Olivier Schneider

Jean Roger-Estrade

Jean-Noël Aubertot
email: aubertot@grignon.inra.fr

Thierry Doré

Effect of seeders and tillage equipment on the vertical distribution of oilseed rape stubble

Keywords:

When the spreading of a disease depends on the proportion of infected residues remaining at soil surface it is of crucial importance to analyse the effects of tillage practices on the vertical distribution of stubble. This is for instance the case with phoma stem canker (blackleg), whose epidemics are initiated in autumn, by air-borne ascospores released from stubble located at the soil surface. In this experiment, we compared the initial vertical distribution of oilseed rape residues to those observed after sowing and various tillage operations (rotary harrowing, stubble disking, chiseling and mouldboard ploughing). Results show that almost 20% of the initially buried residues was brought back to soil surface. Rotary harrow brought 40% of the residues buried in the 0-10 cm layer up to the surface and left unburied about 70% of the surface residues.

Stubble disking appeared to be more efficient for residues burial than chiselling. Finally mouldboard plough was the only tool which buried all the residues. The results were used to develop a simple model that predicts the burial and the return to the soil surface of potentially infected residues as a function of the tillage practices used after harvest.

Simulation of the effects of different tillage sequences showed that the order in which tools were used also affected the location of residues. Our results highlighted the importance of tillage in the cultural control of phoma stem canker and will contribute to the definition of Integrated Pest Management strategies for oilseed rape.

Introduction

When caused by a fungus, the spreading of a disease often depend on the proportion of the residues from the preceding crop left at the soil surface. This is the case for instance with phoma stem canker or blackleg, whose causal fungus is Leptosphaeria maculans, asexual stage Phoma lingam. It is one of the most severe disease damaging oilseed rape crops (Brassica napus) worldwide. This disease occurs in the main growing areas (Australia, Canada and Europe) and can result in major yield losses [START_REF] Hall | Epidemiology of blackleg of oilseed rape[END_REF][START_REF] West | Epidemiology and management of Leptosphaeria maculans (phoma stem canker) on oilseed rape in Australia, Canada and Europe[END_REF]. Soil tillage should improve the control of the pathogen if proper stubble management reduces the risk of contamination in the spreading area [START_REF] Alabouvette | Recherches sur la maladie du colza due a Leptosphaeria maculans (desm.) ces. et de not. Rôle des restes de culture dans la conservation et la dissémination du parasite[END_REF][START_REF] Kharbanda | Integrated management of canola diseases using cultural methods[END_REF][START_REF] Leake | The development of integrated crop management in agricultural crops: comparisons with conventional methods[END_REF]. Epidemics are initiated in autumn by the primary inoculum, i.e. ascospores produced on infected residues. These ascospores that can be spread over several kilometres by wind only appear if the infected debris are totally or partially exposed to solar radiation [START_REF] Lacoste | Action de la température et de l'éclairement sur la reproduction en culture pure de diverses espèces du genre Leptosphaeria Ces. et de Not[END_REF]. Spores can develop on the residues of previous rape crops, when residues are left on the soil surface or brought back to the soil surface by tillage.

Thus, to prevent the development of this disease, stubble management by tillage should examine: (i) the effect of tillage and sowing on residue burial, especially that of stems, which are usually chopped and represent the main residual biomass after harvest and (ii) the effect of tillage and sowing on the return to the soil surface of residues from the 0-10 cm soil layer. Indeed, this layer is the most intensively tilled, due to stubble breaking, seed bed preparation and sowing. These residues in the 0-10 cm layer are either previously buried residues or the upper part of the main roots that are located at this level after harvest. The roots are particularly dangerous because they are the main long-term support for the mycelium, as their high lignin content renders them highly resistant to decomposition [START_REF] Alabouvette | Recherches sur la maladie du colza due a Leptosphaeria maculans (desm.) ces. et de not. Rôle des restes de culture dans la conservation et la dissémination du parasite[END_REF][START_REF] Mcnish | Survival of Leptosphaeria maculans in rapeseed root tissue[END_REF]Turkington et al., 2000b). Thus, both burial and rising to the soil surface should be taken into account when evaluating the effect of tillage. These two processes may explain the conflicting effects of tillage on the risk of ascospore production sometimes reported in the literature. For instance, mouldboard ploughing limits sporulation by incorporating most of the surface residues into the soil and by increasing their decomposition rate (Turkington et al., 2000a). However, ploughing may also induce ascospore production by bringing buried residues back to the soil surface (Turkington et al., 2000b).

Conventional tillage with mouldboard ploughing is being replaced by conservation tillage (where tillage is reduced or suppressed) in many countries. This means that a smaller proportion of surface residues are buried, but this proportion depends on the type of tool used (disks or tines, fixed or not). Furthermore, sowing practices can also modify the location of residues in the soil, near the sowing line.

In this paper, we present the results of a field experiment carried out to analyse the effect of different types of soil tillage on the proportion of surface residues buried and the proportion of residues located in the 0-10 cm layer that rose up back to the surface. For this, we used coloured residues to follow the movement of potentially infected residues in the soil. The results were used to design a simple model of transfer between the two soil compartments: surface and 0-10 cm.

Materials and methods

Site characteristics

In October 2002, a field experiment was carried out at the INRA Experimental Station at Grignon, Yvelines (1°58' E, 48°51' N) in the western Paris region on a silt loamy Calcic Cambisol soil (FAO classification; texture of the 0-30 cm horizon: clay 29%, silt 61% and sand 10%; bulk density: 1.5 g.cm -3 ).

Experiment characteristics

The method used was derived from studies dealing with the effects of soil tillage on the location of weed seeds [START_REF] Cousens | A model of the effects of cultivation on the vertical distribution of weed seeds within the soil[END_REF][START_REF] Colbach | Modelling vertical and lateral seed bank movements during mouldboard ploughing[END_REF] or the vertical distribution of cereal straw [START_REF] Staricka | Spatial variation of crop residue incorporated by tillage[END_REF] in the soil. The residues used in this study were 5 cm long pieces of the main stem of mature oilseed rape plants.

The studied tools were: drill seeder, rotary harrow, stubble disk, chisel and mouldboard plough. The main characteristics of these tools are presented in Table 1.

The experiment was a randomised block design with three replications. However, in each block, tillage operations involving mouldboard ploughing were located side by side due to practical constraints. This was taken into account in the statistical treatment of the data. The size of each elementary plot was 3 x 30 m 2 . Within each experimental plot, we delimited observation quadrats (1 x 2 m 2 ). Residues, sprayed with fluorescent paint to facilitate their recovery, were located either at the soil surface or in the 0-10 cm layer of each quadrat. We placed 20 red pieces horizontally on the soil surface and half buried 20 other red pieces. We considered all residues that were at least partly visible to be surface residues. We placed 20 blue pieces in the 0-10 cm soil layer (5 at 2.5 cm, 5 at 5 cm, 5 at 7.5 cm and 5 at 10 cm), such that no part was visible at the soil surface. These buried residues were located in five 7.5 cm-wide holes per replication. These holes were regularly distributed across the diagonal axis of the observation quadrat. The total number of residues used for this experiment was 900: 5 tool x 3 replications x (5 holes x 4 buried residues + 40 residues at soil surface).

The stubble disk, chisel and mouldboard plough were directly used on untilled soil. Conversely, the soil had to be prepared for the other tools to be used in normal conditions. Thus mouldboard ploughing was performed before rotary harrowing and mouldboard ploughing followed by rotary harrowing was performed before seeding.

The residues were placed just before the last tool was used. At the same time, the mean soil water content of the 0-30 cm horizon was gravimetrically determined in each plot.

