Spectroscopie théorique des quasars et loi de Karlsson.

Jacques Moret-Bailly *

11 mars 2016

Résumé

La loi de Karlsson introduite par la spectroscopie des quasars peu rougis utilise le spectre Lyman de l'atome d'hydrogène. Il faudra comparer les concepts déduits de la spectroscopie classique des quasars, développée ici, à ceux qui sont déduits du Λ-CDM. Une absorption fine et saturée d'une raie spectrale dans un gaz requiert un long parcours sans perturbation comme collisions ou rougissement cosmologique. Ainsi, les spectres de "forêts Lyman" des quasars obéissant à la loi de Karlsson résultent essentiellement d'interactions de la lumière thermique naturelle rayonnée par le quasar avec de l'hydrogène atomique froid sous très basse pression. Ces raies sont produites par trois processus : a) Une absorption usuelle produit un ensemble de raies ; b) Ces raies sont multipliées par absorption après des rougissements fondamentaux 3K (ou 4K), où K est la constante de Karlsson ; 3K (ou 4K) amènent exactement une raie Lyman beta (ou gamma) absorbée sur une raie Lyman alpha : La coincidence d'une raie absorbée rougie, avec Lyman alpha arrête donc quasiment le rougissement et les raies du gaz sont absorbées intensément ; c) Les rougissements se produisent dans des régions où la lumière à la fréquence alpha est faiblement absorbée en raison du rougissement permanent, sauf lorsque l'excitation de l'hydrogène au niveau 2P est suffisante pour qu'un flash superradiant éclate à la fréquence alpha, ce qui provoque une absorption intense des rayons issus du quasar dont la radiance est grande, donc l'inscription d'une raie à la fréquence Lyman alpha courante. Éclairs et pompages produisent des oscillations de relaxation qui inscrivent de nombreuses raies d'absorption. Les rougissements par les atomes H dans les niveaux 2P sont dus à des interactions paramétriques composées d'effets Raman impulsionnels stimulés (ISRS) : Des atomes d'hydrogène excités catalysent des échanges d'énergie entre le rayon observé et des rayons froids du fond thermique, en accord avec la thermodynamique. La description de l'univers devient beaucoup plus simple mais moins merveilleuse.

Theoretical spectroscopy of quasars within Karlsson's law.

Abstract.

The law introduced by Karlsson in spectroscopy of low-redshift quasars involves the Lyman spectrum of hydrogen atoms. Thus, it appears necessary to study the concepts introduced by a standard spectroscopy of quasars, studied here, with those deducted from Λ-CDM. A visible absorption of a sharp and saturated spectral line in a gas requires a long path without perturbations as collisions or cosmological redshift. Spectra of absorbed, saturated lines of quasars obeying Karlsson's law mainly result from interactions of natural, thermal light radiated by quasar with relatively cold, low presure atomic hydrogen. These lines are produced by three processes : a) A conventional absorption in a relatively cold gas produces a set of lines ; b) These lines are multiplied by absorption after fundamental 3K or 4K redshifts, where K is Karlsson's constant : Spectra show that redshifts 3K (or 4K) exactly bring absorbed Lyman beta (or gamma) line on Lyman alpha : redshift almost disappears, and gas lines are intensely absorbed in the absence of alpha absorption ; c) Redshifts occur in regions where light at alpha frequency is poorly absorbed due to permanent redshift, except when excitation of hydrogen to 2P level is sufficient for a superradiant flash emission at alpha frequency. This causes an intense absorption of high radiance rays from quasar, so the absorption of a line at current Lyman alpha frequency. Lightning and pumping produce relaxation oscillations that write many absorption lines. Redshifts by H atoms in 2P levels are due to parametric interactions composed of Impulsive Stimulated Raman Scatterings (ISRS) : excited hydrogen atoms catalyze energy exchanges between observed ray and background cold thermal radiation, in agreement with thermodynamics. Description of Universe becomes much simpler, but less marvelous. 1 Introduction.

Keywords

Karlsson [1] and Burbidge [2] used Karlsson's constant K to represent the so-called quantization of reddshifts of quasars : Most redshifts (relative variations of frequencies of spectral lines) of quasars have a value Z(n) = nK, where n is an integer from the strange serie : 3, 4, 6, .... Note that, with the precision allowed by the calculation of K, deducted from thousands of measures, 3K (resp. 4K) is redshift Z(β, α) (resp. Z(γ, α) which converts frequency of β (resp. γ) line into frequency of α line [3]. Rewriting Karlsson's formula :

Z(p, q) = pZ(β, α) + qZ(γ, α, ) (1)
where p and q are non-negative integers is justified by the absence, in usual interpretation of quasars spectra, of sharp and saturated absorption lines named "redshifted Lyman β (or γ) lines" [4,5]. Calculation of such rays from redshifted rays called "Lyman α" leads to other α lines. Overlay of lines is so perfect that it products no broadning of the observed line.

Karlsson's formula links astronomical redshift to properties of hydrogen atom. The purpose of this study is allowing easy comparison of results deducted from Λ-CDM to those given here by standard spectroscopy.

Karlsson-Burbidge model of a quasar sur-

rounded exclusively by hydrogen.

Generalities, conventions.

In Doppler frequency shifts, emitted and received frequencies verify equation which involves, as parameters, emission and reception speeds V emission and V reception :

ν reception = c-V reception c-V emission * ν emission (2)
In a first approximation we assume that frequency shifts at various light frequencies obey an equation similar to equation ( 2), so that a single parameter is needed to compute frequency shifts at all frequencies, for instance the ratio f 1 /f 0 of a particular shifted frequency over initial frequency :

ν shif ted = f 1 f 0 ν initial (3)
If this approximation appears too bad, it may be corrected by multiplication of ν shif ted by a dispersion function D(ν shif ted ), generally close to one. Using Rydberg's formula, compute redshifts which put Lyman ν β and ν γ frequencies of H atom to ν α frequency :

Z (β,α) = (ν β -ν α )/ν α = [(1 -1/32 -(1 -1/22)]/(1 -1/22)] ≈ 5/27 ≈ 0.1852 ≈ 3 * 0.0617 ≈ 3 * K; (4) Z (γ,α) = (ν γ -ν α )/ν α =[(1-1/42 -(1-1/22)]/(1-1/22) ] = 1/4 = 0,25 = 4*0.0625 ≈ 4K. (5)
Assuming that use of 3 lowest Lyman frequencies of H atom is better than use of K, and that variation of shifts at various frequencies is obtained from a particular shift as in a Doppler redshift, Karlsson's formula becomes simpler : shifted frequencies ν(p, q) depend only on absolute frequency ν 0 , three well known frequencies and two any non-negative integers p and q : ν(p, q) = (ν α /ν β ) p * (ν α /ν γ ) q * ν 0 . ( 6) Genuine Karlsson's formula, applied, for instance, to Lyman beta line with n=3 does not generate exactly Lyman alpha frequency, while formula (6) does. We choose formula (6) because choice of possible values of p and q seems more natural than extraction of n from a less natural serie. Is it only aesthetics ?

Hypothesis and observations.

We assume that "Lyman forest" is built by absorption of a thermal emission of an extremely hot star by pure, relatively cold (2 000-50 000 K), low pressure, unexcited, atomic hydrogen.

Following properties of many lines of "Lyman forest" are directly deduced from Petitjean's paper [4] :

-A-As lines are sharp, widening of lines by collisions must be negligible, pressure of gas must be very low.

-B-To obtain absorption of Lyman lines at several frequencies, a redshift process of electromagnetic waves is necessary. Here, we make only hypothesis of formula (3) about this redshift.

