now will be explained the methodology for calculating the multiphasic solution for achieving a uniform vascular enhancement . the distribution of contrast medium in a blood vessel depends on the circulating blood flow and blood volume of the vessel . although a whole body model provides a complete description of enhancement characteristic in each vessel and each organ , modeling with a limited number of compartments is less complex and more approachable for theoretical analysis of various injection parameters . there are various ways to model body compartments . an optimal model is the one that uses the smallest number of compartments , but adequately describes the underlying pharmacokinetic process . an approach frequently used in prior art studies of drug distribution is to model the whole body with two compartments , whereby contrast medium is introduced into a central plasma compartment , distributed to a peripheral extracellular compartment , and then eliminated from the central plasma compartment by renal excretion . although this scheme is sufficient for describing the late pharmacokinetics of contrast medium ( hours ), it needs further refinement to be applied to the description of early pharmacokinetics ( minutes ). fig2 shows a compartment model which is designed to simulate early contrast enhancement in the aorta . in this model , contrast medium is injected into the antecubital vein and distributed to the right heart , the pulmonary compartment , the left heart , and the aorta . it then recirculates back to the right heart via the systemic circulation . this transport scheme is specifically simplified to focus on early pharmacokinetics of the aortic contrast enhancement , thus reducing the complexity of our analysis . for example , a constant elimination of contrast medium from the central blood compartment by renal excretion ( transport to urine ) is only substantial in late pharmacokinetics and thus not considered in this simple compartment model . the model in fig2 is described mathematically as follows . let cv , cr , cp , cl , and cs be the contrast concentrations in the peripheral vein ( from the antecubital to the right heart , right heart , pulmonary , left heart , and systemic circulation , respectively . vv , vr , vp , vl and vs represent the corresponding compartment ( blood and interstitial ) volumes of the peripheral vein , right heart , pulmonary , left heart , and systemic circulation , respectively . qv is the volumetric flow rate of blood leaving the peripheral vein . qr , qp , ql , and qs are equivalent and represent the cardiac output of the system . cc and qc are the concentration and volumetric flow rate of injected contrast medium , respectively . during contrast injection , all the volumetric blood flow rates ( qv , qr , qp , ql , and qs ) are increased by qc . the governing equations for the model are written from mass balance equations for each compartment ( equations 1 - 5 in appendix a ). the aortic enhancement curves were computer simulated by numerically solving equations 1 - 6 in appendix a . the physiological parameters used in the model for humans include 40 ml for vv ( peripheral vein ), 250 ml each for vr ( right heart ) and for vl ( left heart ), 600 ml for vp ( pulmonary circulation ), and 10 l for vs ( systemic circulation ). associated volumetric blood flow rates are 250 ml / min or 4 . 2 ml / sec for qv and 6 . 5 l / min for the cardiac output . these values were estimated based on published human physiology data for a standard adult . to mimic channels of blood vessels , the peripheral venous compartment and pulmonary compartment are further divided into multiple smaller compartments in series ( 5 subcompartments for the peripheral venous compartment and 30 subcompartments for the pulmonary compartment ). since detailed cardiovascular physiologic data for porcine models are rather lacking compared to human models , the inventors rescaled the above human physiological parameters to determine the physiological parameters for the porcine model . the compartment volumes of the porcine model were estimated by multiplying the compartment volumes of a standard 70 kg human model by the body weight ratio , e . g . for a 25 kg pig , the ratio is 25 : 70 . it is known that the average cardiac output per body weight of pigs is twice as high as that of humans . therefore , the cardiac output for a 25 kg pig corresponds to that of a 50 kg human . although there is some subjectivity in selecting these parameters , they were estimated within available physiologic data and represent simply a set of reference values for simulation to compare with experimental data . a total of 38 ordinary differential equations were used to describe the model in fig2 . these equations were solved using numerical integration programs of fifth - order runge - kutta method . this model was run at a personal computer and took less than a fraction of a second to compute . the contrast concentration curve over time was calculated for each region by solving these differential equations for a given contrast injection protocol . after the contrast concentration in each compartment was computed by solving equations 1 - 6 , it was translated into a ct enhancement value . for a given input injection protocol , the mathematical model described above can be used to predict the output contrast enhancement curve of the aorta . conversely , the model can be used to solve the inverse