During tillage operations, the mean soil water content was 0.16 g.g -1 . After the last tillage or seeding, the working depth of each tool in each treatment was controlled, and the colour and the new vertical location of the individual oilseed rape fragments were recorded. The final vertical position of buried residues was determined by measuring the depth of the centre of the residue to the average soil surface. The percentage of the total amount of oilseed rape residues located in each 2.5 cm layer between the soil surface and a depth of 30 cm was calculated. Transfer parameters between two vertical compartments were calculated by taking the average percentage of each type of residue recovered in the 0-10 cm soil layer and at the soil surface, for the three replications with each tool.

Statistical analysis

Statistical analysis was carried out using the SAS software (SAS Institute Inc., 1990). For each treatment, we calculated the mean percentage of residues recovered at the soil surface and in the 0-10 cm layer (a very small number of them were not recovered); 95% confidence intervals were calculated. Transfer coefficients, which represent the proportions of residues that changed vertical compartment, were deduced from these percentages. As the treatment was not totally randomised, we tested residuals independence with the Durbin-Watson test on the studied variables [START_REF] Bergonzini | Analyse et planification des expériences. Les dispositifs en blocs[END_REF]. 5

Considering the range of variation of the results, data were transformed using the arcsin transformation [START_REF] Gomez | Statistical procedures for agricultural research[END_REF] before carrying out analysis of variance on the whole dataset using the GLM procedure. The differences between treatments were tested using the Student-Newman-Keuls test (p=0.05). 8

Results

Movement of residues to the soil surface

The origin of surface residues depended on the tillage method used. The mouldboard plough was the only tool that did not bring any residue from the 0-10 cm layer up to the surface (Fig. 1a). The rotary harrow brought 40% of the buried residues up to the soil surface. The seeder brought back nearly 20% of the initially buried residues. The chisel and stubble breaker brought up similar proportions of buried residues (ca 15%), even though the working depth of the chisel was greater (18 cm) than that of the stubble breaker (8 cm).

Burial of residues

Burial efficiency also differed from one tool to another (Fig. 1b). The chisel and stubble breaker left similar amounts of residues at the soil surface. Rotary harrowing left 70 % of the surface residues at soil surface. Sowing left almost all the residues at the soil surface. As expected, ploughing buried all the surface residues. Mouldboard ploughing led to the deepest residue position (down to 30 cm) and led to the distribution of residues throughout the whole soil profile (Fig. 2). Very few residues were buried less deep than 15 cm and none were buried above 5 cm; most were located between 15 and 27.5 cm.

Vertical distribution of residues buried by superficial tillage tools

The vertical distribution of residues within the soil profiles differed among the three superficial tillage tools. The chisel (Fig. 3a) buried residues at a maximum depth of 10 cm. Most of the residues were found between 0 and 7.5 cm of depth, even though the working depth of the chisel was 18 cm. The stubble breaker (Fig. 3b) buried residues between 0 and 10 cm. This tool distributed the residues in the whole tilled layer (0-8 cm), and some of them were found slightly deeper than the working depth. Very few residues were found in the upper layer (0-2.5 cm). The rotary harrow (Fig. 3c) left the largest proportion of the residues at the surface and buried the others very superficially, mostly between 0 and 7.5 cm of depth. Sowing (Fig. 3d) buried very few residues.

Simulations

Tillage tools are often combined in usual crop management sequences after harvesting oilseed rape. To simulate the effects of successive use of tillage tools, we determined the transfer coefficients (Table 2) using the results presented in Fig. 1 and2.

As the proportions of residues buried or moved up were not significantly different for chisel and stubble breaker, the same parameters were chosen for these two tools. The simulated effects of various sequences on residue burial and movement are shown in Table 3.

Five tillage sequences were chosen for the simulations. Two of them comprised mouldboard ploughing followed either by stubble breaker/chisel or by rotary harrow before sowing. These sequences are frequent in conventional tillage after oilseed rape harvest. In the third sequence, only the stubble breaker or chisel was used (two passages) before sowing. The fourth and fifth sequences combined the stubble breaker or chisel and the rotary harrow. The latter three sequences reflect the diversity of the soil management sequences used by farmers practising conservation tillage. The sequences including mouldboard ploughing (sequences 1 and 2) were the most efficient at deep residue burial and at preventing the movement of residues to the surfact during secondary tillage. The sequence without the rotary harrow (sequence 3) resulted in a high proportion of the surface residues buried in the 0-10 cm layer (41%) and the movement of only 19% of the residues initially located in the 0-10 cm layer to the soil surface. The effect of the sequences including the rotary harrow differed depending on whether this tool preceded (sequence 4) or followed (sequence 5) the chisel: in the first case, a lower proportion of surface residues was left at soil surface and a higher proportion of the 0-10 cm residues were left in this layer.

Discussion

In this experiment where the conditions (speed, working depth, soil water content) closely reflected those encountered in the area for the rapeseed cropping, the implication of tillage on the vertical distribution of residues was found to differ considerably according to the tool used. This, consequently, has a pronounced effect on the spreading of diseases caused by fungus, like phoma stem canker. Mouldboard ploughing did not bring any residue initially located in the 0-10 cm layer back to the soil surface. This is probably due to the working depth of the plough (27 cm) and the action of the skim coulter [START_REF] Roger-Estrade | Modelling vertical and lateral weed seed movement during mouldboard ploughing with a skim-coulter[END_REF]. Furthermore, mouldboard ploughing buried all the surface residues deep within the tilled layer. Most residues were buried at more than 10 cm of depth, the highest proportion was buried in the 20-22.5 cm layer (almost 25%) and none were buried in the most superficial layers (0-5 cm). Thus, even when followed by secondary tillage, mouldboard ploughing can efficiently prevent further phoma infection from surface residues and superficially buried residues.

Conversely, the rotary harrow brought back a large proportion of residues to the surface, probably because of the vigorous mixing action of this tool. It is possible that the rotary harrow sorts soil and residues. After being hit by the tool, the residues, which are lighter than soil, land on the surface when falling back to the soil. Although the chisel and stubble breaking disk have different working depths, they brought the same proportion of residues back to soil surface. Seeding brought up a large proportion of the more superficially buried residues. Furthermore, the rotary harrow was poorly efficient at residue burial. This tool is appreciated for its fragmentation action, but appears to be very dangerous as it brought back a large proportion of residues to the surface, thus enhancing the production of phoma stem canker primary inoculum.

Stubble breaking and the chisel are known to exert a less efficient fragmentation action on the soil. These tools buried a greater proportion of residues than the rotary harrow, with a slight advantage for stubble breaking as it left very few residues between 0 and 2.5 cm. Thus, stubble disking and the chisel should be preferred to the rotary harrow when trying to prevent sporulation.

Our results also showed that the seeder affects the location of residues, increasing the risk of bringing superficially buried residues back to the soil surface. This is especially dangerous because these residues are exposed at the precise moment when the crop is the most sensitive to phoma disease.

The importance of tillage on crop residue management was outlined by different authors [START_REF] Guérif | A review of tillage effects on crop residue management, seedbed conditions and seedling establishment[END_REF]. Similar effect of the chisel was observed [START_REF] Colvin | Tillage implement effects on corn and soybean residue[END_REF], decreasing of the surface residues proportion by 80%. Conversely, a more pronounced effect of the seeder in bringing back buried residues at soil surface. This was probably due to the type of tool used. Indeed, there is a great variety between seeder types and their effects on residue location need to be further investigated.

Others experiments gave similar results on the vertical distribution of coloured tracers (ceramic spheres or polyethylene chips) or oat residues [START_REF] Staricka | Spatial variation of crop residue incorporated by tillage[END_REF][START_REF] Allmaras | Spatial relation beween oat residue and ceramic spheres when incorporated sequentially by tillage[END_REF]. In these studies the tracers or residues were incorporated by mouldboard ploughing down 27-30 cm depth and by disk or chisel below 10 cm with a general pattern of distribution in the soil similar to the one obtained in our experiment.

