The time-dependent diffusivity in the abdominalganglion of Aplysia californica, comparing experiments

and simulations

Khieu Van Nguyena,b, Denis Le Bihana, Luisa Ciobanua,∗, Jing-Rebecca Lic,∗

aNeurospin, CEA Saclay, 91191 Gif sur Yvette, FrancebUniversity Paris-Sud, XI, 91450 Orsay, France

cINRIA-Saclay, Equipe DEFI, CMAP, Ecole Polytechnique, 91128, Palaiseau, France

Abstract

The nerve cells of the Aplysia are much larger than mammalian neurons. Usingthe Aplysia ganglia to study the relationship between the cellular structure and thediffusion MRI signal can shed light on this relationship for more complex organ-isms. We measured the dMRI signal at several diffusion times in the abdominalganglion and performed simulations of water diffusion in geometries obtainedafter segmenting high resolution T2-weighted images and incorporating knowninformation about the cellular structure from the literature. By fitting the exper-imental signal to the simulated signal for several types of cells in the abdominalganglion at a wide range of diffusion times, we obtained estimates of the intrinsicdiffusion coefficient in the nucleus and the cytoplasm. We also evaluated the re-liability of using an existing formula for the time-dependent diffusion coefficientto estimate cell size.

Keywords: magnetic resonance imaging (MRI), magnetic resonance microscopy(MRM), diffusion MRI (dMRI), time dependent diffusion, Bloch-Torreyequation, simulation, Aplysia

∗Corresponding authorEmail addresses: luisa.ciobanu@cea.fr (Luisa Ciobanu),

jingrebecca.li@inria.fr (Jing-Rebecca Li)

Preprint submitted to Journal ... November 14, 2016

1. Introduction

Diffusion magnetic resonance imaging (dMRI) has shown tremendous promisein a wide range of brain imaging applications. The correlation of dMRI-derivedmetrics with brain micro-structure properties such as axon size, myelin thickness,neurite orientation distribution, is an active area of research.

To explain the correlations and justify the proposed dMRI metrics there havebeen numerous biophysical models (usually subdividing the tissue into compart-ments described by spheres, ellipsoids, cylinders, and the extra-cellular space)[1, 2, 3, 4, 5, 6, 3, 7]. However, it is difficult to connect the geometrical param-eters contained in these models to ground truth values due to the complexity ofbrain tissue.

To get closer to the ground truth, one can construct phantoms [8, 9, 10, 11, 12]or perform histological analysis of the tissue of interest [7, 13, 14, 15, 16, 17]to get independant information of the microstructure of the imaged object. Thisinformation is then compared with model predictions based on the experimentaldMRI data. This comparison procedure can be useful in evaluating the quality andusefulness of the proposed dMRI models.

In this paper we propose another approach to validation, namely, using themuch larger neural cells of the Aplysia californica as a surrogate phantom. Theadvantages of this choice lies in the simple structure of the neural system, con-sisting of large round cells, smaller round cells gathered in bags, and cylindricalbundles of unmylinated axons. In structure, these cell components can be surro-gates for mammalian brain cell components (the soma, axon bundles, dendrites).In particular, the large size of the Aplysia neural cells make it possible to testclaims about short time diffusion imaging (usually implemented with OGSE [18]sequences) in the mammal brain by imaging the Aplysia neural system using thePGSE [19] sequence.

2. Materials and methods

2.1. Animal modelThe neural system of the Aplysia californica consists of five pairs of ganglia:

buccal, cerebral, pleural, pedal, and abdominal ganglia [20].

2

Fig. 1: The Aplysia abdominal ganglion diagram.

The abdominal ganglion was chosen in this imaging study because the cellularnetwork is very well determined in terms of single cell neurons, axonal orienta-tion.

3

giant cellbig cell

bag cells

L. connective n.

Siphon n.

Gential-pericardial n. Branchial n.

R. connective n.

Fig. 2: The Aplysia abdominal ganglion diagram.

There has been numerous works on the structure of the buccal and abdominalganglion [21, 22]. In particular, we have defined the following four types of cellsand cell components of particular interest for this imaging study:

1. Individual neuron cells;there are usually some large neuron cells in the abdominal ganglion witha diameter of at least 150 µm that are visible by inspection in the highresolution (26 µm isotropic) T2w images. Typically, these cells are theL2 to L9, L11, R2 to R8, R14 and R15 (labeled L or R for left or righthemiganglion, e.g. see in [23]). The single cell neurons with diameter lessthan 150 µm are not included for study in this group because we could notidentify them on the T2w images. We note that the sizes of these identifiedneurons are not fixed, varying instead as a function of the age and the weightof the animal. The large cell neurons contain a nucleus, cytoplasm andare probably surrounded by small satellite (glial) cells [24]. The satellitecells are very small cell, 6 µm maximum in diameter, without a nucleus[21, 22, 24].In order to organize the discussion of our results, we further separate theT2w identifiable neurons into two groups, by size:

4

(a) Giant cell neurons;one or two biggest cell neurons of diameter greater than 320µm in eachganglion will be selected for study, typically R2, L6, L7, and L11.

(b) Big cell neurons;two or three big cell neurons of diameter between 150 µm and 300µm that are clearly and easily identifiable from the T2w image will beselected for study.