problem , i . e . to predict an input function for a given output contrast enhancement profile . solving for an input contrast injection algorithm which will generate a prolonged , uniform vascular contrast enhancement is the focus of the present invention . the inverse problem can be solved directly by the laplace transform of governing equations in the model with a given desired constant aortic enhancement and initial conditions . mathematical manipulation for the solution is detailed in appendix a . this solution , i . e . a contrast injection profile , was in turn applied as an input to the mathematical model to simulate and verify the reproducibility of desired constant aortic contrast enhancement . simulation was performed for both porcine and human mathematical models by adjusting the physiological inout values . different injection profiles were tested to study how they affect aortic contrast enhancement . in addition , the effect of reduced cardiac output on the enhancement was investigated . the model was modified by decreasing the cardiac output by 20 % and 40 %. contrast enhancements were simulated in this model with the input injection which , when used in normal cardiac output , would produce a uniform contrast enhancement . the patterns of these enhancements were compared with that from normal cardiac output . in order to test the mathematical solutions , a porcine study was conducted . all animal care and procedures performed were approved by the institutional animal study committee . four pigs weighing initially 24 - 26 kg underwent scanning in two or three separate sessions . each session was separated by at least two days . two pigs had all their sessions within a week , while the other two pigs had their first two and last sessions delays 4 - 5 weeks , which results in an increase in their weight to 35 - 40 kg in their last session . in each session , the pig was anesthetized , intubated , and underwent scanning for three or four sets of images obtained in random order . during scanning , each pig was ventilated with oxygen and low tidal volume to minimize breathing motion artifact . each image set consisted of 27 dynamic ct sections ( 5 mm collimation ) acquired at a fixed mid - abdominal aortic level , following i . v . injection of contrast medium into a peripheral vein . each set of scans were 45 - 60 minutes apart to minimize the effect of prior contrast administration . all ct scanning was performed with a somatom plus - s scanner ( siemens medical systems , iselin , n . j .) using a one - second scanning time and a one - second interscan delay . three types of injection schemes were tested : uniphasic , biphasic , and multiphasic . biphasic injections were performed by a prior art power injector which was used in routine clinical ct scanning while uniphasic and multiphasic injections were conducted with a power injector which was invented for achieving the required protocols . this power injector was capable of delivering contrast medium in various uniphasic or multiphasic injection algorithms , as is explained in greater detail below . the multiphasic injection rate is determined by an initial injection rate and an exponential decay coefficient , as shown in fig3 . the total injected volume of contrast medium corresponds to the integrated sum of the multiphasic injection over injection duration . most injections were performed with the initial injection rate of 2 ml / s . volumes of contrast medium used were 50 , 70 and 90 ml of iothalmate meglumine ( conray 60 ; mallinckrodt medical , st . louis , mo . ; 282 mgi / ml ). three differential exponential decay coefficients ( 0 . 007 , 0 . 017 , 0 . 026 ) were tested . these coefficients were initially designed as ( 0 . 01 , 0 . 02 , 0 . 03 ), respectively , until further testing and verification revealed discrepancies between the design and actual values . these were the three smallest discrete increments allowed in the prototype of the inventive power injector . decay coefficient higher than 0 . 03 was not used because it was evident that a further increase in the coefficient would deviate further away from uniform vascular enhancements . a uniphasic injection was determined as being equivalent to a zero exponential decay coefficient where the injection rate remains constant at an initial injection rate throughout the injection duration . most extensively tested and compared injections were 50 ml total of contrast medium injected by a uniphasic injection of 2 ml / s and by a multiphasic injection of 2 ml / s initial rate having an exponential decay coefficient of 0 . 017 . the same injection methods were repeated but with an increased total contrast medium volume to 70 ml or with both increased injection rate to 3 ml / s and increased volume to 90 ml . other injections studied include biphasic injections of 50 ml ( 2 ml / s for 12 sec and then 1 . 4 ml / s for 18 sec ) and 70 ml of contrast medium ml ( 2 ml / s for 17 sec and then 1 . 0 ml / s for 36 sec ). approximately half of the total contrast volume was injected in each phase of the biphasic injections . the first and second injection rate of the biphasic injections were determined by the initial and final injection rates of the multiphasic injections with an exponential decay coefficient 0 . 