The robustness of the results obtained in this study needs to be further explored.

We propose to analyse the effects of the same tools in other soil conditions, where a different mechanical behaviour may modify the vertical distribution induced by tillage and sowing. Despite these experimental drawbacks, tillage appears to be a very promising method for the control of phoma stem canker because it does not apply a selection pressure to the fungus as do fungicides or host resistance genes. Reducing the number of surface residues is an efficient way of reducing the pathogen population and limiting its sexual reproduction. Therefore, it is a good way of limiting the genetic variability of the fungus. For these reasons, tillage control can reduce the adaptability of the fungus and thus can improve the durability of the genetic resistance of the rapeseed crops [START_REF] Brun | Presence in Leptosphaeria maculans populations of isolates virulent on resistance introgressed into Brassica napus from the B. nigra genome[END_REF][START_REF] Mcdonald | The population genetics of plant pathogens and breeding strategies for durable resistance[END_REF]. Furthermore, Tox 0 phoma strains, recently identified as another species, Leptosphaeria biglobosa [START_REF] Shoemaker | The teleoporphe of the weakly aggressive segregate of Leptosphaeria maculans[END_REF][START_REF] Mendes-Pereira | Molecular phylogeny of the Leptosphaeria maculans-L. biglobosa species complex[END_REF], tend to be located in the upper part of the stem, whereas Tox + phoma strains tend to be located at the bottom of the stem and in the root. As tillage has different effects on the upper part of the stem (located at the soil surface) and on the roots (located in the 0-10 cm layer), the morphological repartition of phoma strains can be greatly modified by tillage, which can in turn modify the ecological balance.

The results presented in this paper were used to develop a simple compartmental model that we used to simulate different tillage sequences. This model, which should be validated before being used to predict the locations of residues, was built with the sole aim of taking into account the effect of tool sequences. It does not take into account residue decomposition rate. This rate decreases the mass of infected residues as a function of numerous parameters like residue size, roots or stem, soil nature (Turkington et al., 2000b). This aspect will have to be included in the model to improve risk assessment. Furthermore, the model did not integrate the movement of residues that were initially buried at more than 10 cm, even though this is necessary to simulate the effect of complex tool sequences. Nevertheless, our simulations showed that different tillage sequences have very different effects on the locations of residues. Conventional tillage appeared to be more efficient at preventing the spread of phoma than did superficial tillage. Thus, these results suggest that the choice of the tillage tools after rape harvest and the order of the sequence have a significant effect on the risk of spreading phoma stem canker.

FIGURE CAPTIONS Figure 1 :

 CAPTIONS1 FIGURE CAPTIONS

Figure 2 :

 2 Figure 2: Location of surface residues (percentage) after burial by mouldboard

Figure 3 :

 3 Figure 3: Stubble burial expressed as the percentage of total amount of residues

Table 1 :

 1 Characteristics of the five tools used in the experiment. 10

	Tool	from S to S	from S to 0-10 cm	from 0-10 cm to S	from 0-10 cm to 0-10 cm
	SBC	32	62	15	50
	RH	78	21	45	37
	S	96	4	20	80
	P	0	4	0	14

Table 2 :

 2 Transfer coefficients in percentage (S: surface); SB: stubble breaking,

	RH: rotary harrowing, C: chisel cultivating, S: sowing, P: ploughing.
	5
		Residues initially located at soil surface	Residues initially located at 0-10 cm of depth
	Tillage sequence	% left at soil surface	% buried at 0-10 cm depth	% buried deeper than 10 cm	% moved to soil surface	% left at 0-10 cm depth	% buried deeper than 10 cm
	P/SBC/S	1	2	97	3	6	91
	P/HR/S	2	1	97	7	4	89
	SBC/SBC/S	29	41	30	19	28	53
	HR/SBC/S	39	48	13	28	38	34
	SBC/HR/S	57	26	17	37	19	44

Table 3 :

 3 Percentage of oilseed rape residues buried, unchanged and brought back

	10	to the soil surface following different combinations of tillage tools after an oilseed rape
		harvest. Codes for the tools: stubble breaking or chisel cultivating (SBC), rotary
		harrowing (RH), sowing (S) and ploughing (P).

Acknowledgements

The authors thank Jacques Troizier, Head of the Experimental Centre at Grignon, and his teams for conducting the field trials. The statistic assistance from Jean-François Fourbet was greatly appreciate. The authors are also grateful to Beatrice Le Fouillen, Véronique Tanneau, Richard Gosse, Dominique Le Floch and Marc Klockiewicz for technical assistance. This work was financed by INRA and CETIOM.


The equilibrium configurations under imposed (varying in dependence of a finite set of parameters) displacement boundary conditions are those configurations which minimize the just introduced discrete Lagrangian deformation energy.

The advantage presented by such an approach when compared with the other one based on the formulation of second gradient continuum model and its discretization with FEM can become evident in many situations. Indeed when the considered fabric does not include too many structural elements then the discretization which we propose here: (i) has a more direct physical meaning, (ii) is still able to effectively capture the global behavior of the system and (iii) is more computationally efficient.

Using the Lagrangian model presented here it has been possible to design experimental setups aimed to show the peculiar features of newly conceived metamaterial. Moreover we could qualitatively and quantitatively predict the elastic behavior of specimens built by means of 3D printing technology also in the regime of large geometrical and material nonlinearities.

In the following sections we show the first available experimental evidence obtained for sheared pantographic specimens: we designed a mechanism transforming imposed elongations into imposed shear deformations (see Fig. 1) and we characterized a large class of deformation states of pantographic sheets. Paralleling their behavior in extensional bias tests (see [START_REF] Dell'isola | Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence[END_REF]) pantographic sheets are extremely tough also under large shear deformations. It can be shown that the code which we have developed is very effective and predictive in the elastic regime: we could get a complete correspondence between experimental data and numerical ones by simply fitting the four material parameters characterizing our model. These parameters are easily related to: (i) extensional fiber stiffness, (ii) bending fiber stiffness, (iii) shear stiffness related to fiber connectivity and (iv) material nonlinearity determining eventually non-quadratic shear deformation energy.

To be able to predict fiber rupture phenomena we subsequently postulated a simple fiber rupture mechanism. We assume that once a fiber experiences an elongation larger than a specific threshold (which is characteristic of the fiber constituting material) then it is simply broken, and the corresponding term in deformation energy is assumed to vanish. Therefore by simply adding a further constitutive parameter to the previously introduced four elastic ones we get a generalized incremental numerical model. When in the considered quasi-static imposed displacement process the rupture criterion is attained, then in the subsequent step the elastic energy is modified by imposing to zero the corresponding energy addend.

This simple model was able to predict very well the observed rupture phenomena, as it is described in the following Section 2. The results obtained and presented in Section 3 strongly motivate further development of proposed numerical and theoretical models: further rupture mechanism have been observed and need to be modeled, involving fiber interconnection rupture or partial or total loss of fiber bending stiffness, see Section 4. The consequence of the further sophistication of the used models will lead to the conception of more complex experimental setups aimed to more careful characterization of the behavior of pantographic sheets.

Lagrangian discrete model for pantographic structures

As discussed in full detail in [START_REF] Turco | Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models[END_REF] Lagrangian discrete models can be a viable option in the description of pantographic sheets: the possibility to predict their shapes in large deformation equilibrium configurations using said discrete models has been the object of the investigations presented in [START_REF] Turco | Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete Lagrangian model[END_REF]. Referring the reader to just cited papers for a detailed discussion of the model and the algorithmic procedure we use to determine equilibrium configurations, we delineate here the main ideas on which we base our solution strategy when dealing with both elastic and rupture deformation phenomena.

