remnants was discovered.2. We were able to determine the time point when a liver switchesfrom partial growth to a normal, evolutionarily developed,growth (i.e., the joining point) in dogs. This result can be usedfor optimizing the size of liver transplants and the fraction ofliver left in the donor.3. The portion of the liver in dogs that participates inregeneration from the very beginning was found.4. We found the functional dependence of the conversion of‘‘passive’’ (with regard to growth) liver parts to ‘‘active’’,growing parts in dogs.5. We discovered apparently stable relationships between the sizeof a fully grown liver and the time point when the liver switchesto normal regeneration (in case of large resections, to a fullregeneration).6. In dogs, the rates of liver growth before the joining point aresimilar.7. In humans, the fact that left-lobe liver remnants grow fasterthan right-lobe remnants is partially due to differences in theirgeometry. We qualitatively described this effect and found thatit may account for about 10% of the difference in growth rates,depending on the initial volume of liver graft relative to thewhole liver.Although we focused on modeling growth of livers, the presentmethod can potentially also be applied to modeling growth ofother organs or whole organisms.Our results show that the growth equation, which is themathematical representation of the general growth law, is anadequate quantitative and phenomenological tool for manypractical applications and theoretical studies. It accuratelydescribes the dynamics of organ growth in quantitative terms,and it allows hypothesizing about the mechanisms underlyingmany effects observed in experimental studies.The proposed method can be used for quantitative estimation ofthe optimal size of liver transplants from the perspective of patientsafety and recovery time. The present method also allowsoptimizing the shape of transplants, and provides quantitativeindications for nutrient supply in safe and fast recovery.A related method, also based on the general growth law, forfinding metabolic characteristics of organisms and their constitu-ents (cells, organs, etc.) has been developed and experimentallyverified using data on liver and liver transplants. It allows findingrates of nutrient consumption for growth and maintenance, andthe total amount of nutrients required for growth. These studiesare presented in a second article.
Acknowledgments
The authors thank A. Y. Shestopaloff for discussions, reviews, and editingefforts, Dr. P. H. Pawlowski (Institute of Biochemistry and Biophysics,Polish Academy of Sciences) for continuous support in the study of thegeneral growth law, and Reviewer for valuable comments. uthor Contributions
Analyzed the data: YS. Wrote the paper: YS. Editing manuscript: YS IS.Conceiving the idea to study liver growth: YS IS. Designed software usedin analysis: YS. Discussions of the results and material: YS IS.
References
Method for Modeling Growth of Organs and Transplants Based on the General Growth Law: Application to the Liver in Dogs and Humans
Yuri K. Shestopaloff , Ivo F. Sbalzarini Figure 4. Geometric model of a human liver.