017 of a corresponding total contrast medium volume , respectively . attenuation values of the aorta were measured from post - contrast scans ( at the same level as the pre - contrast scans ) using a circular region of interest ( roi ) at the center of the aorta . contrast enhancement was calculated as the absolute different in attenuation value between the pre - and post - contrast scans . for the data analysis , the injection duration ( id ), the magnitude of peak aortic enhancement ( pa ), and the uniformity of enhancement : the duration of the enhancement achieved with 90 % of the peak : 90 % dcf . were evaluated . means and standard deviations were also computed . the results are now explained . fig4 shows a simulated aortic enhancement curve generated from the model for a 25 kg pig with 50 ml of 282 mgi / ml contrast medium injected at a uniphasic 2 ml / sec . this curve was in good agreement with an empiric aortic enhancement curve observed in a 25 kg pig ( fig1 a ), including the time to and the magnitude of the peak aortic enhancement ( simulated vs . empiric : 28 vs . 26 sec and 234 vs . 250 hu ). these curves differed notably at the after - peak portion when the recirculation of contrast becomes substantial with the discontinuity of contrast injection . this portion was simplified in the model which mainly focused on the early part of the injection protocol , i . e . the first pass of contrast bolus pharmacokinetics . the contrast injection algorithm that provided a uniform , prolonged vascular enhancement was solved as shown in appendix a . the solution , i . e . contrast injection protocol is expressed as the product of an initial rejection rate and an exponential function of time , as shown in equation 15 . the exponential decay coefficient equals q / vs , the ratio of the cardiac output to the systematic volume of distribution of contrast medium , which is itself proportional to the cardiac output per body weight . fig5 a shows three exponential injection profiles with a 2 ml / s initial injection rate and decay coefficients ( 0 . 01 , 0 . 02 , and 0 . 03 ) for a 120 sec injection duration . the total amount of contrast medium in each injection is represented by the area under each curve . a lower exponential decay resulted in a higher total amount of contrast medium and a higher final injection rate at 120 sec . aortic contrast enhancement curves corresponding to these exponential injection profiles were simulated from the mathematical model ( with porcine physiological parameters ) by solving equations 1 - 6 and are depicted in fig5 b . uniform , plateau aortic enhancement was observed with an exponential decay constant of 0 . 02 ( q / vs = 77 / 3571 = 0 . 021 ). with decay coefficients 0 . 01 or 0 . 03 , contrast enhancement either steadily rises above this plateau level or declines after a peak below the plateau level , respectively . fig6 shows two simulated aortic enhancement curves for a human model using uniphasic or multiphasic injection protocols with 0 . 01 ( q / vs = 108 / 10000 ) exponential decay injections at an initial injection rate of 3 ml / s for a total of 160 ml of contrast medium . a prolonged , uniform contrast enhancement was achieved with the multiphasic injection protocol . notice that this exponential decay coefficient for the human model is approximately half that of the porcine model , reflecting the physiological values used in the model that the average cardiac output per body weight for humans is half that for pigs . the effect of reduced cardiac output on the enhancement was evaluated by reducing the cardiac output by 20 and 40 % in the model . the exponential injection with a decay coefficient 0 . 01 , which generates a uniform enhancement for normal cardiac output cq = 108 ml / s ), was used as the input contrast injection to this model with reduced cardiac output . the output simulated aortic enhancements are shown in fig7 . as shown therein , the contrast enhancement curves become more dome - shaped with an increase in magnitude , as the cardiac output decreases . fig8 demonstrates the empiric porcine aortic enhancement curves obtained for two pigs using multiphasic exponential injections with three different exponential decay coefficients ( 0 . 007 , 0 . 017 , 0 . 026 ). the contrast injection profiles are described in fig3 . exponential injection with a decay constant of 0 . 017 showed the aortic enhancement to be more uniform than with other injection protocols . this result was compatible with the theoretical model prediction that an exponential injection with a decay constant of 0 . 02 provided a plateaued aortic enhancement . injections with lower ( 0 . 007 ) or higher ( 0 . 026 ) decay constant resulted in aortic enhancements steadily rising or declining after a peak , respectively , as predicted by the theoretical model . the magnitude of aortic enhancement in fig8 a was substantially higher than that in fig8 b , reflecting the difference in body weight between two pigs t25 kg vs . 40 kg ). however , the patterns of aortic enhancement produced by three different exponential decay coefficients were consistent . aortic enhancement curves in two pigs experiencing unichasic and multiphasic exponential injections are shown in fig9 for ( a ) 50 ml and ( b ) 70 ml of contrast medium . the unichasic injection used a contrast injection rate of 2 ml / s , while the multiphasic injection started at 2 ml / s but declined exponentially with a decay constant 0 . 