First of all, we limit our attention to quasi-static deformation processes where the control parameters determine a well-determined class of imposed boundary displacements. The principle of virtual work will allow a facile generalization of our formulation to the fully dynamic case. Second we introduce a finite dimensional configuration space to describe the kinematics of considered sheets: it is constituted by the collection of the positions of the pivots interconnecting the two arrays of fibers, each of them being assumed extensible and flexible.

The elastic deformation energy of the sheet is assumed to be given by the sum of terms respectively (see Fig. 2) accounting for: (i) extensional energy between two adjacent pivots, (ii) bending energy of the fibers and (iii) shear energy stored in the interconnecting pivots.

We limit ourselves to consider hard-devices imposing boundary displacements to the sheets: therefore the problem of finding the equilibrium configurations in a quasi-static deformation process reduces to the determination of the minima of deformation energy corresponding to the assigned values of the parameters characterizing the boundary conditions.

The algorithm which is successfully used here (following the ideas proposed in this context in [START_REF] Turco | Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models[END_REF]) handles the strong nonlinearities arising in the aforementioned minimization process by proceeding to calculate sequences of close equilibrium configurations relative to close boundary conditions. The relevant economy of calculation time is exploited also in the simulations we present here for elastic regimes. Not only: we start here from a wellgrounded simplifying assumption. We assume that fiber rupture occur only when a threshold elongation (or bending angle or shear angle) is attained in our quasi-static deformation process. These thresholds are easily estimated in terms of the constitutive parameters of the material used in the 3D printing which produced the specimens we have studied. The simple modification of the elastic energy consequent to a rupture consists in assuming as vanishing the corresponding addend in the deformation energy.

Synoptically, by referring to Fig. 2, strain energies and strain thresholds for each kind of spring are: (i) for an extensional spring whose stiffness is k a

E a = 1 2 k a ( -ε) 2 , ( 1
)
a min ≤ -ε ε ≤ a max , (2)
where and ε are the distances between two pivots in the actual and in the reference configuration, respectively; a min and a max are the minimum and maximum value of the elongation threshold, respectively; (ii) bending springs whose stiffness is k b,i

E b,i = k b,i (1 + cos ϕ i ) , i = 1, 2, (3)
cos ϕ i ≤ b max,i , i = 1, 2, (4)
where ϕ i is the angle, in the actual configuration, between two consecutive fibers along i-direction; b max,i is the maximum admissible value in i-direction; (iii) shear springs whose stiffness is k s

E s,i = 1 2 k s ϕ i -2 2 , i = 1, . . ., 4, (5)
ϕ i -2 2 ≤ s max , i = 1, . . ., 4, (6)
where ϕ i is, in the actual configuration, the angle, in the i-quadrant, between two unaligned fibers which share the same pivot; s max,j is the maximum value of the admissible angle change.

Numerical predictions versus experimental evidence for shear tests

A polyamide specimen, constructed using 3D printing technology, with L = 70 mm and L/ε = 10 √ 2 mm was analyzed. The cross-section of the fibers is rectangular and has the width equal to 0.9 mm and the depth equal to 1.6 mm. Pivots are cylinders whose diameter is 0.9 mm and whose height is 0.5 mm. For the Young's modulus we chose the value 1600 MPa.

The specimen was deformed imposing, gradually, a horizontal displacement on the top of the specimen by using as loading device a MTS TYTRON-250 with a 250 N load cell. The experiment was displacement-controlled, more precisely a displacement-rate was defined and applied to the specimen through a DC-linear motor where an air-film-bearing realized almost frictionless movement. In order to reduce the influences of external vibration the system is arranged horizontally and also based on a massive substructure.

Fig. 3 reports two pictures taken, the first, at the beginning of the test (reference configuration) and, the second, exactly immediately behind the rupture of the first fiber of the specimen, i.e. for a horizontal displacement of 139.96 mm.

The numerical simulation presented below was performed assuming that the stiffnesses of the springs are: k a = 134 N/mm, Fig. 4. Shear test: global structural reaction R (in the direction of the imposed displacement) vs. the non-dimensional displacement parameter , the black and red curves are related to the experiment and to the numerical calculations, respectively. Fig. 4 reports the global structural reaction R (in the direction of the imposed displacement) versus the non-dimensional displacement parameter (which attains a unit value when the maximum imposed displacement is reached); the black and red curves are related to the experiment and to the numerical calculations, respectively. We observe a remarkable closeness between the experimental measurements on structural reaction R and the results of the numerical simulation.

Finally, Fig. 5 reports the deformation of the specimen immediately behind the rupture of the first fiber (depicted in black) and it is highlighted by a green arrow for clarity reasons. The colors in the plot represent the total strain energy level, from lower levels (cool tones, e.g. in blue) to higher ones (warm tones, e.g. in red).

Conclusions

Also in the case of the description of fiber rupture phenomena we could adapt the Lagrangian discrete model proposed in [START_REF] Turco | Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models[END_REF] to get efficiently predictive numerical simulations. The advantage of the presented numerical model can be resumed as follows: (i) it is able to produce easily results in the regime of large deformations and displacements, (ii) it is based on the best fit of a limited number of constitutive parameters (between five and seven) to all of which a clear physical meaning can be attributed, (iii) all aforementioned constitutive parameters are easily related to specific geometrical and mechanical properties of considered fabrics so that their value, in a specific situation and for a specific specimen, can be easily estimated to belong to a well-determined range, (iv) it seems to be easily generalizable to the study of dynamic regimes, where inertial forces may play a relevant role, see some insight in [START_REF] Cazzani | On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation[END_REF][START_REF] Cazzani | On the whole spectrum of Timoshenko beams. Part II: further applications[END_REF].

The results which we present here have a twofold impact: (i) from the experimental point of view we prove that the relevant toughness in extension (proven in [START_REF] Dell'isola | Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence[END_REF]) of pantographic sheets coexists with a similarly relevant toughness in presence of imposed shear deformations, (ii) from the theoretical point of view we prove that used Lagrangian discrete model is flexible enough to be able to incorporate a facile modification allowing for the efficient description of rupture phenomena.

It has to be remarked that even if pantographic structures were synthesized to give an example of second gradient metamaterial, see e.g. [START_REF] Dell'isola | Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response[END_REF][START_REF] Giorgio | Pattern formation in the three-dimensional deformations of fibered sheets[END_REF][START_REF] Scerrato | Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations[END_REF][START_REF] Scerrato | On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets[END_REF][START_REF] D'agostino | Continuum and discrete models for structures including (quasi-) inextensible elasticae with a view to the design and modeling of composite reinforcements[END_REF][START_REF] Placidi | Identification of two-dimensional pantographic structure via a linear d4 orthotropic second gradient elastic model[END_REF][START_REF] Giorgio | Numerical identification procedure between a micro cauchy model and a macro second gradient model for planar pantographic structures[END_REF], modeled as generalized continua, we prove here that, when some specific deformation phenomena are considered, the most suitable models are not continuous. On the other hand, as seen in [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF][START_REF] Dell'isola | At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola[END_REF], when the qualitative description of their behavior is needed then second gradient continuum models may be more suitable.