2. Clusters of neuron cells gathered in the shape of a bag;the bag shaped clusters comprise of hundreds of neurons that are locatedon the rostral end of the abdominal ganglion [21]. The actual number andthe sizes of the neurons depend on the animal age and weight [25]. Forexample, in young Aplysia, there are probably less than 100 small (about10 µm in diameter) cell neurons in each bag; in adult Aplysia the numberof cells grow to 400 in each bag with cell sizes between 40 to 100 µm indiameter and extended dense networks of processes [21, 26, 27, 28].There are two clusters of bag cells in each abdominal ganglion. When theabdominal ganglion is slid inside the small imaging capillary, only one clus-ter of bag cells is easy to identify from the T2w image, the other cluster isusually close to the abdominal body making it harder to identify from theT2w image. Therefore, only the one clearly identifiable cluster of bag cellsin each ganglion is selected for study.

3. Nerve cells;these are groups of (cylinder shaped) axons. There are five groups of nervesin the abdominal ganglion: the left/right connective nerves, siphon nerve,genial-pericardial nerve and branchial nerve [23] (see Fig. 2). The siphonnerve and left/right connective nerves are of interest in this study becausethey remain intact as the ganglion is inserted inside the imaging capillarywhereas other nerves were cut.The information of the axon sizes and distribution for the nerves shown inTable 1 are for the buccal ganglia of Aplysia californica because we couldnot find analogous information about the abdominal ganglia. The group ofaxons (cylinder shaped) with diameter in ranges from very small (less than1 µm) to large axons (greater than 25 µm).

5

Nerve I (> 25µm) II (10− 25µm) III (1− 10µm) IV (≤ 1µm)1b 0 5 261 71442b 2.33 19.41 235 94153b 0 15 67 2191on 1.67 8.67 681 10483rn 0 16 1579 2692

cbc 2 12 196 1591

Table 1: The nerves (1b, 2b, 3b, on, rn, cbc) of buccal ganglia. There are four type of axons withdifferent range of axons diameter: type I , II , III and IV. The distribution of axons type of eachnerve [22] were shown.

We believe that the siphon nerve and left/right connective nerve of abdominalganglionare more close to buccal 2b nerve, 3b nerve, and cbc nerve because of thesimilarity in size and functionality.

2.2. Sample preparationSeven Aplysia californica (National Resource for Aplysia, Miami, FL, USA)

were used in this study. The animals were anaesthetized by injection of an iso-tonic magnesium chloride solution (MgCl2, 360 mM; HEPES, 10 mM; pH = 7.5).All chemicals were purchased from Sigma-Aldrich (Saint Luis, MO, USA). Theabdominal ganglion were fixed by PFA 4% in 10 minutes and then washed threetimes in PBS pH = 7.4. The abdominal ganglion were resected and inserted into a2.0 mm ID glass capillary filled with florinent and then slid inside the transceiverfor imaging.

2.3. Image acquisitionAll experiments were performed at 19C on a 17.2 T system (Bruker BioSpin,

Ettlingen, Germany) equipped with 1 T/m gradients. RF transceivers were home-built micro-coils with inner diameters of 2.4 mm, the design of which has beendescribed elsewhere [29, 30]. Typically, six to eight acquisitions were acquiredfor each sample. A standard, T2 weighted images of the abdominal ganglion ofAplysia californica ware acquired with parameters TR = 2000 ms, TE = 20 ms,RARE AF = 8, isotropic spatial resolution 26 µm, matrix size of 400× 88× 88).Diffusion-weighted images (diffusion prepared FISP pulse sequence [31]) acqui-sition corresponding with parameters TE/TR=1.63/1000 ms, NEX=2, isotropicspatial resolution 52µm, 3 directions (x, y, z) were obtained on the same samplesat five to seven diffusion encoding times (δ = 2.5 ms, ∆ = [5, 7.5, 10, 12, 15, 20, 25]

6

ms) and 8 b-values ([70, 100, 200, · · · , 700] s/mm2), matrix size 200 × 44 × 44.All acquisition were acquired under FOV = 10.4× 2.3× 2.3 mm3.

2.4. Image analysisAll 3D region of interest (ROIs) were manually segmented from the T2w im-

ages (Fig. 3) of seven animals. In total, there are 8 ROIs of giant cell neurons,13 ROIs of the big cell neurons, 16 ROIs of the bag cell neurons clusters, and10 ROIs of the nerve selected for analysis. We note that we cover each bag cellscluster with two ROIs.

The T2w images were manually aligned with the diffusion-weighted imagesand the dMRI signals corresponding to the selected ROIs were processed to com-pute the apparent diffusion coefficient (ADC) using a linear fit of the log of thesignal versus the b-value.

For the nerve cells, the diffusion is anisotropic due to the shape of the cylindri-cal axons. Because for this study, only three diffusion directions were measured,it is not possible to compute the effictive diffusion tensor for the nerve regions.Hence, we average the ADC’s in the three directions to obtain the mean diffusiv-ity [32, 33, 34, 35]:

MD =ADCx + ADCy + ADCz

3.

For giant cell neurons, big cell neurons and bag cells clusters, there may be lesspronouced anisotropy due to the shape of the cells and the shape and position ofthe nucleus. In order to avoid these additional complications, we also average theADC on three gradient directions x, y, z to calculate the mean diffusivity MD forthese types of cells.

The first principle effective diffusion λ1 is also called the axial diffusivity orparallel diffusivity and the diffusivity in the two minor axes λ2, λ3 are often aver-

aged to produce a measure of radial diffusivity [36, 37], λ⊥ =λ2 + λ3

2. There-

fore, we propose to use the averaged ADC on three directions x, y, z of eachexperimental data during this study to compare with the simulation results.