017 . the illustrated results clearly demonstrate that multiphasic injections yielded more prolonged and uniform vascular enhancement than uniphasic injections . performance of the multiphasic compared with the uniphasic injection can be summarized for four pigs as follows . for a 2 ml / s initial injection rate of 50 ml contrast medium , the multiphasic injection increased id by 30 %, reduced pa by a mean of 19 %, and increased 90 % dce by a mean of 81 %. for 70 ml injections with a 2 ml / s initial injection rate , id increased by 51 %, pa decreased by 18 %, and 90 % dce increased by 94 %. fig1 a shows empiric aortic enhancement curves in a 40 kg pig obtained with uniphasic and multiphasic exponential ( decay coefficient 0 . 017 ) injections with a 3 ml / s initial injection rate and 90 ml of contrast medium . in this pig , the multiphasic injection method resulted in more prolonged , uniform but slightly declining aortic enhancement . fig1 b demonstrates three empiric aortic enhancement curves generated by uniphasic , multiphasic and biphasic injections of 70 ml of contrast medium . the uniphasic injection consisted of a 2 ml / s injection for 35 sec , while the multiphasic injection had an initial rate of 2 ml / s with a decay coefficient of 0 . 017 for 53 sec . the biphasic injection was performed with a 2 ml / s rate for 17 sec and then a 1 . 0 ml / s rate for 36 sec . multiphasic injections again yielded more prolonged and uniform vascular enhancement than uniphasic injections . a biphasic injection resulted in more prolonged enhancement than a unichasic injection but generated two enhancement peaks with a valley in between . a prolonged , uniform vascular enhancement is desirable in ct angiography and some chest ct applications where the vessels , not the parenchyma of organs , are the target of interest . this enhancement pattern is useful for the purpose of image processing and display , in which 3d postprocessing is frequently based on a threshold ct attenuation value . it may also provide a longer optimal scanning interval for a given volume of contrast medium than a single - peaked contrast enhancement generated by a conventional uniphasic injection . alternatively , it may enable the use of a lower volume of contrast material for a given scanning duration . prolonged , uniform aortic contrast enhancements can be achieved by multiphasic exponential injections with adequately selected decay coefficients in accordance with the teaching of the present invention . the multiphasic injection protocol was mathematically derived from a physiologically - based pharmacokinetic model , and then a porcine model was used to confirm findings observed in theoretical analyses and computer simulations . although further clinical studies are warranted to validate the findings and injector performance , it is expected from previous experiences in comparative studies and pharmacokinetics that a human model would behave similarly . a simplified compartment model which has a limited number of compartments , instead of a more complex whole body model was used . the current compartment model was designed specifically to solve for a contrast injection profile which generates a prolonged , uniform vascular enhancement . this simple model does not provide a complete description of enhancement characteristics in each organ but can adequately describe the underlying pharmacokinetic process of interest , i . e . first - pass enhancement characteristics of the aorta . in this respect , the simulated results correlated well with the experimental results from the porcine model . since the model equations 1 - 5 in appendix a do not include renal or other clearance from the systemic circulation , the contrast concentration maintains a steady plateau following the cessation of contrast injection . this may not be a significant factor in scans of less than 5 - 1min duration . the fact that a multiphasic exponential injection generates a uniform vascular enhancement can be explained conceptually as follows . contrast enhancement in a system is proportional to the net amount of contrast medium present , i . e . inflow minus outflow contrast medium . aortic enhancement reflects an accumulation of contrast medium in the central blood volume ( i . e . contrast medium injected and recirculated minus medium diffused away from the vessel ). thus , vascular enhancement rises when the rate of contrast material infusion into the central blood volume exceeds the rate at which contrast medium diffuses away . this physiological event explains that aortic enhancement peaks shortly after the completion of the injection with a uniphasic injection , representing the maximal accumulation of contrast medium within the central blood volume compartment . the rate at which contrast medium leaves the central blood compartment to the interstitium compartment is likely proportional to the concentration gradient between the two compartments , i . e . an exponential function of time , because the contrast transport phenomenon is governed by passive diffusion and permeability . thus , when the outflow rate of contrast medium is balanced by the infusion rate of contrast medium by a multiphasic exponential injection protocol , a uniform vascular enhancement occurs . the experimental results showed that proper selection of a decay coefficient in multiphasic exponential injections was crucial to generate uniform vascular enhancement . the decay coefficient was proportional to the cardiac output per body weight . since the cardiac output per body weight in humans is half that of pigs , a 0 . 