-C-A single, unshifted Ly β is observed ; no unshifted Ly γ appears because it does not remain absorbable energy at high frequencies after redshifts of thermal emission profile.

Absence of an absorption line may result from : -a-Absence of emitted energy around frequency of line. -b-A permanent shift of light frequencies dilutes absorption, so that all absorbed (or emitted) lines have width of shift and lines are weak, not observable. Accordingly, absorption (or emission) of sharp lines requires a stop of frequency shifts.

-c-Accurate superposition of observed lines results from the choice of redshift equations : Absence of shifted β and γ lines while sharp α lines are observed, is due to superposition of these lines with α lines.

In Λ-CDM theory, sharpness of saturated absorbed lines requires contradictory conditions :

-Column density of gas must be large for saturation ; -Absorbing gas must be thin to avoid a broadening of lines by frequency shift during absorption.

-Pressure of gas must be low to avoid collisional broadening of line.

Thus gas must be in filaments which are only detected on paths from quasars.

Supposing that redshift is related to a physical property of gas, condition is :

-Light is redshifted except if an absorbed line is at Lyman α frequency. Thus redshift requires a Lyman α absorption, that is generation of 2P atomic hydrogen.

Building a spectrum by Karlsson's redshifts.

Figure 1 represents a canvas of atomic hydrogen spectrum for building any absorption spectrum by addition of lines, in particular able to play the role of Ly β or Ly γ lines.

Rules used to build spectrum taking into account only hydrogen atoms are simple :

-i -At start, close to star, we suppose that α, β and γ lines have been absorbed.

-ii -It appears a frequency shift until an absorbed, shifted line of initial frequency ν reaches ν α frequency. Thus all absorbed frequencies have been multiplied by ν α /ν, coefficent lower than 1. In pure atomic hydrogen ν may be ν β or ν γ , assuming that higher frequency lines are too weak.

-iii -During stop of frequency shift, the three main lines of H could be absorbed, but there remain no energy at Ly α frequency.

-iv -Assume that Ly β absorption produces a very weak redshift. During this negligible redshift, gas lines are visibly absorbed. If weak redshift is able to shift absorbed frequencies off Ly α absorption line before full absorption of light at Ly β frequency, fast redshift restarts, go to -ii -. Else there is no more redshift, absorbed Ly β line is visible, but Ly γ line is probably not because its frequency is probably larger than the shifted high frequency limit of emission of star.

In pure H, total redshift results from several relative shifts Z β,α and Z γ,α which correspond to multiplication of light frequencies by ν α /ν β or ν α /ν γ .

Computed lower frequencies written on figure are not very good. They can be corrected by use of an unique dispersion function D(ν) equal to ν at and in neighborhood of ν α frequency.

Thus, formula 6 becomes :

ν(p, q) = D[(ν α /ν β ) p * (ν α /ν γ ) q * ν 0 ]. ( 7

Introduction of other gas in spectrum building.

Lines found in figure 1 appear in real spectra, after a correction of their frequencies by the chromatic dispersion factor F (ν). But many lines have an other origin : various lines absorbed at frequencies larger than ν α may play the role of Ly β and Ly γ , multiplying density of lines.

Figures much more complex than fig. 1 can be drawn.

Structuring space.

Redshifts stop if frequency of an absorbed line becomes ν α and, for instance, not if absorbed ν γ frequency reaches ν β frequency. This shows that redshift results on generation of 2P hydrogen atoms.

We have supposed that it exists redshifts in some places. Thus, assuming that quasar is far from other stars, space is divided into spherical shells where generated 2P atoms shift light frequencies, and shells in which there is no 2P atoms and no frequency shifts.

As quasars are not alone in space, perturbation of generation of these regions may result from pumpings of atoms by light of other stars. Thus region in which quantized redshifts appear must be small enough to avoid an important lighting by other stars : This explains Burbidge's selection of quasars.

To avoid different pumpings by light coming from different regions of surface of the star, redshift must appear only at a distance much larger than size of the star, that is in a region where pressure of gas is low.

These conditions are evidently not verified for a set of relatively close stars as a galaxy. As the star must produce very high light frequencies, if it is small it must be extremely hot. An hypothesis which seems valuable is an accreting neutron star, a type of stars which were never found in nebulae while they should be bright enough to be seen.

Dynamics of space structures : Absorption of Lyman α frequency by competition of modes.

In model described in 2, we obtained spherical shells of hydrogen atoms either excited by absorption of Lyman ν α frequency, or in ground state. But it is difficult to obtain the large number of lines observed for instance between Ly α and Ly β lines ( Ly α forest ).

In 2.5, we took only ito account the lack of absorption at ν α frequency to explain stops of redshifts. Let us consider a spherical shell in which Lyman alpha line is absorbed, therefore light is redshifted.

The fraction of excited atoms grows without reaching the limit, not very lower than 1/2, which corresponds to the very high temperature of spectral radiance of star at Lyman α frequency. Einstein's coefficient B becomes large, so that intense superradiant emissions can burst. Flash of-excites atoms brutally and causes an intense absorption of other rays at α frequency by competition of modes. Thus an absorption line appears at Lyman α frequency. This process is in conformity with thermodynamics because spectral radiance of stellar ray is higher than that in superradiant mode. After flash, gas restarts an excitation. Process absorption-flash reproduces without any simple computation of relaxation period, therefore of position of new absorbed lines.

It is difficult to discuss on frequency and stability of relaxation oscillations which generate these absorptions, thus on stability of spectra. However, the medium is probably much more homogeneous and stable than the medium which provides aurora borealis. Frequencies must be adjusted from observation. It is also difficult to discuss geometry of flashes, seen as flares, probably induced by rays emitted by the star into directions close to direction of observation of the star.

Physical interpretation.

Propagation of light in 1S and 2P atomic hydrogen.

In 1S state, hydrogen has 1420 MHz quadrupolar resonance frequency, (period T = 0.7 ns ; wavelength λ = 21 cm).

In its first excited state, hydrogen atom has quadrupolar resonance frequencies : 178 MHz (T = 5.6 ns, λ = 1,7 m) in state 2S 1/2 , 59 MHz (T = 17 ns, λ = 5 m) in 2P 1/2 and 24 MHz (T = 42 ns, λ = 12 m) in 2P 3/2 .

A qualitative difference between laboratory and space ISRS is that, in labs, ISRS is observed using femtosecond laser pulses while ordinary incoherent light is made of longer, around 1 nanosecond pulses. For space coherence, hyperfine periods in excited states, must be longer than length of pulses (1ns) in ordinary light as required by conditions of space coherence of "Impulsive Stimulated Raman Scattering" (ISRS) : Length of pulses must be shorter than all involved time constants [6].

Space coherence of incident and scattered light in ISRS allows an interference of exciting and scattered light, which produces an intermediate fre-quency, that is shifts frequency of incident light and preserves the geometry of light beams.

The other condition for ISRS is : Collisional time must be longer than 1 nanosecond, that is pressure must be very low, so that pressure broadening of lines is low.

These conditions make ISRS in space very weak compared with ISRS in labs of chemistry, which uses around 10 femtosecond laser pulses, around k = 10 5 times shorter than light pulses of temporally incoherent light :

-Division of pressure by k divides the shift by k.

-Division of quadrupolar frequency by k divides the shift by k twice :

--Once mixing incident and Raman frequency shifted of hyperfine frequency ;

--Once by division of difference of populations of hyperfine levels, assuming thermal equilibrium.