Further investigations will produce a complete description of all rupture mechanism involving (and observed in) pantographic sheets among which we cite the fibers' detachments. Also the relevant effort of modeling out of plane deformation configurations and related buckling phenomena must be pursued, see the guidelines described, for example, in [START_REF] Gabriele | On the imperfection sensitivity of thin-walled frames[END_REF][START_REF] Pignataro | Effects of warping constraints and lateral restraint on the buckling of thin-walled frames[END_REF][START_REF] Rizzi | Initial postbuckling behavior of thin-walled frames under mode interaction[END_REF][START_REF] Gabriele | A 1D higher gradient model derived from Koiter's shell theory[END_REF][START_REF] Aminpour | A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis[END_REF][START_REF] Gabriele | A 1D nonlinear TWB model accounting for in plane cross-section deformation[END_REF][START_REF] Aminpour | On the continuum modelling of carbon nano tubes[END_REF][START_REF] Gabriele | A one-dimensional nonlinear thin walled beam model derived from Koiter shell theory[END_REF][START_REF] Aminpour | A one-dimensional beam model for single-wall carbon nano tube column buckling[END_REF][START_REF] Rizzi | On the postbuckling analysis of thin-walled frames[END_REF].

It seems somewhat interesting to build macro-springs obtained by using the capabilities of NURBS interpolation, see e.g. [START_REF] Cazzani | Isogeometric analysis of plane curved beams[END_REF][START_REF] Cazzani | Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches[END_REF][START_REF] Cazzani | Constitutive models for strongly curved beams in the frame of isogeometric analysis[END_REF][START_REF] Bilotta | Performance of a high-continuity finite element in three-dimensional elasticity[END_REF][START_REF] Greco | B-Spline interpolation of Kirchhoff-Love space rods[END_REF][START_REF] Greco | An implicit G 1 multi patch B-spline interpolation for Kirchhoff-Love space rod[END_REF][START_REF] Greco | An isogeometric implicit G1 mixed finite element for Kirchhoff space rods[END_REF][START_REF] Cazzani | An analytical assessment of finite elements and isogeometric analysis of the whole spectrum of Timoshenko beams[END_REF] in order to design new and enriched metamaterials. It is also very interesting the extension to the 3D case using the suggestions reported in [START_REF] Steigmann | Variational theory for spatial rods[END_REF].

Furthermore, the Hencky-type model briefly recalled here could be interesting also to model the granular media interactions, see [START_REF] Misra | Granular micromechanics based micromorphic model predicts frequency band gaps[END_REF], or generalized continua, see [START_REF] Dell'isola | Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response[END_REF][START_REF] Altenbach | On generalized Cosserat-type theories of plates and shells: a short review and bibliography[END_REF][START_REF] Eremeyev | Material symmetry group and constitutive equations of micropolar anisotropic elastic solids[END_REF][START_REF] Reis | Construction of micropolar continua from the asymptotic homogenization of beam lattices[END_REF][START_REF] Elnady | Construction of second order gradient continuous media by the discrete asymptotic homogenization method[END_REF] for a general review and [START_REF] Tedesco | Multiscale 3D mixed FEM analysis of historical masonry constructions[END_REF] for an application on masonry structures.

Finally, it deserves attention the identification of the parameters, i.e. the stiffnesses of the springs and the strain thresholds, used to build the numerical model since the performances of the model depend significantly on their values. In this perspective it is definitely helpful a sensitivity analysis on the parameters by using the tools reported, shortly, in the review paper [START_REF] Turco | Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives[END_REF] and, in depth, in [START_REF] Bilotta | A numerical study on the solution of the Cauchy problem in elasticity[END_REF][START_REF] Bilotta | Reconstructing blockages in a symmetric duct via quasi-isospectral horn operators[END_REF][START_REF] Bilotta | Numerical sensitivity analysis of corrosion detection[END_REF][START_REF] Alessandrini | Evaluating the volume of a hidden inclusion in an elastic body[END_REF][START_REF] Alessandrini | Computing volume bounds of inclusions by EIT measurements[END_REF][START_REF] Turco | Identification of axial forces on statically indeterminate pin-jointed trusses by a nondestructive mechanical test[END_REF] and applied in problems concerning huge and innovative structures as those described e.g. in [START_REF] Buffa | The Sardinia Radio Telescope: a comparison between close range photogrammetry and FE models[END_REF][START_REF] Stochino | Sardinia Radio Telescope finite element model updating by means of photogrammetric measurements[END_REF].

Fig. 1 .

 1 Fig. 1. Specimen constructed using 3D printing technology.

Fig. 2 .

 2 Fig. 2. Mechanical model of a pantographic 2D structure.

Fig. 3 .

 3 Fig. 3. Experimental shear test: reference configuration (a) and deformation corresponding to the breakdown onset (b) (the broken fiber is in the lower left).

Fig. 5 .

 5 Fig. 5. Numerical shear test: deformation corresponding to the breakdown onset (the broken fiber is indicated in black and it is highlighted by an arrow). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)


Rutger H J Fick

Madelaine Daianu

Marco Pizzolato

Demian Wassermann

Russell E Jacobs

Paul M Thompson

Terrence Town

Rachid Deriche

Comparison of Biomarkers in Transgenic Alzheimer Rats Using Multi-shell Diffusion MRI

In this study, we assessed the evolution of diffusion MRI (dMRI) derived markers from different white matter models as progressive neurodegeneration occurs in transgenic Alzheimer rats (TgF344-AD) at 10, 15 and 24 months. We compared biomarkers reconstructed from Diffusion Tensor Imaging (DTI), Neurite Orientation Dispersion and Density Imaging (NODDI) and Mean Apparent Propagator (MAP)-MRI in the hippocampus, cingulate cortex and corpus callosum using multi-shell dMRI. We found that NODDI's dispersion and MAP-MRI's anisotropy markers consistently changed over time, possibly indicating that these measures are sensitive to age-dependent neuronal demise due to amyloid accumulation. Conversely, we found that DTI's mean diffusivity, NODDI's isotropic volume fraction and MAP-MRI's restrictionrelated metrics all followed a two-step progression from 10 to 15 months, and from 15 to 24 months. This two-step pattern might be linked with a neuroinflammatory response that may be occuring prior to, or during microstructural breakdown. Using our approach, we are able to provide -for the first time -preliminary and valuable insight on relevant biomarkers that may directly describe the underlying pathophysiology in Alzheimer's disease.

Introduction

Diffusion MRI (dMRI) allows us to non-invasively study microstructural changes caused by neuropathology. Among these pathologies, gaining understanding of Alzheimer's disease (AD) is of particular importance, affecting over one in nine people age 65 and above in the U.S. alone [START_REF] Association | Alzheimers Disease Facts and Figures[END_REF]. Traditionally, dMRI studies have used Diffusion Tensor Imaging (DTI) [START_REF] Basser | MR diffusion tensor spectroscopy and imaging[END_REF] to model the grey and white matter structure abnormalities in AD patients. Only recently, more complex white matter models like Neurite Orientation Dispersion and Density Imaging (NODDI) [START_REF] Zhang | NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain[END_REF] have been explored to classify AD, and have shown greater discriminative power than DTI [START_REF] Nir | Alzheimers disease classification with novel microstructural metrics from diffusion-weighted MRI[END_REF]. This reinforces the importance of exploring white matter models that provide more detailed microstructural information than DTI.

In human studies, it is hard to relate dMRI derived metrics to corresponding microstructural changes for lack of histological validation. As a solution, animal models provide a way to gain understanding on the underlying pathophysiology of AD by allowing dMRI in addition to histological measurements. Mouse models of human tauopathy (rTg4510) have been previously studied at various time points using DTI [START_REF] Sahara | Age-related decline in white matter integrity in a mouse model of tauopathy: an in vivo diffusion tensor magnetic resonance imaging study[END_REF][START_REF] Wells | In vivo imaging of tau pathology using multi-parametric quantitative MRI[END_REF], and at a single time point comparing DTI with NODDI metrics [START_REF] Colgan | Application of neurite orientation dispersion and density imaging (NODDI) to a tau pathology model of Alzheimer's disease[END_REF]. In this latter study, NODDI derived metrics once again appeared more discriminative than those derived from DTI. Further efforts focusing on multi-shell dMRI analysis of transgenic Alzheimer rats (TgF344-AD) have shown that dMRI measurements at higher gradient strengths aid the classication of AD-like pathology [START_REF] Daianu | Multi-Shell Hybrid Diffusion Imaging (HYDI) at 7 Tesla in TgF344-AD Transgenic Alzheimer Rats[END_REF]. However, only anisotropy measures of DTI and hybrid diffusion imaging (HYDI) [START_REF] Wu | Hybrid diffusion imaging[END_REF] were explored.