7

Fig. 3: Abdominal ganglion T2w image: a giant single cell neuron ROI (blue) , a big single cellROI (cyan), a bag cell neurons ROI (red), and a nerve ROI (green). The scale bar represents 300µm

2.5. SimulationsIn diffusion MRI, the complex transverse water proton magnetization M(x, t)

is a function of position x and time t, and depends on the diffusion-encodinggradient magnetic field G(t) = gf(t). The amplitude and direction informationof the diffusion-encoding is contained in the vector g ∈ R3, the time profile ofthe effective gradient magnetic field is f(t). For the PGSE sequence, the effectivetime profile is defined by:

f(t) =

1 ts < t ≤ ts + δ,

−1 ts + ∆ < t ≤ ts + ∆ + δ,

0 otherwise,

where ts is the starting time of the first gradient pulse, δ is the duration of thepulses and ∆ the delay between the start of the pulses. The signal is measured atthe echo time TE, ∆ + δ ≤ TE < 2ts + 2∆.

The complex transverse magnetization is governed by Bloch-Torrey equation[38]

∂

∂tM(x, t) = −ıγf(t)g · xM(x, t) +∇ ·

(Dl∇M(x, t)

), x ∈ ∪Ωl, (1)

where ı is the imaginary unit, Dl is the intrinsic diffusion coefficient in the ge-ometrical compartment Ωl, γ = 42.576 MHz/Tesla is the gyromagnetic ratio of

8

the water proton. We solve the above equation subject to impermeable boundaryconditions on ∂Ωl:

Dl∇M(x, t) · ν = 0, x ∈ ∂Ωl, (2)

where ν is the outward normal vector. We impose the initial condition

M(x, 0) = 1, x ∈ Ωl, (3)

meaning uniform spin density in all Ωl. The numerical method for solving equa-tion (1)-(3) is adapted from [39].

The diffusion MRI signal is the integral of magnetization at TE:

S =∑l

∫x∈Ωl

M(x, TE)dx,

The b-value [40], in case of the PGSE sequence is

b(g, δ,∆) = γ2‖g‖2δ2(∆− δ/3).

The apparent diffusion coefficient (ADC) is:

ADC = − ∂

∂blog

S(b)

S(0)

∣∣∣∣b=0

.

For the giant cell neurons, big cell neurons, and the small cell neurons, theintrinsic diffusivities in the cytoplasm and nucleus, Dc andDn, were chosen froma range described in the literature [24]. Subsequently, the ADC obtained from thenumerically simulated dMRI signal was computed by

ADCsimul = − logS(b = 10)− logS(b = 0)

10,

and compared to the ADC values found from the experimental data.

2.5.1. Giant and big cell neuronsTo match the dMRI signal in the single cell neurons with numerical simu-

lations the cell outline was segmented from the anatomical T2w image. Insidethis cell shape, an irregularly shaped nucleus was manually generated (around 25-30% volume fraction) (Fig. 4). The shape of the nucleus was inspired by the highresolution images in [].

The shape in Fig. 4 was scaled so that the effective cell diameter range from160 µm to 300 µm and 320 µm to 400 µm for big cell neurons and giant cell

9

neurons, respectively. Even though not visible in the T2w images, there may bea small volume (up to 5%) of satellite cells (very small cells, 6µm maximumdiameter, without nucleus [21, 22, 24]) surrounding the single cell neurons. Weallow this possibility by adding the magnetization due to small spherical cells of6 µm in diameter.

Fig. 4: Large single cell neuron geometry: nucleus (red) irregular shape was manually generated,cytoplasm (light green). The cell shape were estimated from the T2w image.

2.5.2. Bag clustersIn order to match the dMRI signal of the bag cell neurons, the small cells were

modelled as small spheres with a smaller concentric spherical nucleus (around25% volume fraction). As all Aplysia used in this study are in late juvenile period,therefore we modelled the cells as spheres with diameters that varied between 20and 60 µm, following a normal distribution with mean = 46 µm and std = 6. Themean and std here were chosen so that under the random with normal distribution,99% cell diameter are between 20 and 60 µm.

2.5.3. Nerve cells: group of axonsThe nerves was modelled by combining axons (cylinders) of different diam-

eters. The groups of axons can be small (modelled by cylinders with 1 µm indiameter) to large (modelled by cylinders 25 µm in diameter), consistent with theliterature [22].

10

The numerical simulation were simulated in 2D perpendicular direction of thenerves and the free diffusion in along direction of the nerve. The final numericalmean diffusivity coefficient was then simulated by combined the ADC in per-pendicular direction andl the free diffusion coefficients in aligned the nerves byfollowing formula:

MD =Da+ 2× ADCT

3, (4)

where Da here mean the free intrinsic diffusion coefficient inside the axons,ADCT mean the ADC value in perpendicular of the nerves (2D simulation). Byusing the numerical mean diffusivity MD to compared with the MD found fromthe experimental data will be extracted the unknown parameter like intrinsic dif-fusivity (Da) of the nerves.

3. Short time ADC

A well-known approximation for the ADC in the short time regime is thefollowing [41, 42]:

ADCshort = D0

(1− 4

√D0

3 dim√π

√∆S

V

), (5)

whereS

Vis the surface to volume ratio. In the above formula the pulse duration

δ is assumed to be very small compared to ∆. A recent correction to the aboveformula taking into account the finite pulse duration δ [43] is the following:

ADCshort = D0

[1− 4

√D0

3 dim√πCδ,∆

S

V

], (6)

where

Cδ,∆ =4

35

(∆ + δ)7/2 + (∆− δ)7/2 − 2(δ7/2 + ∆7/2

)δ2 (∆− δ/3)

=√

∆

(1 +

1

3

δ

∆− 8

35

(δ

∆

)3/2

+ · · ·

).