Thus order of magnitude of path needed to observe ISRS is ≈ 10 15 times longer than in laboratory experiments : astronomical paths are needed for observation.

De-excitation of hyperfine levels.

As pressure is very low, collisions between atoms are negligible. A radiative process is necessary to de-excite hyperfine levels, thus obtain a permanent Raman frequency shift.

Happily, there are cold background electromagnetic waves. Thus, the real process is not a single ISRS, but a set of ISRS such that hyperfine levels of 2P atoms have a constant excitation. Excited atoms H catalyze an exchange of energy between light beams, it is a parametric interaction. Variations of energy produce frequency shifts of light beams such that entropy is increased. This parametric interaction is usually named "Coherent Raman Effects between Incoherent Light beams" (CREIL).

Main applications in astrophysics of Co-

herent Raman Effects on Incoherent Light.

Blueshift of microwaves emitted by Pioneer probes.

Anomalous blueshifts of microwaves exchanged with Pioneer 10 and 11 probes [7] are explained by an "anomalous acceleration". Anomalous increase of frequency happened between 10 and 15 AU from Sun, exactly where solar wind cools into excited atomic hydrogen. It results from a transfer of energy from solar light to weak microwaves.

Dispersion of multiplets emitted by dense regions of quasars.

This dispersion results from dispersion of hyperfine polarizability of hydrogen atoms. There is no need of variation of fine structure constant.

Redshifts of light by excited hydrogen.

Excited hydrogen atoms are abundant close to hot stars and galaxies. Thus Hubble's law exaggerates distances close to these hot objects. This radial increase of distance produces, as angular distances are not changed, an increase of transverse distances. Thus, it appears anisotropic bubbles, well visible on maps of galaxies, bubbles which give a spongious appearance to these maps.

As distances of spiral galaxies are increased, their size is increased and celestial mechanics seems fail. It does not !

Conclusion.

Standard coherent spectroscopy is used to study spectra of quasars : Karlson's formula on redshifts of quasars is demonstrated, widespread and applied to study sets of sharp, saturated absorption spectral lines. Generation of other sharp lines results from a competition of modes during flash, superradiant of-excitements of very low pressure 2P hydrogen atoms.

Interpretation of many astronomical frequency shifts is easy using coherent exchanges of energy between light beams, which increase entropy of the beams. Excited atomic hydrogen, mainly in its 3P state is a powerful catalyst of these exchanges. it is unnecessary to introduce, as in the theory of expanding universe, new concepts to explain some observations.

An other coherent optical effect should be used in astrophysics : Superradiance shows the limbs of Strömgren spheres as possibly dotted circles. Strömgren's sphere of supernova 1987A is strangulated into an hourglass by absorption of energy by planets. Multiphotonic interactions, superradiance and competition of modes explain that the star disappeared when the beautiful rings appeared.

Astrophysicists should learn some coherent spectroscopy or collaborate with spectroscopists.

  Figure1represents a canvas of atomic hydrogen spectrum for building any absorption spectrum by addition of lines, in particular able to play the role of Ly β or Ly γ lines.Rules used to build spectrum taking into account only hydrogen atoms are simple :-i -At start, close to star, we suppose that α, β and γ lines have been absorbed.-ii -It appears a frequency shift until an absorbed, shifted line of initial frequency ν reaches ν α frequency. Thus all absorbed frequencies have been multiplied by ν α /ν, coefficent lower than 1. In pure atomic hydrogen ν may be ν β or ν γ , assuming that higher frequency lines are too weak.-iii -During stop of frequency shift, the three main lines of H could be absorbed, but there remain no energy at Ly α frequency.-iv -Assume that Ly β absorption produces a very weak redshift. During this negligible redshift, gas lines are visibly absorbed. If weak redshift is able to shift absorbed frequencies off Ly α absorption line before full absorption of light at Ly β frequency, fast redshift restarts, go to -ii -. Else there is no more redshift, absorbed Ly β line is visible, but Ly γ line is probably not because its frequency is probably larger than the shifted high frequency limit of emission of star.In pure H, total redshift results from several relative shifts Z β,α and Z γ,α which correspond to multiplication of light frequencies by ν α /ν β or ν α /ν γ .Computed lower frequencies written on figure are not very good. They can be corrected by use of an unique dispersion function D(ν) equal to ν at and in neighborhood of ν α frequency.Thus, formula 6 becomes :ν(p, q) = D[(ν α /ν β ) p * (ν α /ν γ ) q * ν 0 ]. (7)Determination of D function requires either study of a good spectrum or calculation of dispersion of polarizability of H atom in state 2P.

Figure 1 -

 1 Figure 1 -Generation of lines of Lyman forest of a quasar by coincidence of line α of gas with previously absorbed, shifted β or γ. During stop of redshift Ly β and Ly γ lines are absorbed. Lines of other local gas may be also absorbed and may later play the role of Ly β and Ly γ lines if their frequencies are larger than ν α . Written frequencies do not take into account dispersion of hyperfine polarizability of H atom.


Sandrine Selosse
email: sandrine.selosse@mines-paristech.fr

Nadia Maïzi

Exploring the biomass carbon capture solution to climate policy: A water impact analysis with TIAM-FR

Keywords: Energy system, long-term modelling, TIAM-FR, Climate change, CO2 mitigation, Carbon Capture and Storage (CCS), Water impact

A sustainable energy future requires a wide range of different mitigation options that can reduce the CO2 emissions. Particularly, renewables and carbon capture and storage appear as preferred or more largely evocated options. The aim of this study is to analyze alternative development paths of the energy system investigating different constraints on the use of CCS and BECCS, under climate policy context, and using the global multiregional optimization model, TIAM-FR. The analyze also focuses on the increasing pressure involved by the development of carbon capture technologies (fossil and biomass) on the water resources. Water and energy are indeed inextricably linked and interdependent sectors. Water requirements of existing and emerging technologies (such as carbon capture technologies) are so necessary to completely assess the water impacts of a developing decarbonizing economy.

www.modelisation-prospective.org W According to the 5th assessment report from IPCC, all key GHG emissions mitigation options (energy efficiency, renewables, carbon capture and storage -CCS) need to deliver to 2050 on a vast scale. More particularly, the technology 'readiness' of advanced technologies, as the industrial scale of carbon capture and storage (CCS) and the combination of Bioenergy, Carbon Capture and geologic Storage (BECCS) appear more and more as incontrovertible to attain stringent CO2 mitigation targets and reduce future CO2 emissions in line with the consensual limit of 2°C temperature increase. This all the more if we consider that coal fossil fuels will remain the dominant sources of energy over the next decades and that, as a result, CO2 emissions will drastically increase to reach unsustainable levels. Furthermore, between 250 and 300 EJ (a quarter to a third of the world's energy supply in the second half of this century) may need to come from biomass to make the decarbonation of energy system possible. In the transport sector, according to IEA, as key challenges and opportunities, the integration of advanced (2nd generation / 2G) biofuel plants with conventional (1st generation / 1G) biofuel plants can lead to significant synergies and cost savings, especially for bioethanol plants. For biodiesel, conversion of fossil refineries to advanced biofuel production is another promising option as well. In the electricity sector, where electricity generation alone accounts for approximately a third of the global emissions, those climate targets accentuate the need for negative emissions on a large scale. Indeed, BECCS offers a unique opportunity for a net carbon removal from the atmosphere while fulfilling energy needs (Herzog et al., 2005;Azar et al., 2006;van Vuuren et al., 2007;Katofsky et al., 2010) (figure 1)

I -Introduction

Figure 1: Negatives emissions in the bioenergy lifecyle Source: Adapted from Assessment Report from the Global Climate and Energy Project (GCEP) workshop (Milne and Field, 2012) In the case of a fossil fuel facility, adding CCS technology allows to drastically reducing the level of CO2 emissions, it is expected even ideally to lead to a zero net emission as could a dedicated biomass facility. The combination of biomass and CCS could lead to a net reduction in atmospheric CO2 levels from the lifecycle of the process (Milne and Field, 2012, from Henrik Karlsson) as the level of CO2 sequestrated are higher than the one that is released to the atmosphere over a given time. The aim of this study is to analyze alternative development paths of the energy system investigating different constraints on the use of CCS and BECCS, under climate policy context. This analysis is conducted using the global multiregional TIAM-FR optimization model.