In this study, we compare the evolution of dMRI-derived markers from different white matter models as progressive neurodegeneration occurs in transgenic Alzheimer rats (TgF344-AD). In particular, we study the patterns of alteration across three time points in the hippocampus, cingulate cortex and corpus callosum -areas known to be affected in AD. The two grey matter areas were previously shown to manifest age-dependent cerebral amyloidosis that precedes tauopathy, gliosis and apoptotic loss of neurons [START_REF] Cohen | A transgenic Alzheimer rat with plaques, tau pathology, behavioral impairment, oligomeric a, and frank neuronal loss[END_REF], making these cortical regions extremely relevant for understanding the underlying mechanisms in AD. We compare biomarkers derived from DTI, NODDI and Mean Apparent Propagator (MAP)-MRI [START_REF] Özarslan | Mean apparent propagator (MAP) MRI: a novel diffusion imaging method for mapping tissue microstructure[END_REF] using multi-shell data. To the best of our knowledge, this is the first study that investigates multi-shell biomarkers at different time points in AD animal models.

The paper is structured as follows: we first describe the diffusion MRI data and the metrics we derive in Section 2. We provide the results in section 3 and discuss them in section 4. We finally provide our conclusions in section 5.

Materials and Methods

In this section, we first detail the diffusion MRI data acquisition, preprocessing and region of interest selection of the AD rats. We then give a brief overview of the methods we use and their metrics of interest. We detail the fractional anisotropy (FA) and mean diffusivity (MD) of classical DTI, the orientation dispersion index (ODI), neurite density index (NDI) and isotropic volume fraction (IsoVF) of the multi-compartment NODDI model, and finally the formulation of several q-space indices of the MAP-MRI functional basis. We estimated the DTI and MAP-MRI metrics using the diffusion imaging in python (dipy) open source software [START_REF] Garyfallidis | Dipy, a library for the analysis of diffusion MRI data[END_REF] and the NODDI metrics using the NODDI toolbox [START_REF] Zhang | NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain[END_REF].

Processing of Transgenic Alzheimer Rat Data Sets

We use multi-shell dMRI data of three ex-vivo transgenic Alzheimer rats (line TgF344-AD) [START_REF] Cohen | A transgenic Alzheimer rat with plaques, tau pathology, behavioral impairment, oligomeric a, and frank neuronal loss[END_REF], also previously analyzed by Daianu et al. [START_REF] Daianu | Multi-Shell Hybrid Diffusion Imaging (HYDI) at 7 Tesla in TgF344-AD Transgenic Alzheimer Rats[END_REF]. The rats were Fig. 1. Regions of interest for biomarker estimation on the registered FA map of rat 1. We mark the cingulate cortex (green), corpus callosum (blue) and hippocampus (red). euthanized at 10, 15 and 24 months, fixed brains were prepared as described in [START_REF] Daianu | Multi-Shell Hybrid Diffusion Imaging (HYDI) at 7 Tesla in TgF344-AD Transgenic Alzheimer Rats[END_REF], and scanned using a 7 Tesla Bruker Biospin MRI scanner at California Institute of Technology. A high-resolution fast low angle magnetic shot (FLASH) anatomical image with a mix of T1 and T2 weighting (375 × 224 × 160 matrix; voxel size: 0.08×0.08×0.08 mm 3 ) was used. The diffusion MRI data were sampled on 5 shells with b-values {1000,3000,4000,8000,12000} s/mm 2 , all with the same 60 directions and 5 b0 measurements. Other parameters were δ/∆ = 11/16 ms and T E/T R = 34/500 ms. The voxel dimensions were 0.15 × 0.15 × 0.25 mm 3 .

During preprocessing, extra-cerebral tissue was removed using the skullstripping Brain Extraction Tool from BrainSuite (http://brainsuite.org/), for both the anatomical images and the DWIs. We corrected for eddy current distortions using the eddy correct FSL tool (www.fmrib.ox.ac.uk/fsl) for which a gradient table was calculated to account for the distortions. As an image processing step, DWIs were up-sampled to the resolution of the anatomical images (with isotropic voxels) using FSLs flirt function with 9 degrees of freedom; the gradient direction tables were rotated accordingly after each linear registration. For our study, we draw regions of interest (ROIs) in the cingulate cortex, hippocampus and corpus callosum as shown in Figure 1.

DTI Metrics

The classical DTI model [START_REF] Basser | MR diffusion tensor spectroscopy and imaging[END_REF] assumes that the measured diffusion signal belongs to the set of Gaussian distributions. While DTI has well-known limitations with respect to the modeling of crossing tissue configurations and restricted diffusion, its derived metrics FA and MD have been found useful to classify AD patients [START_REF] Nir | Alzheimers disease classification with novel microstructural metrics from diffusion-weighted MRI[END_REF]. Using signal attenuation E(b) = S(b)/S(0), the DTI model describes the diffusion signal as E(b) = exp(-bg T Dg) with D a 3 × 3 symmetric positive-definite matrix and g the gradient direction. Estimating the eigenvalues of D as {λ 1 , λ 2 , λ 3 } the FA and MD are given as

F A = 1 2 (λ 1 -λ 2 ) 2 + (λ 2 -λ 3 ) 2 + (λ 3 -λ 1 ) 2 λ 2 1 + λ 2 2 + λ 2 3 M D = λ 1 + λ 2 + λ 3 3 (1)
In accordance with DTI's Gaussian diffusion assumption, we only use the b0 and b = 1000 s/mm 2 data when fitting DTI. The FA and MD in our slice of interest are shown in Figure 2.

NODDI Metrics

The more advanced multi-compartment NODDI model [START_REF] Zhang | NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain[END_REF] separates the signal contribution of different tissues by fitting a combination of intra-cellular, extracellular and free-water models.

E = (1 -ν iso )(ν ic E ic (ODI) + (1 -ν ic ) * E ec ) + ν iso E iso (2)
The intra-cellular signal E ic is modeled as a set of dispersed sticks, i.e., cylinders of zero radius, to capture the highly restricted nature of diffusion perpendicular to neurites and unhindered diffusion along them. The amount of dispersion is given by the orientation dispersion index (ODI), which is defined by a Watson distribution. The extra-cellular signal E ec is described as a dispersed mixture of Gaussian anisotropic diffusion, and an isotropic Gaussian compartment E iso represents free diffusion. Similarly as in [START_REF] Colgan | Application of neurite orientation dispersion and density imaging (NODDI) to a tau pathology model of Alzheimer's disease[END_REF], we study the ODI, the neurite density index N DI = (1 -ν iso )ν ic and the isotropic volume fraction IsoV F = ν iso .

In accordance with NODDI's recommended acquisition scheme [START_REF] Zhang | NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain[END_REF], we fit NODDI only using the b0 and b = {1000, 3000} s/mm 2 data. Furthermore, as water diffusivity changes in ex-vivo tissue, we set the intra-cellular and isotropic diffusivity to 0.6×10 -9 m 2 s -1 and 2.0×10 -9 m 2 s -1 [START_REF] Alexander | Orientationally invariant indices of axon diameter and density from diffusion MRI[END_REF]. An illustration of the ODI, NDI and IsoVF can be seen in Figure 2.