When δ ∆, the value Cδ,∆ becomes√

∆.For spheres (three dimensions) or disks (two dimensions), the surface to vol-

ume ratioS

Vis

dimR

, where R is the radius. From the experimental ADC formultiple diffusion times we can fit the experimental ADC as the linear function of√

∆ or Cδ,∆, and the radius can be estimated.

11

4. Results

4.1. Time dependent dMRI signal and ADC in four types of cellsThe experimental dMRI signals for multiple diffusion times for some repre-

sentative ROIs are shown in Fig. 5 for the four types of cells. It is clear in therange of b-values shown, the logarithm of the signal is a straight line as a functionof b-value, meaning that the ADC is sufficient to describe the signal in this range,higher order effects such as a Kurtosis term need not to be considered.

0 200 400 600

9

9.5

10

10.5

b value (s/mm2)

log(s

ignal)

Giant cell neurons: Dc=0.80, Dn=1.80

(a)

0 200 400 6009

9.5

10

10.5

b value (s/mm2)

log(s

ignal)

Big cell neurons: Dc=0.80, Dn=1.90

(b)

0 200 400 6009

9.5

10

10.5

b value (s/mm2)

log(s

ignal)

Bag cell neurons: Dc=0.65, Dn=1.85

(c)

0 200 400 600

8

8.5

9

9.5

10

10.5

b value (s/mm2)

log(s

ignal)

Nerve cell: Da = 1.70

(d)

Fig. 5: The dMRI signals for multiple diffusion times in: giant cell neurons (5a), big cell neurons(5b), bag cell neurons (5c), and nerve cell neurons (5d). In each figure, from top to bottom repre-sent the signals for ∆ from 5 ms to 25 ms, respectively. The marker represents the experimentaldMRI signal, the dashed line represents the simulated dMRI signal corresponding with the intrin-sic diffusion coefficient Dc,Dn,Da (under units of µm2/ms) presented on the title. The differentcolor represents the different gradient direction, red, blue, black for x, y, z directions, respectively.

12

The time-dependent measured ADC measured over the ROIs of the four celltypes are shown in Table 2 and Fig. 6. We found that, when increasing the diffu-sion time (δ = 2.5 ms, ∆ from 5 to 25 ms), the mean of ADC were observed todrop by 9.5% in big cell neurons; by 2.5% in giant cell neurons; drop by 21.4%in bag cell neurons; the mean of diffusivity (MD) in nerve cells was observed todrop by 15.01%.

∆ (ms)Giant cell

neurons (N=8)Big cell

neurons (N=13)Bag cell

neurons(N=16)Nerve cell

neurons (N=10)5 0.945± 0.056 0.968±0.085 0.773±0.065 0.851±0.068

7.5 0.927±0.062 0.948±0.076 0.748±0.054 0.829±0.08010 0.934±0.058 0.946±0.070 0.697±0.059 0.786±0.07512 0.942±0.067 0.915±0.079 0.676±0.078 0.779±0.08215 0.927±0.060 0.908±0.068 0.656±0.055 0.756±0.07620 0.926±0.057 0.889±0.072 0.631±0.048 0.744±0.07125 0.918±0.063 0.877±0.078 0.615±0.046 0.725±0.069

ADC drop 2.5% 9.5% 21.4% 15%

Table 2: The experimental ADCs of four different cell types. When increasing the gradient du-ration time, ∆, from 5 to 25 ms, the ADCs were observed to drop by about 2.5%, 9.5%, 21.4%,and 15% for giant cell neurons, big cell neurons, bag cell neurons and nerve cell neurons ROIs,respectively. The results are presented as mean ± SD. The units of ADC value is µm2/ms.

13

2 3 4 50.5

0.6

0.7

0.8

0.9

1

1.1

1/2(ms )

AD

C(µ

m2/m

s)

Fig. 6: The measured diffusion coefficients are shown as colored marker (mean) with error bars(standard deviation) of experimental ADC. The solid lines are shown the linear fit of the measureddata. The bag cell neurons ROIs (squared), the big cell neurons ROIs (circle), the giant cell neuronsROIs (left-pointing triangle), and the nerve ROIs (diamond).

4.2. Comparison with simulationsBy comparing the time-dependent simulated ADC with the experimental ADC,

we extracted the following parameters Dc,Dn for cytoplasm and nucleus intrin-sic diffusivities, respectively,Da for axon intrinsic diffusivity, andDs for satellitecell. For the giant cell neurons we found intrinsic diffusivities ofDc = 0.70−0.80µm2/ms and Dn = 1.65 − 1.80e − 3 µm2/ms for the cytoplasm and nucleus,respectively and a 5% volume fraction of satellite cells with intrinsic diffusiv-ity of approximately Ds = 2.00e − 3 µm2/ms. For the big cell neurons wefound: without satellite cell neurons, intrinsic diffusivities of Dc = 0.70 − 0.75µm2/ms and Dn = 1.85 − 2.00 µm2/ms for cytoplasm and nucleus, respec-tively; with up to 5% percentage satellite cell neurons, intrinsic diffusivities ofDc = 0.70 − 0.90 µm2/ms and Dn = 1.80 − 2.00 µm2/ms for cytoplasm andnucleus, respectively and a 5% volume fraction of satellite cells with intrinsic dif-fusivity of approximately 2.00 µm2/ms. For the bag cell neurons: intrinsic diffu-sivities Dc = 0.65− 0.75 µm2/ms and Dn = 1.60− 1.95 µm2/ms for cytoplasmand nucleus, respectively. For nerves we found depending on the distribution ofthe axons types in the nerve, the intrinsic diffusivity ofDa = 1.35−1.70 µm2/msor Da = 1.65− 1.95 µm2/ms. The results were summarized in Fig. 7. Giant cellneurons: The simulation were ran over the parameters Dc = [0.65 : 0.05 : 1.00]