W Working Paper N°2016-01-19 www.cma.fr Exploring the biomass carbon capture solution to climate policy: A water impact analysis with TIAM-FR Furthermore, the perspective of investments in CCS implies awareness on the water impacts of this climate strategy. Indeed, the global increase of coal power, along with the coal industry's adoption of new technologies, will drive the largest share of water consumption for energy use through 2035, and the amount of fresh water consumed for world energy production should double within the next 25 years, according to the IEA. A transition to a less carbon-intensive electricity sector results in an increase in water consumption per unit of electricity generated, depending on the choice of technologies and cooling systems employed. Particularly, low-carbon emitting technologies that utilize cooling towers as pulverized coal with carbon capture technologies appear as a very high (among the highest) water consumption factors (Macknick et al., 2011). Water needs of the processes have been implemented in this model (Bouckaert et al., 2012(Bouckaert et al., , 2014)). Water and energy are two areas closely related and highly interdependent. Measures taken in one area impact directly and indirectly on the other domain, whether positive or negative. The amount of water required to produce energy depends on the chosen type of energy production. Demand for energy and fresh water will increase considerably in the coming decades. This growth will pose major challenges and will exert strong pressure on the resources of almost all regions, particularly in developing economies and emerging. So, in this study, we also analyze the increasing pressure by the development of carbon capture technologies (fossil and biomass) on the water resources and introduce the question of the plausibility of technological choices in terms of water availability. The paper is organized as follows: section 2 describes the methodology used for the analysis and the constraints scenarios. Section 3 presents the results of the long-term modeling, and the final section concludes with a discussion on CCS options and water impact.

II -Modeling approach and scenario analysis

a/ Modeling approach: The TIMES Integrated Assessment Model (TIAM-FR) TIAM-FR is the French version of the TIMES Integrated Assessment Model, the widely used global multiregional model from the TIMES family models developed by the Energy Technology Systems Analysis Program (ETSAP), under the aegis of the International Energy Agency (IEA) (Loulou and Labriet, 2008). This linear programming model estimates an inter-temporal partial economic equilibrium on energy markets and, in other words, minimizes the total discounted cost of the world energy system over a long time period under environmental, technological and demand constraints. The net present value (NPV) of the total energy system costs for all regions is the sum of all annual costs per region r and year t, ACr,t, discounted at a dr,y general rate:

Where: t0 is the reference year for discounting; T is the set of years for which costs are incurred, which includes all years in the model horizon, plus past years (before the reference year t0) if costs have been defined for past investments, plus a number of years after the end of the planning period where some investment and dismantling costs are still being incurred, as well as the salvage value of fixed capital; R is the set of TIAM-FR regions. An important feature is that global investment decisions are made in each period with full knowledge of the future cost and demand trajectories. In other words, the decision makers are assumed by the model to operate globally with the benefit of full information and perfect foresight (clairvoyance of energy planner) for the calculation period, the described economic sectors, and commodities. TIAM-FR formulates and computes its projection of optimal energy systems based on the Linear Programming approach. It can be summed up as follows: where X is the vector of all variables with associated discounted cost vector c, I the number of demands categories for energy services; Qk,i(t) the capacities of end-use technologies k susceptible of addressing service demand i at time t; Di(t) the exogenous demand for energy service i to be satisfied at time t; B and b vectors or matrixes of exogenous parameters (echoing emission contents and potential caps, energy contents and energy efficiency mandates, technology mandates, etc.). Expression (2) defines the total discounted cost to be minimized. Expression (3) formulates the set of demand satisfaction constraints. Expression (4) synthesizes the set of constraints weighing on the cost minimization, a large number of which express the physical and logical relations that must be satisfied in order to properly depict the energy system (Loulou, 2008).

Cost of the energy system includes investment costs, operation and maintenance costs, costs of imported fuels, incomes of exported fuels, and the residual value of technologies at the end of the horizon. TIAM-FR aims to supply energy services at minimum global cost by simultaneously making decisions on equipment investment, equipment operation, primary energy supply, and energy trade. End-use demands (i.e. energy services) are based on socio-economic assumptions and on external projections of the growth of regional GDP as well as population and volume of various economic sectors (transport, residential, industry, etc.) over the planning horizon. These drivers and IEA statistics for a given base year, in this case 2005, are the basis for future projections of the consumption of different energy such as road passenger transportation, steel demand or residential heating. In order to satisfy the energy services demands, the system includes the extraction, transformation, distribution, end-uses, and trade of various energy forms and materials.

Indeed, TIAM-FR is a technology-rich, bottom-up model which depicts the energy system with a detailed description of different energy forms, resources, processing technologies and end-uses, on a Reference Energy System (RES). Each economic sector is described by means of technologies, each characterized by its economic and technological parameters in all sectors of the energy system (agriculture, industry, commercial, residential and transport; taking into account conversion and the electricity sector (Loulou and Labriet, 2008). Figure 1 The main outputs of the model are future investments and activities of technologies for each time period. Furthermore, the structure of the energy system is given as an output, i.e. type and capacity of the energy technologies, energy consumption by fuel, emissions, energy trade flows between regions, transport capacities, a detailed energy system costs, and marginal costs of environmental measures as GHG reduction targets. The model calculates CO2, CH4, and N2O emissions from fuel combustion and processes and integrates a climate module which allows calculating or constraining atmospheric GHG concentration, radiative forcing and temperature changes. Emission reduction is brought about by technology and fuel substitutions (leading to efficiency improvements and process changes in all sectors), carbon sequestration (including CO2 capture at the power plant and hydrogen plant level, sequestration by forests, and storage in oil/gas fields, oceans, aquifers, etc.). An additional output of the model is the implicit price, or opportunity cost (shadow price), of each energy form, material and emission. Indeed, TIAM-FR integrates several carbon capture and sequestration technologies derived from fossil or bioenergy resources. In the power sector, the model considers two capture technologies for bioplants: precombustion for the biomass gasification process, and post-combustion for the direct combustion process. Biomass co-firing in coal power plants has also been implemented in TIAM-FR, with and without carbon capture technologies (Ricci and Selosse, 2013).

In the model, biomass is characterized by manifold sources -industrial waste, municipal waste, landfill gas, bioenergy crops, and solid biomass resources -and the fact that it is not traded between regions. The maximum amount of available biomass for each region is determined exogenously according to IEA data. The global potential is estimated at 234 EJ per year in 2050. In literature, biomass potential varies greatly given the different assumptions on land use, yield development, food consumption and other criteria of sustainability such as water scarcity and loss in biodiversity (van Vuuren et al, 2009). This potential varies between 100 EJ and 400 EJ per year over the period 2050-2100. To highlight the importance of water impact in the evolution of energy system and discuss the plausibility of future technological options, particularly in climate policy context, water footprints of the different processes have been implemented in the model. So water consumption and withdrawal have been indicated for all processes (Bouckaert et al, 2012(Bouckaert et al, , 2014)).

b/ Scenario analysis: Alternative paths of the future energy system

To analyze possible alternative development paths of a lower carbonated future energy system we investigated alternative scenarios according to different assumptions concerning:

Climate policies 2) and 2) the world carbon storage is bounded to 9,392 GtCO2.

www.cma.fr

Exploring the biomass carbon capture solution to climate policy: A water impact analysis with TIAM-FR

III -Results

a/ What development of CCS in the climate pathways?