01 decay coefficient would be adequate for humans . this value , which is already normalized by body weight , is independent of body weight . for example , a multiphasic injection with a 0 . 017 decay coefficient resulted in a similar uniform vascular enhancement pattern but with a decrease in magnitude in the same pig scanned at its baseline weight of 25 kg and later after gaining 15 - 20 kg . the decay coefficient designed to generate a uniform enhancement for normal cardiac output resulted in more dome - shaped enhancement with increased magnitude when there is a reduced cardiac output , demonstrating the effect of cardiac output on contrast enhancement . in theory , albeit difficult in practice , if the degree of cardiac output reduction is known , the exact same uniform vascular enhancement can be reproduced for patients with reduced cardiac output . this can be achieved by lowering the initial injection rate and decay coefficient calculated for patients with normal cardiac output in an amount proportional to the reduction in cardiac output . however , it is apparent that a multiphasic injection designed to achieve a certain level of vascular enhancement in patients with normal cardiac output will not result in overestimation of contrast medium enhancement in patients with reduced cardiac output . the duration of aortic enhancement can be prolonged either by increasing the volume of contrast medium for a given initial rejection rate or by injecting slowly at a lower initial rate for a given contrast medium volume . with a uniphasic injection , peak magnitude of aortic enhancement depends on three injection factors , i . e . the concentration , injection rate , and total volume of contrast medium . with a multiphasic injection , however , the peak magnitude can be independent of the total volume of contrast medium , provided that the volume is not too small to reach an initial upslope enhancement to a plateau level . thus , a multiphasic injection protocol is advantageous over a uniphasic injection when a prolonged duration is desired , while keeping contrast enhancement from rising , by increasing the volume of contrast medium . although the theoretical analysis indicated that a multiphasic injection should follow an exponential decay to generate a prolonged , uniform vascular enhancement , other functional patterns may be used to approximate an exponential decay . for example , a short segment of an exponential curve can be approximated by a linear function without much disparity . this implies in practice that a linear or ramped injection protocol may be used instead of a strictly exponential injection when the injection duration is not too long and when the decay coefficient is relatively small ( for example , the exponential curve with 0 . 007 decay coefficient in fig3 ). in addition , a subtle discrepancy in enhancement from a slightly different approximation of exponential function may be indiscernible because of intrinsic physiological fluctuations in enhancement caused by vascular pulsation and respirator motion . however , these are all included as part of the present invention in accordance with the teaching herein . the data demonstrates that biphasic injections were not sufficient to generate a uniform vascular enhancement . in the study , multiphasic injections were generated with subsecond temporal resolution by the prototype injector . however , this degree of high temporal resolution may not be necessary . the number and interval of temporal steps required in multiphasic injection depends on the injection duration and exponential decay coefficients . although the effect of temporal resolution on the enhancement produced by multiphasic injections has not been fully explored , multiphasic temporal resolution of 2 - 3 seconds appears sufficient to generate uniform enhancement because of intrinsic physiological fluctuation . this factor presents another reason why strict adherence to all exponential decay function is not necessary in order to achieve clinically satisfactory and uniform vascular enhancement . the particular contrast injector or delivery system which is thought by the inventors to be particularly useful for implementation of the present invention includes a computer , or other digitally programmable control , for providing operator input and control of the injection protocol . in particular , a liebel - flarsheim model ct 9000 adv contrast delivery system , as depicted in fig1 , represents such a contrast injector . as shown therein , the contrast delivery system 20 includes a power head 22 for accepting the syringe containing the contrast medium , a control console 24 which may be a lcd display to provide for operator input and control of the injector , and a stand 26 with a base 28 containing the computer or other digital controller . in the subject injector , the injection parameters are entered in phases . each phase has a constant flow , a volume and an optional delay . these parameters are displayed on the injector control console 24 and the operator can change the values . typically , an operator can enter up to four phases . the injector then performs the injection by executing these phases in sequential order . for the injector of the present invention , the parameters of the 2nd phase were changed and the console screen redesigned to allow the operator to enter the parameters for an injection with an exponentially decaying flow rate . the