14

Giant cell

Big cellBag cell

Nerve cell0.6

0.7

0.8

0.9

1.6

1.7

1.8

1.9

2

Fig. 7: The intrinsic diffusion coefficient extracted from simulation for giant cell neurons, big cellneurons, bag cell neurons and nerve cell. The top represents theDn for giant cell neurons, big cellneurons and bag cell neurons, and Da for the nerve cell. The bottom represents the Dc. The unitof y-axis is µm2/ms.

µm2/ms for cytoplasm and Dn = [1.60 : 0.05 : 2.00] µm2/ms for nucleus andintrinsic diffusivity Ds = [0.80 : 0.05 : 2.00] µm2/ms in satellite cell. The bestfitting between ADC from simulation with and without small percentage satellitecell volume fraction shown in Fig. 8.We found that, without satellite cell simula-tion, the intrinsic diffusivity best fitting in range Dc = 0.65 − 0.75 µm2/ms andDn = 1.85 − 2.00 µm2/ms for cytoplasm and nucleus respectively; with smallvolume fraction of satellite cells (up to 5%), the intrinsic diffusivity best fitting inrange Dc = 0.70− 0.90 µm2/ms and Dn = 1.80− 2.00 µm2/ms for cytoplasmand nucleus respectively, and free diffusivity (D0 = 2.00 µm2/ms) for satellitecell.

15

2 3 4 50.5

0.6

0.7

0.8

0.9

1

1.1

AD

C(µ

m2/m

s)

(ms1/2

)2 3 4 5

0.5

0.6

0.7

0.8

0.9

1

1.1

(ms1/2

)

AD

C(µ

m2/m

s)

Fig. 8: The experimental ADC of giant cell neurons (marker with errors bar as the standard de-viation) and the best simulation fitting ADC (dashed line) of equivalent radius from 160 − 200µm without (left) and with (right) satellite cells. Left: Dc = 0.70 − 0.75 µm2/ms andDn = 1.65 − 1.80 µm2/ms for cytoplasm and nucleus, respectively. Right: Dc = 0.75 − 0.80µm2/ms and Dn = 1.65− 1.70 µm2/ms for cytoplasm and nucleus respectively, and D0 = 2.00µm2/ms for satellite cell.

Big cell neurons: The simulation were ran with Dc = [0.65 : 0.05 : 1.00]µm2/ms for cytoplasm and Dn = [1.60 : 0.05 : 2.00] µm2/ms for nucleus.The simulation results can be add up to 5% volume fraction of satellite cells withradius from 2−4 µm without nucleus inside and intrinsic diffusivityDs = [0.80−2.00] µm2/ms. The best fitting between ADC from simulation with and withoutsmall percentage satellite cells volume fraction shown in Fig. 9. We found that,without satellite cell, the intrinsic diffusivity best fitting in rangeDc = 0.65−0.75µm2/ms and Dn = 1.85− 2.00 µm2/ms for cytoplasm and nucleus respectively;with up to 5% satellite cells volume fraction, the intrinsic diffusivity best fitting inrange Dc = 0.70− 0.90 µm2/ms and Dn = 1.80− 2.00 µm2/ms for cytoplasmand nucleus respectively, and free diffusivity (Ds = 2.00 µm2/ms) for satellitecell.

Bag cell neurons: The simulation were ran with Dc = [0.50 : 0.05 : 1.00]µm2/ms for cytoplasm and Dn = [1.30 : 0.05 : 2.00] µm2/ms for nucleus. Thesimulation results were shown in Fig. 10. The best intrinsic diffusivities for bagcell neurons are Dc = 0.65 − 0.75 µm2/ms and Dn = 1.60 − 1.95 µm2/ms forcytoplasm and nucleus, respectively.

Nerve cells: The simulation ran over the range of Da = [0.85 : 0.05 : 2.00]µm2/ms. The best fits between the MDs from the numerical simulation and thosefrom experimental data of the nerve are shown in Fig. 11. In Table 3, shownthe range of intrinsic diffusivity which ones good fitting results of some kind of

16

2 3 4 50.5

0.6

0.7

0.8

0.9

1

1.1

(ms1/2)

ADC(µm2/ms)

2 3 4 50.5

0.6

0.7

0.8

0.9

1

1.1

ADC(µm2/ms)

(ms1/2)

Fig. 9: The experimental ADC of big cell neurons (marker with errors bar as the standard devi-ation) and the best simulation fitting ADC (dashed line) of equivalent radius from 80 − 150 µmwithout (left) and with (right) glial cells. Left: Dc = 0.65− 0.75 µm2/ms and Dn = 1.85− 2.00µm2/ms for cytoplasm and nucleus respectively, without satellite cell. Right: Dc = 0.70 − 0.90µm2/ms and Dn = 1.80− 2.00 µm2/ms for cytoplasm and nucleus respectively, and Ds = 2.00µm2/ms for satellite cell.

the nerves and the mean drop of MDs simulation from these fitting parameters.As mentioned above, the size and structure of the abdominal siphon nerve andleft/right connective nerves more precise closer to the buccal 1b, 2b, 3b and cbcnerves than on, rn nerves. Which this assume and from Table 3, we can extractedthe intrinsic diffusivity for the abdominal siphon or left/right connectivity nervesby about 1.35− 1.70 µm2/ms or 1.65− 1.95 µm2/ms depending on the structureof the nerves.