The analysis of the results focuses on the effects of the assumed various constraints (environmental, resources and technological) on the power mix and on the future technological investments. The water impact of the latter, in terms of water needs, is also analyzed, insofar as the fresh water consumptions for energy production are assessed by the model. This study constitutes a first analysis of the water impact resulting from technological options under climate pathways.

Indeed, the investigated climate policy scenarios highlight the importance of technological improvements and lead to a noticeable expansion of renewable energy and CCS technologies in the power sector, fossil and/or BECCS according to technological constraints (figure 3). In 2050, in the RF_2p6 scenario, renewables (hydro excluded) represent 32.8% of the electricity production. This contribution reaches 37.6% in 2100 in the same scenario. The large development of renewables appears later in the RF_3p7 scenario, these electricity sources representing 33.7% of the power generation in 2100 but 7.8% in 2050. It is interesting to note an increase of the nuclear production of electricity at the end of the period, but whatever the scenario. In a general manner, CCS from fossil fuel is mainly deployed in fast developing countries that are well endowed with coal. The remaining fossil fuel use is indeed in combination with carbon capture and storage. Gas and CCS is mainly deployed in developing countries in 2050 as BECCS in the RF_2p6 scenario.

In 2100, in both scenarios, coal is the most important source of CCS power plants, even if BECCS is highly distributed in developing countries. However, biomass resources are widely available in all regions and it is interesting to note that BECCS tends to replace CCS at the end of the period in both scenarios.

The assumptions according to carbon storage capacities and BECCS investments involve different sets of mitigation options across regions, with varying shares of renewable energy, CCS, gas, and biomass (figure 4). We focus here on the stronger climate scenario, the RF_2p6 scenario and investigate an analysis of the power mix according to different options of development of the CCS technologies. In the RF_2p6_NoBECCS scenario, we assume that no investment is made in bioplants with CCS. Note that the co-firing associating coal and biomass is allowed, that is not the case in the RF_2p6_NoBECCSCF scenario where investments are only possible in fossil power plants with CCS. In the RF_2p6_NoCCS scenario, no investment is made in the CCS option. In this case, the electricity mix is dominated by renewables up to 55% in 2050 and 59% in 2100. Nuclear represents then 27 and 31% respectively in 2050 and 2100 and hydroelectricity 15% and 9% at the same periods. The situation is quite the same in 2100 in the RF_3p7_NoCCS scenario with a mix composed by 45%, 41% and 11% of renewables, nuclear and hydro respectively. Biomass is however developed up to 3%. In 2050, the climate constraint is not strong enough and the mix remains dominated by coal (55%). Nuclear represents then 22% of the electricity generation and hydro 11%. Renewables account for only 9% of the production. Exploring the biomass carbon capture solution to climate policy: A water impact analysis with TIAM-FR What is interesting to note, if we compare RF_2p6_NoBECCS and RF_2p6_NoBECCSCF, is the level of investment on CCS in 2100. Indeed, while the share of coal, hydro and nuclear in the electricity production is similar in both scenarios during the time period, while the share of renewables and CCS is similar in 2050, with around 35% and 30% respectively, the level of development of CCS in 2100 decreases a little in the RF_2p6_NoBECCS but decrease sharply in the RF_2p6_NoBECCSCF. Plants with CCS then represent 27% of the power generation in the RF_2p6_NoBECCS scenario but it falls to 5% in the RF_2p6_NoBECCSCF scenario. Conversely renewables increase drastically and represent 54% of the power generation in the RF_2p6_NoBECCSCF scenario by comparison with 35% in RF_2p6_NoBECCS, the same level than as in 2050. The unavailability of negative emissions permitted by the combustion of biomass makes it more competitive, given the strong climate constraint, to develop renewables, compared to CCS on power plants using fossil fuels.

b/ The water impact analysis of the CCS development

The water analysis allows us to introduce the dependence between water and the energy production system and, by the way, to put in perspective the technological solutions to climate and environmental constraints. In the context of our study with a climate constraint of strict control of emissions (RCP2.6; equivalent to a limitation of GHG atmospheric concentration to 450ppme) and of relatively moderate control of emissions (RCP3.7; equivalent to a limitation of GHG atmospheric concentration to 550ppme), fresh water consumption increases drastically, particularly at the end of the period where the constraint is the highest. In 2100, fresh water consumption is 2.4 times higher than the level of BAU in the RF_2p6 scenario, and 1.8 times higher in the RF_3p7 scenario. In 2050, the level is 2.1 and 1.5 times higher than the level of BAU, respectively in the RF_2p6 and the RF_3p7 scenario (figure 7). In 2050 and 2100, the fresh water consumption is around 1.3 times higher in the RF_2p6 scenario than in the RF_3p7 scenario.

www.modelisation-prospective.org If one puts into perspective the increase in water consumption and switches operated in the electricity mix, it appears clearly that the choice in technologies to decarbonize the energy system induce serious consequences on water resources. Indeed, the power sector is highly dependent on water resources. In USA for example, it is the largest user of water in the nation (Macknick et al., 2011). More precisely, the deployment of renewable energies and CCS technologies can be heavy on fresh water consumption.

Figure 8 focuses on strong climate context and highlights the decreasing consumption of fresh water gradually as the use of CCS technologies is reduced. Fresh water consumption remains nonetheless important when CCS is not developed, due to the investment on renewables energies, even if it is 1.6 times higher than the BAU level in 2100 in RF_2p6_NoCCS against 2.4 times higher than the BAU level in 2100 in RF_2p6. In 2020, when the impact of the climate constraint is lowest, it is interesting to note a limited consequence on the fresh water consumption. Exploring the biomass carbon capture solution to climate policy: A water impact analysis with TIAM-FR

IV -Discussion

A sustainable energy future requires strategies to facilitate the use of energy resources that enable the reduction of concentration of CO2 in the atmosphere whilst supplying energy needs over the years. A wide range of different mitigation options can be presented to reduce these CO2 emissions and particularly renewables and carbon capture and storage as a preferred option. Fossil energies would constitute a large share of the energy mix due to the important growth of primary energy of emerging countries probably resting heavily on the use of coal and the size of remaining reserves. That the reason why CCS/BECCS technologies appear as an increasing option to transit toward a low-carbon energy system, especially for fast developing countries and developing countries. The capture and storage of CO2 offers the potential for near-zero CO2 emissions from fossil-based power plants and negative CO2 emissions from biomass-based power plants.

