Bloodstain Pattern Analysis: implementation of a fluid dynamic model for position determination of victims

Bloodstain Pattern Analysis is a forensic discipline in which, among others, the position of victims can be determined at crime scenes on which blood has been shed. To determine where the blood source was investigators use a straight-line approximation for the trajectory, ignoring effects of gravity and drag and thus overestimating the height of the source. We determined how accurately the location of the origin can be estimated when including gravity and drag into the trajectory reconstruction. We created eight bloodstain patterns at one meter distance from the wall. The origin’s location was determined for each pattern with: the straight-line approximation, our method including gravity, and our method including both gravity and drag. The latter two methods require the volume and impact velocity of each bloodstain, which we are able to determine with a 3D scanner and advanced fluid dynamics, respectively. We conclude that by including gravity and drag in the trajectory calculation, the origin’s location can be determined roughly four times more accurately than with the straight-line approximation. Our study enables investigators to determine if the victim was sitting or standing, or it might be possible to connect wounds on the body to specific patterns, which is important for crime scene reconstruction.


Blood (dried)
-1274±3 ---We used human blood which was obtained with the same procedure as 8 . For each blood sample collected, the hematocrit (Hct) value (the percentile amount of red blood cells in blood) was determined by means of a capillary centrifuge (Haematokrit 210, Hittich Zentrifugen, germany). The density of liquid blood (ρ blood ) was determined by means of weighing a (empty and blood filled) volumetric flask of 10 ml (Hirschmann, Germany). In addition, the density of dried blood was measured using a gas expansion pycnometer (Micromeritics Multivolume Pycnometer 1305, USA). The viscosity of blood was determined by using a rheometer (MCR 302, Anton Paar, Austria) with a cone and plate geometry, using a shear rate sweep, (1 s -1 < ̇ < 1000 s -1 ). Blood is shear thinning but viscosity reaches a plateau value for high shear rates 9 , see Figure S1. The high shear rate viscosity η ∞ was determined by fitting the phenomenological equation: where k [Pa·s n ] and n are fit parameters. Figure S1 | Viscosity as a function of shear rate, ̇, of blood at 22 °C (circles). For high shear rates, the viscosity reaches a constant, which was determined by means of fitting (̇) = ∞ + •̇− 1 to the data points (red line), from which we obtain a high shear rate viscosity of η ∞ = 4.8 mPa·s; k = 0.048 mPa·s n and n = 0.31 are fitting constants.
To determine the hematocrit dependency for the drying ratio, blood with varying hematocrit values were required. To do so, multiple blood samples (4 ml) were obtained from one volunteer. The samples were centrifuged to separate the plasma from the red blood cells.
Accordingly, plasma was drawn from one sample, thus effectively increasing the Hct value and added to another sample, decreasing its Hct value. In this manner, nine blood samples with Hct values between 23% and 54% were created.

S2. Volume estimation
To determine the impact velocity of the droplet we require the original droplet volume 5-7,30 .
After a bloodstain has dried, it leaves a residue, consisting of dried red blood cells, the amount of which is dependent on the hematocrit value 8 . It is possible to deduce the original volume of a bloodstain by measuring the volume of the dried stain and divide it by the volume ratio of dried and fresh blood. To measure the volume we use a 3D surface scanner.

Material and methods
The AreaScan3D (VRMagic, Germany) records the deformation of a regular projected light pattern. The scanner consists of a camera which is positioned perpendicular to the surface, at a distance of 25 cm. Next to the camera there is a projector, which is inclined with respect to the surface under an angle of approximately 80°, which projects several light patterns onto the surface. Due to the inclination and height variations of the surface, the camera records distorted lines from which height information can be obtained. The scanner has a lateral range of 1.8 by 1.2 cm (748x480 pixels) with step sizes of 24.064 µm/pixel and 25 µm/pixel, respectively. The axial range is one centimeter, with a height resolution in the order of a micrometer. Each scan is in the form of a point cloud with x, y and z coordinates. By means of a software program (written in Java), we select the object (bloodstain) based on the height difference of the object relative to the surface, from which we directly determined the volume. Figure S2 shows a single bloodstain from an impact pattern together with the 3D scan ( Figure   S2b, intensity graph). In addition, from the cross-section of the bloodstain ( Figure S2c) it is possible to see the difference between the irregular surface and the bloodstain.
Surface irregularities are taken into account by determining the mean in height deviations over a large area of the surface, without the object. The total volume V tot is determined by selecting the object and accumulating the height differences with respect to the surface. The selected area is multiplied with the mean of the surface irregularities which results in V surface .
The volume of the object V netto is determined by subtracting V surface from V tot . To validate the volume measurements using the AreaScan3D, we created small aluminum cylinders with predefined volumes, ranging from 1 to 140 µl, as a reference. The volume obtained from these scans we compared to the volume obtained from the weight and density (2701 kg/m 3 ) 31 of the aluminum cylinders. In addition, we created bloodstains of different sizes (5 to 50 µl) by means of a pipette, and bloodstains (7 µl) by means of a needle and deposited from several heights (10 to 100 cm). Each bloodstain was created on a separate surface and weighed (immediately after deposition and after drying) to determine the volume by means of the density of blood (table S1), to compare this to the volume obtained from the AreaScan3D. After creation of a single bloodstain or a bloodstain pattern, the blood was dried for a minimum of three hours. After this time period the blood was completely dried after which a 3D scan was made of the bloodstains.