17

2 3 4 50.5

0.6

0.7

0.8

0.9

1

1.1

1/2

AD

C(µ

m2/m

s)

(ms )

Fig. 10: The experimental ADC of bag cell neurons (marker with errors bar as the standard devi-ation) and some good fitting simulation of ADC (dashed line). Dc = 0.65 − 0.75 µm2/ms andDn = 1.60− 1.95 µm2/ms for cytoplasm and nucleus, respectively.

NerveD0 fitting Avg. drop

MD (%)min max best1b 1.95 2.00 2.00 12.462b 1.65 1.95 1.80 12.483b 1.35 1.60 1.50 16.75on 1.85 2.00 2.00 14.87rn 1.80 2.00 2.00 18.73

cbc 1.45 1.70 1.60 18.76

Table 3: The simulation results for some nerves (1b, 2b, 3b, on, rn, cbc). There are four type ofaxons with different range of axons diameter: type I (> 25µm), II (10−25µm), III(1−10µm) andIV(≤ 1µm). The distribution of axons type of each nerve [22] were shown in columns second tofifth. The intrinsic diffusivity Da (µm2/ms) range which one good fitting of the mean diffusivity(MD) with the experimental MD were shown in columns sixth to eighth. The squared errors of thebest fitting MD described in column ninth. The last column shown the mean of the drop of MD.

18

2 3 4 50.5

0.6

0.7

0.8

0.9

1

1.1

AD

C(µ

m2/m

s)

(ms1/2

)

Fig. 11: The experimental MD of nerves (marker with errors bar as the standard deviation) andthe best fitting MD of some nerve types. Nerve 1b (cyan, D0 = 2.0µm2/ms), nerve 2b (blue,D0 = 1.8µm2/ms), nerve 3b (red, D0 = 1.5µm2/ms), nerve on (green, D0 = 2.0µm2/ms),nerve rn (pink, D0 = 2.0µm2/ms) and nerve cbc (blue violet, D0 = 1.6µm2/ms)

4.3. Fitting short time ADC formula for S/VFrom experimental ADC, by fitting ADC = A

√∆ + B (corresponding with

formula (5)) or ADC = ACδ,∆ + B (corresponding with formula (6)), we can

find A and B coefficients. By compared with short time ADC formula forS

Vfrom

both equation (5) and (6), we found that A = −D04√D0

3√π

1

R, and B = D0. It

implies that, R = −D04√D0

3√π

1

A. The estimated cell radius R for the three types

of cells are shown in table 4, when the cells shape are supposed spherical.

Giant cell neurons Big cell neurons Bag cell neuronsReal cell radius >160 µm 75-150 µm 10-50 µmR/D0 estimated

by Mitra formulaR = 93.94D0 = 0.96

R = 24.11D0 = 1.05

R = 10.78D0 = 0.896

R/D0 estimatedby Simona formula

R = 89.95D0 = 0.96

R = 23.44D0=1.058

R = 10.68D0 = 0.918

Table 4: The cell radius and intrinsic diffusivities estimated by using Mitra formula (5) and Simonaformula (6). The units of R is µm and the units of D0 is µm2/ms.

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5. Discussion

From the approximation cell’s radius R results above using the short diffusiontime formula, by compared approximation cell’s radius with the real range of cellradius, we found that the approximation cell’s radius are too smaller than the realrange of cell radius. It gives the idea that the single sphere as approximation ofthe large cell neurons is not a good model for dMRI simulation.

6. Conclusion

We have validated certain relationships between the cellular geometry and thedMRI signal in the Aplysia neuronal network for bag cell neurons, large singlecell neurons and nerves. We found necessary to include a nucleus region of highintrinsic diffusivity embedded in a cytoplasmic region with lower intrinsic diffu-sivity in order to fit the large drop in ADC observed when varying the diffusiontime from 5ms to 25ms. Further work combining simulations and experimenta-tion in the Aplysia is needed to make the relationship between the geometry andthe dMRI signal more precise.

Acknowledgement

This work was funded by grant ANR-13-BSV5-0014-01 (project ANImE) andby the doctoral school EOBE, University Paris Sud, XI, 91405 Orsay, France.

References

[1] Y. Assaf, P. J. Basser, Composite hindered and restricted model of diffusion(charmed) MR imaging of the human brain, NeuroImage 27 (1) (2005)48 – 58. doi:http://dx.doi.org/10.1016/j.neuroimage.2005.03.042.URL http://www.sciencedirect.com/science/article/pii/S1053811905002259

[2] Y. Assaf, T. Blumenfeld-Katzir, Y. Yovel, P. J. Basser, Axcaliber: A methodfor measuring axon diameter distribution from diffusion mri, Magnetic Res-onance in Medicine 59 (6) (2008) 1347–1354. doi:10.1002/mrm.21577.URL http://dx.doi.org/10.1002/mrm.21577

[3] H. Zhang, P. L. Hubbard, G. J. Parker, D. C. Alexander, Axondiameter mapping in the presence of orientation dispersion withdiffusion MRI, NeuroImage 56 (3) (2011) 1301 – 1315.