While biomass with CCS technology may be difficult, it is scientifically feasible. So why isn't it being deployed on a larger scale? A major issue related to the deployment of CCS and BECCS is their economic viability and the cost/advantages comparison according to alternatives. Due to the low carbon price, investors are more incited to build cheaper power plants with a high level of emission than add expensive CCS to co-fired plants. As regards BECCS, this option cannot compete in matter of price with other energy sources. Indeed, the role of CCS and BECCS in mitigating climate change partly depends on their ability to reduce costs and, by consequence, their commercialization on an industrial scale in the marketplace (technology learning), as well as sustained Research, Development & Demonstration. In addition, covering the distance to CO2 capture and storage sites will involve developing and financing infrastructure for transporting CO2. Safety problems and social acceptability must also be considered in terms of risks and concerns for long-term CO2 geological storage. A regulatory system is required to supervise the selection of appropriate sites, long-term ownership and liabilities, and a monitoring program to detect problems. For example, in the case of carbon leaking back into the atmosphere, methods should be developed to stop or control CO2 releases. Concerning biomass energy, being directly tied to forests, food and other ecosystems, its use induces environmental and social impacts, both positive and negative. That the reason why biomass has to be sustainably produced. When accompanied by incentives, BECCS appears as an option to satisfy climate constraints and, at the same time, as fulfilling energy demands (consistent with Obersteiner et al, 2001). This analysis is consistent with the necessity of large scale biomass deployment to meet the maximum temperature change of 2°C. The potential for fossil and biomass exists but regulatory barriers have to be discussed and removed as knowledge gaps are filled. The increasing need to limit CO2 emissions and the current limits of alternative technologies constitute assets in this sense. For example, renewable options should in no way be excluded from the debate, and in that sense renewables appears as an important solution to reach the climate constraint, with or without CCS, but they need to be thought out in terms of the structural costs of investments in the power network required to integrate intermittent energies. Therefore, a complete and complex chain of processes and procedures has to be thought through and determined in the design of future energy policies. Another important issue, less developed in the energy system literature, consists in the increase pressure these energy system's technologies involve on the water resources. Energy and water security occupy a central place in the human and economic development. Both resources are now more and more interdependent than ever. Whether hydroelectricity, of course, but also the cooling of thermal power stations or the extraction and processing of fuels, almost all energy production processes, require significant amounts of water. Water and energy are then inextricably linked and interdependent sectors. While these resources are subject to an increase in global demand, water scarcity threatens the long term viability of energy projects, with serious consequences for development. The global community has also to consider that climate change will affect fresh water availability (IPCC, 2007a) leaving energy and water resources under unprecedented pressure, and subject to increasing competition on the part of the people, industries, ecosystems and economies booming. When the world population will reach 9 billion people, agricultural production will have increased by 15%. By 2030, global energy consumption will increase by 35% while the water consumption of the energy sector will increase by 85% according to projections of the International Energy Agency (IEA). It is so important to understand and estimate the dependency between water and energy systems, particularly according to the possibility of reduced water availability in the future due to climate change and the increased energy and water demands. The availability of water in certain regions may limit the penetration of low carbon technologies and can impact the choices. This could have important implications for policy makers (Siddiqi and Diaz Anadon, 2011). Decisions affecting the power sector's impact on the climate may need to include water considerations to avoid negative unintended environmental consequences on water resources. 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V -

  Figure 1: Overview of the reference energy system of the TIAM-FR model Source: Adapted from Loulou and Labriet, 2008

Figure 2 .

 2 Figure 2. The regional distribution of the TIAM-FR model

Figure 3 :

 3 Figure 3: Biomass potentials (EJ/year)

Figure

  Figure 4: World power generation (TWh)

Figure 5 :

 5 Figure 5: World power generation (TWh) according to carbon capture technologies availability

Figure 6 :

 6 Figure 6: Power plants with CCS by resources (%)

Figure 7 :

 7 Figure 7: Fresh water consumption (km3)

Figure 8 :

 8 Figure 8: Fresh water consumption (km3) according to carbon capture technologies availability

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Acknowledgements

This research was supported by the Chair Modeling for sustainable development, driven by MINES ParisTech, Ecole des Ponts ParisTech and AgroParisTech, supported by ADEME, EDF, GRTgaz, and SCHNEIDER ELECTRIC and the French Ministry of Ecology, Sustainable Development and Energy.


Andrei Cândea

Halanay

I Rodica Rȃdulescu

Doina Cândea

Andrei Halanay

A Control Delay Differential Equations Model of Evolution of Normal and Leukemic Cell Populations Under Treatment

Keywords: leukemia, asymmetric division, competition, optimal control, treatment AMS Classification: 34K35, 37N25, 92C50, 93C23

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

For the description of biological processes implied in hematopoiesis, a mathematical model that includes time delays will be used. It is based on the mass action principle, in the spirit of [START_REF] Adimy | Stability of Limit Cycles in a Pluripotent Stem Cell Dynamics Model[END_REF][START_REF] Adimy | A mathematical study of the hematopoiesis process with application to chronic myelogenous leukemia[END_REF][START_REF] Beckman | Asymmetric cell division within the human hematopoietic stem and progenitor cell compartment: identification of asymetrically segregating proteins[END_REF][START_REF] Halanay | Periodic Solutions in Mathematical Models for the Treatment of Chronic Myelogenous Leukemia[END_REF][START_REF] Mackey | Periodic oscillations of blood cell population in chronic myelogenous leukemia[END_REF] and [START_REF] Radulescu | Stability and bifurcation in a model for the dynamics of stem-like cells in leukemia under treatment[END_REF]. Other authors ( [START_REF] Kim | Dynamics and Potential Impact of the Immune Response to Chronic Myelogenous Leukemia[END_REF][START_REF] Kim | A theory of immunodominance and adaptive regulation[END_REF]) used the more simple model from [START_REF] Michor | Dynamics of chronic myeloid leukemia[END_REF] for the dynamics of hematopoietic stem cells (HSC).

Chronic Myelogenous Leukemia (CML), also known as Chronic Granulocytic Leukemia, is a cancer of white blood cells. It is a clonal marrow stem cells disorder in which the main characteristic is the proliferation of granulocytes (neutrophils, eosinophils and basophils) and of their precursors in the bone marrow and the accumulation of these cells in the blood. It is a type of myeloproliferative disease associated to a chromosomal translocation called the Philadelphia chromosome (see also [START_REF] Colijn | A mathematical model of hematopoiesis -I. Periodic chronic myelogenous leukemia[END_REF][START_REF] Moore | A mathematical model for chronic myelogenous leukemia (CML) and T -cell interaction[END_REF]) presenting the oncogene BCR-ABL that encodes a tyrosine kinase protein. Tyrosine kinases are enzymes that play an important role in tumor development by supporting cell growth through phosphorylation of signaling proteins [START_REF] Gottschalk | Imatinib (STI571)-Mediated Changes in Glucose Metabolism in Human Leukemia BCR-ABL-Positive Cells[END_REF]. Understanding the molecular mechanism of CML permitted the development of specific tyrosine kinase inhibitors (TKIs) as imatinib (Gleevec), dasatinib or nilotinib. The standard first line therapy is nowadays imatinib, which acts through competitive inhibition at the ATP-binding site of the BCR-ABL enzyme, leading to the inhibition of tyrosine phosphorylation of proteins involved in BCR-ABL signal transduction [START_REF] Deiniger | The tyrosine kinase inhibitor CGP57148B selectively inhibits the growth of BCRABL positive cells[END_REF]. The molecular effect of imatinib is mainly the inhibition of cell proliferation of BCR-ABL-positive cells but, there are experimental evidences that, in the mature cell lines, the inhibition of cell proliferation is followed by apoptosis [START_REF] Gottschalk | Imatinib (STI571)-Mediated Changes in Glucose Metabolism in Human Leukemia BCR-ABL-Positive Cells[END_REF]. Although imatinib has a very good successful rate, there are many experimental evidences attesting it does not affect quiescent stem cells deep in the bone marrow, and the consequence is the disease reapers after the treatment is stopped.