operator enters the exponential coefficient and an elapsed time in order to define the injection protocol . the initial flow rate for the 2nd phase is the flow rate of the 1st phase . this arrangement allows the operator flexibility to experiment with different injection profiles that may include steady state flows before and / or after the exponential decaying flow . in the prototype injector , the 1st flow was used to allow the selected flow rate to reach steady state ( approximately 2 ml of volume ) and the 3rd phase was not used . the flow chart for , and the particular software program used in , the prototype injector are included as appendix b . also included as part of appendix b are the validation data which includes the data obtained by measuring during an injection the syringe volume versus time using a linear position transducer . a plot of this data is included , and compared with a calculated graph of the exponential curve desired to be obtained . as shown , the actual result curve is a close approximation of the theoretical curve . while the invention has been disclosed and described in the form of a preferred embodiment , the inventors contemplate that various changes and variations may be envisioned by those of ordinary skill in the art without departing from the invention . for example , the digital control and vagaries of actual devices may well result in injection protocols which are not truly exponential curves . furthermore , as mentioned above , other factors may serve to limit how close an injection rate may approximate an exponential curve . however , such variations are included within the teaching of the present invention as well as other modifications including changes to the particular injector . for example , virtually any logically controlled injector would be able to perform a multiphasic injection protocol and the invention should not be considered to be limited to a computer controlled , or even a digitally controlled injector . while operator input is usually considered as desirable , a preprogrammed or instructed or wired injector which is set up to perform a multiphasic injection protocol is also considered to be part of the present invention . indeed , the invention should be considered as being limited only by the scope of the claims appended hereto , and their legal equivalents . to develop governing equations for the model in fig2 mass balance equations are written for each compartment with input and output contrast flow . ## equ1 ## subject to the initial conditions at time t = 0 : where cv ( t ), cr ( t ), cp ( t ), cl ( t ), and cs ( t ) represent the concentration of the contrast with respect to time , t , within the peripheral venous , right heart , pulmonary , left heart ( aorta ), and systemic compartments respectively . the vv , vr , vp , vl , and vs are : the volumes of the various compartments which are assumed constant . these blood flow and volume of each compartment are determined from known physiologic data . the qv , qr , qp , ql , and qs represent the flows . qc ( t ) and cc are the time - dependent flow and concentration of the injected contrast material . the flow in each compartment equals the cardiac output , q : equations 1 - 6 can be numerically solved to predict and simulate an aortic enhancement curve for a given contrast injection condition . an example in a porcine model with 2 ml / s uniphasic injection is shown in fig4 . conversely , these differential equations can be used to estimate an input contrast injection algorithm that generates a uniform , prolonged vascular enhancement , i . e . solving an inverse problem , as shown below . because the initial contrast concentrations in the body compartments equal zero , taking the laplace transform of equations [ 1 ]-[ 5 ] yields ## equ2 ## where cv ( s ), cr ( s ), cp ( s ), cl ( s ), cs ( s ), and qc ( s ) are the laplace transforms of cv ( t ), cr ( t ), cp ( t ), cl ( t ), cs ( t ), and qc ( t ) respectively . a uniform aortic enhancement profile , i . e . constant cl ( t ), can be modeled effectively by letting ## equ3 ## where a is a scaling constant and h ( s ) is the laplace transform of the heaviside step - function , h ( t ): ## equ4 ## combining [ 12 ] and [ 7 ]-[ 11 ], ## equ5 ## the inverse laplace transform of [ 13 ] gives the desired result : ## equ6 ## equation [ 14 ] can be approximated by eliminating the terms involving the dirac delta function , d ( t ), and its derivatives since these terms contribute only to the impulse rise in contrast concentration immediately following t = 0 and not to the steady - state behavior . without these terms , [ 14 ] simplifies to ## equ7 ## the parameters forming multiplication terms outside the exponential term in this equation are independent of time and normalized by normalization constant ` α ` to set as the initial injection rate .
a contrast medium injector for injecting a patient with contrast medium for a ct scan is programmed to provide an injection protocol that is a representation of an ideally defined exponential curve with an initial injection rate decaying at an exponential rate . this has been found to produce a uniform vascular enhancement for the scanning of a patient &# 39 ; s blood vessels . the particular exponential decay coefficient which has been found to be optimal is directly proportional to the cardiac output divided by the patient &# 39 ; s weight and is approximated at 0 . 01 for a typical human .