20

doi:http://dx.doi.org/10.1016/j.neuroimage.2011.01.084.URL http://www.sciencedirect.com/science/article/pii/S1053811911001376

[4] D. C. Alexander, P. L. Hubbard, M. G. Hall, E. A. Moore, M. Ptito, G. J.Parker, T. B. Dyrby, Orientationally invariant indices of axon diameter anddensity from diffusion MRI, NeuroImage 52 (4) (2010) 1374 – 1389.doi:http://dx.doi.org/10.1016/j.neuroimage.2010.05.043.URL http://www.sciencedirect.com/science/article/pii/S1053811910007755

[5] E. Fieremans, J. H. Jensen, J. A. Helpern, White matter characterizationwith diffusional kurtosis imaging, NeuroImage 58 (1) (2011) 177 – 188.doi:http://dx.doi.org/10.1016/j.neuroimage.2011.06.006.URL http://www.sciencedirect.com/science/article/pii/S1053811911006148

[6] E. Panagiotaki, T. Schneider, B. Siow, M. G. Hall, M. F. Lythgoe, D. C.Alexander, Compartment models of the diffusion MR signal in brainwhite matter: A taxonomy and comparison, NeuroImage 59 (3) (2012) 2241– 2254. doi:http://dx.doi.org/10.1016/j.neuroimage.2011.09.081.URL http://www.sciencedirect.com/science/article/pii/S1053811911011566

[7] S. N. Jespersen, C. D. Kroenke, L. stergaard, J. J. Ackerman, D. A. Yablon-skiy, Modeling dendrite density from magnetic resonance diffusion measure-ments, NeuroImage 34 (4) (2007) 1473 – 1486.

[8] O. Dietrich, A. Hubert, S. Heiland, Imaging cell size and permeability in bi-ological tissue using the diffusion-time dependence of the apparent diffusioncoefficient, Physics in Medicine and Biology 59 (12) (2014) 3081.

[9] I. Lavdas, K. C. Behan, A. Papadaki, D. W. McRobbie, E. O. Aboagye, Aphantom for diffusion-weighted mri (dw-mri), Journal of Magnetic Reso-nance Imaging 38 (1) (2013) 173–179. doi:10.1002/jmri.23950.URL http://dx.doi.org/10.1002/jmri.23950

[10] J. S. Campbell, K. Siddiqi, V. V. Rymar, A. F. Sadikot, G. B. Pike, Flow-based fiber tracking with diffusion tensor and q-ball data: Validation andcomparison to principal diffusion direction techniques, NeuroImage 27 (4)(2005) 725–736.URL http://www.sciencedirect.com/science/article/pii/S1053811905003411

[11] A. Moussavi-Biugui, B. Stieltjes, K. Fritzsche, W. Semmler, F. B.Laun, Novel spherical phantoms for q-ball imaging under in vivo

21

conditions, Magnetic Resonance in Medicine 65 (1) (2011) 190–194.doi:10.1002/mrm.22602.URL http://dx.doi.org/10.1002/mrm.22602

[12] P. Pullens, A. Roebroeck, R. Goebel, Ground truth hardware phantoms forvalidation of diffusion-weighted mri applications, Journal of Magnetic Res-onance Imaging 32 (2) (2010) 482–488. doi:10.1002/jmri.22243.URL http://dx.doi.org/10.1002/jmri.22243

[13] K. Schilling, V. Janve, Y. Gao, I. Stepniewska, B. A. Landman, A. W.Anderson, Comparison of 3d orientation distribution functions measuredwith confocal microscopy and diffusion MRI, NeuroImage 129 (2016)185 – 197. doi:http://dx.doi.org/10.1016/j.neuroimage.2016.01.022.URL http://www.sciencedirect.com/science/article/pii/S1053811916000288

[14] I. O. Jelescu, M. Zurek, K. V. Winters, J. Veraart, A. Rajaratnam, N. S.Kim, J. S. Babb, T. M. Shepherd, D. S. Novikov, S. G. Kim, E. Fieremans,In vivo quantification of demyelination and recovery using compartment-specific diffusion MRI metrics validated by electron microscopy, NeuroIm-age 132 (2016) 104 – 114.

[15] S. N. Jespersen, L. A. Leigland, A. Cornea, C. D. Kroenke, Determination ofaxonal and dendritic orientation distributions within the developing cerebralcortex by diffusion tensor imaging, IEEE Transactions on Medical Imaging31 (1) (2012) 16–32. doi:10.1109/TMI.2011.2162099.

[16] A. R. Khan, A. Cornea, L. A. Leigland, S. G. Kohama, S. N. Jespersen,C. D. Kroenke, 3d structure tensor analysis of light microscopy datafor validating diffusion MRI, NeuroImage 111 (2015) 192 – 203.doi:http://dx.doi.org/10.1016/j.neuroimage.2015.01.061.URL http://www.sciencedirect.com/science/article/pii/S1053811915000889

[17] H. Wang, C. Lenglet, T. Akkin, Structure tensor analysis of serial opti-cal coherence scanner images for mapping fiber orientations and tractog-raphy in the brain, Journal of Biomedical Optics 20 (3) (2015) 036003.doi:10.1117/1.JBO.20.3.036003.URL http://dx.doi.org/10.1117/1.JBO.20.3.036003

[18] M. D. Does, E. C. Parsons, J. C. Gore, Oscillating gradient measurements ofwater diffusion in normal and globally ischemic rat brain, Magnetic Reso-nance in Medicine 49 (2) (2003) 206–215.

22

[19] E. O. Stejskal, J. E. Tanner, Spin diffusion measurements: Spin echoes in thepresence of a timedependent field gradient, The Journal of Chemical Physics42 (1) (1965) 288–292.