In this paper, we study an optimal control delay differential equation model of four cell populations, namely two healthy and two leukemic. For these classes of cells, we consider a population of mature cells which lost their self-renew ability and a population of stem-like cells involving a larger category consisting of proliferating stem and progenitor cells with self-renew capacity. The emphasis in this optimal control model is on establishing treatment strategies, considering the competition of healthy vs. CML cell populations and three types of division that a stem-like cell can exhibit: self-renew, asymmetric division and differentiation ([4, 17, 20, 21, 24]).

Of course, besides a correct mathematical model for the time evolution of the studied cell populations, it is very important to model the treatment effect as accurate as one can. Obviously, different drugs have different effects: some affect not only leukemic populations but also healthy ones (the cytotoxic ones), some kill the cells while others only delay or stop the division process (the cytostatic ones). We take into account here the standard treatment protocol with imatinib which, as we specified, acts by inhibiting the BCR-ABL signal transduction. In this way, it restrains the proliferative advantage of the CML cells and healthy cells regain their advantage. Moreover, it is straightforward to say that imatinib restores most of the abnormal functions of the CML cells, and the most important function affected by the drug is the division process of CML cell population. However, it is uncertain to what extent it affects all three kinds of division and therefore, we consider here the hypothesis that imatinib influences self-renew, asymmetric division and differentiation equally.

Description of the model

In the present paper, is assumed that the hematopoietic stem cells that are considered are in the proliferative phase or spend a short time into the resting phase. These cells are called, following [START_REF] Marciniak-Czochra | Modeling of replicative senescence in hematopoietic development[END_REF], Short-Term Hematopoietic Stem Cells (ST-HSC). In what follows x 1 denotes the density of short-term stem-like healthy cells , x 2 the density of mature healthy cells, x 3 the density of short-term stem-like leukemic cells , x 4 the density of mature leukemic cells.

The time necessary for a ST-HSC to complete a cycle of self-renewal, asymmetric division or differentiation is τ 1l for leukemic cells and τ 1h for the healthy ones, while the time necessary for the maturation of leukocytes is denoted by τ 2l in the case of leukemic cells and τ 2h for the healthy ones.

As we mentioned in the introduction, experimental evidences attest that the imatinib therapy affects primarily the proliferation rate and secondary, the apoptosis rate. In view of this fact, we consider the treatment functions

f u = 1 1 -u and f 1a = (γ 1h -γ 1l ) u 1 , f 2a = cγ 2h u 2 , with u, u 1 , u 2 : [0, T ] → [0, 1],
where u(t), u 1 (t), u 2 (t) are the treatment effects. The action of treatment on the proliferation rate will be considered through f u in the function of self-renew β l and in the function of differentiation or asymmetric division k l . Note that, in this way, both β l and k l became decreasing functions of u. If no drug is given (i.e. u(t) = 0) then β l ((

x 1 + y 1 )f u ) = β l (x 1 + y 1 ), k l ((x 1 + y 1 )f u ) = k l (x 2 + y 2 )
and also, a maximal effect happens for u(t) = 1 when the process of division essentially stops (β l ≡ k l ≡ 0). Treatment will be consider to act only on the leukemic stem cells compartment. The treatment acts on the apoptosis of mature CML cells through f 2a and on the apoptosis of stem-like CML cells through f 1a , restoring this rate to a value closed to the mortality rate of healthy cells. From the law of the mass, we have f1a = t t-τ 1l u 1 (s)ds.. The optimal control model is

ẋ1 = f 1 (x 1 , x 2 , y 1 , y 2 , x 1τ 1h , x 2τ 1h , y 1τ 1h , y 2τ 1h ) ẋ2 = f 2 (x 2 , x 1τ 2h , x 2τ 2h , y 2τ 2h ) ẏ1 = f 3 (t, x 1 , x 2 , y 1 , y 2 , x 1τ 1l , x 2τ 1l , y 1τ 1l , y 2τ 1l , u 1 , u, u τ 1l ) ẏ2 = f 4 (y 2 , x 2τ 2l , y 1τ 2l , y 2τ 2l , u 2 , u τ 2l ) (1)
where

f 1 = -γ 1h x 1 -(η 1h + η 2h )k h (x 2 + y 2 )x 1 -(1 -η 1h -η 2h )β h (x 1 + y 1 )x 1 + +2e -b 1h τ 1h (1 -η 1h -η 2h )β h (x 1τ 1h + y 1τ 1h )x 1τ 1h + +η 1h e -b 1h τ 1h k h (x 2τ 1h + y 2τ 1h )x 1τ 1h f 2 = -γ 2h x 2 + A h k h (x 2τ 2h + y 2τ 2h )x 1τ 2h f 3 = -(γ 1l + f 1a )y 1 -[(η 1l + η 2l )k l ((x 2 + y 2 ) f u ) + (1 -η 1l -η 2l )β l ((x 1 + y 1 ) f u )] y 1 + +[2e -b 1l τ 1l -f1a (1 -η 1l -η 2l )β l ((x 1τ 1l + y 1τ 1l ) f uτ 1l )+ +η 1l e -b 1l τ 1l -f1a k l ((x 2τ 1l + y 2τ 1l ) f uτ 1l )]y 1τ 1l f 4 = -(γ 2l + f 2a ) y 2 + A l k l ((x 2τ 2l + y 2τ 2l ) f uτ 2l )y 1τ 2l subject to minimization of the cost functional min J(u), (2)
where

J(u) = ay 1 (T ) + by 2 (T ) + T 0 u 2 (t) + u 2 1 (t) + u 2 2 (t) dt
with g(y(T )) = ay 1 (T ) + by 2 (T ) the weighted sum of the final tumor population and L(u(t))

= T 0 u 2 (t) + u 2 1 (t) + u 2 2 (t)
dt, the cumulative drug toxicity. In this paper we denote X τ = X(t -τ ), where X = (x 1 , x 2 , y 1 , y 2 ). The history of the state variables is given by

X(θ) = ϕ(θ), θ ∈ [-τ max , 0], τ max = max(τ 1h , τ 2h , τ 1l , τ 2l ).
It is not difficult to see that if the initial conditions have all components positive functions, the solutions of the system will have positive components on all the interval of existence. Indeed, if x 1 (θ) > 0 for any θ ∈ [-τ, 0] and there exists T > 0 such that x 1 (T ) = 0 the derivative ẋ1 (T ) will be positive and this leads to a contradiction. The same argument works for the other components, too.

We assume that: a percentage η 1α , α = h, l of stem-like cells population is supposed to undergo asymmetric division; a percentage η 2α , α = h, l of the population differentiate symmetrically and the percentage (1-η 1α -η 2α ), α = h, l of the population is supposed to self-renew (see also [START_REF] Halanay | Stability analysis of equilibria in a delay differential equations model of CML including asymmetric division and treatment[END_REF]).

Furthermore,it is assumed that homeostatic mechanisms mantain the hematopoietic stem cell population at a constant level. In this respect, the rate of selfrenewal is given by a Hill function

β α (X) = β 0α θ m 1 θ m 1 + X m , α = h,
l and the rate of differentiation, through symmetric or asymmetric division is supposed to be dictated, through a feedback law, by

k α (X) = k 0α θ n 2 θ n 2 + X n , α = h, l.
Because in this paper we consider competition between healthy and CML cell populations, both this rates will depend on the sum of stem-like respectively mature populations (similar approaches on competition were modeled in [START_REF] Tang | Selection pressure exerted by imatinib therapy leads to disparate outcomes of imatinib discontinuation trials[END_REF][START_REF] Stiehl | Mathematical Modeling of Leukemogenesis and Cancer Stem Cell Dynamics[END_REF]).