Results
There is a very high correlation between the volume measured by means of the AreaScan3D and the volume obtained from the weight/density measurement, of the aluminum cylinders ( Figure S3). By fitting a line to the data points (y = 0.979 · x), we see there is only a small variation of 2.1% between the measured volumes. It is clear that the AreaScan3D has a very high accuracy for determining volumes. However, the question remains if we are able to determine the volume of bloodstains just as accurately. By plotting the dried weight of the bloodstains as a function of the relative dried volume ( Figure S4) we are able to determine the density of the dried bloodstains (1546 kg/m 3 ) as measured by our system. However, the actual density of dried blood is lower, namely 1274 kg/m 3 . The volume we measure with the AreaScan3D is thus too low. However, this density deviation is a systematic error, which we can account for by means of a calibration curve (Figure 2, main article).
The systematic error of the AreaScan3D is caused by light absorption of the blood. The AreaScan3D projects white light onto the sample and captures the diffuse reflectance with a camera. Blood itself is highly absorbent in the white light range 32 . Accordingly, less light is reflected back into the camera and loss of information occurs, which causes the measured systematic error for which we are able to correct with our calibration curve.

S3. Trajectory reconstruction
We made eight bloodstain impact patterns at a distance of one meter from the wall at a height of 63.7 cm, using a hammer on a spring setup, described in de Bruin et al. 4  Eq. 2 can be solved for the motion in the x, y and z plane separately: y(t) = y 0 + v y t with v y = v 0 sin cos (S4) with impact angle α, the directional angle γ and the impact velocity v 0 .
To take drag into account the equations of motion have to be solved numerically as they cannot be solved analytically. The trajectories of the blood droplets were modeled as described in Kabaliuk et al. 19 using a Lagrangian approach for Newton's second law of motion:  33 . For this study, the blood drops were modeled as solid spheres (no deformation and a constant drag coefficient equal to 0.5). This model was found to be the most suited for this practical situation.

Point of Origin
The intersections between the trajectories were determined within the x-y plane ( Figure S5

Region of origin
The region of origin is determined by taking the standard deviation into account, which was determined from the smallest distance between the point of origin and each trajectory. The standard deviation given in Eq. 4 (main article) is determined by means of the following method. First of all, the PO is determined by means of Eq. S10. Next we define the distance between the PO and each trajectory (index p) as a function of time.
( ) = √(PO ( ) − ( )) 2 + (PO( ) − ( )) 2 + (PO( ) − ( )) 2 S11 here x p (t), y p (t) and z p (t) are the time dependent trajectories of the droplets given by equations S3, S4 and S5, respectively. To determine the shortest distance (for the straightline method and gravity included method) we solve the time derivative for t of Eq. S11 equaling zero, yielding a t min , which is filled into Eq. S11 for each trajectory. For the method including both gravity and drag it is not possible to determine the derivative of Eq. S11 because the trajectories are numerically solved. Thus, Eq. S11 is directly numerically calculated by determining S p (t) for each time step (Δt=0.1 ms). Accordingly, the shortest distance of each trajectory is determined. Next, the standard deviation of the minimum distances is calculated for each pattern.
where L is the number of trajectories.

S5. Supplementary video
The supplementary video shows a high speed recording of a hammer striking a volume of liquid blood (~2 ml). The video was recorded with 3000 frames per second, using a high speed camera (Photron). As the hammer hits the blood, sheets and ligaments of blood disperse away from the impact area, which in turn, breaks up in droplets several centimeters away from the impact.