MAP-MRI Metrics

The MAP-MRI approach [START_REF] Özarslan | Mean apparent propagator (MAP) MRI: a novel diffusion imaging method for mapping tissue microstructure[END_REF] uses a functional basis to represent the 3D diffusion signal with as little assumptions as possible. It then analytically reconstructs the 3D diffusion propagator by only assuming the short gradient pulse approximation (δ ≈ 0). In this way, it accurately estimates the diffusion propagator in the presence of both non-Gaussian diffusion and crossing tissue configurations.

MAP-MRI represents the discretely measured signal attenuation E(q) using a set of continuous orthogonal basis functions representing the space Ê(q; c), where the signal is now represented in terms of basis coefficients c and the qspace wave vector q = |q|g with g the gradient direction is related to the b-value as |q| = b/(∆ -δ/3)/2π. Without going into the formulation of MAP-MRI's basis functions, we detail the estimation of basis coefficients c in Eq. (3). In short, we regularize the fitting of c such that Ê(q; c) smoothly interpolates between the measured q-space points by using Laplacian regularization [START_REF] Fick | MAPL: Tissue microstructure estimation using Laplacianregularized MAP-MRI and its application to HCP data[END_REF], where regularization weight λ is set using voxel-wise generalized cross-validation. We also constrain the estimated diffusion Propagator P (R; c) to be positive using quadratic programming [START_REF] Özarslan | Mean apparent propagator (MAP) MRI: a novel diffusion imaging method for mapping tissue microstructure[END_REF]. Once c is known, the MAP-MRI basis simultaneously represents the 3D dMRI signal and 3D diffusion propagator. We estimate the q-space indices Return-To-Origin, Return-To-Axis and Return-To-Plane Probability (RTOP, RTAP and RTPP), which in theory are related to the volume, surface and length of a cylindrical pore [START_REF] Özarslan | Mean apparent propagator (MAP) MRI: a novel diffusion imaging method for mapping tissue microstructure[END_REF]. We also estimate the non-Gaussianity (NG), which describes the ratio between the Gaussian and non-Gaussian volume of the signal. Finally we estimate the propagator anisotropy (PA), which is a normalized metric that describes the anisotropy of the 3D diffusion propagator. As MAP-MRI is designed to represent the entire 3D diffusion signal, we estimate all metrics using the entire 5 shell data up to a b-value of 12000 s/mm 2 , using a radial order of 6, resulting in 50 estimated coefficients. We illustrate these metrics in Figure 3.

argmin c Data Fidelity R 3 E(q) -Ê(q; c) 2 dq + Smoothness λ R 3 ∇ 2 Ê(q; c)

Results

In Figure 4 we show the evolution of the mean with 0.5 standard deviation of all dMRI-derived metrics in the ROIs shown in Figure 1. We use the same colors for the hippocampus (red), corpus callosum (blue) and cingulate cortex (green). The only metric that consistently increases over time is NODDI's ODI and consistently decreases is MAP-MRI's PA, with the exception of the cortex.

It is also apparent that FA, NDI, RTOP, RTAP and RTPP follow a different, 2-step pattern, first decreasing and then slightly increasing. Inversely, for MD, IsoVF and NG we first find an increase and then a decrease. We provide the raw data values in Table 1. We also produce correlation plots for dispersion and anisotropy measures in Figure 5 and for the 2-step metrics in Figure 6. It can be seen that ODI is negatively correlated with FA and PA, and that IsoVF is positively correlated with MD and negatively with RTOP.

Discussion

In this work, we have shown that different metrics of DTI, NODDI and MAP-MRI appear to be sensitive to different processes as age-dependent cerebral amyloidosis manifests in both grey and white matter in the Alzheimer rats.

DTI findings: We find a significant drop in FA in all ROIs from 10 to 15 months and a small increase from 15 to 24 months. This corresponds with previous findings in the hippocampus using data up to b = 1000 s/mm 2 [START_REF] Daianu | Multi-Shell Hybrid Diffusion Imaging (HYDI) at 7 Tesla in TgF344-AD Transgenic Alzheimer Rats[END_REF]. While a comparison of using different b-values in the DTI estimation was outside of the scope of this study, it was shown that when higher b-values are included, the FA trend consistently decreases over time [START_REF] Daianu | Multi-Shell Hybrid Diffusion Imaging (HYDI) at 7 Tesla in TgF344-AD Transgenic Alzheimer Rats[END_REF]. Nonetheless, it has been argued that compared to FA, MD lends itself better to the assessment of cortical and subcortical grey matter, where net diffusion may not be expected to conform to any one specific direction [START_REF] Chiapponi | Cortical grey matter and subcortical white matter brain microstructural changes in schizophrenia are localised and age independent: a casecontrol diffusion tensor imaging study[END_REF]. When we assess MD, we consistently find an increase from 10 to 15 months and a decrease from 15 to 24 months. This may suggest that FA and MD are sensitive to different processes taking place in AD.

NODDI findings: Several studies have suggested that NODDI metrics, in particular ODI, have better AD classifying potential due to NODDI's ability to delineate signal contributions from different tissue compartments [START_REF] Nir | Alzheimers disease classification with novel microstructural metrics from diffusion-weighted MRI[END_REF][START_REF] Colgan | Application of neurite orientation dispersion and density imaging (NODDI) to a tau pathology model of Alzheimer's disease[END_REF]. While we cannot do a classification study using our data, we find that ODI consistently increases in areas where tau pathology increases in our rat model [START_REF] Cohen | A transgenic Alzheimer rat with plaques, tau pathology, behavioral impairment, oligomeric a, and frank neuronal loss[END_REF]; the hippocampus, cingulate cortex and corpus callosum. We also find that IsoVF shows an increase from 10 to 15 months and a decrease from 15 to 24 months in all areas, following the same trend as DTI's MD. Though, it should be mentioned that fitting NODDI requires presetting the intra-cellular and isotropic diffusiv- ity, which influences obtained metric values. Fitting NODDI on the selected b max = 3000s/mm 2 or the full data does not significantly impact our findings.

MAP-MRI findings:

To the best of our knowledge, this is the first study that estimates MAP-MRI metrics on data from an AD model. We find that all metrics except PA follow a two-stage progression pattern similar to DTI's MD. The decrease-increase of return-to-origin, return-to-axis and return-to-plane probability (RTOP, RTAP and RTPP) makes sense with the increase-decrease of MD, as an increased diffusivity means that spins are able to move away farther, reducing the chance they return to their origin, axis or plane. Interestingly, this does not make the signal more Gaussian, as the Non-Gaussianity follows an increasedecrease pattern in all ROIs. The exception to this trend is the RTPP in the corpus callosum, which increases monotonically, indicating a steady increase in restriction parallel to the axon direction. Finally, PA consistently decreases in all areas except the cortex, where a small increase is found, followed by a larger decrease. This decreasing trend in anisotropy measures when using higher gradients strengths was also reported with DTI's FA or HYDI's NQA [START_REF] Daianu | Multi-Shell Hybrid Diffusion Imaging (HYDI) at 7 Tesla in TgF344-AD Transgenic Alzheimer Rats[END_REF]. We note that while we fitted MAP-MRI to the full data with 300 DWIs, it was shown that its metrics are stable under subsampling to less than 100 DWIs [START_REF] Fick | MAPL: Tissue microstructure estimation using Laplacianregularized MAP-MRI and its application to HCP data[END_REF] or could even be fitted directly on a NODDI acquisition scheme.

Biological explanation for biomarker trends: The trends of all derived metrics can be divided into two groups: those that consistently decrease or increase and those that show a 'decrease-increase' or 'increase-decrease' pattern.