For α = h, l, the other parameter are defined as follows: b 1α accounts for the death rate of stem cells and a positive K α for the loss rate due to differentiation into other cell lines -the resulting loss rate is denoted as γ 1α = K α + b 1α ; β 0α and k 0α represent the maximal rate of self-renewal, respectively of asymmetric division or differentiation into leukocyte line; θ i , i = 1, 2, is the value for which β α , respectively k α attains half of their maximum value; γ 2α is the mortality of mature cells; A α is an amplification factor of mature cells due to differentiation; m is the parameter controlling the sensitivity of the mitotic re-entry rate β α to changes in the size of G 0 and n is the parameter controlling the sensitivity of the asymmetric division or differentiation rate k α to changes in the size of mature population.

Existence of an optimal control The existence of an optimal control results from transforming the problem into an optimal control problem for a system of ordinary differential equations (see next section) whose solutions will be bounded together with their derivatives on compact intervals (see [START_REF] Benosman | Control of the dynamics of Chronic Myeloid Leukemia by Imatinib[END_REF]).

Discretization of the optimal control problem

In this section, we apply the numerical procedure from Gollmann et al. [START_REF] Gollmann | Theory and applications of optimal control problems with multiple time-delays[END_REF], in order to solve the delay optimal control problem (1)+(2) (see also [START_REF] Gollmann | Optimal control problems with control and state delays and applications to growth processes[END_REF][START_REF] Gollmann | Optimal control problems with delays in state and control variables subject to mixed control state constraints[END_REF]). For that matter, we write the cost functional in the Mayer form

J(u, y) = h(y(T )), y = (y 1 , y 2 ) ∈ R 2 .
In our case, the reduction of the more general cost functional (2) to Mayer form, proceeds by the introduction of the additional state variable z through the delayed equation

ż(t) = L(u(t)) so ż(t) = u 2 (t) + u 2 1 (t) + u 2 2 (t), z(0) = 0.
Then, the cost functional ( 2) is rewritten as

J(u, y, z) = g(y(T )) + z(T ).
In the following, let τ > 0 such that

τ 1h = k 1 τ, τ 2h = k 2 τ, τ 1l = k 3 τ, τ 2l = k 4 τ, k i ∈ N * , i = 1, 4
, T = N τ and use the Euler integration method with a uniform step size τ > 0. Of course, τ can be refined in order to obtain an appropriate smaller step-size. Using the grid points t i = iτ , i = 0, N and the approximations x

1 (t i ) x 1i ∈ R, x 2 (t i ) x 2i ∈ R, y 1 (t i ) y 1i ∈ R, y 2 (t i ) y 2i ∈ R, u(t i ) u i , u 1 (t i ) u 1i
and u 2 (t i ) u 2i , the treatment function f 1a becomes k3 j=1 u 1i-j τ and the delay control problem (1)+( 2) is transformed into the nonlinear programming problem (NLP)

M inimize J = g(x N , y N ) + z N (3) subject to            x 1i -x 1i+1 + τ f 1 (x 1i , x 2i , y 1i , y 2i , x 1i-k1 , x 2i-k1 , y 1i-k1 , y 2i-k1 ) = 0 x 2i -x 2i+1 + τ f 2 (x 2i , x 1i-k2 , x 2i-k2 , y 2i-k2 ) = 0 y 1i -y 1i+1 + τ f 3 (x 1i , x 2i , y 1i , y 2i , x 1i-k3 , x 2i-k3 , y 1i-k3 , y 2i-k3 , u i , u 1i , u 1i-1 , .., u 1 i-k 3 , u i-k3 ) = 0 y 2i -y 2i+1 + τ f 4 (y 2i , x 2i-k4 , y 1i-k4 , y 2i-k4 , u 2i , u i-k4 ) = 0 z i -z i+1 + τ u 2 i + u 2 1i + u 2 2i = 0 (4) -u i ≤ 0, u i -1 ≤ 0, ( 5
) -u 1i ≤ 0, u 1i -1 ≤ 0, -u 2i ≤ 0, u 2i -1 ≤ 0. i = 0, N -1.
Herein, the initial value profile ϕ 1 ,ϕ 2 , ϕ 3 and ϕ 4 gives the values

x 1-i : = ϕ 1 (-iτ ), i = 0, k 1 x 2-i : = ϕ 2 (-iτ ), i = 0, k 2 y 1-j : = ϕ 3 (-jτ ), j = 0, k 3 y 2-j : = ϕ 4 (-jτ ), j = 0, k 4
The variable to be optimized is represented by the vector

w = u 0 , u 10 , u 20 , x 11 , x 21 , y 11 , y 21 , z 1 , ..., u N -1 , u 1 N -1 , u 2 N -1 , x 1 N , x 2 N , y 1 N , y 2 N , z N ∈ R 8N .
The numerical procedure described above is applied in the next section and the graphs for states and control for different sets of parameters are obtained.

Numerical results and simulations

In the following figures, we plotted the trajectories of the healthy, respectively CML cell populations for the competition system, showing a comparison between the dynamics of a system without treatment and the dynamics of a system subject to optimal control of treatment. To solve the problem of optimal control the Matlab solver for NLP problems fmincon was used, selecting the 'interiorpoint' solver.

In all figures, for the healthy cell populations, we choose the same set of parameters value: η 1h = 0.7, η 2h = 0.1, τ 1h = 2, τ 2h = 4, γ 1h = 0.1, γ 2h = 2.4, A h = 922, β 0h = 1.77, k 0h = 0.1 , θ 1h = 0.5 • 10 6 , θ 2h = 0.36 • 10 8 . For leukemic cell populations, we consider alteration of the value of the following parameters:

-smaller percent of asymmetric division (η 1l ); -bigger percent of self-renewal (1 -η 1l -η 2l ); -lower rate of apoptosis of leukemic stem cells (γ 1l ); -lower rate of apoptosis of leukemic mature cells(γ 2l ); -enhanced differentiation (A l ).

In the following example (Figures 1,2) all these features were modified.

If we maintain the configuration of parameters from the previous example but consider that the percentage of self renewal of leukemic cells is the same as the percentage of self renewal of healthy cells, we see another manifestation of the disease, for which the treatment dose is lower than in the previous case (Figures 3,4).

Discussion

The plots of optimal controls (Figures 2 and4) exhibit an optimal control effect almost constant at 0.5 respectively at 0.25 until the 90th day for all controls. One can intuitively expect that the treatment effect is proportional with the cell concentration of imatinib; however, in order to study the influence of the prescribed drug concentration on the population's time-evolution, imatinib pharmacokinetics (PK) and pharmacodynamics (PD) need to be taken into account (see also [START_REF] Radulescu | Stability and bifurcation in a model for the dynamics of stem-like cells in leukemia under treatment[END_REF]). Nevertheless, for an optimal effect of treatment, the prescribed dose should be adapted in view of the disease parameters of a certain patient. In that respect, a competition model of healthy vs. CML cell population that takes into account the influence of PK and PD of imatinib, is subject of further research.

1 .Fig. 1 . 1 .Fig. 2 . 1 .Fig. 3 . 1 .Fig. 4 .

 11121314 Fig.1. Comparison between the dynamics of a system without treatment and with with optimal control of treatment. The results correspond to the best local minimal solution obtained for different initial guesses ( the value of cost functinal was improved from 1597 to 901)

This work was supported by CNCS-ROMANIA Grant ID-PCE-2011-3-0198.