[20] E. R. Kandel, , I. Kupfermann, The functional organization of inverte-brate ganglia, Annual Review of Physiology 32 (1) (1970) 193–258, pMID:4906119.

[21] P. Conn, L. Kaczmarek, The bag cell neurons of aplysia. a model for thestudy of the molecular mechanisms involved in the control of prolonged an-imal behaviors, Molecular neurobiology 3 (4) (1989) 237273.

[22] C. Musio, C. Bedini, Fine structure and axonal organization in the buccalganglia nerves ofaplysia (mollusca, gastropoda), Zoomorphology 110 (1)(1990) 17–26.

[23] I. Kupfermann, T. J. Carew, E. R. Kandel, Local, reflex, and central com-mands controlling gill and siphon movements in aplysia., Journal of Neuro-physiology 37 (5) (1974) 996–1019.

[24] C. H. Lee, J. J. Flint, B. Hansen, S. J. Blackband, Investigation of the subcel-lular architecture of L7 neurons of aplysia californica using magnetic reso-nance microscopy (MRM) at 7.8 microns, Scientific Reports 5 (2015) 11147EP –, article.

[25] J. E. Blankenship, R. E. Coggeshall, The abdominal ganglion of aplysiabrasiliana: A comparative morphological and electrophysiological study,with notes on a. dactylomela, Journal of Neurobiology 7 (5) (1976) 383–405.

[26] R. E. Coggeshall, A light and electron microscope study of the abdominalganglion of Aplysia californica, J. Neurophysiol. 30 (6) (1967) 1263–1287.

[27] W. T. Frazier, E. R. Kandel, I. Kupfermann, R. Waziri, R. E. Coggeshall,Morphological and functional properties of identified neurons in the ab-dominal ganglion of Aplysia californica, Journal of Neurophysiology 30 (6)(1967) 1288–1351.

[28] J. T. Haskins, C. H. Price, J. E. Blankenship, A light and electron micro-scopic investigation of the neurosecretory bag cells of Aplysia, J. Neurocy-tol. 10 (5) (1981) 729–747.

23

[29] I. O. Jelescu, R. Nargeot, D. L. Bihan, L. Ciobanu, Highlighting manganesedynamics in the nervous system of aplysia californica using MEMRI at ultra-high field, NeuroImage 76 (0) (2013) 264 – 271.

[30] G. Radecki, R. Nargeot, I. O. Jelescu, D. Le Bihan, L. Ciobanu, Functionalmagnetic resonance microscopy at single-cell resolution in Aplysia califor-nica, Proceedings of the National Academy of Sciences 111 (23) (2014)8667–8672.

[31] L. Lu, B. Erokwu, G. Lee, V. Gulani, M. A. Griswold, K. M. Dell, C. A.Flask, Diffusion-prepared fast imaging with steady-state free precession(DP-FISP): A rapid diffusion mri technique at 7 T, Magnetic Resonance inMedicine 68 (3) (2012) 868–873.

[32] D. Le Bihan, J.-F. Mangin, C. Poupon, C. A. Clark, S. Pappata, N. Molko,H. Chabriat, Diffusion tensor imaging: Concepts and applications, Journalof Magnetic Resonance Imaging 13 (4) (2001) 534–546.

[33] S. Mori, J. Zhang, Diffusion Tensor Imaging (DTI) , in: L. R. Squire (Ed.),Encyclopedia of Neuroscience, Academic Press, Oxford, 2009, pp. 531 –538.

[34] P. Basser, J. Mattiello, D. LeBihan, MR diffusion tensor spectroscopy andimaging, Biophysical Journal 66 (1) (1994) 259 – 267.

[35] P. B. Kingsley, Introduction to diffusion tensor imaging mathematics: Part I.tensors, rotations, and eigenvectors, Concepts in Magnetic Resonance PartA 28A (2) (2006) 101–122.

[36] S.-K. Song, S.-W. Sun, M. J. Ramsbottom, C. Chang, J. Russell, A. H. Cross,Dysmyelination revealed through MRI as increased radial (but unchangedaxial) diffusion of water, NeuroImage 17 (3) (2002) 1429 – 1436.

[37] C. A. Wheeler-Kingshott, M. Cercignani, About axial and radial diffusivi-ties, Magnetic Resonance in Medicine 61 (5) (2009) 1255–1260.

[38] H. C. Torrey, Bloch equations with diffusion terms, Phys. Rev. 104 (1956)563–565.

[39] J.-R. Li, D. Calhoun, C. Poupon, D. L. Bihan, Numerical simulation ofdiffusion MRI signals using an adaptive time-stepping method, Physics inMedicine and Biology 59 (2) (2013) 441–454.

24

[40] D. L. Bihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis, M. Laval-Jeantet, MR imaging of intravoxel incoherent motions: application to diffu-sion and perfusion in neurologic disorders., Radiology 161 (2) (1986) 401–407.

[41] P. P. Mitra, P. N. Sen, L. M. Schwartz, P. Le Doussal, Diffusion propagatoras a probe of the structure of porous media, Phys. Rev. Lett. 68 (1992) 3555–3558.

[42] P. P. Mitra, P. N. Sen, L. M. Schwartz, Short-time behavior of the diffusioncoefficient as a geometrical probe of porous media, Phys. Rev. B 47 (1993)8565–8574.

[43] S. Schiavi, H. Haddar, J.-R. Li, Correcting the short time ADC formula toaccount for finite pulses, ISMRM Workshop, 2016.

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List of changes

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