The first group could point towards the accelerating cerebral amyloidosis as age increases in these rats [START_REF] Cohen | A transgenic Alzheimer rat with plaques, tau pathology, behavioral impairment, oligomeric a, and frank neuronal loss[END_REF]. Over time, this "amyloid burden" results in agedependent neuronal demise that is likely owed to oligomeric Aβ accumulation. In turn, this neuronal demise could result in a more dispersed, less anisotropic diffusion signal. This corresponds with the observed correlations between dispersion and anisotropy measures in Figure 5.

The second group may indicate an inflammatory response to amyloid accumulation, occurring prior to (or coincident with and obscuring) the onset of microstructural breakdown and macrostructural atrophy [START_REF] Weston | Diffusion imaging changes in grey matter in Alzheimers disease: a potential marker of early neurodegeneration[END_REF]. At 15 months TgF344-AD rats have heavy plaque burden and strong neuroinflammation, whereas by 24 months most of the inflammatory reaction to the plaques has passed. This corresponds to what we see when MD and IsoVF increase-decrease and RTOP, RTAP and RTPP decrease-increase (except RTPP at corpus callosum). The correlations between MD, IsoVF and RTOP in Figure 6 therefore makes sense. Though, the increase-decrease in NG indicates that while the inflammatory response increases diffusivity, it also increases the non-Gaussian portion of the signal at higher b-values.

Difficulties of comparing our findings with previous animal studies: There have been several previous dMRI studies using Alzheimer animal models. However, different species and disease expressions make comparisons of dMRI metrics difficult. For instance, our TgF344-AD rat model was made to drive cerebral amyloid and downstream tauopathy and neuronal loss, also known as the "amyloid cascade hypothesis" of John Hardy [START_REF] Hardy | Alzheimer's disease: the amyloid cascade hypothesis[END_REF]. In contrast, the Tg4510 mouse model used by Colgan et al. [START_REF] Colgan | Application of neurite orientation dispersion and density imaging (NODDI) to a tau pathology model of Alzheimer's disease[END_REF] was developed to only assess tauopathy; and not the amyloid cascade hypothesis. For this reason, it is hard to make claims about differences in biomarker trends found between this study and theirs.

Limitations of the study: As we did not have healthy rats to statistically test for changes with disease progression -which means there is room for improvement -we used the youngest rat (10 months old) as a control subject to compare against suggestive changes at later time points. Another limitation is the low number of experimental subjects that also prevents us from statistically differentiating between the disease stages of the transgenic Alzheimer rat model.

Conclusion

We presented a unique study on transgenic Alzheimer rats at 10, 15 and 24 months, comparing DTI, NODDI and MAP-MRI-derived metrics, in grey and white matter areas known to manifest age-dependent cerebral amyloidosis that precedes neurofibrillary tangles and apoptotic loss of neurons. We found that NODDI's ODI and MAP-MRI's PA metrics uniformly changed over time, likely indicating that they are sensitive to age-dependent neuronal demise due to amyloid accumulation. It is relevant to note that both of these metrics require bvalues higher than 1000 s/mm 2 . Conversely, we found that DTI's MD, NODDI's IsoVF and MAPMRI's RTOP, RTAP, RTPP and NG all follow a two-step progression from 10 to 15 to 24 months -either an increase-decrease or a decreaseincrease -likely indicating sensitivity to the neuroinflammatory response at 15 months and potentially, atrophy of the microstructure at 24 months. While this study does not have enough subjects to statistically differentiate between the different disease stages, it does provide valuable insight on which biomarkers and models come closest to explaining the biological changes in the cerebral tissue.

2 dqFig. 2 .

 22 Fig. 2. Illustrations of a DTI and NODDI metrics in the same coronal slice for the three time points.

Fig. 3 .

 3 Fig.3. Illustrations of MAP-MRI's q-space indices in the same coronal slice for the three time points. To visualize RTOP, RTAP and RTPP in the same unit (mm -1 ) we show the cubed root of RTOP and squared root of RTAP.

Fig. 4 .

 4 Fig. 4. DTI, NODDI and MAP-MRI metrics for the same time points in the hippocampus (red), corpus callosum (blue) and cingulate cortex (green).

Fig. 5 .

 5 Fig.5. Scatter plots of FA, ODI and PA for the rats of ages 10 months (blue), 15 months (green) and 24 months (red) in the hippocampus. It can be seen that ODI is negatively correlated with both FA and PA.

Fig. 6 .

 6 Fig.6. Scatter plots of MD, IsoVF and RTOP for the rats of ages 10 months (blue), 15 months (green) and 24 months (red) in the hippocampus. It can be seen that IsoVF is positively correlated with MD and negatively with RTOP.

Table 1 .

 1 Mean and standard deviation of DTI, NODDI and MAP-MRI metrics for the three time points in each region of interest.

	DTI Metrics		Age
	Metric	ROI	10 months 15 months 24 months
	FA	Hippocampus 0.29±0.08 0.19±0.05 0.20±0.06
		C. Callosum 0.51±0.15 0.27±0.08 0.30±0.09
		C. Cortex	0.28±0.08 0.20±0.05 0.22±0.08
	MD (×10 3 )	Hippocampus 0.32±0.02 0.39±0.03 0.29±0.02
		C. Callosum 0.19±0.05 0.30±0.05 0.21±0.02
		C. Cortex	0.31±0.04 0.49±0.06 0.23±0.04
	NODDI Metrics		Age
	Metric	ROI	10 months 15 months 24 months
	ODI	Hippocampus 0.39±0.11 0.48±0.10 0.55±0.11
		C. Callosum 0.39±0.08 0.48±0.09 0.53±0.09
		C. Cortex	0.44±0.11 0.47±0.10 0.63±0.10
	NDI	Hippocampus 0.46±0.03 0.45±0.04 0.54±0.05
		C. Callosum 0.93±0.09 0.86±0.07 0.93±0.08
		C. Cortex	0.58±0.04 0.60±0.10 0.74±0.10
	IsoVF	Hippocampus 0.03±0.02 0.11±0.03 0.03±0.01
		C. Callosum 0.02±0.03 0.11±0.05 0.02±0.02
		C. Cortex	0.08±0.05 0.28±0.06 0.02±0.03
	MAP-MRI Metrics		Age
	Metric	ROI	10 months 15 months 24 months
	RTOP (×10 7 )	Hippocampus 0.68±0.08 0.45±0.10 0.76±0.14
		C. Callosum 1.03±0.12 0.85±0.10 0.94±0.10
		C. Cortex	1.04±0.27 0.90±0.27 1.58±0.56
	RTAP (×10 5 )	Hippocampus 0.38±0.03 0.31±0.04 0.41±0.05
		C. Callosum 0.93±0.09 0.86±0.07 0.93±0.08
		C. Cortex	0.51±0.08 0.45±0.08 0.65±0.12
	RTPP (×10 3 )	Hippocampus 0.16±0.01 0.15±0.01 0.17±0.01
		C. Callosum 0.21±0.02 0.22±0.02 0.23±0.02
		C. Cortex	0.18±0.01 0.17±0.01 0.21±0.02
	NG	Hippocampus 0.43±0.03 0.49±0.02 0.45±0.02
		C. Callosum 0.51±0.02 0.55±0.01 0.52±0.01
		C. Cortex	0.51±0.03 0.57±0.02 0.49±0.02
	PA	Hippocampus 0.27±0.09 0.2±0.07 0.15±0.06
		C. Callosum 0.78±0.11 0.64±0.11 0.53±0.13
		C. Cortex	0.30±0.12 0.33±0.16 0.17±0.10

Acknowledgments

This work was partly supported by ANR "MOSIFAH" under ANR-13-MONU-0009-01, the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665: CoBCoM) and NIH grants U54 EB020403 and R01NS076794.